Finite Group Databasebeta

← #10801136

Subgroups — #10801136

Order 768.

1322 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
1C4×#1007C211yesmaximal, characteristic
2C3×#2327C211yesmaximal, characteristic, complemented
3C2×#5095C211yesmaximal, characteristic
4C4×#942C212yesmaximal, complemented
5#71784C211yesmaximal, characteristic, complemented
6C4×#836C211yesmaximal, characteristic, complemented
7#69693C212yesmaximal, complemented
8#69692C211yesmaximal, characteristic
9#69672C211yesmaximal, characteristic, complemented
10#69562C212yesmaximal, complemented
11#69430C212yesmaximal, complemented
12#21476—31nomaximal, Sylow
13C2×#1007C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17#5812C2211yescharacteristic
18C2×#942C2212yes—
19#5574—22no—
20C4×#239—22no—
21#5572—22no—
22C4×D3×Q8—22no—
23#5504—22no—
24#5504C2212yes—
25C4×#235—22no—
26C4×#235C2212yes—
27C3×#391C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#387—22no—
30C3×#387—22no—
31C3×#382C2211yescharacteristic
32C3×#376C2211yescharacteristic
33C2×#836C2211yescharacteristic
34C2×#836C2211yescharacteristic
35#5225C2212yes—
36#5095C418yescomplemented
37#5030—22no—
38C4×#216—22no—
39#5026—22no—
40C4×#215—22no—
41#4966C2212yes—
42#4964C2212yes—
43C2×#805C2212yes—
44C2×#805C2212yes—
45C4×#208C2212yes—
46#4530C2212yes—
47#4523C2212yes—
48#4523C2212yes—
49C4×2−1+4—31no—
50C2×#456—31no—
51C4×#137—32no—
52#2327D311yescharacteristic, complemented
53#1886—31no—
54#1882—32no—
55#1860—32no—
56#1857—31no—
57#1793—31no—
58#1759—32no—
59C4×#101—31no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62#1007—22no—
63#1007—24no—
64#1007C2×C414yes—
65C2×#239—22no—
66C2×D3×Q8—22no—
67C2×#235—22no—
68C2×#235C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8C2×C414yescomplemented
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4D412yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C42—21no—
81C2×C3×C42D412yes—
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85Q8×Dic3—22no—
86Q8×Dic3—22no—
87C2×#216—22no—
88#942—24no—
89#942C2×C418yescomplemented
90C2×#215—22no—
91C4×#67—22no—
92C4×#67—24no—
93C4×#67—22no—
94#926—22no—
95#926C2312yes—
96C2×#208C2312yes—
97C2×#208C2312yes—
98#893—24no—
99#893—22no—
100#892—24no—
101#892—22no—
102D3×C4⋊C4—24no—
103D3×C4⋊C4—22no—
104#887—24no—
105#887—22no—
106C4×D12—22no—
107C4×D12—22no—
108C4×D12D412yes—
109C4×D12D412yes—
110#872—24no—
111#872—22no—
112C42×D3—24no—
113C42×D3—22no—
114C4×Dic6—22no—
115C4×Dic6—22no—
116C4×Dic6D412yes—
117C4×Dic6D412yes—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—22no—
121C3×#95—22no—
122C3×#95—22no—
123#836—22no—
124#836C2×C414yescomplemented
125#835—22no—
126#835—22no—
127#810—22no—
128#809—22no—
129#808—22no—
130#807—22no—
131#805—24no—
132#805C2×C418yescomplemented
133#803—22no—
134C4×C3⋊C8—22no—
135C2×2−1+4—31no—
136C2×#137—32no—
137C2×C4⋊Q8C2×D311yescharacteristic
138#520—32no—
139#520—62no—
140#520—62no—
141#519—31no—
142#519—62no—
143C4×#41—32no—
144C4×#41—62no—
145C4×#41—62no—
146C2×C4×Q8—62no—
147C2×C4×Q8C2×D311yescharacteristic
148C2×C4×Q8C2×D311yescharacteristic
149#456—38no—
150#441—62no—
151#440—62no—
152C4×Q16—62no—
153C4×SD16—62no—
154#428—32no—
155C2×#104—32no—
156C2×#104—32no—
157#419—32no—
158#417—32no—
159C2×#101—31no—
160C2×#101—31no—
161C4×M16—32no—
162#391C2×D311yescharacteristic
163#391C2×D311yescharacteristic
164#387—22no—
165#387—22no—
