Finite Group Databasebeta

← #10950870

Subgroups — #10950870

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79725C211yesmaximal, characteristic, complemented
2#79715C211yesmaximal, characteristic
3C3×#2176C211yesmaximal, characteristic, complemented
4#71504C211yesmaximal, characteristic, complemented
5#71501C211yesmaximal, characteristic, complemented
6#70913C211yesmaximal, characteristic, complemented
7#70909C211yesmaximal, characteristic, complemented
8#69347C211yesmaximal, characteristic
9#69341C211yesmaximal, characteristic, complemented
10#69228C211yesmaximal, characteristic
11#69227C211yesmaximal, characteristic, complemented
12#69218C211yesmaximal, characteristic, complemented
13#69217C211yesmaximal, characteristic, complemented
14#69077C211yesmaximal, characteristic, complemented
15D3×#346C211yesmaximal, characteristic, complemented
16#33747—31nomaximal, Sylow
17#5580—21no—
18#5578—21no—
19C2×C3×Q32C2211yescharacteristic
20C3×C4×Q16C2211yescharacteristic
21C3×C4×Q16C2211yescharacteristic
22#5191C2211yescharacteristic
23#5188C2211yescharacteristic
24#5188C2211yescharacteristic
25#5183C2211yescharacteristic
26C2×#829C2211yescharacteristic
27C2×#828C2211yescharacteristic
28C4×#216—21no—
29C4×#215—21no—
30C2×#800C2211yescharacteristic
31C2×D3×C16C2211yescharacteristic
32C2×D3×C16C2211yescharacteristic
33#4887C2211yescharacteristic
34#4883C2211yescharacteristic
35#4878C2211yescharacteristic
36#4874C2211yescharacteristic
37#4813—21no—
38#4807—21no—
39#4800—21no—
40#4792—21no—
41C4×Dic12C2211yescharacteristic
42C4×D24C2211yescharacteristic
43C4×D3×C8C2211yescharacteristic
44C3×#367C2211yescharacteristic
45C3×#358C2211yescharacteristic
46C3×#358C2211yescharacteristic
47C3×#346C2211yescharacteristic
48#4564C2211yescharacteristic
49#4564C2211yescharacteristic
50#4509C2211yescharacteristic
51#4504C2211yescharacteristic
52#4503C2211yescharacteristic
53#4502C2211yescharacteristic
54#4502C2211yescharacteristic
55#4492C2211yescharacteristic
56#4491C2211yescharacteristic
57#4490C2211yescharacteristic
58#4489C2211yescharacteristic
59#4461C2211yescharacteristic
60#3170—31no—
61#3168—31no—
62#2255—31no—
63#2251—31no—
64#2245—31no—
65#2243—31no—
66#2217—31no—
67#2217—31no—
68#2216—31no—
69#2216—31no—
70#2176—31no—
71#2176D311yescharacteristic, complemented
72#2175—31no—
73#2175—31no—
74C2×#346—31no—
75C2×C3×Q16C2311yescharacteristic
76C2×C3×Q16C2311yescharacteristic
77C3×C4×Q8—21no—
78C3×C4×Q8—21no—
79#947—21no—
80#946—21no—
81#946—21no—
82#944—21no—
83C2×#216—21no—
84C2×#215—21no—
85C2×Dic12C2311yescharacteristic
86C2×D24C2311yescharacteristic
87C2×D3×C8D411yescharacteristic
88C2×D3×C8C2311yescharacteristic
89#898—41no—
90#897—21no—
91#896—41no—
92#895—21no—
93#894—41no—
94#890—21no—
95#889—41no—
96#888—21no—
97C4×D12—21no—
98C4×D12—41no—
99#872—41no—
100#872—41no—
101C42×D3D411yescharacteristic
102C4×Dic6—21no—
103C4×Dic6—41no—
104C3×Q32—22no—
105C2×C3×C16C2311yescharacteristic
106C2×C3×C16C2311yescharacteristic
107C3×#107C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×#103—21no—
110C3×#103—21no—
111C3×C4×C8C2311yescharacteristic
112#835—21no—
113#835—21no—
114#829—22no—
115#828—22no—
116#821—21no—
117#818C2311yescharacteristic
118#818C2311yescharacteristic
119#816—21no—
120C8×Dic3D411yescharacteristic
121C8×Dic3C2311yescharacteristic
122C2×C3⋊C16C2311yescharacteristic
123C2×C3⋊C16C2311yescharacteristic
124#810—21no—
125#809—21no—
126#808—21no—
127#807—21no—
