Finite Group Databasebeta

← #33055764

Subgroups — #33055764

Order 18688.

724 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618894C211yesmaximal, characteristic, complemented
2C4×#14556615C211yesmaximal, characteristic
3#436×D73C211yesmaximal, characteristic, complemented
4#22608845C211yesmaximal, characteristic, complemented
5#22608842C211yesmaximal, characteristic, complemented
6#22608832C211yesmaximal, characteristic, complemented
7#22608830C211yesmaximal, characteristic, complemented
8#22608823C211yesmaximal, characteristic, complemented
9#22608819C211yesmaximal, characteristic, complemented
10#22608814C211yesmaximal, characteristic, complemented
11#22608682C211yesmaximal, characteristic, complemented
12#22608320C211yesmaximal, characteristic, complemented
13#22608315C211yesmaximal, characteristic, complemented
14C8×#11924011C211yesmaximal, characteristic, complemented
15C73×#1589C211yesmaximal, characteristic, complemented
16C2×#14556615C411yescharacteristic
17C2×M16×D73—21no—
18C4×#11924027C2211yescharacteristic
19C4×D4×D73C2211yescharacteristic
20#17988717C2211yescharacteristic, complemented
21#17988714C2211yescharacteristic, complemented
22#17988699C411yescharacteristic
23#17988696C411yescharacteristic
24#17988686C411yescharacteristic
25#17988684C411yescharacteristic
26C4×#11924017C2211yescharacteristic
27C4×#11924016C2211yescharacteristic
28C4×#11924013C2211yescharacteristic
29#17988666C2211yescharacteristic
30#17988665C2211yescharacteristic
31#17988665C2211yescharacteristic, complemented
32#17988664C2211yescharacteristic
33#17988663—21no—
34#17988662—21no—
35#17988661—21no—
36#17988656—21no—
37#17988639C2211yescharacteristic
38#17988638C2211yescharacteristic
39#17988637—21no—
40#17988628C411yescharacteristic
41C4⋊C8×D73C2211yescharacteristic
42#17988590C411yescharacteristic
43#17988585C411yescharacteristic
44C22⋊C8×D73—21no—
45M16×Dic73—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988339C2211yescharacteristic, complemented
49#17988338C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988319C2211yescharacteristic, complemented
52#17988317C2211yescharacteristic, complemented
53#17988316C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic, complemented
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic, complemented
58#17988308C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60C4×D8×C73C2211yescharacteristic
61#436×C73C2211yescharacteristic
62#436×C73C2211yescharacteristic, complemented
63#335×C73C2211yescharacteristic, complemented
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic, complemented
66C82×C73C2211yescharacteristic, complemented
67C2×#11924027C2×C411yescharacteristic
68C2×D4×D73C2×C411yescharacteristic
69C22×C4×D73—21no—
70#14556615—24no—
71C2×#11924017C2×C411yescharacteristic
72C2×#11924016C2×C411yescharacteristic
73C2×#11924013C2×C411yescharacteristic
74C2×#11924012—21no—
75C2×#11924012—21no—
76C2×#11924011—21no—
77C2×#11924011—21no—
78C2×#11924011C2×C411yescharacteristic
79C2×#11924011C2×C411yescharacteristic
80M16×D73—44no—
81C2×C8×D73—21no—
82C2×C8×D73D411yescharacteristic
83C2×C8×D73D411yescharacteristic
84C4×Dic146C2311yescharacteristic
85#14556556C2×C411yescharacteristic
86C4×D292C2311yescharacteristic
87D4×Dic73C2×C411yescharacteristic
88D4×Dic73C2×C411yescharacteristic
89#14556553C2×C411yescharacteristic
90C4×C22⋊D73—21no—
91C4×C22⋊D73—21no—
92#14556551—21no—
93#14556551—21no—
94#14556549C2×C411yescharacteristic
95#14556547—21no—
96C4⋊C4×D73C2×C411yescharacteristic
97C22⋊C4×D73—21no—
98C2×C4×Dic73—21no—
99C42×D73C2311yescharacteristic
