Finite Group Databasebeta

← #10804136

Subgroups — #10804136

Order 768.

1554 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#525C211yesmaximal, characteristic
2#84027C211yesmaximal, characteristic, complemented
3C2×#5211C211yesmaximal, characteristic, complemented
4#72139C212yesmaximal, complemented
5#71705C214yesmaximal, complemented
6#71670C214yesmaximal, complemented
7#71594C212yesmaximal, complemented
8#21748—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C24×C3×C4C2211yescharacteristic
11C3×#525—22no—
12C3×#525—22no—
13C3×#525—22no—
14C3×#525—22no—
15C2×C3×#123C2212yes—
16C2×C3×C22⋊Q8C412yescomplemented
17C2×C3×#121C412yescomplemented
18C2×C3×#120C2212yes—
19C2×C3×C4×D4C414yescomplemented
20C2×C3×#117C2211yescharacteristic
21C22×#217C2212yes—
22C2×#926C2212yes—
23#5247C2212yes—
24C2×#834C2218yes—
25#5212—22no—
26#5211—22no—
27#5211C2212yes—
28#5211C2212yes—
29#5211C2212yes—
30C4×#217—22no—
31C2×C3⋊#95—22no—
32#4526C2218yes—
33C2×#525D311yescharacteristic
34#2280—31no—
35C2×#381—31no—
36#1899—34no—
37#1898—34no—
38#1820—32no—
39#1782—32no—
40C25×C3C2311yescharacteristic
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—44no—
44C22×C3×Q8C2×C411yescharacteristic, complemented
45C22×C3×D4C2×C414yes—
46C22×C3×D4C2×C411yescharacteristic, complemented
47C22×C3×D4C2312yes—
48C23×C3×C4—44no—
49C23×C3×C4C2×C412yes—
50C23×C3×C4C2×C412yes—
51C23×C3×C4C2×C412yescomplemented
52C23×C3×C4D412yes—
53C23×C3×C4D412yes—
54C23×C3×C4D412yes—
55C23×C3×C4D412yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—22no—
59C3×#123—24no—
60C3×#123—22no—
61C3×#123—22no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—22no—
66C3×#121—22no—
67C3×#121—22no—
68C3×#121—22no—
69C3×#121—22no—
70C3×#120—22no—
71C3×#120—24no—
72C3×#120—22no—
73C3×C4×D4—24no—
74C3×C4×D4—24no—
75C3×C4×D4—24no—
76C3×C4×D4—24no—
77C3×#117—22no—
78C3×#117—22no—
79C2×C3×C4⋊C4C2×C412yescomplemented
80C2×C3×C4⋊C4C2×C414yescomplemented
81C2×C3×C22⋊C4C2×C414yes—
82C2×C3×C22⋊C4C2×C414yescomplemented
83C2×C3×C22⋊C4C2312yes—
84C2×C3×C42C2×C412yescomplemented
85C2×#217—24no—
86C2×#217—22no—
87C2×#217—44no—
88C2×#217C2314yes—
89C2×#217C2314yes—
90#926—44no—
91C2×C4⋊Dic3—22no—
92C2×C4×Dic3—22no—
93#834—48no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C22×#41C2×D311yescharacteristic
98C24×C4C2×D311yescharacteristic
99#525—22no—
100#525—22no—
101#525—22no—
102#525—22no—
103C2×#123C2×D312yes—
104C2×C22⋊Q8Dic312yescomplemented
105C2×#121Dic312yescomplemented
106C2×#120C2×D312yes—
107C2×C4×D4Dic314yescomplemented
108C2×#117—32no—
109C2×#117C2×D311yescharacteristic
110C22×C22⋊C4—32no—
111C2×#99—38no—
112#381—32no—
113#381—32no—
114#381—32no—
115#381—62no—
116#380—62no—
117#379—32no—
118C4×C22⋊C4—62no—
119C2×#95—62no—
120#342—38no—
121C24×C3—24no—
122C24×C3—44no—
123C24×C3C22×C412yes—
124C24×C3C22×C412yes—
125C24×C3C2×D411yescharacteristic
126C24×C3C2×D412yes—
127C24×C3C2×D411yescharacteristic
128C24×C3C2411yescharacteristic
129C3×#41—48no—
130C3×#41—48no—
131C2×C3×Q8—22no—
132C2×C3×Q8—42no—
133C2×C3×D4—24no—
134C2×C3×D4—24no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—28no—
138C2×C3×D4—22no—
139C2×C3×D4—42no—
140C2×C3×D4—48no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—44no—
145C22×C3×C4—44no—
146C22×C3×C4—44no—
