Finite Group Databasebeta

← #33057512

Subgroups — #33057512

Order 18688.

720 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556616C211yesmaximal, characteristic
2M16×D292C211yesmaximal, characteristic, complemented
3#22608831C211yesmaximal, characteristic, complemented
4#22608821C211yesmaximal, characteristic, complemented
5#22608795C211yesmaximal, characteristic, complemented
6#22608786C211yesmaximal, characteristic, complemented
7#22608765C211yesmaximal, characteristic, complemented
8#22608760C211yesmaximal, characteristic, complemented
9#22608753C211yesmaximal, characteristic, complemented
10C8×#11924013C211yesmaximal, characteristic, complemented
11#22608699C211yesmaximal, characteristic, complemented
12#22608373C211yesmaximal, characteristic, complemented
13#22608370C211yesmaximal, characteristic, complemented
14#22608234C211yesmaximal, characteristic, complemented
15C73×#2970C211yesmaximal, characteristic, complemented
16C2×#14556616C411yescharacteristic
17C2×M16×D73—21no—
18C2×C4×D292C2211yescharacteristic
19#17988697C411yescharacteristic
20#17988697C411yescharacteristic
21#17988685C411yescharacteristic
22#17988685C411yescharacteristic
23C4×#11924015C2211yescharacteristic
24C4×#11924015C2211yescharacteristic
25C4×#11924013C2211yescharacteristic
26C4×#11924013C2211yescharacteristic
27#17988666C2211yescharacteristic
28#17988666C2211yescharacteristic
29#17988660—21no—
30#17988660—21no—
31C8×D292C2211yescharacteristic
32C8×D292C2211yescharacteristic
33#17988630C411yescharacteristic
34#17988611C2211yescharacteristic
35#17988591C411yescharacteristic
36#17988588C411yescharacteristic
37#17988550—21no—
38C4×#11924012C2211yescharacteristic
39C4×#11924012C2211yescharacteristic
40#17988339C2211yescharacteristic, complemented
41#17988339C2211yescharacteristic, complemented
42#17988337C2211yescharacteristic, complemented
43#17988337C2211yescharacteristic, complemented
44Q8⋊#10596831C2211yescharacteristic, complemented
45Q8⋊#10596831C2211yescharacteristic, complemented
46D4⋊#10596831C2211yescharacteristic, complemented
47D4⋊#10596831C2211yescharacteristic, complemented
48#17988314C2211yescharacteristic
49#17988314C2211yescharacteristic
50#17988314C2211yescharacteristic
51#17988314C2211yescharacteristic
52#17988309C2211yescharacteristic, complemented
53#17988309C2211yescharacteristic, complemented
54#17988308C2211yescharacteristic, complemented
55C8×#10596831C2211yescharacteristic, complemented
56C4×#41×C73C2211yescharacteristic
57#446×C73C2211yescharacteristic, complemented
58#446×C73C2211yescharacteristic, complemented
59#435×C73—21no—
60#435×C73—21no—
61C8×D4×C73C2211yescharacteristic, complemented
62C8×D4×C73C2211yescharacteristic, complemented
63#432×C73—21no—
64#432×C73C2211yescharacteristic
65C4×M16×C73C2211yescharacteristic
66C2×#97×C73—21no—
67C22×D292C2×C411yescharacteristic
68C22×C4×D73—21no—
69#14556616—24no—
70C2×#11924015C2×C411yescharacteristic
71C2×#11924015C2×C411yescharacteristic
72C2×#11924013C2×C411yescharacteristic
73C2×#11924013C2×C411yescharacteristic
74C2×#11924012—21no—
75C2×#11924012—21no—
76C2×#11924012C2×C411yescharacteristic
77C2×#11924012C2×C411yescharacteristic
78C2×#11924011—21no—
79C2×#11924011—21no—
