Finite Group Databasebeta

← #11714567

Subgroups — #11714567

Order 768.

1531 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1531 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×S4C212yesmaximal, complemented
2A4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#85506C212yesmaximal, complemented
4#85311C214yesmaximal, complemented
5#85308C212yesmaximal
6#85303C212yesmaximal, complemented
7C4×C4⋊S4C211yesmaximal, characteristic
8C4×A4⋊Q8C211yesmaximal, characteristic, complemented
9#59127—31nomaximal, Sylow
10C2×Q8×A4C2212yes—
11Q8×S4—28no—
12C2×C4⋊S4C2211yescharacteristic
13C2×C4×S4C2212yes—
14C2×C4×S4C2214yes—
15C2×A4⋊Q8C2214yes—
16C2×A4⋊Q8C2211yescharacteristic
17#5727—41nomaximal
18A4×C4⋊C4C2212yes—
19A4×C4⋊C4C2212yes—
20C42×A4C2211yescharacteristic
21(C2×S4)⋊C4C2214yes—
22(C2×S4)⋊C4C2212yes—
23C4⋊SL(2,ℤ/4)C2211yescharacteristic
24C4⋊SL(2,ℤ/4)C2211yescharacteristic
25SL(2,ℤ/4)⋊C4C2214yes—
26SL(2,ℤ/4)⋊C4C2212yes—
27C4×SL(2,ℤ/4)C2212yes—
28C2×D4×Q8—31no—
29C2×D4×Q8—32no—
30C22×C4⋊Q8D311yescharacteristic, complemented
31#2677—34no—
32#2551—32no—
33#2546—31no—
34#2542—31no—
35#2452—32no—
36#2449—34no—
37#2440—32no—
38#2434—32no—
39#2429—32no—
40#2409—32no—
41#2365—31no—
42#2361—32no—
43C4×C22⋊Q8—31no—
44C4×#121—31no—
45C22×S4C2312yes—
46C2×D3×Q8—42no—
47Q8×A4—24no—
48C2×C4×A4C2314yes—
49C2×C4×A4C2311yescharacteristic
50C2×C4×A4C2311yescharacteristic
51C2×C4×A4C2311yescharacteristic
52C2×SL(2,ℤ/4)C2312yes—
53C2×SL(2,ℤ/4)C2312yes—
54C2×SL(2,ℤ/4)C2312yes—
55C4⋊S4Q814yescomplemented
56C4×S4—24no—
57C4×S4—28no—
58A4⋊Q8—28no—
59A4⋊Q8D414yescomplemented
60C3×C4⋊Q8—41no—
61#946—42no—
62#896—44no—
63#893—42no—
64#888—42no—
65C4×D12—41no—
66C4×Dic6—41no—
67C23×Q8—32no—
68C23×Q8C2×D312yes—
69D4×Q8—68no—
70D4×Q8—68no—
71D4×Q8—62no—
72D4×Q8—62no—
73D4×Q8—62no—
74D4×Q8—61no—
75D4×Q8—61no—
76C2×C4⋊Q8—31no—
77C2×C4⋊Q8—32no—
78C2×C4⋊Q8—32no—
79C2×C4⋊Q8—32no—
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8—34no—
82C2×C4⋊Q8—32no—
83C2×C4⋊Q8—32no—
84C2×C22⋊Q8—31no—
85C2×C22⋊Q8—32no—
86C2×C22⋊Q8—34no—
87C2×C22⋊Q8—34no—
88C2×C22⋊Q8—32no—
89C2×C22⋊Q8—34no—
90C2×C22⋊Q8—32no—
91C2×#121—31no—
92C2×C4×Q8—32no—
93C2×C4×Q8—31no—
94C2×C4×D4—32no—
95C2×C4×D4—31no—
96C2×C4×D4—34no—
97C2×C4×D4—32no—
98C22×C4⋊C4C2×D312yes—
99C22×C4⋊C4C2×D312yes—
100C22×C42C2×D311yescharacteristic
101#399—34no—
102#398—34no—
103#398—34no—
104#396—34no—
105#395—34no—
106#395—34no—
107#395—34no—
108#393—34no—
109#391—32no—
110#391—32no—
111#391—31no—
112#390—32no—
113#390—31no—
114#389—32no—
115#389—32no—
116#389—31no—
