Finite Group Databasebeta

← #10848876

Subgroups — #10848876

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85850C211yesmaximal, characteristic
2C3×#3039C211yesmaximal, characteristic, complemented
3#83252C211yesmaximal, characteristic, complemented
4#83203C211yesmaximal, characteristic, complemented
5#72110C214yesmaximal, complemented
6C4×#837C211yesmaximal, characteristic, complemented
7#70524C211yesmaximal, characteristic, complemented
8#70512C211yesmaximal, characteristic, complemented
9#70348C214yesmaximal, complemented
10#25615—31nomaximal, Sylow
11C2×C3×#137C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C3×C4×#41C2211yescharacteristic
14#5833C2212yes—
15C2×#942C2214yes—
16#5584—21no—
17#5579—21no—
18#5565—22no—
19#5563—21no—
20#5552—22no—
21#5551—21no—
22#5534—22no—
23#5523C2212yes—
24#5477—21no—
25#5477C2211yescharacteristic
26#5470—21no—
27C2×#869C2211yescharacteristic
28C3×#462—21no—
29C3×#462—21no—
30C3×#461—21no—
31C3×#461—21no—
32C3×#423C2211yescharacteristic
33C3×#419C2212yes—
34C2×#837C2212yes—
35#5229C2214yes—
36C2×#825C2214yes—
37#5092—21no—
38#5086—22no—
39#5085—21no—
40#4977—22no—
41#4973—22no—
42C42×Dic3C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4554C2214yes—
45#4532C2212yes—
46#3511—31no—
47#3152—31no—
48#3039D311yescharacteristic, complemented
49#2862—31no—
50#2007—34no—
51#1997—31no—
52#1962—31no—
53#1882—34no—
54C4×#104—31no—
55C2×C3×#41C2311yescharacteristic
56C22×C3×Q8C2312yes—
57C2×#235—22no—
58C2×#235C2312yes—
59C22×Dic6D412yes—
60C3×#137—44no—
61C2×C3×Q16—22no—
62C2×C3×SD16—22no—
63C2×C3×M16C2312yes—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4×Q8D411yescharacteristic
69C3×C4×Q8D411yescharacteristic
70C3×C4×D4—21no—
71C3×C4×D4—21no—
72C3×C4×D4D411yescharacteristic
73C3×C4×D4D411yescharacteristic
74C3×#117—21no—
75C3×#117C2311yescharacteristic
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80#946—22no—
81C2×#216—24no—
82#942—28no—
83#942—44no—
84C2×#215—24no—
85#939—24no—
86#938—22no—
87#930—22no—
88#930—21no—
89#930—21no—
90#930—22no—
91#926—22no—
92C2×C4⋊Dic3D412yes—
93C2×C4⋊Dic3C2312yes—
94#924—22no—
95#924—22no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3—21no—
98C2×C4×Dic3C2312yes—
99C2×#208C2312yes—
100#898—24no—
101#896—24no—
102#892—22no—
103D3×C4⋊C4—22no—
104#885—24no—
105#883—24no—
106#67⋊C4—24no—
107#875—22no—
108#875D412yes—
109#874D411yescharacteristic
110C4×D12—22no—
111C4×D12—22no—
112C4×D12—21no—
113C4×D12—21no—
114#870—22no—
115#869—21no—
116#869—21no—
117#869—21no—
118#869—21no—
119#869D411yescharacteristic
120C4×Dic6—22no—
121C4×Dic6—22no—
122C3×C4⋊C8—21no—
123C3×C4⋊C8—21no—
124C3×#103—22no—
125C3×#103—22no—
126#837—24no—
127C3⋊#95—28no—
128#825—44no—
129#803—21no—
130C4×C3⋊C8—21no—
131C2×#137—34no—
132C2×#137C2×D312yes—
133#556—61no—
134#555—61no—
135#554—61no—
136#553—62no—
137#548—62no—
138#547—61no—
139#542—62no—
140#537—32no—
141#533—61no—
142#532—31no—
143#532—61no—
144C2×C4⋊Q8—31no—
145C2×C4⋊Q8C2×D311yescharacteristic
146#520—32no—
147C4×#41C2×D311yescharacteristic
148#497—62no—
149#494—61no—
150#492—61no—
151#462—21no—
152#462—21no—
153#461—21no—
154#461—21no—
155#423C2×D311yescharacteristic
156C2×#104—32no—
157C2×#104—34no—
158#419C2×D312yes—
159C4×M16—31no—
160#391—34no—
161#391—62no—
162#382—62no—
163C43—31no—
164#343—34no—
165#339—32no—
166C3×#41—22no—
167C3×#41—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—24no—
171C2×C3×Q8—24no—
172C2×C3×Q8C2×D411yescharacteristic
