Finite Group Databasebeta

← #10896450

Subgroups — #10896450

Order 768.

604 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83329C211yesmaximal, characteristic, complemented
2#78990C211yesmaximal, characteristic
3C3×#2031C211yesmaximal, characteristic, complemented
4#71842C211yesmaximal, characteristic, complemented
5#70405C211yesmaximal, characteristic, complemented
6#70375C211yesmaximal, characteristic, complemented
7#70374C211yesmaximal, characteristic
8#70044C211yesmaximal, characteristic, complemented
9#70042C211yesmaximal, characteristic
10#69034C211yesmaximal, characteristic, complemented
11#69015C211yesmaximal, characteristic, complemented
12#68773C211yesmaximal, characteristic, complemented
13#68748C211yesmaximal, characteristic, complemented
14#68715C211yesmaximal, characteristic, complemented
15#68699C211yesmaximal, characteristic, complemented
16#29271—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C2×#950C2211yescharacteristic
19#5819C2211yescharacteristic
20#5745C2211yescharacteristic
21C3×#421C2211yescharacteristic
22C2×C3×#104C2211yescharacteristic
23C3×#410C2211yescharacteristic
24#5236C2211yescharacteristic
25#5234C2211yescharacteristic
26#5129C2211yescharacteristic
27#5125C2211yescharacteristic
28#5124C2211yescharacteristic
29C2×#822C2211yescharacteristic
30C2×#822C2211yescharacteristic
31#5119C2211yescharacteristic
32#5024—22no—
33#5024—22no—
34#5006C2211yescharacteristic
35#5005C2211yescharacteristic
36#5005C2211yescharacteristic
37#4999C2211yescharacteristic
38#4999C2211yescharacteristic
39#4999C2211yescharacteristic
40#4998C2211yescharacteristic
41#4997C2211yescharacteristic
42#4997C2211yescharacteristic
43C2×#805C2211yescharacteristic
44#4924C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4824—22no—
47#4823—22no—
48C3×#344C2211yescharacteristic
49C3×#339C2211yescharacteristic
50C3×H(3,ℤ/4)C2211yescharacteristic
51#4559C2211yescharacteristic
52#4533C2211yescharacteristic
53#4533C2211yescharacteristic
54#4531C2211yescharacteristic
55#4523C2211yescharacteristic
56#2961—31no—
57#2957—31no—
58#2098—31no—
59#2098—31no—
60#2031—31no—
61#2031D311yescharacteristic, complemented
62#1976—31no—
63#1975—31no—
64#1953—31no—
65#1946—31no—
66#1868—31no—
67#1868—31no—
68#1861—31no—
69#1861—31no—
70#1833—31no—
71C2×C3×#41C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C2×C3×M16C2311yescharacteristic
74C2×C3×M16C2311yescharacteristic
75C3×C4⋊Q8—21no—
76C3×#125—21no—
77C3×C22⋊Q8—22no—
78C3×C4×Q8—22no—
79C3×#117C2311yescharacteristic
80C3×#117C2311yescharacteristic
81C3×#117C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83#950—21no—
84#950—21no—
85#950—21no—
86#950—21no—
87#948—22no—
88#948—41no—
89C2×#216Q811yescharacteristic, complemented
90C2×#215Q811yescharacteristic, complemented
91C2×#214Q811yescharacteristic, complemented
92C2×#213Q811yescharacteristic, complemented
93#926C2311yescharacteristic
94C2×#208—41no—
95C2×#208C2311yescharacteristic
96C2×#208C2311yescharacteristic
97C2×#208C2311yescharacteristic
98C2×#208C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101#907—22no—
102D3×M16—41no—
103#901—41no—
104C3×#104—22no—
105C3×#104—21no—
106C3×#104—21no—
107C3×#99—22no—
108#837—41no—
109#830—41no—
110#822—21no—
111#822—21no—
112#822—21no—
113#822—21no—
114#822—41no—
115(C2×D3)⋊C8—41no—
116#819—41no—
117Dic3⋊C8—41no—
118#808—21no—
119#807—21no—
120#806—22no—
121#805—22no—
122#805—21no—
123#805—21no—
124#805—41no—
125C4⋊(C3⋊C8)—22no—
126C4⋊(C3⋊C8)—22no—
127C4⋊(C3⋊C8)—22no—
128C4⋊(C3⋊C8)—41no—
129#803—21no—
130#803—21no—
131#803—21no—
132#803D412yes—
133C4×C3⋊C8—21no—
134C4×C3⋊C8D412yes—
135C2×#135—31no—
136#568—31no—
137#568—31no—
138#533C2×D311yescharacteristic
139#446—62no—
140#446—62no—
141#444—62no—
142#443—62no—
143#430—31no—
144#430—31no—