117#389—34no—
118#389—32no—
119#389—31no—
120#388—34no—
121#388—32no—
122#387—34no—
123#387—32no—
124#386—34no—
125#386—32no—
126#385—34no—
127#385—34no—
128#385—34no—
129#385—32no—
130#384—32no—
131#380—31no—
132#380—31no—
133C4×C4⋊C4—32no—
134C4×C4⋊C4—31no—
135C4×C4⋊C4—31no—
136C4×C4⋊C4—32no—
137C4×C22⋊C4—32no—
138C4×C22⋊C4—32no—
139C22×A4C2411yescharacteristic
140C2×S4—22no—
141C2×S4—22no—
142C2×S4C2×Q814yescomplemented
143C2×C3×Q8—42no—
144D3×Q8—88no—
145C2×D12—41no—
146C2×C4×D3—42no—
147C2×C4×D3—44no—
148C2×Dic6—41no—
149C2×Dic6—44no—
150C4×A4—24no—
151C4×A4—21no—
152C4×A4C2×D412yes—
153C4×A4C2×Q812yescomplemented
154SL(2,ℤ/4)—22no—
155SL(2,ℤ/4)—22no—
156SL(2,ℤ/4)C2×D414yes—
157C3×C4⋊C4—42no—
158C3×C4⋊C4—42no—
159C3×C42—41no—
160(C2×D3)⋊C4—42no—
161(C2×D3)⋊C4—44no—
162C4⋊Dic3—41no—
163C4⋊Dic3—41no—
164Dic3⋊C4—42no—
165Dic3⋊C4—44no—
166C4×Dic3—42no—
167C22×Q8—24no—
168C22×Q8—31no—
169C22×Q8—32no—
170C22×Q8—32no—
171C22×Q8—34no—
172C22×Q8—34no—
173C22×Q8—64no—
174C22×Q8—68no—
175C22×Q8—64no—
176C22×Q8—62no—
177C22×Q8—62no—
178C22×Q8—124no—
179C22×D4—31no—
180C22×D4—32no—
181C23×C4—32no—
182C23×C4—34no—
183C23×C4C22×D314yes—
184C23×C4C22×D311yescharacteristic
185C23×C4C22×D311yescharacteristic
186C23×C4C22×D311yescharacteristic
187C4⋊Q8—32no—
188C4⋊Q8—32no—
189C4⋊Q8—32no—
190C4⋊Q8—31no—
191C4⋊Q8—61no—
192C4⋊Q8—61no—
193C4⋊Q8—61no—
194C4⋊Q8—64no—
195C4⋊Q8—616no—
196C4⋊Q8—64no—
197C4⋊Q8—62no—
198C4⋊Q8—68no—
199C4⋊Q8—62no—
200C4⋊Q8S411yescharacteristic, complemented
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203C22⋊Q8—68no—
204C22⋊Q8—68no—
205C22⋊Q8—68no—
206C22⋊Q8—616no—
207C22⋊Q8—64no—
208C22⋊Q8—64no—
209C22⋊Q8—64no—
210C22⋊Q8—62no—
211C22⋊Q8—64no—
212C22⋊Q8—62no—
213C22⋊Q8—64no—
214C22⋊Q8—124no—
215#121—34no—
216#121—34no—
217C4×Q8—61no—
218C4×Q8—61no—
219C4×Q8—61no—
220C4×Q8—68no—
221C4×Q8—61no—
222C4×D4—61no—
223C4×D4—61no—
224C4×D4—61no—
225C4×D4—68no—
226C4×D4—68no—
227C4×D4—64no—
228C4×D4—62no—
229C4×D4—64no—
230C4×D4—64no—
231C4×D4—62no—
232C4×D4—61no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—32no—
237C2×C4⋊C4—32no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—34no—
240C2×C4⋊C4—34no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—34no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—34no—
249C2×C4⋊C4—34no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—34no—
255C2×C22⋊C4—32no—
256C2×C22⋊C4—32no—
257C2×C22⋊C4—34no—
258C2×C22⋊C4—32no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—34no—
261C2×C22⋊C4—32no—
262C2×C22⋊C4—32no—
263C2×C22⋊C4—32no—