147C22×C3×C4—44no—
148C22×C3×C4—44no—
149C22×C3×C4—44no—
150C22×C3×C4—44no—
151C22×C3×C4C22⋊C412yes—
152C22×C3×C4C22⋊C412yes—
153C22×C3×C4C22⋊C412yes—
154C22×C3×C4C22⋊C412yes—
155C22×C3×C4C22⋊C412yes—
156C22×C3×C4C22⋊C412yes—
157C22×C3×C4C22⋊C412yes—
158C22×C3×C4C22⋊C412yes—
159C22×C3×C4C22×C412yes—
160C22×C3×C4C22×C411yescharacteristic
161C22×C3×C4C22×C412yes—
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C22×C414yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×Dic3—24no—
170C22×Dic3—24no—
171C22×Dic3—44no—
172C22×Dic3—48no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—24no—
183C3×C42—24no—
184#217—28no—
185#217—28no—
186#217—48no—
187#217—44no—
188C4⋊Dic3—44no—
189C4×Dic3—44no—
190C25C22×D311yescharacteristic
191C2×#41—22no—
192C2×#41—22no—
193C2×#41—44no—
194C22×Q8C2×Dic311yescharacteristic, complemented
195C22×D4C2×Dic314yes—
196C22×D4C2×Dic311yescharacteristic, complemented
197C22×D4C22×D312yes—
198C23×C4—44no—
199C23×C4—62no—
200C23×C4C2×Dic312yes—
201C23×C4C2×Dic312yes—
202C23×C4C2×Dic312yescomplemented
203C23×C4#6712yes—
204C23×C4#6712yes—
205C23×C4#6712yes—
206C23×C4#6712yes—
207C23×C4C22×D311yescharacteristic
208C23×C4C22×D311yescharacteristic
209#123—24no—
210#123—22no—
211#123—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—22no—
220#120—22no—
221#120—24no—
222#120—22no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—24no—
227#117—22no—
228#117—22no—
229#117—124no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4C2×Dic312yescomplemented
232C2×C4⋊C4C2×Dic314yescomplemented
233C2×C22⋊C4—34no—
234C2×C22⋊C4—34no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—124no—
238C2×C22⋊C4C2×Dic314yes—
239C2×C22⋊C4C2×Dic314yescomplemented
240C2×C22⋊C4C22×D312yes—
241C2×C42—62no—
242C2×C42C2×Dic312yescomplemented
243#99—128no—
244#95—64no—
245#95—64no—
246#95—64no—
247C23×C3—22no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—22no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—24no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—416no—
261C23×C3—44no—
262C23×C3C2×C22⋊C412yes—
263C23×C3C2×C22⋊C412yes—
264C23×C3C23×C411yescharacteristic
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D411yescharacteristic
267C23×C3H(5,𝔽2)14yes—
268C3×Q8—44no—
269C3×D4—44no—
270C3×D4—48no—
271C3×D4—44no—
272C3×D4—44no—
273C3×D4—48no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—48no—
294C2×C3×C4—44no—
295C2×C3×C4C2×C22⋊C412yes—
296C2×C3×C4C2×C22⋊C412yes—
297C2×C3×C4C2×C22⋊C412yes—
298C2×C3×C4C2×C22⋊C412yes—
299C2×Dic3—48no—
300C2×Dic3—48no—
301C2×Dic3—44no—
302C2×Dic3—44no—
303C2×Dic3—88no—
304C24—24no—
305C24—44no—
306C24C22×Dic312yes—
307C24C22×Dic312yes—
308C24C2×#6711yescharacteristic
309C24C2×#6712yes—
310C24C2×#6711yescharacteristic
311C24C23×D311yescharacteristic
312#41—48no—
313#41—48no—
314C2×Q8—22no—
315C2×Q8—42no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—28no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—42no—
323C2×D4—48no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—64no—
335C22×C4—64no—
336C22×C4—124no—
337C22×C4—128no—
338C22×C4#21712yes—
339C22×C4#21712yes—
340C22×C4#21712yes—
341C22×C4#21712yes—
342C22×C4#21712yes—
343C22×C4#21712yes—
344C22×C4#21712yes—
345C22×C4#21712yes—
346C22×C4C22×Dic312yes—