80M16×D73—44no—
81C2×C8×D73—21no—
82C2×C8×D73—21no—
83C4×D292—21no—
84C4×D292—21no—
85C4×D292C2×C411yescharacteristic
86C4×D292C2×C411yescharacteristic
87C4×D292C2311yescharacteristic
88C4×D292C2311yescharacteristic
89C2×C4⋊Dic73C2×C411yescharacteristic
90C2×#11924005—21no—
91#14556524C2×C411yescharacteristic
92#14556524C2×C411yescharacteristic
93#14556522C2×C411yescharacteristic
94#14556522C2×C411yescharacteristic
95C4⋊#10596831—21no—
96C4⋊#10596831C2311yescharacteristic
97C4⋊#10596831C2311yescharacteristic
98#14556515C2×C411yescharacteristic
99#14556515C2×C411yescharacteristic
100#14556512C2×C411yescharacteristic
101#14556512C2×C411yescharacteristic
102#14556511C2×C411yescharacteristic
103#14556511C2×C411yescharacteristic
104#14556511C2×C411yescharacteristic
105#14556511C2×C411yescharacteristic
106#14556503—21no—
107#14556503—21no—
108#14556503—21no—
109#14556503—21no—
110#14556502—21no—
111#14556502C2×C411yescharacteristic
112#14556502C2×C411yescharacteristic
113C4×#10596831C2311yescharacteristic
114C4×#10596831C2311yescharacteristic
115C4×#10596831C2311yescharacteristic
116C4×#10596831C2311yescharacteristic
117C2×#41×C73C2×C411yescharacteristic
118C2×M16×C73D411yescharacteristic
119C2×M16×C73D411yescharacteristic
120C22×C8×C73—21no—
121C22×C8×C73—21no—
122C4×Q8×C73C2×C411yescharacteristic
123C4×Q8×C73C2311yescharacteristic
124C4×D4×C73—21no—
125C4×D4×C73—21no—
126C4×D4×C73C2×C411yescharacteristic
127C4×D4×C73C2311yescharacteristic
128#117×C73—21no—
129#117×C73C2×C411yescharacteristic
130C2×C42×C73—21no—
131C2×C42×C73C2311yescharacteristic
132C4⋊C8×C73—21no—
133C4⋊C8×C73—21no—
134C4⋊C8×C73C2311yescharacteristic
135C4⋊C8×C73C2311yescharacteristic
136C22⋊C8×C73—21no—
137C22⋊C8×C73—21no—
138C22⋊C8×C73—21no—
139#97×C73—21no—
140#97×C73—21no—
141#97×C73—21no—
142#97×C73D411yescharacteristic
143#97×C73D411yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C4×C8×C73C2311yescharacteristic
146C23×D73—21no—
147C2×D292—22no—
148C2×D292—21no—
149C2×D292—41no—
150C2×D292C22×C411yescharacteristic
151C2×D292C22×C411yescharacteristic
152C22×Dic73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73—42no—
158C2×C4×D73—42no—
159#11924015—22no—
160#11924015—22no—
161#11924013—22no—
162#11924013M1614yescomplemented
163#11924012—22no—
164#11924012—41no—
165#11924012—42no—
166#11924012—42no—
167#11924011—42no—
168#11924011—42no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—21no—
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C22×C411yescharacteristic
175C2×#10596831C22×C411yescharacteristic
176C2×#10596831C22×C411yescharacteristic
177C8×D73—42no—
178C8×D73—42no—
179C4⋊Dic73—21no—
180C4⋊Dic73C22×C411yescharacteristic
181C4⋊Dic73C22×C411yescharacteristic
182#11924005—21no—
183#11924005—21no—
184#11924005—21no—
185#11924005—21no—
186#41×C73—22no—
187#41×C73—41no—
188C2×Q8×C73C22×C411yescharacteristic
189C2×D4×C73—21no—
190C2×D4×C73C22×C411yescharacteristic
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73C22×C411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73#4111yescharacteristic
197M16×C73—22no—
198M16×C73—41no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—42no—
202C2×C8×C73—42no—
203C2×C8×C73C2×D411yescharacteristic
