Finite Group Databasebeta

← #33055786

Subgroups — #33055786

Order 18688.

684 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619046C211yesmaximal, characteristic, complemented
2#22618969C211yesmaximal, characteristic, complemented
3C4×#14556615C211yesmaximal, characteristic
4#22608845C211yesmaximal, characteristic, complemented
5#22608843C211yesmaximal, characteristic, complemented
6#22608833C211yesmaximal, characteristic, complemented
7#22608832C211yesmaximal, characteristic, complemented
8#22608824C211yesmaximal, characteristic, complemented
9#22608823C211yesmaximal, characteristic, complemented
10#22608813C211yesmaximal, characteristic, complemented
11#22608682C211yesmaximal, characteristic, complemented
12#22608310C211yesmaximal, characteristic, complemented
13D4⋊C8×D73C211yesmaximal, characteristic, complemented
14C8×#11924011C211yesmaximal, characteristic, complemented
15C73×#1589C211yesmaximal, characteristic, complemented
16C2×#14556615C411yescharacteristic
17C4×#11924027C2211yescharacteristic
18C4×D4×D73C2211yescharacteristic
19#17988717C2211yescharacteristic, complemented
20#17988714C2211yescharacteristic, complemented
21#17988699C411yescharacteristic
22#17988696C411yescharacteristic
23#17988686C411yescharacteristic
24#17988684C411yescharacteristic
25C4×#11924017C2211yescharacteristic
26C4×#11924016C2211yescharacteristic
27C4×#11924013C2211yescharacteristic
28#17988666C2211yescharacteristic
29#17988665C2211yescharacteristic
30#17988665C2211yescharacteristic, complemented
31#17988664C2211yescharacteristic
32#17988663—21no—
33#17988662—21no—
34#17988661—21no—
35#17988656—21no—
36#17988641—21no—
37#17988639C2211yescharacteristic
38#17988638C2211yescharacteristic
39#17988628C411yescharacteristic
40#17988610—21no—
41C4⋊C8×D73C2211yescharacteristic
42#17988590C411yescharacteristic
43#17988585C411yescharacteristic
44#17988550—21no—
45#17988549—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
80C2×C8×D73—21no—
81C2×C8×D73—21no—
82C4×Dic146C2311yescharacteristic
83#14556556C2×C411yescharacteristic
84C4×D292C2311yescharacteristic
85D4×Dic73C2×C411yescharacteristic
86D4×Dic73C2×C411yescharacteristic
87#14556553C2×C411yescharacteristic
88C4×C22⋊D73—21no—
89C4×C22⋊D73—21no—
90#14556551—21no—
91#14556551—21no—
92#14556549C2×C411yescharacteristic
93#14556547—21no—
94C4⋊C4×D73C2×C411yescharacteristic
95C22⋊C4×D73—21no—
96C2×C4×Dic73—21no—
97C42×D73C2311yescharacteristic
98#14556524C2×C411yescharacteristic
99#14556523C2×C411yescharacteristic
100#14556522C2×C411yescharacteristic
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520D411yescharacteristic
107#14556520D411yescharacteristic
108#14556516C2×C411yescharacteristic
109#14556514C2×C411yescharacteristic
110#14556513C2×C411yescharacteristic
111#14556512C2×C411yescharacteristic
112#14556512C2×C411yescharacteristic
113#14556511C2×C411yescharacteristic
114C22⋊#10596831—21no—
115C22⋊#10596831—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503D411yescharacteristic
120#14556503D411yescharacteristic
121#14556501—21no—
122#14556501—21no—
123#14556501C2×C411yescharacteristic
124#14556501C2×C411yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C8×Dic73—21no—
129C8×Dic73—21no—
130C2×D8×C73C2×C411yescharacteristic
131C2×M16×C73—21no—
132C2×M16×C73—21no—
133C4×D4×C73C2311yescharacteristic
134C4×D4×C73C2311yescharacteristic