166#382C2×D311yescharacteristic
167#376C2×D311yescharacteristic
168#343—32no—
169#339—32no—
170#339—32no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8C22×C412yes—
177C2×C3×Q8C22×C412yes—
178C2×C3×Q8#4112yes—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C2×#67—22no—
189#239—28no—
190#239—28no—
191D3×Q8—28no—
192D3×Q8—28no—
193#235—28no—
194#235—28no—
195#235—22no—
196#235—22no—
197#235C22×C414yes—
198#235#4114yes—
199C2×D12—22no—
200C2×D12C2×D412yes—
201C2×C4×D3—24no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—24no—
205C2×C4×D3—22no—
206C2×Dic6—22no—
207C2×Dic6C2×D412yes—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—41no—
213C3×C4⋊C4—41no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42—21no—
217C3×C42—22no—
218C3×C42—42no—
219C3×C42C22×C412yes—
220C3×C42C2×D412yes—
221C3×C42C2×D412yes—
222#217—22no—
223#216—28no—
224#215—28no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—22no—
230C4⋊Dic3C2×D412yes—
231C4⋊Dic3C2×D412yes—
232Dic3⋊C4—24no—
233Dic3⋊C4—22no—
234Dic3⋊C4—22no—
235C4×Dic3—24no—
236C4×Dic3—24no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239C4×Dic3—22no—
240#208—22no—
241#208—22no—
242#208—22no—
243#208C22×C414yes—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
2462−1+4—34no—
2472−1+4—64no—
2482−1+4—62no—
249C2×#41—32no—
250C2×#41—62no—
251C2×#41—62no—
252C22×Q8—62no—
253C22×Q8C22×D311yescharacteristic
254C22×Q8C22×D311yescharacteristic
255#137—38no—
256#137—64no—
257C2×Q16—62no—
258C2×SD16—62no—
259C2×M16—32no—
260C2×M16—32no—
261C4⋊Q8—22no—
262C4⋊Q8C4×D314yescomplemented
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—32no—
268C4×Q8—32no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×Q8—64no—
273C4×Q8—62no—
274C4×Q8—62no—
275C4×D4—32no—
276C4×D4—32no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—64no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—64no—
284#117—32no—
285#117—64no—
286#117—62no—
287#117—64no—
288#117—62no—
289#117—62no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—64no—
294C2×C4⋊C4#6712yes—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C42—21no—
298C2×C42—64no—
299C2×C42—62no—
300C2×C42#6712yes—
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303C2×C42C22×D311yescharacteristic
304#107—62no—
305#106—62no—
306#104—38no—
307#104—64no—
308#103—62no—
309#103—62no—
310#103—62no—
311#102—62no—
312#101—34no—
313#101—62no—
314#97—62no—
315C4×C8—62no—
316#95—21no—
317#95—21no—
318#95—22no—
319#95—22no—
320#95—22no—
321C23×C3C22×D411yescharacteristic
322C22×D3—22no—
323C3×Q8—24no—
324C3×Q8—24no—
325C3×Q8—24no—
326C3×Q8—42no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4C4×D414yes—
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4#12012yes—
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4C23×C411yescharacteristic
349C2×C3×C4C22×D412yes—
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4112yes—
352#67—24no—
353#67—24no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357D12—28no—
358D12—22no—
359D12C4×D414yes—
360C4×D3—216no—
361C4×D3—24no—
362C4×D3—24no—
363C4×D3—44no—
364C4×D3—82no—
365C4×D3—84no—
366Dic6—28no—
367Dic6—22no—
368Dic6C4×D414yes—
369C3⋊C8—24no—
370C3⋊C8—42no—
371#41—34no—
372#41—34no—
373#41—68no—
374#41—68no—
375#41—68no—
376#41—68no—
377#41—62no—
378#41—62no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—32no—
385C2×Q8—62no—
386C2×Q8—68no—
387C2×Q8—68no—
388C2×Q8C2×C4×D312yes—
