Finite Group Databasebeta

← #34018524

Subgroups — #34018524

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#22626854C211yesmaximal, characteristic, complemented
2#22626794C211yesmaximal, characteristic, complemented
3#22625581C211yesmaximal, characteristic, complemented
4#22625579C211yesmaximal, characteristic, complemented
5#22625573C211yesmaximal, characteristic, complemented
6#22625572C211yesmaximal, characteristic, complemented
7#22625568C211yesmaximal, characteristic, complemented
8#22625566C211yesmaximal, characteristic, complemented
9#22625564C211yesmaximal, characteristic, complemented
10#22625562C211yesmaximal, characteristic, complemented
11#22625483C211yesmaximal, characteristic, complemented
12#22625223C211yesmaximal, characteristic, complemented
13#22625217C211yesmaximal, characteristic, complemented
14#22625129C211yesmaximal, characteristic, complemented
15#22618742C211yesmaximal, characteristic
16#17989637C2211yescharacteristic, complemented
17#17989637C2211yescharacteristic, complemented
18#17989622C2211yescharacteristic
19#17989622C2211yescharacteristic
20#17989622C2211yescharacteristic, complemented
21#17989622C2211yescharacteristic, complemented
22#17989621C2211yescharacteristic
23#17989621C2211yescharacteristic
24#17989620—21no—
25#17989620—21no—
26#17989619—21no—
27#17989619—21no—
28#17989607—21no—
29#17989606C2211yescharacteristic
30#17989605C2211yescharacteristic
31#17989590—21no—
32#17989588C2211yescharacteristic
33#17989559—21no—
34#17989558—21no—
35C4×#11924036C2211yescharacteristic
36#17989451C2211yescharacteristic, complemented
37#17989450C2211yescharacteristic, complemented
38#17989449C2211yescharacteristic, complemented
39#17989448C2211yescharacteristic, complemented
40#17989437C2211yescharacteristic, complemented
41#17989437C2211yescharacteristic, complemented
42#17989435C2211yescharacteristic
43#17989435C2211yescharacteristic
44D4⋊#10596833C2211yescharacteristic
45D4⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic, complemented
47D4⋊#10596833C2211yescharacteristic, complemented
48#17989428C2211yescharacteristic, complemented
49#17989428C2211yescharacteristic, complemented
50C8×#10596833C2211yescharacteristic, complemented
51C8×#10596833C2211yescharacteristic, complemented
52C2×#14556615C411yescharacteristic
53C4×#11924027C2211yescharacteristic
54C4×D4×D73C2211yescharacteristic
55#17988697C411yescharacteristic
56#17988689C411yescharacteristic
57#17988687C411yescharacteristic
58#17988678C2211yescharacteristic
59#17988674C2211yescharacteristic
60D8×Dic73C2211yescharacteristic
61#17988667C2211yescharacteristic
62#17988626C411yescharacteristic
63#17988590C411yescharacteristic
64#17988585C411yescharacteristic
65#17988539C2211yescharacteristic
66#442×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924038—21no—
69C2×#11924037—21no—
70C2×#11924037—21no—
71C2×#11924037—21no—
72C2×#11924037—21no—
73C2×#11924036—21no—
74C2×#11924036—21no—
75#14556658—21no—
76#14556658D411yescharacteristic
77#14556658D411yescharacteristic
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648—21no—
86#14556648—21no—
87#14556648D411yescharacteristic
88#14556648D411yescharacteristic
89C22⋊#10596833—21no—
90C22⋊#10596833—21no—
91C22⋊#10596833—21no—
92C22⋊#10596833—21no—
93#14556646—21no—
94#14556646—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×#11924027C2×C411yescharacteristic
102C2×D4×D73C2×C411yescharacteristic
103C22×C4×D73—21no—
104#14556615—24no—
105C2×#11924017C2×C411yescharacteristic
106C2×#11924016C2×C411yescharacteristic
107C2×#11924013C2×C411yescharacteristic
108C2×#11924011C2×C411yescharacteristic
109C2×#11924011C2×C411yescharacteristic
110C4×Dic146C2×C411yescharacteristic
111#14556556C2311yescharacteristic
112C4×D292C2×C411yescharacteristic
113D4×Dic73C2311yescharacteristic
114D4×Dic73C2311yescharacteristic
115#14556553C2311yescharacteristic
116C4×C22⋊D73—21no—
117C4×C22⋊D73—21no—
118#14556551—21no—
119#14556551—21no—
120#14556549C2×C411yescharacteristic
121#14556547—21no—
122C4⋊C4×D73C2×C411yescharacteristic
123C22⋊C4×D73—21no—
124C2×C4×Dic73—21no—
125C42×D73C2311yescharacteristic
126#14556525C2×C411yescharacteristic
127#14556524C2×C411yescharacteristic
