Finite Group Databasebeta

← #34019060

Subgroups — #34019060

Order 18688.

566 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626873C211yesmaximal, characteristic
2#22626835C211yesmaximal, characteristic, complemented
3#22625561C211yesmaximal, characteristic, complemented
4#22625555C211yesmaximal, characteristic, complemented
5#22625554C211yesmaximal, characteristic, complemented
6#22625548C211yesmaximal, characteristic, complemented
7#22625545C211yesmaximal, characteristic, complemented
8#22625543C211yesmaximal, characteristic, complemented
9#22625540C211yesmaximal, characteristic, complemented
10#22625534C211yesmaximal, characteristic, complemented
11#22625495C211yesmaximal, characteristic, complemented
12#22625282C211yesmaximal, characteristic
13#22625279C211yesmaximal, characteristic, complemented
14#22625138C211yesmaximal, characteristic, complemented
15#22620530C211yesmaximal, characteristic
16#17989639—21no—
17#17989638C2211yescharacteristic
18#17989637—21no—
19#17989637C2211yescharacteristic
20#17989637C2211yescharacteristic
21#17989636C2211yescharacteristic
22#17989636C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24#17989622—21no—
25#17989622C2211yescharacteristic
26#17989618—21no—
27D4×#10596833C2211yescharacteristic
28#17989611—21no—
29#17989611—21no—
30#17989611—21no—
31#17989608C2211yescharacteristic
32#17989603C2211yescharacteristic
33#17989593—21no—
34#17989590C2211yescharacteristic
35C2×C4×#10596833C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42#17989437C2211yescharacteristic
43#17989437C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989429C2211yescharacteristic
51#17989429C2211yescharacteristic
52#17989139C2211yescharacteristic
53#17989118C2211yescharacteristic
54#17988995C411yescharacteristic
55#17988978C411yescharacteristic
56#17988954C411yescharacteristic
57#17988945C411yescharacteristic
58#17988845C411yescharacteristic
59#17988843C411yescharacteristic
60#17988831C411yescharacteristic
61#17988681C2211yescharacteristic
62#17988679C2211yescharacteristic
63#17988678C2211yescharacteristic
64SD16×Dic73C2211yescharacteristic
65M16×Dic73C2211yescharacteristic
66#468×C73C411yescharacteristic
67C2×#11924037—21no—
68C2×#11924037—21no—
69C22×#10596833—21no—
70C22×#10596833—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658—21no—
75C4⋊#10596833—21no—
76C4⋊#10596833—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833D411yescharacteristic
79C4⋊#10596833D411yescharacteristic
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C22⋊#10596833—21no—
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833D411yescharacteristic
88C22⋊#10596833D411yescharacteristic
89#14556646—21no—
90#14556646—21no—
91#14556646—21no—
92#14556646—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×Dic292C2×C411yescharacteristic
102C2×#11924018C2×C411yescharacteristic
103C2×#11924016C2×C411yescharacteristic
104C2×#11924012C2×C411yescharacteristic
105#14556590C2×C411yescharacteristic
106#14556589C2×C411yescharacteristic
107#14556584—21no—
108#14556582C2×C411yescharacteristic
109#14556581—21no—
110#14556578C2×C411yescharacteristic
111#14556572—21no—
112#14556570C2×C411yescharacteristic
113#14556570C2×C411yescharacteristic
114#14556568—21no—
115Q8×Dic73C2311yescharacteristic
116#14556556—21no—
117#14556556C2311yescharacteristic
118#14556556C2311yescharacteristic
119D4×Dic73—21no—
120D4×Dic73C2311yescharacteristic
121#14556547—21no—
122#14556547—21no—
123C2×C4×Dic73C2311yescharacteristic
124#14556524C2×C411yescharacteristic
125#14556523C2×C411yescharacteristic
126#14556522C2×C411yescharacteristic
127#14556516C2×C411yescharacteristic
128#14556516C2×C411yescharacteristic
129#14556515C2×C411yescharacteristic
130#14556514C2×C411yescharacteristic
131#14556514C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556501C2×C411yescharacteristic
134#14556501C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C2×SD16×C73C2×C411yescharacteristic
138C2×M16×C73C2×C411yescharacteristic
139C22⋊Q8×C73C2×C411yescharacteristic
140#121×C73C2×C411yescharacteristic