100#14556524C2×C411yescharacteristic
101#14556523C2×C411yescharacteristic
102#14556522C2×C411yescharacteristic
103C4⋊#10596831C2311yescharacteristic
104C4⋊#10596831C2311yescharacteristic
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556516C2×C411yescharacteristic
110#14556514C2×C411yescharacteristic
111#14556513C2×C411yescharacteristic
112#14556512C2×C411yescharacteristic
113#14556512C2×C411yescharacteristic
114#14556511C2×C411yescharacteristic
115C22⋊#10596831—21no—
116C22⋊#10596831—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503—21no—
121#14556501—21no—
122#14556501—21no—
123#14556501C2×C411yescharacteristic
124#14556501C2×C411yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C8×Dic73—21no—
129C8×Dic73D411yescharacteristic
130C8×Dic73D411yescharacteristic
131C2×D8×C73C2×C411yescharacteristic
132C2×M16×C73—21no—
133C2×M16×C73—21no—
134C4×D4×C73C2311yescharacteristic
135C4×D4×C73C2311yescharacteristic
136#107×C73C2×C411yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138C4⋊C8×C73C2311yescharacteristic
139#102×C73C2×C411yescharacteristic
140#102×C73C2×C411yescharacteristic
141C22⋊C8×C73—21no—
142C22⋊C8×C73—21no—
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C4×C8×C73C2311yescharacteristic
146C23×D73—21no—
147#11924027—22no—
148#11924027—22no—
149C2×Dic146C22×C411yescharacteristic
150C2×D292C22×C411yescharacteristic
151C2×C22⋊D73—21no—
152C2×C22⋊D73—21no—
153D4×D73—22no—
154D4×D73—22no—
155C22×Dic73—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160C2×C4×D73—21no—
161C2×C4×D73—42no—
162C2×C4×D73—42no—
163C2×C4×D73C22×C411yescharacteristic
164C2×C4×D73C2×D411yescharacteristic
165C2×C4×D73#4111yescharacteristic
166#11924017—22no—
167#11924016—22no—
168#11924013—22no—
169#11924012—42no—
170#11924012—42no—
171#11924011—22no—
172#11924011—42no—
173#11924011—42no—
174#11924011M1614yescomplemented
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831—21no—
178C2×#10596831C22×C411yescharacteristic
179C2×#10596831C22×C411yescharacteristic
180C2×#10596831C2×D411yescharacteristic
181C2×#10596831C2×D411yescharacteristic
182C8×D73—22no—
183C8×D73—41no—
184C8×D73—42no—
185C4⋊Dic73C22×C411yescharacteristic
186C4⋊Dic73C22×C411yescharacteristic
187#11924007—21no—
188#11924007—21no—
189#11924007—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192#11924005—21no—
193#11924005—21no—
194#11924005—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73C22×C411yescharacteristic
199C4×Dic73C2×D411yescharacteristic
200C4×Dic73#4111yescharacteristic
201C2×D4×C73C22×C411yescharacteristic
202C2×D4×C73C22×C411yescharacteristic
203C22×C4×C73—21no—
204C22×C4×C73—21no—
205D8×C73—22no—
206M16×C73—42no—
207M16×C73—42no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73C22×C411yescharacteristic
212C2×C8×C73C22×C411yescharacteristic
213C2×C8×C73C2×D411yescharacteristic
214C2×C8×C73C2×D411yescharacteristic
215C4⋊C4×C73C22×C411yescharacteristic
216C4⋊C4×C73C22×C411yescharacteristic
217C22⋊C4×C73—21no—
218C22⋊C4×C73—21no—
219C42×C73C2411yescharacteristic
220C23×C73—21no—
221C23×C73—21no—
222C22×D73—21no—
223C22×D73—42no—
224C22×D73—42no—
225C22×D73—42no—
226C22×D73C4×D411yescharacteristic
227D4×C73—21no—
228D4×C73—21no—
229D4×C73—21no—
230D4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73C23×C411yescharacteristic
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C22⋊D73—42no—
241C22⋊D73—42no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73C4×D411yescharacteristic
249C2×Dic73C4×D411yescharacteristic
250C2×Dic73C4×D411yescharacteristic
251D292—21no—