204C2×C8×C73C2×D411yescharacteristic
205C2×C8×C73C2×D411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73C22×C411yescharacteristic
210C4⋊C4×C73C22×C411yescharacteristic
211C22⋊C4×C73—21no—
212C22⋊C4×C73—21no—
213C22⋊C4×C73—21no—
214C42×C73—21no—
215C42×C73—21no—
216C42×C73—21no—
217C42×C73C22×C411yescharacteristic
218C42×C73C2×D411yescharacteristic
219C42×C73#4111yescharacteristic
220C42×C73C2411yescharacteristic
221C23×C73—21no—
222C23×C73C4×D411yescharacteristic
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—42no—
226C22×D73—42no—
227C22×D73—81no—
228Q8×C73—21no—
229Q8×C73—21no—
230D4×C73—21no—
231D4×C73—42no—
232D4×C73C2×M1612yescomplemented
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—41no—
241C2×C4×C73—42no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—81no—
245C2×C4×C73C4×D411yescharacteristic
246C2×C4×C73C4×D411yescharacteristic
247C2×C4×C73C4×D411yescharacteristic
248C2×C4×C73C23×C411yescharacteristic
249C2×C4×C73C22×D411yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—42no—
254D292—21no—
255D292—21no—
256D292—21no—
257D292—42no—
258D292C2×M1612yes—
259C4×D73—42no—
260C4×D73—42no—
261C4×D73—41no—
262C4×D73—41no—
263C4×D73—41no—
264C4×D73—41no—
265C8×C73—21no—
266C8×C73—21no—
267C8×C73—21no—
268C8×C73—41no—
269C8×C73—41no—
270C8×C73AGL(1,ℤ/8)12yescomplemented
271#10596831—22no—
272#10596831—21no—
273#10596831—21no—
274#10596831—21no—
275#10596831—41no—
276#10596831—41no—
277#10596831—41no—
278#10596831C2×M1612yescomplemented
279C22×C73—21no—
280C22×C73—22no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73C2×C4×D411yescharacteristic
286C2×D73—22no—
287C2×D73—41no—
288C2×D73—41no—
289C2×D73—41no—
290C2×D73—41no—
291C2×D73—41no—
292C2×D73—84no—
293C4×C73—21no—
294C4×C73—21no—
295C4×C73—21no—
296C4×C73—21no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73—41no—
300C4×C73C22×M1611yescharacteristic
301C4×C73#56811yescharacteristic
302C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C73#57511yescharacteristic
304Dic73—41no—
305Dic73—41no—
306#18046—731nomaximal, Sylow
307C2×C73—41no—
308C2×C73—42no—
309C2×C73D4×M1611yescharacteristic
310C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
311C2×C73#295611yescharacteristic
312D73—42no—
313D73—81no—
314#2970D7311yescharacteristic, complemented
315C4×AGL(1,ℤ/8)—731no—
316D4×M16—731no—
317#1592—731no—
318#1590—731no—
319#1586—731no—
320#1585—731no—
321#1582—731no—
322#1581—731no—
323#1580—731no—
324C8×D8—731no—
325#1569—731no—
326#1490—731no—
327#1489—731no—
328#1455—731no—
329C73#1804611yescharacteristic, complemented, Sylow
330C2×AGL(1,ℤ/8)—731no—
331C22×M16—1461no—
332C4×#41C2×D7311yescharacteristic
333C2×C4×D4—731no—
334#446C2×D7311yescharacteristic, complemented
335#446C2×D7311yescharacteristic, complemented
336#442—731no—
337#442—731no—
338#440—731no—
339#440—731no—
340C4×SD16—731no—
341C4×SD16—731no—
342C4×D8—731no—
343C4×D8—731no—
344#436—731no—
345#436—731no—
346#435—21no—
347#435—21no—
348#435—1461no—
349#435—1461no—
350C8×D4—731no—
351C8×D4—731no—
352C8×D4C2×D7311yescharacteristic, complemented
353C8×D4C2×D7311yescharacteristic, complemented
354#432—21no—
355#432C2×D7311yescharacteristic
356#428—731no—