264C2×C22⋊C4—34no—
265C2×C42—31no—
266C2×C42—31no—
267C2×C42—31no—
268C2×C42—31no—
269C2×C42—32no—
270C2×C42—32no—
271C2×C42—34no—
272C2×C42—32no—
273#95—32no—
274#95—32no—
275#95—32no—
276#95—32no—
277#95—32no—
278#95—32no—
279#95—34no—
280#95—32no—
281#95—32no—
282#95—34no—
283#95—34no—
284#95—34no—
285C22×D3—42no—
286C2×A4C22×D411yescharacteristic
287C2×A4C22×Q811yescharacteristic
288C2×A42−1+411yescharacteristic
289S4—24no—
290C3×Q8—84no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—44no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298D12—44no—
299C4×D3—88no—
300C4×D3—84no—
301Dic6—44no—
302Dic6—88no—
303C24—32no—
304C24C23×D311yescharacteristic
305C2×Q8—34no—
306C2×Q8—34no—
307C2×Q8—32no—
308C2×Q8—62no—
309C2×Q8—64no—
310C2×Q8—64no—
311C2×Q8—61no—
312C2×Q8—62no—
313C2×Q8—64no—
314C2×Q8—68no—
315C2×Q8—64no—
316C2×Q8—68no—
317C2×Q8—64no—
318C2×Q8—61no—
319C2×Q8—64no—
320C2×Q8—124no—
321C2×Q8—124no—
322C2×Q8—128no—
323C2×Q8—1216no—
324C2×Q8—128no—
325C2×Q8C2×S412yes—
326C2×D4—34no—
327C2×D4—34no—
328C2×D4—34no—
329C2×D4—61no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—61no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—32no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—32no—
347C22×C4—32no—
348C22×C4—31no—
349C22×C4—34no—
350C22×C4—34no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—61no—
356C22×C4—62no—
357C22×C4—64no—
358C22×C4—68no—
359C22×C4—64no—
360C22×C4—64no—
361C22×C4—64no—
362C22×C4—64no—
363C22×C4—62no—
364C22×C4—124no—
365C22×C4D3×D412yes—
366C22×C4D3×Q812yescomplemented
367C4⋊C4—34no—
368C4⋊C4—34no—
369C4⋊C4—38no—
370C4⋊C4—34no—
371C4⋊C4—32no—
372C4⋊C4—34no—
373C4⋊C4—32no—
374C4⋊C4—62no—
375C4⋊C4—62no—
376C4⋊C4—62no—
377C4⋊C4—64no—
378C4⋊C4—61no—
379C4⋊C4—64no—
380C4⋊C4—68no—
381C4⋊C4—61no—
382C4⋊C4—68no—
383C4⋊C4—62no—
384C4⋊C4—68no—
385C4⋊C4—68no—
386C4⋊C4—64no—
387C4⋊C4—62no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—61no—
391C4⋊C4—64no—
392C4⋊C4—124no—
393C4⋊C4C2×S412yes—
394C4⋊C4C2×S412yes—
395C22⋊C4—34no—
396C22⋊C4—38no—
397C22⋊C4—34no—
398C22⋊C4—64no—
399C22⋊C4—64no—
400C22⋊C4—68no—
401C22⋊C4—62no—
402C22⋊C4—64no—
403C22⋊C4—62no—
404C22⋊C4—62no—
405C22⋊C4—62no—
406C22⋊C4—62no—
407C22⋊C4—64no—
408C22⋊C4—62no—
409C22⋊C4—124no—
410C42—31no—
411C42—31no—
412C42—31no—
413C42—61no—
414C42—64no—
415C42—62no—
416C42—64no—
417C42—68no—
418C42—62no—
419C42C2×S411yescharacteristic
420C22×C3—41no—
421C2×D3—44no—
422C2×D3—82no—
423C2×D3—82no—
424A4D4×Q811yescharacteristic, complemented
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—84no—