389C2×Q8C2×C4×D312yes—
390C2×Q8#23512yes—
391C2×D4—32no—
392C2×D4—62no—
393C2×D4—62no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—64no—
399C22×C4—64no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4C2×#6712yes—
404C22×C4C2×#6712yes—
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C23×D311yescharacteristic
408Q16—68no—
409SD16—68no—
410M16—34no—
411M16—62no—
412M16—62no—
413M16—62no—
414C2×C8—62no—
415C2×C8—62no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—32no—
421C4⋊C4—32no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—64no—
425C4⋊C4—62no—
426C4⋊C4—62no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C22⋊C4—64no—
430C22⋊C4—62no—
431C22⋊C4—62no—
432C22⋊C4—62no—
433C42—22no—
434C42—22no—
435C42—21no—
436C42—22no—
437C42—42no—
438C42—64no—
439C42—64no—
440C42—62no—
441C42—62no—
442C42—62no—
443C42C2×C4×D312yes—
444C42D3×D412yes—
445C42D3×D412yes—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3C2×C4×D411yescharacteristic
449C22×C3C2×#12011yescharacteristic
450C22×C3#52511yescharacteristic
451C2×D3—24no—
452C2×D3—42no—
453C2×D3—42no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—22no—
457C3×C4—24no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—81no—
462C3×C4—81no—
463C3×C4C2×C4×D412yes—
464C3×C4#52512yes—
465Dic3—24no—
466Dic3—42no—
467Dic3—42no—
468C23—62no—
469C23C22×#6711yescharacteristic
470Q8—24no—
471Q8—24no—
472Q8—24no—
473Q8—34no—
474Q8—42no—
475Q8—68no—
476Q8—62no—
477D4—34no—
478D4—68no—
479D4—64no—
480D4—64no—
481D4—62no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—616no—
500C2×C4—62no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—124no—
506C2×C4—242no—
507C2×C4—244no—
508C2×C4C4×#6714yes—
509C2×C4#93711yescharacteristic
510C2×C4#93711yescharacteristic
511C2×C4#93712yes—
512C2×C4C22×C4×D311yescharacteristic
513C2×C4C2×#23511yescharacteristic
514C2×C4C2×D3×D412yes—
515C2×C4D3×#4112yes—
516C8—64no—
517C8—122no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3C4×#12011yescharacteristic
521C2×C3#302011yescharacteristic
522C2×C3#302211yescharacteristic
523D3—44no—
524C22—21no—
525C22—21no—
526C22—64no—
527C22—122no—
528C22—122no—
529C22C2×C4×#6711yescharacteristic
530C22C2×#93711yescharacteristic
531C22#582811yescharacteristic
532C4—21no—
533C4—21no—
534C4—22no—
535C4—24no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—64no—
540C4—81no—
541C4—81no—
542C4—122no—
543C4—122no—
544C4C4×D3×D412yes—
545C4#554712yes—
546C3#2147611yescharacteristic, complemented, Sylow
547C2—21no—
548C2—21no—
549C2—124no—
550C2C4×#93711yescharacteristic
551C2#8341511yescharacteristic
552C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2327C2
socleC22×C3C2×#120
prime radicalC2×#1007C22
cosocleC22×C3×C4C24
hypercenter#2327D3
hypocenterC3#21476
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
2. C3×#2327C2semidirect
4. C4×#942C2semidirect
5. #71784C2semidirect
6. C4×#836C2semidirect
7. #69693C2semidirect
9. #69672C2semidirect
10. #69562C2semidirect
11. #69430C2semidirect
36. #5095C4semidirect
52. #2327D3semidirect
70. C3×C4⋊Q8C2×C4semidirect
89. #942C2×C4semidirect
124. #836C2×C4semidirect
132. #805C2×C4semidirect
262. C4⋊Q8C4×D3semidirect
546. C3#21476semidirect

Maximal subgroups

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

subgroupindex
C4×#10072
C3×#23272
C2×#50952
C4×#9422
#717842
C4×#8362
#696932
#696922
#696722
#695622
#694302
#214763