357#423—731no—
358#418—731no—
359#417—731no—
360#407—1461no—
361C4×M16—731no—
362C4×M16—731no—
363C4×M16C2×D7311yescharacteristic
364C2×#97—21no—
365#335—731no—
366#335—731no—
367#334—731no—
368#334—731no—
369Q8⋊C8—731no—
370Q8⋊C8—731no—
371D4⋊C8—731no—
372D4⋊C8—731no—
373D4⋊C8—731no—
374D4⋊C8—731no—
375D4⋊C8—731no—
376D4⋊C8—731no—
377#322—731no—
378#322—731no—
379#322—731no—
380C82—731no—
381C2×#41C4×D7311yescharacteristic
382C22×D4—731no—
383C23×C4—1461no—
384AGL(1,ℤ/8)—1464no—
385C2×SD16—731no—
386C2×SD16—731no—
387C2×D8—731no—
388C2×D8—731no—
389C2×M16—2924no—
390C2×M16—731no—
391C2×M16—731no—
392C2×M16—1461no—
393C2×M16—1461no—
394C2×M16—1461no—
395C2×M16—1461no—
396C2×M16C22⋊D7311yescharacteristic
397C2×M16C22⋊D7311yescharacteristic
398C22×C8—21no—
399C22×C8—21no—
400C22×C8—1461no—
401C22×C8—1461no—
402C4×Q8C4×D7311yescharacteristic
403C4×Q8C22×D7311yescharacteristic
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—731no—
407C4×D4—731no—
408C4×D4—731no—
409C4×D4—731no—
410C4×D4—1461no—
411C4×D4—1461no—
412C4×D4C4×D7311yescharacteristic
413C4×D4C22×D7311yescharacteristic
414#117—21no—
415#117C4×D7311yescharacteristic
416C2×C4⋊C4—731no—
417C2×C22⋊C4—1461no—
418C2×C42—21no—
419C2×C42C22×D7311yescharacteristic
420#107—731no—
421#107—731no—
422#106—731no—
423#106—731no—
424C4⋊C8—21no—
425C4⋊C8—21no—
426C4⋊C8—731no—
427C4⋊C8—731no—
428C4⋊C8—1461no—
429C4⋊C8C22×D7311yescharacteristic
430C4⋊C8C22×D7311yescharacteristic
431#103—731no—
432#103—731no—
433#102—731no—
434#102—731no—
435#102—731no—
436#102—731no—
437#102—731no—
438#102—731no—
439C22⋊C8—21no—
440C22⋊C8—21no—
441C22⋊C8—21no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444C22⋊C8—1461no—
445C22⋊C8—1461no—
446#97—21no—
447#97—21no—
448#97—21no—
449#97—731no—
450#97—731no—
451#97—1461no—
452#97C22⋊D7311yescharacteristic
453#97C22⋊D7311yescharacteristic
454C4×C8—731no—
455C4×C8—731no—
456C4×C8—731no—
457C4×C8—731no—
458C4×C8C22×D7311yescharacteristic
459C4×C8C22×D7311yescharacteristic
460C24—1461no—
461#41—22no—
462#41—41no—
463C2×Q8C2×C4×D7311yescharacteristic
464C2×D4—21no—
465C2×D4—2921no—
466C2×D4—731no—
467C2×D4—731no—
468C2×D4—1462no—
469C2×D4—1461no—
470C2×D4C2×C4×D7311yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—2922no—
475C22×C4—2922no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4C2×C4×D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4#1192402811yescharacteristic
484SD16—1462no—
485SD16—1462no—
486D8—734no—
487D8—1462no—
488M16—22no—
489M16—41no—
490M16—2922no—
491M16—2922no—
492M16—2922no—
493M16—2922no—
494M16—2921no—
495M16—1462no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8—2922no—
501C2×C8—2922no—
502C2×C8—731no—
503C2×C8—731no—
504C2×C8—731no—
505C2×C8—731no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8C2×C22⋊D7311yescharacteristic
511C2×C8C2×C22⋊D7311yescharacteristic
512C2×C8C2×C22⋊D7311yescharacteristic
513C2×C8C2×C22⋊D7311yescharacteristic
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—731no—
517C4⋊C4—731no—
518C4⋊C4—1461no—
519C4⋊C4C2×C4×D7311yescharacteristic
520C4⋊C4C2×C4×D7311yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—1461no—
525C22⋊C4—1461no—
526C22⋊C4—1461no—