252D292—21no—
253C4×D73—21no—
254C4×D73—21no—
255C4×D73—41no—
256C4×D73—42no—
257C4×D73—42no—
258C4×D73—41no—
259C4×D73—41no—
260C4×D73C2×M1612yes—
261C4×D73#13112yes—
262Dic146—21no—
263Dic146—21no—
264C8×C73—21no—
265C8×C73—21no—
266C8×C73—41no—
267C8×C73—41no—
268C8×C73—41no—
269C8×C73C2×M1612yescomplemented
270C8×C73AGL(1,ℤ/8)12yescomplemented
271#10596831—21no—
272#10596831—21no—
273#10596831—21no—
274#10596831—41no—
275#10596831—41no—
276#10596831—41no—
277#10596831C2×M1612yescomplemented
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73C2×C4×D411yescharacteristic
285C2×D73—21no—
286C2×D73—21no—
287C2×D73—41no—
288C2×D73—41no—
289C2×D73—41no—
290C2×D73—84no—
291C2×D73#43612yes—
292C4×C73—21no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73C22×M1611yescharacteristic
296C4×C73C2×#13111yescharacteristic
297C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
298C4×C73#57511yescharacteristic
299Dic73—21no—
300Dic73—21no—
301Dic73—41no—
302Dic73—41no—
303Dic73—41no—
304Dic73#43612yes—
305#17891—731nomaximal, Sylow
306C2×C73—81no—
307C2×C73—81no—
308C2×C73C2×#43611yescharacteristic
309C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
310C2×C73#295511yescharacteristic
311D73—22no—
312D73—81no—
313#2965—731no—
314C4×AGL(1,ℤ/8)—731no—
315C2×#436—731no—
316#1594—731no—
317#1594—731no—
318#1592—731no—
319#1592—731no—
320#1590—731no—
321#1590—731no—
322#1589—731no—
323#1589D7311yescharacteristic, complemented
324#1565—731no—
325#1478—731no—
326#1477—731no—
327C8×M16—731no—
328C73#1789111yescharacteristic, complemented, Sylow
329C2×AGL(1,ℤ/8)—731no—
330C22×M16—1461no—
331C4×#41—731no—
332C2×C4×D4—731no—
333#446—731no—
334#446—731no—
335#442—731no—
336#442—731no—
337#440—731no—
338#440—731no—
339C4×SD16—731no—
340C4×SD16—731no—
341C4×D8—731no—
342C4×D8C2×D7311yescharacteristic
343#436—731no—
344#436—731no—
345#436—731no—
346#436—731no—
347#436—1461no—
348#436—1461no—
349#436C2×D7311yescharacteristic
350#436C2×D7311yescharacteristic, complemented
351#435—1461no—
352#435—1461no—
353#431—731no—
354#431—731no—
355#431—1461no—
356#428—731no—
357C2×C4⋊C8—731no—
358#418—731no—
359#417—731no—
360C2×C22⋊C8—1461no—
361C4×M16—731no—
362C4×M16—1461no—
363C2×C4×C8—731no—
364#335—731no—
365#335C2×D7311yescharacteristic, complemented
366#334—731no—
367#334—731no—
368Q8⋊C8—731no—
369Q8⋊C8—731no—
370D4⋊C8—731no—
371D4⋊C8—731no—
372D4⋊C8—731no—
373D4⋊C8—731no—
374D4⋊C8C2×D7311yescharacteristic
375D4⋊C8C2×D7311yescharacteristic, complemented
376#322—731no—
377#322—731no—
378C82—731no—
379C82C2×D7311yescharacteristic, complemented
380C2×#41—731no—
381C22×D4—731no—
382C23×C4—1461no—
383AGL(1,ℤ/8)—1464no—
384C2×SD16—731no—
385C2×SD16—731no—
386C2×D8—731no—
387C2×D8C4×D7311yescharacteristic
388C2×M16—21no—
389C2×M16—21no—
390C2×M16—2924no—
391C2×M16—731no—
392C2×M16—731no—
393C2×M16—1461no—
394C2×M16—1461no—
395C2×M16—1461no—
396C2×M16—1461no—
397C22×C8—731no—
398C22×C8—731no—
399C22×C8—1461no—
400C4×Q8—731no—
401C4×Q8—731no—
402C4×D4—731no—
403C4×D4—731no—
404C4×D4—731no—
405C4×D4—731no—
406C4×D4—1461no—
407C4×D4—1461no—
408C4×D4—1461no—
409C4×D4—1461no—
410C4×D4C22×D7311yescharacteristic
411C4×D4C22×D7311yescharacteristic
412#117—731no—
413#117—1461no—
414C2×C4⋊C4—731no—
415C2×C22⋊C4—1461no—
416C2×C42—731no—
417C2×C42—1461no—
418#107—731no—
419#107C4×D7311yescharacteristic
420#106—731no—
421#106—731no—
422C4⋊C8—731no—
423C4⋊C8—731no—
424C4⋊C8—1461no—
425C4⋊C8—1461no—
426C4⋊C8—1461no—
427C4⋊C8—1461no—
428C4⋊C8C22×D7311yescharacteristic
429C4⋊C8C22×D7311yescharacteristic
430#103—731no—
431#103—731no—