128#14556522C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556514C2×C411yescharacteristic
131#14556513C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134#14556511C2×C411yescharacteristic
135#14556502C2×C411yescharacteristic
136C8×Dic73C2311yescharacteristic
137C8×Dic73C2311yescharacteristic
138C2×D8×C73C2×C411yescharacteristic
139C4×D4×C73C2×C411yescharacteristic
140C4×D4×C73C2×C411yescharacteristic
141#106×C73C2×C411yescharacteristic
142#102×C73C2×C411yescharacteristic
143#102×C73C2×C411yescharacteristic
144#97×C73C2×C411yescharacteristic
145#11924038—81no—
146#11924038—81no—
147#11924037—81no—
148#11924037—81no—
149#11924037—81no—
150#11924037—81no—
151#11924036—42no—
152#11924036—81no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833—21no—
157C2×#10596833—21no—
158C2×#10596833—21no—
159C2×#10596833C2×D411yescharacteristic
160C2×#10596833C2×D411yescharacteristic
161C2×#10596833C2×D411yescharacteristic
162C2×#10596833C2×D411yescharacteristic
163C23×D73—21no—
164#11924027—22no—
165#11924027—22no—
166C2×Dic146C22×C411yescharacteristic
167C2×D292C22×C411yescharacteristic
168C2×C22⋊D73—21no—
169C2×C22⋊D73—21no—
170D4×D73—22no—
171D4×D73—22no—
172C22×Dic73—21no—
173C22×Dic73—21no—
174C22×Dic73—21no—
175C2×C4×D73—21no—
176C2×C4×D73—21no—
177C2×C4×D73—21no—
178C2×C4×D73—81no—
179C2×C4×D73—81no—
180C2×C4×D73C22×C411yescharacteristic
181C2×C4×D73C2×D411yescharacteristic
182C2×C4×D73#4111yescharacteristic
183#11924017—22no—
184#11924016—22no—
185#11924013—22no—
186#11924011—22no—
187#11924011M1614yescomplemented
188C2×#10596831C22×C411yescharacteristic
189C2×#10596831C22×C411yescharacteristic
190C4⋊Dic73C22×C411yescharacteristic
191C4⋊Dic73C22×C411yescharacteristic
192#11924007—21no—
193#11924007—21no—
194#11924007—21no—
195C22⋊Dic73—21no—
196C22⋊Dic73—21no—
197#11924005—21no—
198#11924005—21no—
199#11924005—21no—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73—21no—
203C4×Dic73C2×D411yescharacteristic
204C4×Dic73#4111yescharacteristic
205C4×Dic73C2411yescharacteristic
206C2×D4×C73C22×C411yescharacteristic
207C2×D4×C73C22×C411yescharacteristic
208C22×C4×C73—21no—
209C22×C4×C73—21no—
210D8×C73—22no—
211C2×C8×C73C22×C411yescharacteristic
212C2×C8×C73C22×C411yescharacteristic
213C4⋊C4×C73C22×C411yescharacteristic
214C4⋊C4×C73C22×C411yescharacteristic
215C22⋊C4×C73—21no—
216C22⋊C4×C73—21no—
217C42×C73C22×C411yescharacteristic
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—81no—
232C2×C4×C73—81no—
233C2×C4×C73C4×D411yescharacteristic
234C2×C4×C73C4×D411yescharacteristic
235C2×C4×C73C23×C411yescharacteristic
236C22⋊D73—42no—
237C22⋊D73—42no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—42no—
246C2×Dic73C4×D411yescharacteristic
247C2×Dic73C22×D411yescharacteristic
248C2×Dic73C2×#4111yescharacteristic
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—
262#10596833—22no—
263#10596833—21no—
264#10596833—21no—
265#10596833—21no—
266#10596833—21no—
267#10596833—41no—
268#10596833—41no—
269#10596833—41no—
270#10596833—41no—
271#10596833—41no—
272C8×C73—21no—
273C8×C73C2×M1612yes—
274#10596831—21no—
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—21no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73C22×M1611yescharacteristic
295C4×C73C2×#13111yescharacteristic
296Dic73—21no—
297Dic73—41no—
298Dic73—41no—
299Dic73—41no—
300Dic73#43612yes—
301Dic73#57611yescharacteristic
302Dic73#57711yescharacteristic
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#1589—731no—
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#6440811yescharacteristic
334#440—731no—
335#440—731no—
336C4×SD16—731no—
337C4×SD16—731no—
338C4×D8—731no—
339C4×D8—731no—
340#436—731no—
341#436—731no—
342#436—731no—
343#436—731no—
344#436—731no—
345#436—731no—
346#436—1461no—
347#436—1461no—
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#335—731no—
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⋊C8—731no—
373D4⋊C8—731no—
374#322—731no—
375#322—731no—
376C82—731no—
377C82—731no—
378C2×#41—731no—
379C22×D4—731no—
380C23×C4—1461no—
381AGL(1,ℤ/8)—1464no—
382C2×SD16—731no—
383C2×SD16—731no—
384C2×D8—731no—
385C2×D8C2×#6440811yescharacteristic
386C2×M16—731no—
387C2×M16—731no—
388C2×M16—1461no—
389C2×M16—1461no—
