Finite Group Databasebeta

← #11712751

Subgroups — #11712751

Order 768.

843 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 843 conjugacy classes of subgroups — too many to draw. Only its 79 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×#5912C211yesmaximal, characteristic
2A4×#117C211yesmaximal, characteristic, complemented
3C2×#5409C211yesmaximal, characteristic, complemented
4#73046C214yesmaximal, complemented
5#20979—31nomaximal, Sylow
6C22×C4×A4C2211yescharacteristic
7C2×GL(2,ℤ/4)—21no—
8#5912—21no—
9#5912—21no—
10#5912C412yescomplemented
11#5912C2212yes—
12C2×C4⋊S4C411yescharacteristic, complemented
13C2×C4×S4—21no—
14C2×A4⋊Q8C411yescharacteristic, complemented
15A4×C4⋊C4—21no—
16A4×C22⋊C4—21no—
17C42×A4C2212yescomplemented
18#5409—21no—
19#5409C2212yescomplemented
20C2×A4⋊C8—21no—
21#5005—41nomaximal
22C2×#525—31no—
23C22×#117D311yescharacteristic, complemented
24C2×#421—31no—
25C2×#421—31no—
26C2×#407—31no—
27#1897—34no—
28#1896—34no—
29#1733—31no—
30#1733—31no—
31C23×A4D411yescharacteristic
32C22×S4—21no—
33C2×#235—41no—
34C2×C4×A4—22no—
35C2×C4×A4C2×C411yescharacteristic
36C2×C4×A4C2×C411yescharacteristic
37C2×C4×A4D411yescharacteristic
38C2×C4×A4D411yescharacteristic
39C2×C4×A4D411yescharacteristic
40C2×C4×A4C2311yescharacteristic
41GL(2,ℤ/4)—22no—
42GL(2,ℤ/4)—22no—
43C2×SL(2,ℤ/4)—21no—
44C4⋊S4—21no—
45C4⋊S4C2×C412yes—
46C4×S4—22no—
47C4×S4—22no—
48A4⋊Q8—21no—
49A4⋊Q8C2×C412yes—
50C3×#117—41no—
51C2×#208—41no—
52A4⋊C8—22no—
53#805—44no—
54C22×#41—31no—
55C24×C4C2×D311yescharacteristic
56C22×M16—31no—
57C22×M16—31no—
58#525—32no—
59#525—32no—
60#525—32no—
61#525—32no—
62#525—61no—
63#525—61no—
64#525—61no—
65#525—61no—
66C2×#123—61no—
67C2×C22⋊Q8—31no—
68C2×C22⋊Q8—31no—
69C2×#121—31no—
70C2×#121—31no—
71C2×#120—61no—
72C2×C4×D4—61no—
73C2×C4×D4—61no—
74C2×#117—31no—
75C2×#117—31no—
76C2×#117—32no—
77C2×#117—31no—
78C2×#117—61no—
79C2×#117—61no—
80C22×C4⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42C2×D312yescomplemented
83#421—62no—
84#421—62no—
85#421—61no—
86#421—62no—
87#421—61no—
88C2×#104—34no—
89C2×#104—34no—
90#407—32no—
91#407—32no—
92#407—61no—
93#407—61no—
94C2×C22⋊C8—61no—
95C2×C22⋊C8—61no—
96#339—34no—
97#339—34no—
98C22×A4—41no—
99C22×A4C22⋊C411yescharacteristic
100C22×A4C22⋊C411yescharacteristic
101C22×A4C2×D411yescharacteristic
102C2×S4—22no—
103C2×S4—42no—
104C22×C3×C4—41no—
105C2×#67—81no—
106#235—42no—
107#235—42no—
108#235—81no—
109#235—81no—
110C2×D12—41no—
111C2×C4×D3—81no—
112C2×Dic6—41no—
113C4×A4—42no—
114C4×A4C22⋊C411yescharacteristic
115C4×A4C22⋊C411yescharacteristic
116C4×A4C22×C411yescharacteristic
117C4×A4C2×D411yescharacteristic
118SL(2,ℤ/4)—22no—
119C3×C4⋊C4—81no—
120C3×C22⋊C4—81no—
121C3×C42—42no—
122#208—42no—
123#208—81no—
124C2×C3⋊C8—81no—
125C25#6711yescharacteristic
126C2×#41—32no—
127C2×#41—32no—
128C2×#41—61no—
129C2×#41—61no—
130C2×#41—61no—
131C2×#41—61no—
132C2×#41—61no—
133C2×#41—61no—
134C2×#41—61no—
135C2×#41—61no—
136C2×#41—61no—
137C2×#41—61no—
138C22×Q8—31no—
139C22×D4—31no—
140C22×D4—61no—
141C22×D4—61no—
142C22×D4—61no—
143C23×C4—22no—
144C23×C4—31no—
145C23×C4—31no—
146C23×C4—31no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4—31no—
150C23×C4—31no—
151C23×C4—31no—
152C23×C4—61no—
153C23×C4C4×D311yescharacteristic
154C23×C4C4×D311yescharacteristic
155C23×C4D1211yescharacteristic
156C23×C4D1211yescharacteristic
157C23×C4#6711yescharacteristic