428C3×C4—81no—
429Dic3—44no—
430Dic3—82no—
431Dic3—82no—
432C23—31no—
433C23—31no—
434C23—31no—
435C23—31no—
436C23—34no—
437C23—62no—
438C23—62no—
439C23—62no—
440C23—62no—
441C23—62no—
442C23—62no—
443C23C2×D3×D411yescharacteristic
444C23C2×D3×Q811yescharacteristic
445C23#100711yescharacteristic
446Q8—24no—
447Q8—68no—
448Q8—64no—
449Q8—64no—
450Q8—64no—
451Q8—128no—
452Q8—124no—
453Q8—128no—
454D4—64no—
455D4—64no—
456D4—64no—
457D4—64no—
458D4—64no—
459D4—64no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—32no—
463C2×C4—32no—
464C2×C4—32no—
465C2×C4—32no—
466C2×C4—32no—
467C2×C4—32no—
468C2×C4—34no—
469C2×C4—34no—
470C2×C4—34no—
471C2×C4—32no—
472C2×C4—31no—
473C2×C4—34no—
474C2×C4—64no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—64no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—64no—
487C2×C4—64no—
488C2×C4—61no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—128no—
502C2×C4—124no—
503C2×C4—128no—
504C2×C4—124no—
505C2×C4C22×S411yescharacteristic
506C2×C4C22×S414yes—
507C2×C4C22×S411yescharacteristic
508C2×C4C22×S411yescharacteristic
509C2×C3—41no—
510C2×C3—41no—
511C2×C3—41no—
512D3—84no—
513C22—31no—
514C22—31no—
515C22—31no—
516C22—31no—
517C22—31no—
518C22—31no—
519C22—31no—
520C22—31no—
521C22—31no—
522C22—61no—
523C22—64no—
524C22—64no—
525C22—64no—
526C22—122no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22#572711yescharacteristic, complemented
531C22C23×S411yescharacteristic
532C4—24no—
533C4—21no—
534C4—32no—
535C4—32no—
536C4—64no—
537C4—64no—
538C4—64no—
539C4—64no—
540C4—61no—
541C4—122no—
542C4—122no—
543C4—122no—
544C4—122no—
545C4D4×S412yes—
546C4Q8×S412yescomplemented
547C3—41noSylow
548C2—31no—
549C2—31no—
550C2—31no—
551C2—31no—
552C2—124no—
553C2C2×D4×S411yescharacteristic
554C2C2×Q8×S411yescharacteristic
555C2#8748311yescharacteristic
subgroupG/H
center, FrattiniC22C23×S4
commutator, cosocleC22×A4C24
FittingC22×C4⋊Q8D3
socleC24C23×D3
prime radicalC2×C4⋊S4C22
hypercenterC4⋊Q8S4
hypocenterA4D4×Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×S4C2semidirect
2. A4×C4⋊Q8C2semidirect
3. #85506C2semidirect
4. #85311C2semidirect
6. #85303C2semidirect
8. C4×A4⋊Q8C2semidirect
30. C22×C4⋊Q8D3semidirect
55. C4⋊S4Q8semidirect
59. A4⋊Q8D4semidirect
142. C2×S4C2×Q8semidirect
153. C4×A4C2×Q8semidirect
200. C4⋊Q8S4semidirect
366. C22×C4D3×Q8semidirect
424. A4D4×Q8semidirect
530. C22#5727semidirect
546. C4Q8×S4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×S42
A4×C4⋊Q82
#855062
#853112
#853082
#853032
C4×C4⋊S42
C4×A4⋊Q82
#591273
#57274