527C22⋊C4—1461no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42C2×C4×D7311yescharacteristic
532C42C2×C22⋊D7311yescharacteristic
533C42#1192402811yescharacteristic
534C42C23×D7311yescharacteristic
535C23—21no—
536C23—2922no—
537C23—2922no—
538C23—5841no—
539C23—1461no—
540C23—1461no—
541C23C4×C22⋊D7311yescharacteristic
542Q8—21no—
543Q8—21no—
544D4—21no—
545D4—42no—
546D4—2922no—
547D4—732no—
548D4—1461no—
549D4—1461no—
550D4—1461no—
551D4M16×D7312yescomplemented
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—81no—
564C2×C4—2922no—
565C2×C4—2922no—
566C2×C4—2922no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4C4×C22⋊D7311yescharacteristic
574C2×C4C4×C22⋊D7311yescharacteristic
575C2×C4C4×C22⋊D7311yescharacteristic
576C2×C4C22×C4×D7311yescharacteristic
577C2×C4C22×C22⋊D7311yescharacteristic
578C2×C4C2×#1192402811yescharacteristic
579C8—21no—
580C8—21no—
581C8—21no—
582C8—41no—
583C8—41no—
584C8—2921no—
585C8—2921no—
586C8—2921no—
587C8—732no—
588C8—1462no—
589C8—1461no—
590C8—1461no—
591C8—1461no—
592C8#1455661612yescomplemented
593C22—21no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—41no—
598C22—41no—
599C22—2921no—
600C22—2921no—
601C22—2921no—
602C22—2921no—
603C22—2921no—
604C22—5844no—
605C22—1462no—
606C22C2×C4×C22⋊D7311yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—2921no—
615C4—2921no—
616C4C2×M16×D7311yescharacteristic
617C4#1798935311yescharacteristic
618C4C2×#1455661611yescharacteristic
619C4#1798938511yescharacteristic
620C2—41no—
621C2—42no—
622C2—2922no—
623C2—5841no—
624C2M16×C22⋊D7311yescharacteristic
625C2C4×#1455661611yescharacteristic
626C2#2261883811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#2970C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556616C4
cosocleC42×C73C24
hypercenter#2970D73
hypocenterC73#18046
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. M16×D292C2semidirect
3. #22608831C2semidirect
4. #22608821C2semidirect
5. #22608795C2semidirect
6. #22608786C2semidirect
7. #22608765C2semidirect
8. #22608760C2semidirect
9. #22608753C2semidirect
10. C8×#11924013C2semidirect
11. #22608699C2semidirect
12. #22608373C2semidirect
13. #22608370C2semidirect
14. #22608234C2semidirect
15. C73×#2970C2semidirect
40. #17988339C22semidirect
41. #17988339C22semidirect
42. #17988337C22semidirect
43. #17988337C22semidirect
44. Q8⋊#10596831C22semidirect
45. Q8⋊#10596831C22semidirect
46. D4⋊#10596831C22semidirect
47. D4⋊#10596831C22semidirect
52. #17988309C22semidirect
53. #17988309C22semidirect
54. #17988308C22semidirect
55. C8×#10596831C22semidirect
57. #446×C73C22semidirect
58. #446×C73C22semidirect
61. C8×D4×C73C22semidirect
62. C8×D4×C73C22semidirect
162. #11924013M16semidirect
232. D4×C73C2×M16semidirect
270. C8×C73AGL(1,ℤ/8)semidirect
278. #10596831C2×M16semidirect
314. #2970D73semidirect
329. C73#18046semidirect
334. #446C2×D73semidirect
335. #446C2×D73semidirect
352. C8×D4C2×D73semidirect
353. C8×D4C2×D73semidirect
551. D4M16×D73semidirect
592. C8#14556616semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#145566162
M16×D2922
#226088312
#226088212
#226087952
#226087862
#226087652
#226087602
#226087532
C8×#119240132
#226086992
#226083732
#226083702
#226082342
C73×#29702
#1804673