173C2×C3×D4—21no—
174C2×C3×D4C2×D411yescharacteristic
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C2×#67—22no—
183C2×#67—24no—
184C22×Dic3—22no—
185C22×Dic3C2×D414yes—
186#235—24no—
187#235—24no—
188#235—84no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12C2×D412yes—
192C2×C4×D3—22no—
193C2×C4×D3—24no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—44no—
197C2×Dic6C2×D412yes—
198C3×Q16—44no—
199C3×SD16—44no—
200C3×M16—24no—
201C2×C3×C8—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—41no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4C2×D411yescharacteristic
209C3×C4⋊C4C2×D411yescharacteristic
210C3×C22⋊C4—21no—
211C3×C22⋊C4—21no—
212C3×C22⋊C4—21no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42C2×D411yescharacteristic
217C3×C42C2×D411yescharacteristic
218C3×C42C2×D411yescharacteristic
219C3×C42C2×D411yescharacteristic
220#217—24no—
221#216—48no—
222#215—48no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—22no—
226(C2×D3)⋊C4—22no—
227C4⋊Dic3—21no—
228C4⋊Dic3—22no—
229C4⋊Dic3—22no—
230C4⋊Dic3—21no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233Dic3⋊C4—24no—
234Dic3⋊C4—24no—
235C4×Dic3—24no—
236C4×Dic3—24no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239C4×Dic3—42no—
240#208—22no—
241#208#4114yescomplemented
242C2×C3⋊C8—22no—
243C2×#41—32no—
244C2×#41—62no—
245C2×#41C22×D311yescharacteristic
246C22×Q8—32no—
247C22×Q8C22×D312yes—
248#137—44no—
249#137—68no—
250#137—124no—
251C2×Q16—22no—
252C2×Q16—64no—
253C2×SD16—22no—
254C2×SD16—64no—
255C2×M16—32no—
256C2×M16C22×D312yes—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—31no—
262C4⋊Q8—61no—
263C4⋊Q8—61no—
264C4⋊Q8—61no—
265C4⋊Q8—61no—
266#127—31no—
267#126—64no—
268#125—62no—
269#124—32no—
270#124—62no—
271#124—64no—
272#123—64no—
273C22⋊Q8—64no—
274C22⋊Q8—62no—
275C22⋊Q8—62no—
276C22⋊Q8—62no—
277#121—62no—
278#121—62no—
279#121—64no—
280#121—62no—
281#121—61no—
282#121—61no—
283C4×Q8—62no—
284C4×Q8—62no—
285C4×Q8#6711yescharacteristic
286C4×Q8#6711yescharacteristic
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—62no—
290C4×D4—62no—
291C4×D4—64no—
292C4×D4—61no—
293C4×D4—61no—
294C4×D4#6711yescharacteristic
295C4×D4#6711yescharacteristic
296#117—21no—
297#117—62no—
298#117—62no—
299#117C22×D311yescharacteristic
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—62no—
305C2×C42—21no—
306C2×C42—32no—
307C2×C42—61no—
308C2×C42—61no—
309C2×C42C22×D311yescharacteristic
310C4⋊C8—21no—
311C4⋊C8—21no—
312#104—64no—
313#104—124no—
314#103—22no—
315#103—22no—
316#97—61no—
317C4×C8—61no—
318#95—68no—
319C23×C3—21no—
320C23×C3C22×D411yescharacteristic
321C22×D3—22no—
322C3×Q8—21no—
323C3×Q8—21no—
324C3×Q8—42no—
325C3×Q8—42no—
326C3×Q8—44no—
327C3×D4—21no—
328C3×D4—21no—
329C3×D4—42no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—44no—
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C2×#4112yes—
347C2×C3×C4H(5,𝔽2)12yes—
348#67—44no—
349#67—84no—
350C2×Dic3—24no—
351C2×Dic3—22no—
352C2×Dic3—24no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—44no—
356D12—24no—
357D12—42no—
358D12—42no—
359C4×D3—42no—
360C4×D3—42no—
361C4×D3—84no—
362Dic6—24no—
363Dic6—44no—
364Dic6—82no—
365C3×C8—42no—
366C3⋊C8—24no—
367#41—22no—
368#41—22no—
369#41—64no—
370#41—64no—
371#41—244no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—32no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8—124no—
380C2×Q8C2×#6711yescharacteristic
381C2×D4—21no—