158C23×C4C22×D311yescharacteristic
159C2×M16—32no—
160C2×M16—32no—
161C2×M16—61no—
162C2×M16—61no—
163C2×M16—61no—
164C2×M16—61no—
165C2×M16—61no—
166C2×M16—61no—
167C2×M16—61no—
168C2×M16—61no—
169C2×M16—61no—
170C2×M16—61no—
171C22×C8—61no—
172C22×C8—61no—
173#123—62no—
174#123—62no—
175#123—62no—
176#123—62no—
177C22⋊Q8—32no—
178C22⋊Q8—32no—
179C22⋊Q8—32no—
180C22⋊Q8—32no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C22⋊Q8—61no—
185#121—32no—
186#121—32no—
187#121—32no—
188#121—32no—
189#121—61no—
190#121—61no—
191#121—61no—
192#121—61no—
193#120—62no—
194#120—62no—
195#120—62no—
196#120—62no—
197C4×D4—62no—
198C4×D4—62no—
199C4×D4—62no—
200C4×D4—62no—
201C4×D4—62no—
202C4×D4—62no—
203C4×D4—62no—
204C4×D4—62no—
205#117—32no—
206#117—32no—
207#117—31no—
208#117—32no—
209#117—32no—
210#117—32no—
211#117—61no—
212#117—64no—
213#117—61no—
214#117—61no—
215#117—61no—
216#117—61no—
217#117S411yescharacteristic, complemented
218C2×C4⋊C4—31no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—61no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—61no—
233C2×C42—32no—
234C2×C42—32no—
235C2×C42—62no—
236C2×C42—61no—
237#104—68no—
238#104—64no—
239#104—64no—
240C22⋊C8—62no—
241C22⋊C8—62no—
242C22⋊C8—62no—
243C22⋊C8—62no—
244C23×C3—41no—
245C22×D3—81no—
246C2×A4—21no—
247C2×A4—21no—
248C2×A4—21no—
249C2×A4C2×C22⋊C411yescharacteristic
250S4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—82no—
258#67—82no—
259#67—82no—
260C2×Dic3—81no—
261D12—42no—
262D12—81no—
263C4×D3—82no—
264C4×D3—82no—
265Dic6—42no—
266Dic6—81no—
267C3⋊C8—82no—
268C24—31no—
269C24—31no—
270C24—31no—
271C24—31no—
272C24—41no—
273C24—61no—
274C24—121no—
275C24(C2×D3)⋊C411yescharacteristic
276C24(C2×D3)⋊C411yescharacteristic
277C24C2×#6711yescharacteristic
278#41—62no—
279#41—62no—
280#41—62no—
281#41—62no—
282#41—62no—
283#41—62no—
284#41—62no—
285#41—62no—
286#41—121no—
287#41—121no—
288#41—121no—
289#41—121no—
290#41—121no—
291#41—121no—
292#41—121no—
293#41—121no—
294C2×Q8—32no—
295C2×Q8—61no—
296C2×Q8—61no—
297C2×Q8—61no—
298C2×Q8—61no—
299C2×Q8—61no—
300C2×D4—32no—
301C2×D4—61no—
302C2×D4—61no—
303C2×D4—61no—
304C2×D4—61no—
305C2×D4—62no—
306C2×D4—62no—
307C2×D4—62no—
308C2×D4—62no—
309C2×D4—62no—
310C2×D4—62no—
311C2×D4—61no—
312C2×D4—122no—
313C2×D4—121no—
314C2×D4—121no—
315C2×D4—121no—
316C2×D4—122no—
317C2×D4—124no—
318C2×D4—121no—
319C22×C4—31no—
320C22×C4—31no—
321C22×C4—31no—
322C22×C4—31no—
323C22×C4—31no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—42no—
351C22×C4—62no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—61no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—62no—
365C22×C4—121no—
366C22×C4—121no—
367C22×C4—121no—
368C22×C4—121no—
369C22×C4—122no—
370C22×C4(C2×D3)⋊C411yescharacteristic
371C22×C4(C2×D3)⋊C411yescharacteristic
372C22×C4C2×C4×D311yescharacteristic
373C22×C4C2×D1211yescharacteristic
374C22×C4C2×S411yescharacteristic
375M16—62no—
376M16—62no—
377M16—62no—
378M16—62no—
379M16—121no—
380M16—121no—
381M16—121no—
382M16—121no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—121no—
386C2×C8—121no—
387C2×C8—121no—
388C2×C8—121no—
389C4⋊C4—21no—
390C4⋊C4—32no—
391C4⋊C4—32no—
392C4⋊C4—62no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—61no—
401C22⋊C4—21no—
402C22⋊C4—62no—
403C22⋊C4—62no—
404C22⋊C4—62no—
405C22⋊C4—62no—
406C22⋊C4—62no—
407C22⋊C4—62no—
408C22⋊C4—62no—
409C22⋊C4—62no—
410C22⋊C4—62no—
411C22⋊C4—62no—