145C2×#108—31no—
146C2×#108—31no—
147#427—31no—
148#424—31no—
149#424—31no—
150#424—31no—
151#424—31no—
152#423—31no—
153#421—31no—
154#421—31no—
155#421—31no—
156#421C2×D311yescharacteristic
157C2×#104—31no—
158C2×#104C2×D311yescharacteristic
159#410—31no—
160#410C2×D311yescharacteristic
161#408—31no—
162#408—31no—
163#405—31no—
164#405—31no—
165C2×#97—31no—
166#344—31no—
167#344—31no—
168#344—31no—
169#344C2×D311yescharacteristic
170#339—31no—
171#339C2×D311yescharacteristic
172H(3,ℤ/4)—31no—
173H(3,ℤ/4)C2×D311yescharacteristic
174C3×#41—21no—
175C3×#41—21no—
176C3×#41—21no—
177C3×#41—21no—
178C2×C3×Q8—22no—
179C2×C3×Q8C2×Q811yescharacteristic
180C2×C3×D4—21no—
181C2×C3×D4C2×Q811yescharacteristic
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4C2411yescharacteristic
185C2×#67—21no—
186#235—21no—
187#235—21no—
188#235—21no—
189#235—21no—
190C2×D12C2×Q811yescharacteristic
191C2×C4×D3—21no—
192C2×Dic6C2×Q811yescharacteristic
193C3×M16—21no—
194C3×M16—21no—
195C3×M16—21no—
196C3×M16—21no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C22⋊C4—21no—
205C3×C22⋊C4—21no—
206C3×C22⋊C4#4112yes—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42C2×D412yes—
211C3×C42#4111yescharacteristic
212C3×C42#4111yescharacteristic
213#217—21no—
214#216—21no—
215#216—21no—
216#215—21no—
217#215—21no—
218#214—21no—
219#214—21no—
220#213—21no—
221#213—21no—
222C4⋊Dic3—21no—
223C4×Dic3—21no—
224#208—21no—
225#208—21no—
226#208—21no—
227#208—21no—
228#208—21no—
229#208—21no—
230#208—21no—
231#208—21no—
232#208—42no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—42no—
240C2×C3⋊C8C2×D412yes—
241C2×C3⋊C8C2×D411yescharacteristic
242C2×C3⋊C8C2×D411yescharacteristic
243C2×C3⋊C8C2×Q811yescharacteristic
244C2×C3⋊C8C2×Q811yescharacteristic
245C2×C3⋊C8#4111yescharacteristic
246C2×C3⋊C8#4111yescharacteristic
247ΓL(1,𝔽25)—42no—
248D3×C8—42no—
249C2×#41—31no—
250C2×#41C22×D311yescharacteristic
251#135—61no—
252#135—61no—
253#135—61no—
254#135—61no—
255C2×Q16—31no—
256C2×SD16—31no—
257C2×SD16—31no—
258C2×D8—31no—
259#131—62no—
260#131—62no—
261#131—121no—
262#131—121no—
263C2×M16—31no—
264C2×M16—31no—
265C2×M16—31no—
266C2×M16—31no—
267C2×M16—121no—
268C2×M16—121no—
269C2×M16C22×D311yescharacteristic
270C2×M16C22×D311yescharacteristic
271C22×C8—31no—
272C22×C8—31no—
273C4⋊Q8—21no—
274#125—21no—
275C22⋊Q8—22no—
276C4×Q8—22no—
277#117—31no—
278#117C22×D311yescharacteristic
279#117C22×D311yescharacteristic
280#117C22×D311yescharacteristic
281C2×C42C22×D311yescharacteristic
282#108—61no—
283#108—61no—
284#108—61no—
285#108—61no—
286#108—121no—
287#108—121no—
288#107—61no—
289#106—61no—
290C4⋊C8—62no—
291C4⋊C8—62no—
292C4⋊C8—62no—
293C4⋊C8—121no—
294C4⋊C8—121no—
295#104—22no—
296#104—21no—
297#104—21no—
298#104—62no—
299#104—61no—
300#104—61no—
301#104—121no—
302#104—121no—
303#99—22no—
304#99—62no—
305C22⋊C8—121no—
306C22⋊C8—121no—
307#97—32no—
308#97—61no—
309#97—61no—
310#97—61no—
311C4×C8—32no—
312C4×C8—61no—
313C23×C3—21no—
314C23×C3C2×#4111yescharacteristic
315C22×D3—21no—
316C3×Q8—21no—
317C3×Q8—21no—
318C3×Q8—82no—
319C3×D4—21no—
320C3×D4—21no—
321C3×D4—41no—
322C3×D4—41no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—81no—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×Q811yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4H(5,𝔽2)11yescharacteristic
337C2×C3×C4H(5,𝔽2)11yescharacteristic
338#67—41no—
339#67—41no—
340C2×Dic3—21no—
341C2×Dic3—41no—
342D12—21no—
343D12—21no—
344C4×D3—41no—
345C4×D3—41no—
346Dic6—21no—
347Dic6—21no—
348C3×C8—41no—
349C3×C8—41no—
350C3⋊C8—22no—
351C3⋊C8—41no—
352C3⋊C8—41no—
353C3⋊C8—41no—
354C3⋊C8—41no—
355C3⋊C8C22⋊Q811yescharacteristic
356C3⋊C8C22⋊Q811yescharacteristic
357C3⋊C8C22⋊Q811yescharacteristic
358C3⋊C8C22⋊Q811yescharacteristic
359#41—21no—
360#41—21no—