128C4×C3⋊C8D411yescharacteristic
129#800—22no—
130D3×C16—22no—
131D3×C16—22no—
132#533—61no—
133#529—61no—
134C2×Q32—31no—
135C2×Q32C2×D311yescharacteristic
136C2×SD32—31no—
137C2×SD32—31no—
138C22×C16—31no—
139C22×C16—31no—
140#500—31no—
141#499—31no—
142#495—31no—
143#494—31no—
144#488—61no—
145#487—61no—
146#467—31no—
147#467—31no—
148#467—31no—
149#467—61no—
150#466—31no—
151#465—61no—
152C4×Q16—31no—
153C4×Q16—61no—
154C4×Q16C2×D311yescharacteristic
155C4×Q16C2×D311yescharacteristic
156C4×SD16—61no—
157C4×D8—31no—
158C2×C4×C8—31no—
159#367—31no—
160#367C2×D311yescharacteristic
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#358—31no—
166#358—31no—
167#358C2×D311yescharacteristic
168#358C2×D311yescharacteristic
169#357—31no—
170#357—31no—
171#346—31no—
172#346—31no—
173#346—31no—
174#346C2×D311yescharacteristic
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×D12—21no—
178C2×D12—41no—
179C2×C4×D3—41no—
180C2×C4×D3D811yescharacteristic
181C2×C4×D3D811yescharacteristic
182C2×C4×D3C2×D411yescharacteristic
183C2×Dic6—21no—
184C2×Dic6—41no—
185C3×Q16—21no—
186C3×Q16—21no—
187C3×Q16—21no—
188C3×Q16—21no—
189C2×C3×C8C2×D411yescharacteristic
190C2×C3×C8C2411yescharacteristic
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—41no—
195C3×C42—41no—
196C3×C42—41no—
197C3×C42C2×D411yescharacteristic
198#216—42no—
199#215—42no—
200(C2×D3)⋊C4—41no—
201(C2×D3)⋊C4—41no—
202(C2×D3)⋊C4—41no—
203(C2×D3)⋊C4—41no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4—41no—
206(C2×D3)⋊C4—41no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—41no—
210C4⋊Dic3—41no—
211Dic3⋊C4—41no—
212Dic3⋊C4—41no—
213Dic3⋊C4—41no—
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218C4×Dic3—41no—
219C4×Dic3D811yescharacteristic
220C4×Dic3D811yescharacteristic
221C4×Dic3C2×D411yescharacteristic
222C2×C3⋊C8C2×D411yescharacteristic
223C2×C3⋊C8C2×D411yescharacteristic
224Dic12—21no—
225Dic12—21no—
226D24—21no—
227D24—21no—
228D3×C8—21no—
229D3×C8—21no—
230D3×C8—22no—
231C3×C16—21no—
232C3×C16#4112yes—
233C3⋊C16—21no—
234C3⋊C16#4112yes—
235C2×Q16—31no—
236C2×Q16—61no—
237C2×Q16C22×D311yescharacteristic
238C2×Q16C22×D311yescharacteristic
239C2×SD16—61no—
240C2×D8—31no—
241C22×C8—31no—
242C22×C8—31no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245#126—121no—
246#125—61no—
247#125—121no—
248#124—61no—
249#123—121no—
250C22⋊Q8—61no—
251C22⋊Q8—61no—
252C22⋊Q8—61no—
253C22⋊Q8—121no—
254#121—61no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—61no—
258C4×Q8—121no—
259C4×D4—61no—
260C4×D4—121no—
261#117—121no—
262#117—121no—
263C2×C42—31no—
264Q32—22no—
265Q32—62no—
266SD32—62no—
267SD32—62no—
268C2×C16—31no—
269C2×C16—31no—
270C2×C16—62no—
271C2×C16—62no—
272C2×C16C22×D311yescharacteristic
273C2×C16C22×D311yescharacteristic
274#107—31no—
275#107—31no—
276#107—61no—
277#107C22×D311yescharacteristic
278#107C22×D311yescharacteristic
279#106—61no—
280#103—21no—
281#103—21no—
282#103—61no—
283#103—61no—
284#103—61no—
285#103—61no—
286#102—61no—
287#102—61no—
288C4×C8—31no—
289C4×C8—31no—
290C4×C8—31no—
291C4×C8C22×D311yescharacteristic
292C22×D3—41no—
293C22×D3C2×D811yescharacteristic
294C3×Q8—41no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4C2×D811yescharacteristic
303C2×C3×C4C2×D811yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3C2×D811yescharacteristic
309C2×Dic3C2×D811yescharacteristic
310C2×Dic3C2×D811yescharacteristic
311D12—41no—
312D12—41no—
313D12—82no—
