Finite Group Databasebeta

← #10950835

Subgroups — #10950835

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3176C211yesmaximal, characteristic, complemented
2#78568C211yesmaximal, characteristic, complemented
3#71355C211yesmaximal, characteristic, complemented
4#71351C211yesmaximal, characteristic, complemented
5#71292C211yesmaximal, characteristic, complemented
6#71288C211yesmaximal, characteristic, complemented
7#71074C211yesmaximal, characteristic, complemented
8#71073C211yesmaximal, characteristic, complemented
9#71072C211yesmaximal, characteristic, complemented
10#71071C211yesmaximal, characteristic, complemented
11C4×#827C211yesmaximal, characteristic, complemented
12#70885C211yesmaximal, characteristic, complemented
13#70884C211yesmaximal, characteristic, complemented
14C4×#826C211yesmaximal, characteristic, complemented
15#69881C211yesmaximal, characteristic, complemented
16#33744—31nomaximal, Sylow
17C3×#540—21no—
18#5534—21no—
19C3×#499C2211yescharacteristic
20C3×#493C2211yescharacteristic
21C3×#490—21no—
22C3×#469—21no—
23C3×#466C2211yescharacteristic
24C3×#466C2211yescharacteristic
25C3×#442—21no—
26C3×C4×D8C2211yescharacteristic
27C3×C4×D8C2211yescharacteristic
28C3×#405C2211yescharacteristic
29C2×#827C2211yescharacteristic
30C2×#827C2211yescharacteristic
31C2×#826C2211yescharacteristic
32C2×#826C2211yescharacteristic
33#5134—21no—
34#5116C2211yescharacteristic
35#5115C2211yescharacteristic
36C2×#812C2211yescharacteristic
37C22×C3⋊C16C2211yescharacteristic
38#4714—21no—
39#4710—21no—
40C4×Dic12C2211yescharacteristic
41#4696C2211yescharacteristic
42C4×D24C2211yescharacteristic
43#4682C2211yescharacteristic
44#4562C2211yescharacteristic
45#4562C2211yescharacteristic
46#4562C2211yescharacteristic
47#4562C2211yescharacteristic
48#4492C2211yescharacteristic
49#4492C2211yescharacteristic
50#4491C2211yescharacteristic
51#4491C2211yescharacteristic
52#4490C2211yescharacteristic
53#4490C2211yescharacteristic
54#4489C2211yescharacteristic
55#4489C2211yescharacteristic
56#4461C2211yescharacteristic
57#4461C2211yescharacteristic
58C4×C3⋊C16C2211yescharacteristic
59C4×C3⋊C16C2211yescharacteristic
60#3176D311yescharacteristic, complemented
61#3175—31no—
62#2255—31no—
63#2250—31no—
64#2242—31no—
65#2240—31no—
66#2220—31no—
67#2218—31no—
68#2217—31no—
69#2215—31no—
70#2177—31no—
71#2175—31no—
72C4×SD32—31no—
73C4×D16—31no—
74#2108—31no—
75C2×C3×D8—21no—
76C2×C3×D8C2311yescharacteristic
77C2×C3×D8C2311yescharacteristic
78C2×C3×M16D411yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×#127—21no—
81C3×#125—21no—
82C3×#123—41no—
83C3×#123—41no—
84C3×#121—21no—
85C3×#121—21no—
86C3×#121—41no—
87C3×#121—41no—
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×#117D411yescharacteristic
91#930—21no—
92#924—21no—
93C2×Dic12C2311yescharacteristic
94C2×D24C2311yescharacteristic
95C2×#204—21no—
96#898—41no—
97#896—41no—
98#885—41no—
99#883—41no—
100#875—21no—
101C4×D12—21no—
102#869—21no—
103C4×Dic6—21no—
104C3×#107C2311yescharacteristic
105C3×#107C2311yescharacteristic
106C3×#106—21no—
107C3×#102—21no—
108C3×#102—21no—
109C3×#102—21no—
110C3×#102—21no—
111C3×#97D411yescharacteristic
112C3×#97D411yescharacteristic
113C3×C4×C8C2311yescharacteristic
114C3×C4×C8C2311yescharacteristic
115#827—22no—
116#827—22no—
117#826—22no—
118#826—22no—
119#821—21no—
120#821—21no—
121#818—21no—
122#818C2311yescharacteristic
123#818C2311yescharacteristic
124#816—21no—
125#816—21no—
126#812—22no—
127C2×C3⋊C16—22no—
128C2×C3⋊C16C2311yescharacteristic
129C2×C3⋊C16C2311yescharacteristic
130C2×C3⋊C16C2311yescharacteristic