135#107×C73C2×C411yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#102×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73—21no—
141C22⋊C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C23×D73—21no—
146#11924027—22no—
147#11924027—22no—
148C2×Dic146C22×C411yescharacteristic
149C2×D292C22×C411yescharacteristic
150C2×C22⋊D73—21no—
151C2×C22⋊D73—21no—
152D4×D73—22no—
153D4×D73—22no—
154C22×Dic73—21no—
155C22×Dic73—21no—
156C22×Dic73—21no—
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160C2×C4×D73—81no—
161C2×C4×D73—81no—
162C2×C4×D73C22×C411yescharacteristic
163C2×C4×D73C2×D411yescharacteristic
164C2×C4×D73#4111yescharacteristic
165#11924017—22no—
166#11924016—22no—
167#11924013—22no—
168#11924012—81no—
169#11924012—81no—
170#11924011—22no—
171#11924011—81no—
172#11924011—81no—
173#11924011M1614yescomplemented
174C2×#10596831—21no—
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831C22×C411yescharacteristic
178C2×#10596831C22×C411yescharacteristic
179C2×#10596831C2×D411yescharacteristic
180C2×#10596831C2×D411yescharacteristic
181C8×D73—42no—
182C8×D73—81no—
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188C22⋊Dic73—21no—
189C22⋊Dic73—21no—
190#11924005—21no—
191#11924005—21no—
192#11924005—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73C22×C411yescharacteristic
197C4×Dic73C2×D411yescharacteristic
198C4×Dic73#4111yescharacteristic
199C2×D4×C73C22×C411yescharacteristic
200C2×D4×C73C22×C411yescharacteristic
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203D8×C73—22no—
204M16×C73—81no—
205M16×C73—81no—
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73C22×C411yescharacteristic
210C2×C8×C73C22×C411yescharacteristic
211C2×C8×C73C2×D411yescharacteristic
212C2×C8×C73C2×D411yescharacteristic
213C4⋊C4×C73C22×C411yescharacteristic
214C4⋊C4×C73C22×C411yescharacteristic
215C22⋊C4×C73—21no—
216C22⋊C4×C73—21no—
217C42×C73C2411yescharacteristic
218C23×C73—21no—
219C23×C73—21no—
220C22×D73—21no—
221C22×D73—42no—
222C22×D73—81no—
223C22×D73—81no—
224C22×D73C4×D411yescharacteristic
225D4×C73—21no—
226D4×C73—21no—
227D4×C73—21no—
228D4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73C23×C411yescharacteristic
236C2×C4×C73C22×D411yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C22⋊D73—42no—
239C22⋊D73—42no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—42no—
244C2×Dic73—81no—
245C2×Dic73—81no—
246C2×Dic73C4×D411yescharacteristic
247C2×Dic73C4×D411yescharacteristic
248C2×Dic73C4×D411yescharacteristic
249D292—21no—
250D292—21no—
251C4×D73—21no—
252C4×D73—21no—
253C4×D73—41no—
254C4×D73—41no—
255C4×D73—41no—
256C4×D73—81no—
257C4×D73—81no—
258C4×D73C2×M1612yes—
259C4×D73#13112yes—
260Dic146—21no—
261Dic146—21no—
262C8×C73—22no—
263C8×C73—21no—
264C8×C73—21no—
265C8×C73—21no—
266C8×C73—41no—
267C8×C73—41no—
268C8×C73C2×M1612yes—
269#10596831—21no—
270#10596831—21no—
271#10596831—21no—
272#10596831—41no—
273#10596831—41no—
274#10596831—41no—
275#10596831C2×M1612yes—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73C2×C4×D411yescharacteristic
283C2×D73—21no—
284C2×D73—21no—
285C2×D73—41no—
286C2×D73—41no—
287C2×D73—41no—
288C2×D73—84no—
289C2×D73#43612yes—
290C4×C73—21no—
291C4×C73—41no—
292C4×C73—41no—
293C4×C73C22×M1611yescharacteristic
294C4×C73C2×#13111yescharacteristic
295C4×C73#57611yescharacteristic
296C4×C73#57711yescharacteristic
297Dic73—21no—
298Dic73—21no—
299Dic73—41no—
300Dic73—41no—
301Dic73—41no—