314C4×D3—22no—
315C4×D3—21no—
316C4×D3—21no—
317C4×D3—41no—
318C4×D3—82no—
319Dic6—41no—
320Dic6—41no—
321Dic6—82no—
322C3×C8—21no—
323C3×C8C2×#4111yescharacteristic
324C3×C8C2×#4111yescharacteristic
325C3⋊C8—21no—
326C3⋊C8#13512yes—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—61no—
330C2×Q8—121no—
331C2×D4—61no—
332C2×D4—121no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—121no—
337Q16—21no—
338Q16—21no—
339Q16—21no—
340Q16—21no—
341Q16—61no—
342Q16—61no—
343Q16—122no—
344SD16—122no—
345D8—61no—
346D8—61no—
347C2×C8—31no—
348C2×C8—31no—
349C2×C8—62no—
350C2×C8—61no—
351C2×C8—61no—
352C2×C8D3×D411yescharacteristic
353C2×C8C23×D311yescharacteristic
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—121no—
361C4⋊C4—121no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C4⋊C4—121no—
368C4⋊C4—121no—
369C22⋊C4—121no—
370C22⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—121no—
376C42—31no—
377C42—31no—
378C42—31no—
379C42—41no—
380C42—41no—
381C42—121no—
382C42D3×D411yescharacteristic
383C16—21no—
384C16—32no—
385C16—61no—
386C16#23512yes—
387C22×C3C22×D811yescharacteristic
388C2×D3—21no—
389C2×D3—21no—
390C2×D3—81no—
391C2×D3—81no—
392C2×D3—81no—
393C2×D3#51012yes—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—81no—
397C3×C4—81no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4#53211yescharacteristic
401C3×C4C2×#13511yescharacteristic
402Dic3—21no—
403Dic3—21no—
404Dic3—21no—
405Dic3—81no—
406Dic3—81no—
407Dic3—81no—
408C23—31no—
409C23—121no—
410Q8—41no—
411Q8—41no—
412Q8—41no—
413Q8—41no—
414Q8—121no—
415Q8—121no—
416Q8—242no—
417D4—121no—
418D4—121no—
419D4—242no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—31no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—62no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—242no—
435C2×C4D3×D811yescharacteristic
436C2×C4D3×D811yescharacteristic
437C2×C4C2×D3×D411yescharacteristic
438C8—21no—
439C8—32no—
440C8—61no—
441C8C2×#23511yescharacteristic
442C8D3×#4111yescharacteristic
443C2×C3#316911yescharacteristic
444C2×C3C2×#51011yescharacteristic
445C2×C3#341411yescharacteristic
446D3—22no—
447D3—161no—
448C22—32no—
449C22—61no—
450C22—61no—
451C22—241no—
452C22—241no—
453C22—241no—
454C22C2×D3×D811yescharacteristic
455C4—21no—
456C4—21no—
457C4—61no—
458C4—61no—
459C4—61no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—241no—
465C4—241no—
466C4—241no—
467C4#554811yescharacteristic
468C4D3×#13511yescharacteristic
469C3#3374711yescharacteristic, complemented, Sylow
470C2—62no—
471C2—481no—
472C2#7959811yescharacteristic
473C2D3×#51011yescharacteristic
474C2#8203911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2176C2
socleC22×C3C22×D8
prime radical#4883C22
hypercenter#2176D3
hypocenterC3#33747
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2176

Complements

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

NH ≅ G/Ntype
1. #79725C2semidirect
3. C3×#2176C2semidirect
4. #71504C2semidirect
5. #71501C2semidirect
6. #70913C2semidirect
7. #70909C2semidirect
9. #69341C2semidirect
11. #69227C2semidirect
12. #69218C2semidirect
13. #69217C2semidirect
14. #69077C2semidirect
15. D3×#346C2semidirect
71. #2176D3semidirect
469. C3#33747semidirect

Maximal subgroups

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

subgroupindex
#797252
#797152
C3×#21762
#715042
#715012
#709132
#709092
#693472
#693412
#692282
#692272
#692182
#692172
#690772
D3×#3462
#337473