412C22⋊C4—62no—
413C22⋊C4—62no—
414C22⋊C4—61no—
415C42—34no—
416C42—32no—
417C42—62no—
418C42—62no—
419C42—62no—
420C42C2×S412yescomplemented
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—161no—
425C2×D3—82no—
426C2×D3—162no—
427A4#42111yescharacteristic, complemented
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—41no—
432C3×C4—162no—
433Dic3—82no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—31no—
438C23—31no—
439C23—31no—
440C23—31no—
441C23—31no—
442C23—31no—
443C23—31no—
444C23—31no—
445C23—31no—
446C23—61no—
447C23—61no—
448C23—61no—
449C23—61no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23—122no—
461C23—122no—
462C23—121no—
463C23C2×(C2×D3)⋊C411yescharacteristic
464C23GL(2,ℤ/4)11yescharacteristic
465Q8—62no—
466Q8—62no—
467Q8—121no—
468Q8—121no—
469D4—62no—
470D4—62no—
471D4—121no—
472D4—122no—
473D4—124no—
474D4—122no—
475D4—122no—
476D4—124no—
477D4—124no—
478D4—122no—
479D4—122no—
480D4—122no—
481D4—121no—
482C2×C4—22no—
483C2×C4—31no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—31no—
487C2×C4—31no—
488C2×C4—31no—
489C2×C4—31no—
490C2×C4—31no—
491C2×C4—31no—
492C2×C4—31no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—31no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—31no—
500C2×C4—31no—
501C2×C4—31no—
502C2×C4—31no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—121no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—121no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4C4×S411yescharacteristic
532C2×C4C4×S411yescharacteristic
533C2×C4C4⋊S411yescharacteristic
534C2×C4C4⋊S411yescharacteristic
535C2×C4GL(2,ℤ/4)11yescharacteristic
536C2×C4C22×S411yescharacteristic
537C8—122no—
538C8—122no—
539C2×C3—41no—
540C2×C3—81no—
541C2×C3—81no—
542C2×C3—81no—
543D3—162no—
544C22—31no—
545C22—31no—
546C22—32no—
547C22—31no—
548C22—31no—
549C22—31no—
550C22—41no—
551C22—61no—
552C22—61no—
553C22—61no—
554C22—61no—
555C22—61no—
556C22—61no—
557C22—61no—
558C22—61no—
559C22—61no—
560C22—61no—
561C22—61no—
562C22—121no—
563C22—121no—
564C22—121no—
565C22—122no—
566C22—122no—
567C22—122no—
568C22—122no—
569C22—242no—
570C22—242no—
571C22#500511yescharacteristic, complemented
572C22(C2×S4)⋊C411yescharacteristic
573C22(C2×S4)⋊C411yescharacteristic
574C22C2×GL(2,ℤ/4)11yescharacteristic
575C4—31no—
576C4—31no—
577C4—31no—
578C4—31no—
579C4—42no—
580C4—122no—
581C4—122no—
582C4—122no—
583C4—122no—
584C4(C2×S4)⋊C411yescharacteristic
585C4(C2×S4)⋊C411yescharacteristic
586C4C2×C4×S411yescharacteristic
587C4C2×C4⋊S411yescharacteristic
588C3—41noSylow
589C2—21no—
590C2—21no—
591C2—21no—
592C2—31no—
593C2—31no—
594C2—61no—
595C2—61no—
596C2—61no—
597C2—242no—
598C2C2×(C2×S4)⋊C411yescharacteristic
subgroupG/H
centerC4C2×C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#117D3
socleC23C2×(C2×D3)⋊C4
prime radicalC2×#5912C2
cosocleC2×C4×A4C23
hypercenter#117S4
hypocenterA4#421
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. A4×#117C2semidirect
3. C2×#5409C2semidirect
4. #73046C2semidirect
10. #5912C4semidirect
12. C2×C4⋊S4C4semidirect
14. C2×A4⋊Q8C4semidirect
17. C42×A4C22semidirect
19. #5409C22semidirect
23. C22×#117D3semidirect
82. C22×C42C2×D3semidirect
217. #117S4semidirect
420. C42C2×S4semidirect
427. A4#421semidirect
571. C22#5005semidirect

Maximal subgroups

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

subgroupindex
C2×#59122
A4×#1172
C2×#54092
#730462
#209793
#50054