302Dic73#43612yes—
303#17893—731nomaximal, Sylow
304C2×C73—81no—
305C2×C73—81no—
306C2×C73C2×#43611yescharacteristic
307C2×C73#294611yescharacteristic
308C2×C73#295611yescharacteristic
309D73—22no—
310D73—81no—
311#2989—731no—
312#2975—731no—
313C4×AGL(1,ℤ/8)—731no—
314#1594—731no—
315#1594—731no—
316#1592—731no—
317#1592—731no—
318#1590—731no—
319#1590—731no—
320#1589—731no—
321#1589D7311yescharacteristic, complemented
322#1565—731no—
323#1476—731no—
324C2×D4⋊C8—731no—
325C8×M16—731no—
326C73#1789311yescharacteristic, complemented, Sylow
327C2×AGL(1,ℤ/8)—731no—
328C4×#41—731no—
329C2×C4×D4—731no—
330#446—731no—
331#446—731no—
332#442—731no—
333#442—731no—
334#440—731no—
335#440—731no—
336C4×SD16—731no—
337C4×SD16—731no—
338C4×D8—731no—
339C4×D8C2×D7311yescharacteristic
340#436—731no—
341#436—731no—
342#436—731no—
343#436—731no—
344#436—1461no—
345#436—1461no—
346#436C2×D7311yescharacteristic
347#436C2×D7311yescharacteristic, complemented
348#435—1461no—
349#435—1461no—
350#432—1461no—
351#431—731no—
352#431—731no—
353#428—731no—
354#423—1461no—
355C2×C4⋊C8—731no—
356#418—731no—
357#417—731no—
358#407—1461no—
359#407—1461no—
360C4×M16—731no—
361C2×C4×C8—731no—
362#335—731no—
363#335C2×D7311yescharacteristic, complemented
364#334—731no—
365#334—731no—
366Q8⋊C8—731no—
367Q8⋊C8—731no—
368D4⋊C8—731no—
369D4⋊C8—731no—
370D4⋊C8—731no—
371D4⋊C8—731no—
372D4⋊C8C2×D7311yescharacteristic
373D4⋊C8C2×D7311yescharacteristic, complemented
374#322—731no—
375#322—731no—
376C82—731no—
377C82C2×D7311yescharacteristic, complemented
378C2×#41—731no—
379C22×D4—731no—
380C23×C4—1461no—
381AGL(1,ℤ/8)—1464no—
382C2×SD16—731no—
383C2×SD16—731no—
384C2×D8—731no—
385C2×D8C4×D7311yescharacteristic
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—731no—
389C2×M16—731no—
390C2×M16—1461no—
391C2×M16—1461no—
392C2×M16—1461no—
393C2×M16—1461no—
394C22×C8—1461no—
395C22×C8—1461no—
396C4×Q8—731no—
397C4×Q8—731no—
398C4×D4—731no—
399C4×D4—731no—
400C4×D4—731no—
401C4×D4—731no—
402C4×D4—1461no—
403C4×D4—1461no—
404C4×D4—1461no—
405C4×D4—1461no—
406C4×D4C22×D7311yescharacteristic
407C4×D4C22×D7311yescharacteristic
408#117—731no—
409#117—1461no—
410C2×C4⋊C4—731no—
411C2×C22⋊C4—1461no—
412C2×C42—731no—
413C2×C42—1461no—
414#107—731no—
415#107C4×D7311yescharacteristic
416#106—731no—
417#106—731no—
418C4⋊C8—731no—
419C4⋊C8—731no—
420C4⋊C8—731no—
421C4⋊C8—731no—
422C4⋊C8—1461no—
423C4⋊C8—1461no—
424C4⋊C8—1461no—
425C4⋊C8C22×D7311yescharacteristic
426C4⋊C8C22×D7311yescharacteristic
427#103—731no—
428#103—731no—
429#102—731no—
430#102—731no—
431#102—731no—
432#102—731no—
433#102C4×D7311yescharacteristic
434#102C4×D7311yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—21no—
437C22⋊C8—731no—
438C22⋊C8—731no—
439C22⋊C8—1461no—
440C22⋊C8—1461no—
441C22⋊C8—1461no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444#97—731no—
445#97—731no—
446#97—1461no—
447#97—1461no—
448C4×C8—731no—
449C4×C8—731no—
450C4×C8—731no—
451C4×C8—1461no—
452C4×C8—1461no—
453C4×C8C22×D7311yescharacteristic
454C4×C8C22×D7311yescharacteristic
455C4×C8C22×D7311yescharacteristic
456C24—1461no—
457#41—1462no—
458#41—1462no—
459C2×Q8—731no—
460C2×D4—731no—
461C2×D4—1462no—
462C2×D4—1462no—
463C2×D4—1461no—
464C2×D4—1461no—
465C2×D4C2×C4×D7311yescharacteristic
466C2×D4C2×C4×D7311yescharacteristic
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—5841no—
470C22×C4—5841no—