131C2×C3⋊C16C2311yescharacteristic
132#542—61no—
133#540—21no—
134C2×SD32—31no—
135C2×SD32—31no—
136C2×D16—31no—
137C2×D16—31no—
138C2×M32—31no—
139C22×C16—31no—
140#500—31no—
141#499C2×D311yescharacteristic
142#495—31no—
143#493C2×D311yescharacteristic
144#490—21no—
145#489—61no—
146#469—21no—
147#468—61no—
148#467—31no—
149#466—31no—
150#466C2×D311yescharacteristic
151#466C2×D311yescharacteristic
152#442—21no—
153#440—61no—
154C4×Q16—31no—
155C4×D8—31no—
156C4×D8C2×D311yescharacteristic
157C4×D8C2×D311yescharacteristic
158#405C2×D311yescharacteristic
159#367—31no—
160#367—31no—
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#357—31no—
166#357—31no—
167#357—31no—
168#357—31no—
169#357—31no—
170#357—31no—
171#346—31no—
172#346—31no—
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—41no—
178C2×C3×D4—41no—
179C2×C3×D4—41no—
180C2×C3×D4—41no—
181C22×C3×C4—41no—
182C22×C3×C4—41no—
183C22×C3×C4C2×D411yescharacteristic
184C2×#67—41no—
185C2×D12—21no—
186C2×C4×D3—41no—
187C2×Dic6—21no—
188C2×Dic6—41no—
189C3×D8—21no—
190C3×D8—21no—
191C3×D8—21no—
192C3×D8—21no—
193C3×D8—42no—
194C3×M16—22no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8C2×D411yescharacteristic
199C2×C3×C8C2×D411yescharacteristic
200C2×C3×C8C2411yescharacteristic
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—41no—
204C3×C4⋊C4—41no—
205C3×C4⋊C4D811yescharacteristic
206C3×C4⋊C4D811yescharacteristic
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4—41no—
210C3×C22⋊C4—41no—
211C3×C22⋊C4D811yescharacteristic
212C3×C22⋊C4D811yescharacteristic
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215#217—41no—
216(C2×D3)⋊C4—41no—
217(C2×D3)⋊C4—41no—
218(C2×D3)⋊C4—41no—
219(C2×D3)⋊C4—41no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—41no—
223Dic3⋊C4—41no—
224Dic3⋊C4—41no—
225C4×Dic3—41no—
226Dic12—21no—
227Dic12—21no—
228D24—21no—
229D24—21no—
230#204—42no—
231C3⋊C16—21no—
232C3⋊C16—21no—
233C3⋊C16#4112yes—
234C3⋊C16#4112yes—
235C2×Q16—31no—
236C2×SD16—61no—
237C2×D8—21no—
238C2×D8—31no—
239C2×D8C22×D311yescharacteristic
240C2×D8C22×D311yescharacteristic
241C2×M16#6711yescharacteristic
242C22×C8C22×D311yescharacteristic
243C4⋊Q8—61no—
244#127—21no—
245#126—121no—
246#125—21no—
247#124—61no—
248#124—121no—
249#123—41no—
250#123—41no—
251#123—121no—
252C22⋊Q8—61no—
253C22⋊Q8—121no—
254#121—21no—
255#121—21no—
256#121—41no—
257#121—41no—
258#121—61no—
259C4×Q8—61no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—61no—
263#117#6711yescharacteristic
264SD32—62no—
265SD32—62no—
266D16—62no—
267D16—62no—
268M32—62no—
269C2×C16—31no—
270C2×C16—31no—
271C2×C16—31no—
272C2×C16—31no—
273C2×C16—62no—
274#107—31no—
275#107—31no—
276#107—61no—
277#107C22×D311yescharacteristic
278#107C22×D311yescharacteristic
279#106—21no—
280#103—61no—
281#103—61no—
282#102—21no—
283#102—21no—
284#102—21no—
285#102—21no—
286#102—61no—
287#102—61no—
288#97#6711yescharacteristic
289#97#6711yescharacteristic
290C4×C8C22×D311yescharacteristic
291C4×C8C22×D311yescharacteristic
292C23×C3—41no—
293C23×C3—41no—
294C23×C3C2×D811yescharacteristic
295C22×D3—41no—
296C3×D4—41no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—41no—
300C3×D4—161no—
301C3×D4—161no—
302C3×D4—161no—
303C3×D4—161no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—161no—
309C2×C3×C4—161no—
310C2×C3×C4C2×D811yescharacteristic
311C2×C3×C4C2×D811yescharacteristic
312C2×C3×C4C2×D811yescharacteristic
313C2×C3×C4C2×D811yescharacteristic
314C2×C3×C4C2×D811yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316#67—161no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320D12—41no—