141#107×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#102×C73C2×C411yescharacteristic
144#11924037—81no—
145#11924037—81no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—42no—
153C2×#10596833—81no—
154C2×#10596833C2×D411yescharacteristic
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×Dic146—21no—
159C2×Dic146C22×C411yescharacteristic
160C2×Dic146C22×C411yescharacteristic
161C22×Dic73—21no—
162C22×Dic73C2×D411yescharacteristic
163C22×Dic73#4111yescharacteristic
164Dic292—41no—
165#11924018—41no—
166#11924016—41no—
167#11924012—41no—
168C2×#10596831C22×C411yescharacteristic
169C2×#10596831C22×C411yescharacteristic
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73C22×C411yescharacteristic
173C4⋊Dic73C22×C411yescharacteristic
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182C22⋊Dic73—21no—
183C22⋊Dic73—21no—
184C22⋊Dic73—21no—
185C22⋊Dic73—21no—
186C22⋊Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73C22×C411yescharacteristic
191C4×Dic73C2×D411yescharacteristic
192C4×Dic73#4111yescharacteristic
193C4×Dic73C2411yescharacteristic
194C2×Q8×C73C22×C411yescharacteristic
195C2×D4×C73—21no—
196C2×D4×C73C22×C411yescharacteristic
197C22×C4×C73C22×C411yescharacteristic
198SD16×C73—41no—
199M16×C73—41no—
200C2×C8×C73C22×C411yescharacteristic
201C2×C8×C73C22×C411yescharacteristic
202C4⋊C4×C73—21no—
203C4⋊C4×C73C22×C411yescharacteristic
204C4⋊C4×C73C22×C411yescharacteristic
205C22⋊C4×C73—21no—
206C22⋊C4×C73—21no—
207C23×C73—21no—
208C23×C73C4×D411yescharacteristic
209Q8×C73—21no—
210Q8×C73—21no—
211D4×C73—21no—
212D4×C73—21no—
213D4×C73—81no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73C4×D411yescharacteristic
220C2×C4×C73C23×C411yescharacteristic
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—41no—
228C2×Dic73—41no—
229C2×Dic73—41no—
230C2×Dic73C4×D411yescharacteristic
231C2×Dic73C4×D411yescharacteristic
232C2×Dic73C22×D411yescharacteristic
233C2×Dic73C2×#4111yescharacteristic
234Dic146—21no—
235Dic146—21no—
236Dic146—21no—
237Dic146—21no—
238Dic146—81no—
239#10596833—22no—
240#10596833—21no—
241#10596833—21no—
242#10596833—21no—
243#10596833—21no—
244#10596833—41no—
245#10596833—41no—
246#10596833—41no—
247#10596833—41no—
248#10596833—41no—
249C8×C73—21no—
250C8×C73—21no—
251#10596831—21no—
252#10596831—21no—
253C22×C73—21no—
254C22×C73—21no—
255C22×C73—21no—
256C22×C73—41no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73C2×C4×D411yescharacteristic
260C4×C73—21no—
261C4×C73—41no—
262C4×C73—41no—
263C4×C73—41no—
264C4×C73#56811yescharacteristic
265C4×C73#56811yescharacteristic
266Dic73—21no—
267Dic73—21no—
268Dic73—21no—
269Dic73—41no—
270Dic73—41no—
271Dic73—41no—
272Dic73—41no—
273Dic73#57711yescharacteristic
274Dic73#57811yescharacteristic
275#17988—731nomaximal, Sylow
276C2×C73—41no—
277C2×C73—81no—
278C2×C73#293211yescharacteristic
279C2×C73#294711yescharacteristic
280C2×C73#295611yescharacteristic
281#3172—731no—
282#2993—731no—
283#2985—731no—
284#1588—731no—
285#1587—731no—
286#1587—731no—
287#1585—731no—
288#1584—731no—
289#1584—731no—
290#1583—731no—
291#1582—731no—
292#1569—731no—
293#1494—731no—
294#1493—731no—
295#1451—731no—
296C73#1798811yescharacteristic, complemented, Sylow
297#533—731no—
298#529—731no—
299#501—731no—
300#497—731no—
301#491—731no—
302#489—731no—
303#470—731no—
304#470—731no—
305#468—731no—
306#468#6440811yescharacteristic
307#446—731no—
308#446—731no—
309#446—731no—
310#446—1461no—
311#446—1461no—
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—731no—
315C4×Q16—731no—
316C4×Q16—731no—
317C4×SD16—731no—
318C4×SD16—731no—
319#436—731no—
320#436—1461no—
321#435—1461no—
322C8×D4—731no—
323#433—1461no—
324#433—1461no—
325#433—1461no—
326#432—731no—
327#431—731no—
328#424—1461no—
329#423—731no—
330C4×M16—731no—
331C2×C4×C8—731no—
332#335—731no—
333#335—731no—
334#334—731no—
335#334—731no—
336Q8⋊C8—731no—
337Q8⋊C8—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342D4⋊C8—731no—
343D4⋊C8—731no—
344#322—731no—
345#322—731no—
346#322—731no—
347#322—731no—
348C2×Q16—731no—
349C2×Q16—731no—
350C2×SD16—731no—
351C2×SD16C2×#6440811yescharacteristic
352C2×M16—731no—