361#41—21no—
362#41—21no—
363#41—61no—
364#41—61no—
365#41—61no—
366#41—61no—
367C2×Q8—22no—
368C2×Q8—31no—
369C2×Q8D3×Q811yescharacteristic
370C2×D4—21no—
371C2×D4—31no—
372C2×D4—61no—
373C2×D4D3×Q811yescharacteristic
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—61no—
377C22×C4C23×D311yescharacteristic
378Q16—61no—
379Q16—61no—
380SD16—61no—
381SD16—61no—
382SD16—61no—
383SD16—61no—
384D8—61no—
385D8—61no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—21no—
390M16—61no—
391M16—61no—
392M16—61no—
393M16—61no—
394M16—61no—
395M16—61no—
396M16—61no—
397M16—61no—
398M16—122no—
399M16—122no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—31no—
403C2×C8—31no—
404C2×C8—31no—
405C2×C8—31no—
406C2×C8—31no—
407C2×C8—31no—
408C2×C8—32no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—122no—
416C2×C8—122no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—61no—
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—61no—
426C22⋊C4#23712yes—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—61no—
431C42D3×D412yes—
432C42#23711yescharacteristic
433C42#23711yescharacteristic
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—81no—
437C22×C3#52511yescharacteristic
438C22×C3#55411yescharacteristic
439C22×C3#55911yescharacteristic
440C2×D3—41no—
441C2×D3—81no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—81no—
448C3×C4C2×C22⋊Q811yescharacteristic
449C3×C4#53511yescharacteristic
450C3×C4#55411yescharacteristic
451C3×C4#55911yescharacteristic
452Dic3—41no—
453Dic3—81no—
454C23—21no—
455C23—61no—
456C23C2×#23711yescharacteristic
457Q8—21no—
458Q8—21no—
459Q8—61no—
460Q8—61no—
461Q8—82no—
462D4—21no—
463D4—21no—
464D4—41no—
465D4—41no—
466D4—61no—
467D4—61no—
468D4—121no—
469D4—121no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—61no—
479C2×C4—81no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4C2×D3×D411yescharacteristic
484C2×C4C2×#23711yescharacteristic
485C2×C4C2×D3×Q811yescharacteristic
486C2×C4D3×#4111yescharacteristic
487C2×C4#100911yescharacteristic
488C2×C4#100911yescharacteristic
489C8—31no—
490C8—31no—
491C8—31no—
492C8—31no—
493C8—41no—
494C8—41no—
495C8—62no—
496C8—121no—
497C8—121no—
498C8—121no—
499C8—121no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—81no—
504C2×C3#258011yescharacteristic
505D3—81no—
506C22—41no—
507C22—41no—
508C22—81no—
509C22—121no—
510C22—241no—
511C22#558611yescharacteristic
512C22#563511yescharacteristic
513C22#568111yescharacteristic
514C4—41no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—41no—
519C4—81no—
520C4—121no—
521C4—241no—
522C4D3×C22⋊Q811yescharacteristic
523C4#563511yescharacteristic
524C4#567411yescharacteristic
525C4#568111yescharacteristic
526C3#2927111yescharacteristic, complemented, Sylow
527C2—21no—
528C2—21no—
529C2—21no—
530C2—81no—
531C2—241no—
532C2#7680211yescharacteristic
subgroupG/H
centerC4#5674
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2031C2
socleC2×C3#2580
prime radicalC2×#950C22
hypercenter#2031D3
hypocenterC3#29271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2031

Complements

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

NH ≅ G/Ntype
1. #83329C2semidirect
3. C3×#2031C2semidirect
4. #71842C2semidirect
5. #70405C2semidirect
6. #70375C2semidirect
8. #70044C2semidirect
10. #69034C2semidirect
11. #69015C2semidirect
12. #68773C2semidirect
13. #68748C2semidirect
14. #68715C2semidirect
15. #68699C2semidirect
61. #2031D3semidirect
89. C2×#216Q8semidirect
90. C2×#215Q8semidirect
91. C2×#214Q8semidirect
92. C2×#213Q8semidirect
526. C3#29271semidirect

Maximal subgroups

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

subgroupindex
#833292
#789902
C3×#20312
#718422
#704052
#703752
#703742
#700442
#700422
#690342
#690152
#687732
#687482
#687152
#686992
#292713