321D12—41no—
322C4×D3—161no—
323Dic6—41no—
324Dic6—41no—
325Dic6—161no—
326C3×C8—21no—
327C3×C8—21no—
328C3×C8#13512yes—
329C3×C8C2×#4111yescharacteristic
330C3×C8C2×#4111yescharacteristic
331C2×Q8—61no—
332C2×Q8—121no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—61no—
340C2×D4—121no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—121no—
344C22×C4C2×#6711yescharacteristic
345Q16—61no—
346Q16—61no—
347SD16—122no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—21no—
352D8—42no—
353D8—61no—
354D8—61no—
355M16—22no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8C2×#6711yescharacteristic
359C2×C8C2×#6711yescharacteristic
360C2×C8C2×#6711yescharacteristic
361C2×C8C23×D311yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—121no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4#21311yescharacteristic
372C4⋊C4#21311yescharacteristic
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4#21311yescharacteristic
383C22⋊C4#21311yescharacteristic
384C42—121no—
385C42C2×#6711yescharacteristic
386C42C2×#6711yescharacteristic
387C16—32no—
388C16—32no—
389C16—61no—
390C16—61no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3C22×D811yescharacteristic
401C2×D3—81no—
402C2×D3—81no—
403C2×D3—81no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4#53211yescharacteristic
412C3×C4C2×#13511yescharacteristic
413Dic3—81no—
414Dic3—81no—
415Dic3—81no—
416C23—41no—
417C23—41no—
418C23—121no—
419C23C2×#21311yescharacteristic
420Q8—121no—
421Q8—121no—
422Q8—481no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—121no—
428D4—121no—
429D4—161no—
430D4—161no—
431D4—161no—
432D4—161no—
433D4—481no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—161no—
442C2×C4—161no—
443C2×C4—481no—
444C2×C4C2×#21311yescharacteristic
445C2×C4C2×#21311yescharacteristic
446C2×C4C2×#21311yescharacteristic
447C2×C4C2×#21311yescharacteristic
448C2×C4C2×#21311yescharacteristic
449C2×C4C22×#6711yescharacteristic
450C8—21no—
451C8—21no—
452C8#95012yes—
453C8D3×#4111yescharacteristic
454C8D3×#4111yescharacteristic
455C2×C3—41no—
456C2×C3—161no—
457C2×C3—161no—
458C2×C3#316911yescharacteristic
459C2×C3#341511yescharacteristic
460C2×C3#341611yescharacteristic
461D3—161no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—241no—
472C22—241no—
473C22—241no—
474C22C22×#21311yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—81no—
481C4—81no—
482C4—241no—
483C4—241no—
484C4—241no—
485C4C2×#95011yescharacteristic
486C4#583211yescharacteristic
487C3#3374411yescharacteristic, complemented, Sylow
488C2—41no—
489C2—161no—
490C2—161no—
491C2—481no—
492C2#8329111yescharacteristic
493C2#8400711yescharacteristic
494C2#8400811yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3176C2
socleC22×C3C22×D8
hypercenter#3176D3
hypocenterC3#33744
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3176

Complements

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

NH ≅ G/Ntype
1. C3×#3176C2semidirect
2. #78568C2semidirect
3. #71355C2semidirect
4. #71351C2semidirect
5. #71292C2semidirect
6. #71288C2semidirect
7. #71074C2semidirect
8. #71073C2semidirect
9. #71072C2semidirect
10. #71071C2semidirect
11. C4×#827C2semidirect
12. #70885C2semidirect
13. #70884C2semidirect
14. C4×#826C2semidirect
15. #69881C2semidirect
60. #3176D3semidirect
487. C3#33744semidirect

Maximal subgroups

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

subgroupindex
C3×#31762
#785682
#713552
#713512
#712922
#712882
#710742
#710732
#710722
#710712
C4×#8272
#708852
#708842
C4×#8262
#698812
#337443