353C2×M16—1461no—
354C2×M16—1461no—
355C2×M16C2×#6440811yescharacteristic
356C22×C8—1461no—
357C22×C8—1461no—
358C4⋊Q8—731no—
359C4⋊Q8—731no—
360#126—1461no—
361#125—731no—
362#125—1461no—
363#124—731no—
364#123—1461no—
365C22⋊Q8—731no—
366C22⋊Q8—731no—
367C22⋊Q8—1461no—
368C22⋊Q8C2×#6440811yescharacteristic
369#121C2×#6440811yescharacteristic
370C4×Q8—731no—
371C4×Q8—731no—
372C4×Q8—731no—
373C4×Q8—1461no—
374C4×D4—731no—
375C4×D4—1461no—
376#117—1461no—
377#117—1461no—
378C2×C42—731no—
379#107—731no—
380#107C2×#6440811yescharacteristic
381#106—731no—
382#106—731no—
383C4⋊C8—731no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—731no—
388C4⋊C8—731no—
389C4⋊C8—1461no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396#103—731no—
397#103—731no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103C2×#6440811yescharacteristic
402#102—731no—
403#102C2×#6440811yescharacteristic
404C22⋊C8—731no—
405C22⋊C8—731no—
406C22⋊C8—1461no—
407C22⋊C8—1461no—
408C22⋊C8—1461no—
409#97—731no—
410#97—731no—
411#97—1461no—
412#97—1461no—
413#97—1461no—
414#97—1461no—
415C4×C8—731no—
416C4×C8—731no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—1461no—
422C4×C8—1461no—
423C4×C8—1461no—
424C4×C8—1461no—
425C2×Q8—731no—
426C2×Q8—731no—
427C2×Q8—1461no—
428C2×Q8C22×#6440811yescharacteristic
429C2×D4—21no—
430C2×D4C22×#6440811yescharacteristic
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—1461no—
434C22×C4C22×#6440811yescharacteristic
435Q16—2921no—
436Q16—2921no—
437SD16—41no—
438SD16—2921no—
439M16—41no—
440M16—2921no—
441M16—5841no—
442M16—5841no—
443C2×C8—2922no—
444C2×C8—5841no—
445C2×C8—731no—
446C2×C8—731no—
447C2×C8—731no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8C22×#6440811yescharacteristic
458C2×C8C22×#6440811yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—731no—
461C4⋊C4—731no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4C22×#6440811yescharacteristic
473C4⋊C4C22×#6440811yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1461no—
477C22⋊C4—1461no—
478C22⋊C4—1461no—
479C22⋊C4—1461no—
480C22⋊C4—1461no—
481C42—731no—
482C42—731no—
483C42—731no—
484C42—731no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C23—21no—
489C23D4×#6440811yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—5841no—
493Q8—1461no—
494Q8—1461no—
495Q8—1461no—
496Q8—1461no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—2921no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—731no—
509C2×C4—731no—
510C2×C4—731no—
511C2×C4—731no—
512C2×C4—1461no—
513C2×C4—1461no—
514C2×C4—1461no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4D4×#6440811yescharacteristic
519C2×C4C23×#6440811yescharacteristic
520C8—21no—
521C8—21no—
522C8—2921no—
523C8—2921no—
524C8—2921no—
525C8—2921no—
526C8—2921no—
527C8—1461no—
528C8—1461no—
529C8—1461no—
530C8—1461no—
531C8—1462no—
532C8—1461no—
533C8—1461no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22C2×D4×#6440811yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—2921no—
546C4—2921no—
547C4—2921no—
548C4—2921no—
549C4—731no—
550C4—731no—
551C4—1461no—
552C4—1461no—
553C4—1461no—
554C4#1798966511yescharacteristic
555C4#1798966611yescharacteristic
556C2—41no—
557C2—81no—
558C2#2262665911yescharacteristic
559C2#2262668911yescharacteristic
560C2#2262670511yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#468×C73C4
socleC22×C73C2×C4×D4
prime radical#121×C73C2×C4
cosocleC4×Dic73C24
hypercenter#468#64408
hypocenterC73#17988
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #22626835C2semidirect
3. #22625561C2semidirect
4. #22625555C2semidirect
5. #22625554C2semidirect
6. #22625548C2semidirect
7. #22625545C2semidirect
8. #22625543C2semidirect
9. #22625540C2semidirect
10. #22625534C2semidirect
11. #22625495C2semidirect
13. #22625279C2semidirect
14. #22625138C2semidirect
296. C73#17988semidirect

Maximal subgroups

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

subgroupindex
#226268732
#226268352
#226255612
#226255552
#226255542
#226255482
#226255452
#226255432
#226255402
#226255342
#226254952
#226252822
#226252792
#226251382
#226205302
#1798873