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×D4C2×#6440811yescharacteristic
407C4×D4C2×#6440811yescharacteristic
408#117—731no—
409#117—1461no—
410C2×C4⋊C4—731no—
411C2×C22⋊C4—1461no—
412C2×C42—731no—
413C2×C42—1461no—
414#107—731no—
415#107—731no—
416#106—731no—
417#106C2×#6440811yescharacteristic
418C4⋊C8—731no—
419C4⋊C8—731no—
420C4⋊C8—731no—
421C4⋊C8—731no—
422C4⋊C8—731no—
423C4⋊C8—731no—
424C4⋊C8—1461no—
425C4⋊C8—1461no—
426C4⋊C8—1461no—
427#103—731no—
428#103—731no—
429#102—731no—
430#102—731no—
431#102—731no—
432#102—731no—
433#102C2×#6440811yescharacteristic
434#102C2×#6440811yescharacteristic
435C22⋊C8—731no—
436C22⋊C8—731no—
437C22⋊C8—1461no—
438C22⋊C8—1461no—
439C22⋊C8—1461no—
440C22⋊C8—1461no—
441C22⋊C8—1461no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444#97—731no—
445#97—1461no—
446#97—1461no—
447#97C2×#6440811yescharacteristic
448C4×C8—731no—
449C4×C8—731no—
450C4×C8—731no—
451C4×C8—731no—
452C4×C8—731no—
453C4×C8—731no—
454C4×C8—1461no—
455C4×C8—1461no—
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×D4C22×#6440811yescharacteristic
466C2×D4C22×#6440811yescharacteristic
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—5841no—
485M16—5841no—
486M16—5841no—
487M16—5841no—
488M16—5841no—
489M16—5841no—
490M16—734no—
491M16—1462no—
492C2×C8—2922no—
493C2×C8—5841no—
494C2×C8—731no—
495C2×C8—731no—
496C2×C8—731no—
497C2×C8—731no—
498C2×C8—731no—
499C2×C8—731no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8C22×#6440811yescharacteristic
507C2×C8C22×#6440811yescharacteristic
508C4⋊C4—731no—
509C4⋊C4—731no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4C22×#6440811yescharacteristic
514C4⋊C4C22×#6440811yescharacteristic
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—
528C42C22×#6440811yescharacteristic
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—81no—
549C2×C4—81no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2922no—
557C2×C4—2921no—
558C2×C4—5841no—
559C2×C4—5841no—
560C2×C4—731no—
561C2×C4—731no—
562C2×C4—732no—
563C2×C4—732no—
564C2×C4—731no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4D4×#6440811yescharacteristic
571C2×C4D4×#6440811yescharacteristic
572C2×C4C23×#6440811yescharacteristic
573C8—21no—
574C8—2921no—
575C8—2921no—
576C8—2921no—
577C8—2921no—
578C8—2921no—
579C8—732no—
580C8—1461no—
581C8—1461no—
582C8—1461no—
583C8—1461no—
584C8—1462no—
585C8—1461no—
586C8D73⋊M1612yes—
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—
600C22C2×D4×#6440811yescharacteristic
601C4—21no—
602C4—21no—
603C4—41no—
604C4—41no—
605C4—2921no—
606C4—2921no—
607C4—2921no—
608C4—732no—
609C4—731no—
610C4—731no—
611C4—1461no—
612C4C2×D73⋊M1611yescharacteristic
613C4D73⋊#13111yescharacteristic
614C2—81no—
615C2—81no—
616C2—5841no—
617C2—1462no—
618C2D73⋊#43611yescharacteristic
619C2#2262668811yescharacteristic
620C2#2262670511yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#442×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#14556615C4
cosocleC4×Dic73C24
hypercenter#442#64408
hypocenterC73#17893
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #22626854C2semidirect
2. #22626794C2semidirect
3. #22625581C2semidirect
4. #22625579C2semidirect
5. #22625573C2semidirect
6. #22625572C2semidirect
7. #22625568C2semidirect
8. #22625566C2semidirect
9. #22625564C2semidirect
10. #22625562C2semidirect
11. #22625483C2semidirect
12. #22625223C2semidirect
13. #22625217C2semidirect
14. #22625129C2semidirect
16. #17989637C22semidirect
17. #17989637C22semidirect
20. #17989622C22semidirect
21. #17989622C22semidirect
36. #17989451C22semidirect
37. #17989450C22semidirect
38. #17989449C22semidirect
39. #17989448C22semidirect
40. #17989437C22semidirect
41. #17989437C22semidirect
46. D4⋊#10596833C22semidirect
47. D4⋊#10596833C22semidirect
48. #17989428C22semidirect
49. #17989428C22semidirect
50. C8×#10596833C22semidirect
51. C8×#10596833C22semidirect
187. #11924011M16semidirect
326. C73#17893semidirect

Maximal subgroups

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

subgroupindex
#226268542
#226267942
#226255812
#226255792
#226255732
#226255722
#226255682
#226255662
#226255642
#226255622
#226254832
#226252232
#226252172
#226251292
#226187422
#1789373