382C2×D4—32no—
383C2×D4—62no—
384C2×D4—64no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4C2×#6711yescharacteristic
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—34no—
393C22×C4—64no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4D3×D411yescharacteristic
397C22×C4D3×D411yescharacteristic
398C22×C4C23×D311yescharacteristic
399Q16—44no—
400Q16—128no—
401SD16—44no—
402SD16—128no—
403M16—24no—
404M16—34no—
405M16—62no—
406C2×C8—22no—
407C2×C8—62no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—64no—
415C4⋊C4—64no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—62no—
421C4⋊C4—62no—
422C4⋊C4C2×#6711yescharacteristic
423C4⋊C4C2×#6711yescharacteristic
424C22⋊C4—21no—
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—64no—
428C22⋊C4—64no—
429C22⋊C4—64no—
430C22⋊C4—62no—
431C22⋊C4—62no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—64no—
436C42—64no—
437C42—62no—
438C42—62no—
439C42—122no—
440C42D3×D411yescharacteristic
441C42D3×D411yescharacteristic
442C42C2×#6711yescharacteristic
443C42C2×#6711yescharacteristic
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3C2×#12011yescharacteristic
450C22×C3#55012yes—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—42no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—41no—
459C3×C4—82no—
460C3×C4#45512yes—
461C3×C4#45612yes—
462C3×C4#53211yescharacteristic
463C3×C4#53511yescharacteristic
464C3×C4#54612yes—
465Dic3—44no—
466Dic3—42no—
467Dic3—44no—
468C23—21no—
469C23—62no—
470C23C2×D3×D411yescharacteristic
471Q8—21no—
472Q8—21no—
473Q8—42no—
474Q8—42no—
475Q8—44no—
476Q8—64no—
477Q8—124no—
478Q8—242no—
479D4—21no—
480D4—21no—
481D4—42no—
482D4—64no—
483D4—124no—
484D4—122no—
485D4—122no—
486D4—244no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—64no—
498C2×C4—62no—
499C2×C4—64no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—124no—
505C2×C4—244no—
506C2×C4#93711yescharacteristic
507C2×C4#93711yescharacteristic
508C2×C4#93711yescharacteristic
509C2×C4#93711yescharacteristic
510C2×C4C2×D3×D411yescharacteristic
511C2×C4#100412yes—
512C2×C4D3×#4112yes—
513C2×C4C22×#6711yescharacteristic
514C8—42no—
515C8—64no—
516C2×C3—22no—
517C2×C3—81no—
518C2×C3#243111yescharacteristic
519C2×C3C2×#45511yescharacteristic
520C2×C3C2×#45611yescharacteristic
521D3—82no—
522C22—21no—
523C22—21no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22#559912yes—
531C22C2×#93711yescharacteristic
532C4—21no—
533C4—21no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—82no—
538C4—124no—
539C4—124no—
540C4—122no—
541C4#520212yes—
542C4#520512yes—
543C4#561212yes—
544C4#580511yescharacteristic
545C4#583211yescharacteristic
546C3#2561511yescharacteristic, complemented, Sylow
547C2—22no—
548C2—81no—
549C2—242no—
550C2#8265211yescharacteristic
551C2C2×#520211yescharacteristic
552C2C2×#520511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3039C2
socleC22×C3C2×#120
prime radical#85850C2
hypercenter#3039D3
hypocenterC3#25615
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
2. C3×#3039C2semidirect
3. #83252C2semidirect
4. #83203C2semidirect
5. #72110C2semidirect
6. C4×#837C2semidirect
7. #70524C2semidirect
8. #70512C2semidirect
9. #70348C2semidirect
48. #3039D3semidirect
241. #208#41semidirect
546. C3#25615semidirect

Maximal subgroups

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

subgroupindex
#858502
C3×#30392
#832522
#832032
#721102
C4×#8372
#705242
#705122
#703482
#256153