347C22×C4C22×Dic311yescharacteristic
348C22×C4C22×Dic312yes—
349C22×C4C22×Dic311yescharacteristic
350C22×C4C22×Dic314yes—
351C22×C4C2×#6712yes—
352C22×C4C2×#6712yes—
353C22×C4C2×#6712yes—
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6711yescharacteristic
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—124no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—68no—
368C22⋊C4—68no—
369C22⋊C4—124no—
370C22⋊C4—128no—
371C42—24no—
372C42—124no—
373C22×C3—28no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—44no—
383C22×C3—48no—
384C22×C3—48no—
385C22×C3—44no—
386C22×C3—42no—
387C22×C3—44no—
388C22×C3—84no—
389C22×C3#41014yes—
390C22×C3C22×C22⋊C411yescharacteristic
391C22×C3#51812yes—
392C22×C3#53412yes—
393C22×C3#53412yes—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—42no—
399C3×C4—44no—
400Dic3—84no—
401Dic3—84no—
402C23—24no—
403C23—22no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—24no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—416no—
416C23—44no—
417C23C2×#21712yes—
418C23C2×#21712yes—
419C23#100414yes—
420C23C23×Dic311yescharacteristic
421C23C22×#6711yescharacteristic
422C23C22×#6711yescharacteristic
423Q8—44no—
424D4—44no—
425D4—48no—
426D4—48no—
427D4—44no—
428D4—44no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—128no—
451C2×C4—128no—
452C2×C4—124no—
453C2×C4—124no—
454C2×C4—248no—
455C2×C4C2×#21712yes—
456C2×C4C2×#21712yes—
457C2×C4C2×#21712yes—
458C2×C4C2×#21712yes—
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—24no—
462C2×C3—22no—
463C2×C3—42no—
464C2×C3—82no—
465C2×C3#230811yescharacteristic
466C2×C3C2×#41012yes—
467C22—28no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—44no—
477C22—48no—
478C22—48no—
479C22—44no—
480C22—42no—
481C22—44no—
482C22—84no—
483C22#524014yes—
484C22#579812yes—
485C22#580412yes—
486C22#580412yes—
487C22C22×#21711yescharacteristic
488C4—22no—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—42no—
493C4—44no—
494C4—244no—
495C4—244no—
496C3#2174811yescharacteristic, complemented, Sylow
497C2—22no—
498C2—22no—
499C2—24no—
500C2—22no—
501C2—42no—
502C2—82no—
503C2#8404311yescharacteristic
504C2C2×#524012yes—
subgroupG/H
centerC22C22×#217
commutatorC23×C3C23×C4
FrattiniC24C23×D3
Fitting, prime radicalC2×C3×#525C2
socleC22×C3C22×C22⋊C4
cosocleC24×C3C24
hypercenterC2×#525D3
hypocenterC3#21748
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
2. #84027C2semidirect
3. C2×#5211C2semidirect
4. #72139C2semidirect
5. #71705C2semidirect
6. #71670C2semidirect
7. #71594C2semidirect
16. C2×C3×C22⋊Q8C4semidirect
17. C2×C3×#121C4semidirect
19. C2×C3×C4×D4C4semidirect
44. C22×C3×Q8C2×C4semidirect
46. C22×C3×D4C2×C4semidirect
51. C23×C3×C4C2×C4semidirect
79. C2×C3×C4⋊C4C2×C4semidirect
80. C2×C3×C4⋊C4C2×C4semidirect
82. C2×C3×C22⋊C4C2×C4semidirect
84. C2×C3×C42C2×C4semidirect
104. C2×C22⋊Q8Dic3semidirect
105. C2×#121Dic3semidirect
107. C2×C4×D4Dic3semidirect
194. C22×Q8C2×Dic3semidirect
196. C22×D4C2×Dic3semidirect
202. C23×C4C2×Dic3semidirect
231. C2×C4⋊C4C2×Dic3semidirect
232. C2×C4⋊C4C2×Dic3semidirect
239. C2×C22⋊C4C2×Dic3semidirect
242. C2×C42C2×Dic3semidirect
496. C3#21748semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5252
#840272
C2×#52112
#721392
#717052
#716702
#715942
#217483