Finite Group Databasebeta

← #10950850

Subgroups — #10950850

Order 768.

496 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3177C211yesmaximal, characteristic, complemented
2#78568C211yesmaximal, characteristic
3#71355C211yesmaximal, characteristic, complemented
4#71351C211yesmaximal, characteristic, complemented
5#71314C211yesmaximal, characteristic, complemented
6#71310C211yesmaximal, characteristic, complemented
7#71084C211yesmaximal, characteristic
8#71083C211yesmaximal, characteristic, complemented
9#71082C211yesmaximal, characteristic
10#71081C211yesmaximal, characteristic, complemented
11#70913C211yesmaximal, characteristic, complemented
12#70909C211yesmaximal, characteristic, complemented
13C4×#829C211yesmaximal, characteristic, complemented
14C4×#828C211yesmaximal, characteristic, complemented
15#69881C211yesmaximal, characteristic, complemented
16#33745—31nomaximal, Sylow
17C3×#541—21no—
18#5534—21no—
19C3×#499C2211yescharacteristic
20C3×#494C2211yescharacteristic
21C3×#491—21no—
22C3×#470—21no—
23C3×#467C2211yescharacteristic
24C3×#467C2211yescharacteristic
25C3×#441—21no—
26C3×C4×Q16C2211yescharacteristic
27C3×C4×Q16C2211yescharacteristic
28C3×#405C2211yescharacteristic
29C2×#829C2211yescharacteristic
30C2×#829C2211yescharacteristic
31C2×#828C2211yescharacteristic
32C2×#828C2211yescharacteristic
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#4564C2211yescharacteristic
45#4564C2211yescharacteristic
46#4564C2211yescharacteristic
47#4564C2211yescharacteristic
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#3177D311yescharacteristic, complemented
61#3175—31no—
62#2255—31no—
63#2251—31no—
64#2242—31no—
65#2241—31no—
66#2220—31no—
67#2219—31no—
68#2217—31no—
69#2216—31no—
70#2176—31no—
71#2175—31no—
72C4×Q32—31no—
73C4×SD32—31no—
74#2108—31no—
75C2×C3×Q16—21no—
76C2×C3×Q16C2311yescharacteristic
77C2×C3×Q16C2311yescharacteristic
78C2×C3×M16D411yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C4⋊Q8—21no—
81C3×#126—41no—
82C3×#126—41no—
83C3×#125—21no—
84C3×#125—41no—
85C3×#125—41no—
86C3×C22⋊Q8—21no—
87C3×C22⋊Q8—21no—
88C3×C4×Q8—21no—
89C3×C4×Q8—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×#103—21no—
108C3×#103—21no—
109C3×#103—21no—
110C3×#103—21no—
111C3×#97D411yescharacteristic
112C3×#97D411yescharacteristic
113C3×C4×C8C2311yescharacteristic
114C3×C4×C8C2311yescharacteristic
115#829—22no—
116#829—22no—
117#828—22no—
118#828—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#541—21no—
134C2×Q32—31no—
135C2×Q32—31no—
136C2×SD32—31no—
137C2×SD32—31no—
138C2×M32—31no—
139C22×C16—31no—
140#500—31no—
141#499C2×D311yescharacteristic
142#495—31no—
143#494C2×D311yescharacteristic
144#491—21no—
145#489—61no—
146#470—21no—
147#468—61no—
148#467—31no—
149#467C2×D311yescharacteristic
150#467C2×D311yescharacteristic
151#466—31no—
152#441—21no—
153#440—61no—
154C4×Q16—31no—
155C4×Q16C2×D311yescharacteristic
156C4×Q16C2×D311yescharacteristic
157C4×D8—31no—
158#405C2×D311yescharacteristic
159#367—31no—
160#367—31no—
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#358—31no—
166#358—31no—
167#358—31no—
168#358—31no—
169#357—31no—
170#357—31no—
171#346—31no—
172#346—31no—
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C22×C3×C4C2×D411yescharacteristic
178C2×#67—41no—
179C2×D12—21no—
180C2×C4×D3—41no—
181C2×Dic6—21no—
182C2×Dic6—41no—
183C3×Q16—21no—
184C3×Q16—21no—
185C3×Q16—21no—
186C3×Q16—21no—
187C3×Q16—42no—
188C3×M16—22no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2411yescharacteristic
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—41no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204C3×C4⋊C4—41no—
205C3×C4⋊C4D811yescharacteristic
206C3×C4⋊C4D811yescharacteristic
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4D811yescharacteristic
210C3×C22⋊C4D811yescharacteristic
211C3×C42—41no—
212C3×C42—41no—
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—21no—
236C2×Q16—31no—
237C2×Q16C22×D311yescharacteristic
238C2×Q16C22×D311yescharacteristic
239C2×SD16—61no—
240C2×D8—31no—
241C2×M16#6711yescharacteristic
242C22×C8C22×D311yescharacteristic
243C4⋊Q8—21no—
244C4⋊Q8—61no—
245#126—41no—
246#126—41no—
247#126—121no—
248#125—21no—
249#125—41no—
250#125—41no—
251#124—61no—
252#124—121no—
253#123—121no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—61no—