432#102—731no—
433#102—731no—
434#102—731no—
435#102—731no—
436#102C4×D7311yescharacteristic
437#102C4×D7311yescharacteristic
438C22⋊C8—21no—
439C22⋊C8—21no—
440C22⋊C8—1461no—
441C22⋊C8—1461no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444C22⋊C8—1461no—
445C22⋊C8—1461no—
446#97—731no—
447#97—731no—
448#97—1461no—
449#97—1461no—
450C4×C8—731no—
451C4×C8—731no—
452C4×C8—731no—
453C4×C8—731no—
454C4×C8—731no—
455C4×C8—1461no—
456C4×C8C22×D7311yescharacteristic
457C4×C8C22×D7311yescharacteristic
458C4×C8C22×D7311yescharacteristic
459C24—1461no—
460#41—1462no—
461#41—1462no—
462C2×Q8—731no—
463C2×D4—731no—
464C2×D4—1462no—
465C2×D4—1462no—
466C2×D4—1461no—
467C2×D4—1461no—
468C2×D4C2×C4×D7311yescharacteristic
469C2×D4C2×C4×D7311yescharacteristic
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—2922no—
473C22×C4—2922no—
474C22×C4—731no—
475C22×C4—731no—
476C22×C4—731no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483SD16—1462no—
484SD16—1462no—
485D8—22no—
486D8—1462no—
487M16—42no—
488M16—42no—
489M16—2922no—
490M16—2922no—
491M16—2922no—
492M16—2922no—
493M16—734no—
494M16—1462no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—2921no—
499C2×C8—2922no—
500C2×C8—731no—
501C2×C8—731no—
502C2×C8—731no—
503C2×C8—731no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1462no—
508C2×C8C2×C4×D7311yescharacteristic
509C2×C8C2×C4×D7311yescharacteristic
510C2×C8D4×D7311yescharacteristic
511C2×C8D4×D7311yescharacteristic
512C4⋊C4—731no—
513C4⋊C4—731no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4C2×C4×D7311yescharacteristic
518C4⋊C4C2×C4×D7311yescharacteristic
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—1461no—
522C22⋊C4—1461no—
523C22⋊C4—1461no—
524C22⋊C4—1461no—
525C22⋊C4—1461no—
526C42—731no—
527C42—731no—
528C42—731no—
529C42—1461no—
530C42—1461no—
531C42—1461no—
532C42C23×D7311yescharacteristic
533C23—21no—
534C23—21no—
535C23—2922no—
536C23—2922no—
537C23—2922no—
538C23—731no—
539C23—1461no—
540Q8—1461no—
541Q8—1461no—
542D4—21no—
543D4—21no—
544D4—21no—
545D4—21no—
546D4—2922no—
547D4—2922no—
548D4—1461no—
549D4—1461no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—2921no—
557C2×C4—2922no—
558C2×C4—2922no—
559C2×C4—2922no—
560C2×C4—2922no—
561C2×C4—2922no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—731no—
565C2×C4—732no—
566C2×C4—732no—
567C2×C4—731no—
568C2×C4—731no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4C22×C4×D7311yescharacteristic
575C2×C4C2×D4×D7311yescharacteristic
576C2×C4#41×D7311yescharacteristic
577C8—21no—
578C8—21no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—2921no—
583C8—2921no—
584C8—2921no—
585C8—732no—
586C8—1461no—
587C8—1461no—
588C8—1461no—
589C8M16×D7312yescomplemented
590C8#1455661512yescomplemented
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—2921no—
598C22—2921no—
599C22—2921no—
600C22—5844no—
601C22—732no—
602C22—1461no—
603C22—1461no—
604C22C4×D4×D7311yescharacteristic
605C4—21no—
606C4—41no—
607C4—41no—
608C4—2921no—
609C4—2921no—
610C4—2921no—
611C4—732no—
612C4—1461no—
613C4—1461no—
614C4C2×M16×D7311yescharacteristic
615C4#131×D7311yescharacteristic
616C4C2×#1455661511yescharacteristic
617C4#1798938411yescharacteristic
618C2—81no—
619C2—81no—
620C2—5841no—
621C2—1462no—
622C2#436×D7311yescharacteristic
623C2C4×#1455661511yescharacteristic
624C2#2261882911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1589C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556615C4
cosocleC42×C73C24
hypercenter#1589D73
hypocenterC73#17891
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1589