471C22×C4—731no—
472C22×C4—731no—
473C22×C4—731no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480SD16—1462no—
481SD16—1462no—
482D8—22no—
483D8—1462no—
484M16—81no—
485M16—81no—
486M16—5841no—
487M16—5841no—
488M16—5841no—
489M16—5841no—
490M16—734no—
491M16—1462no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—2922no—
496C2×C8—5841no—
497C2×C8—731no—
498C2×C8—731no—
499C2×C8—731no—
500C2×C8—731no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8C2×C4×D7311yescharacteristic
505C2×C8C2×C4×D7311yescharacteristic
506C2×C8D4×D7311yescharacteristic
507C2×C8D4×D7311yescharacteristic
508C4⋊C4—731no—
509C4⋊C4—731no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4C2×C4×D7311yescharacteristic
514C4⋊C4C2×C4×D7311yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—1461no—
518C22⋊C4—1461no—
519C22⋊C4—1461no—
520C22⋊C4—1461no—
521C22⋊C4—1461no—
522C42—731no—
523C42—731no—
524C42—731no—
525C42—1461no—
526C42—1461no—
527C42—1461no—
528C42C23×D7311yescharacteristic
529C23—21no—
530C23—21no—
531C23—2922no—
532C23—5841no—
533C23—5841no—
534C23—731no—
535C23—1461no—
536Q8—1461no—
537Q8—1461no—
538D4—21no—
539D4—21no—
540D4—21no—
541D4—21no—
542D4—2922no—
543D4—2922no—
544D4—1461no—
545D4—1461no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—2921no—
553C2×C4—2922no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—5841no—
557C2×C4—5841no—
558C2×C4—5841no—
559C2×C4—5841no—
560C2×C4—731no—
561C2×C4—732no—
562C2×C4—732no—
563C2×C4—731no—
564C2×C4—731no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4C22×C4×D7311yescharacteristic
571C2×C4C2×D4×D7311yescharacteristic
572C2×C4#41×D7311yescharacteristic
573C8—22no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—2921no—
580C8—2921no—
581C8—2921no—
582C8—732no—
583C8—1461no—
584C8—1461no—
585C8—1461no—
586C8M16×D7312yes—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—2921no—
594C22—2921no—
595C22—2921no—
596C22—5844no—
597C22—732no—
598C22—1461no—
599C22—1461no—
600C22C4×D4×D7311yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—2921no—
605C4—2921no—
606C4—2921no—
607C4—732no—
608C4—1461no—
609C4—1461no—
610C4C2×M16×D7311yescharacteristic
611C4#131×D7311yescharacteristic
612C4#1798938811yescharacteristic
613C4#1798939411yescharacteristic
614C2—81no—
615C2—81no—
616C2—5841no—
617C2—1462no—
618C2#436×D7311yescharacteristic
619C2#2261875611yescharacteristic
620C2#2261883611yescharacteristic
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#17893
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. #22619046C2semidirect
2. #22618969C2semidirect
4. #22608845C2semidirect
5. #22608843C2semidirect
6. #22608833C2semidirect
7. #22608832C2semidirect
8. #22608824C2semidirect
9. #22608823C2semidirect
10. #22608813C2semidirect
11. #22608682C2semidirect
12. #22608310C2semidirect
13. D4⋊C8×D73C2semidirect
14. C8×#11924011C2semidirect
15. C73×#1589C2semidirect
19. #17988717C22semidirect
20. #17988714C22semidirect
30. #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
173. #11924011M16semidirect
321. #1589D73semidirect
326. C73#17893semidirect
347. #436C2×D73semidirect
363. #335C2×D73semidirect
373. D4⋊C8C2×D73semidirect
377. C82C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226190462
#226189692
C4×#145566152
#226088452
#226088432
#226088332
#226088322
#226088242
#226088232
#226088132
#226086822
#226083102
D4⋊C8×D732
C8×#119240112
C73×#15892
#1789373