257C22⋊Q8—121no—
258#121—61no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—61no—
262C4×D4—61no—
263#117#6711yescharacteristic
264Q32—62no—
265Q32—62no—
266SD32—62no—
267SD32—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—21no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—61no—
285#103—61no—
286#102—61no—
287#102—61no—
288#97#6711yescharacteristic
289#97#6711yescharacteristic
290C4×C8C22×D311yescharacteristic
291C4×C8C22×D311yescharacteristic
292C23×C3C2×D811yescharacteristic
293C22×D3—41no—
294C3×Q8—41no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—41no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4C2×D811yescharacteristic
305C2×C3×C4C2×D811yescharacteristic
306C2×C3×C4C2×D811yescharacteristic
307C2×C3×C4C2×D811yescharacteristic
308C2×C3×C4C2×D811yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310#67—161no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314D12—41no—
315D12—41no—
316C4×D3—161no—
317Dic6—41no—
318Dic6—41no—
319Dic6—161no—
320C3×C8—21no—
321C3×C8—21no—
322C3×C8#13512yes—
323C3×C8C2×#4111yescharacteristic
324C3×C8C2×#4111yescharacteristic
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—61no—
328C2×Q8—121no—
329C2×D4—61no—
330C2×D4—121no—
331C22×C4—121no—
332C22×C4C2×#6711yescharacteristic
333Q16—21no—
334Q16—21no—
335Q16—21no—
336Q16—21no—
337Q16—42no—
338Q16—61no—
339Q16—61no—
340SD16—122no—
341D8—61no—
342D8—61no—
343M16—22no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8C2×#6711yescharacteristic
347C2×C8C2×#6711yescharacteristic
348C2×C8C2×#6711yescharacteristic
349C2×C8C23×D311yescharacteristic
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—61no—
361C4⋊C4—61no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4#21311yescharacteristic
366C4⋊C4#21311yescharacteristic
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—121no—
370C22⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4#21311yescharacteristic
375C22⋊C4#21311yescharacteristic
376C42—41no—
377C42—41no—
378C42—121no—
379C42C2×#6711yescharacteristic
380C42C2×#6711yescharacteristic
381C16—32no—
382C16—32no—
383C16—61no—
384C16—61no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3C22×D811yescharacteristic
389C2×D3—81no—
390C2×D3—81no—
391C2×D3—81no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—81no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4—81no—
401C3×C4#53211yescharacteristic
402C3×C4C2×#13511yescharacteristic
403Dic3—81no—
404Dic3—81no—
405Dic3—81no—
406C23—121no—
407C23C2×#21311yescharacteristic
408Q8—41no—
409Q8—41no—
410Q8—41no—
411Q8—41no—
412Q8—121no—
413Q8—121no—
414Q8—481no—
415D4—121no—
416D4—121no—
417D4—481no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—481no—
428C2×C4C2×#21311yescharacteristic
429C2×C4C2×#21311yescharacteristic
430C2×C4C2×#21311yescharacteristic
431C2×C4C2×#21311yescharacteristic
432C2×C4C2×#21311yescharacteristic
433C2×C4C22×#6711yescharacteristic
434C8—21no—
435C8—21no—
436C8#95012yes—
437C8D3×#4111yescharacteristic
438C8D3×#4111yescharacteristic
439C2×C3—41no—
440C2×C3#316911yescharacteristic
441C2×C3#341611yescharacteristic
442C2×C3#341711yescharacteristic
443D3—161no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22C22×#21311yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—241no—
461C4—241no—
462C4—241no—
463C4C2×#95011yescharacteristic
464C4#583211yescharacteristic
465C3#3374511yescharacteristic, complemented, Sylow
466C2—41no—
467C2—481no—
468C2#8329111yescharacteristic
469C2#8401611yescharacteristic
470C2#8401711yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3177C2
socleC22×C3C22×D8
prime radical#5115C22
hypercenter#3177D3
hypocenterC3#33745
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
1. C3×#3177C2semidirect
3. #71355C2semidirect
4. #71351C2semidirect
5. #71314C2semidirect
6. #71310C2semidirect
8. #71083C2semidirect
10. #71081C2semidirect
11. #70913C2semidirect
12. #70909C2semidirect
13. C4×#829C2semidirect
14. C4×#828C2semidirect
15. #69881C2semidirect
60. #3177D3semidirect
465. C3#33745semidirect

Maximal subgroups

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

subgroupindex
C3×#31772
#785682
#713552
#713512
#713142
#713102
#710842
#710832
#710822
#710812
#709132
#709092
C4×#8292
C4×#8282
#698812
#337453