Complements

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

NH ≅ G/Ntype
1. #22618894C2semidirect
3. #436×D73C2semidirect
4. #22608845C2semidirect
5. #22608842C2semidirect
6. #22608832C2semidirect
7. #22608830C2semidirect
8. #22608823C2semidirect
9. #22608819C2semidirect
10. #22608814C2semidirect
11. #22608682C2semidirect
12. #22608320C2semidirect
13. #22608315C2semidirect
14. C8×#11924011C2semidirect
15. C73×#1589C2semidirect
20. #17988717C22semidirect
21. #17988714C22semidirect
31. #17988665C22semidirect
48. #17988339C22semidirect
49. #17988338C22semidirect
50. #17988337C22semidirect
51. #17988319C22semidirect
52. #17988317C22semidirect
55. D4⋊#10596831C22semidirect
57. #17988309C22semidirect
58. #17988308C22semidirect
59. C8×#10596831C22semidirect
62. #436×C73C22semidirect
63. #335×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. C82×C73C22semidirect
174. #11924011M16semidirect
269. C8×C73C2×M16semidirect
270. C8×C73AGL(1,ℤ/8)semidirect
277. #10596831C2×M16semidirect
323. #1589D73semidirect
328. C73#17891semidirect
350. #436C2×D73semidirect
365. #335C2×D73semidirect
375. D4⋊C8C2×D73semidirect
379. C82C2×D73semidirect
589. C8M16×D73semidirect
590. C8#14556615semidirect

Maximal subgroups

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

subgroupindex
#226188942
C4×#145566152
#436×D732
#226088452
#226088422
#226088322
#226088302
#226088232
#226088192
#226088142
#226086822
#226083202
#226083152
C8×#119240112
C73×#15892
#1789173