Finite Group Databasebeta

← #10943962

Subgroups — #10943962

Order 768.

3866 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 3866 conjugacy classes of subgroups — too many to draw. Only its 251 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×#5964C211yesmaximal, characteristic, complemented
2C3×#2404C211yesmaximal, characteristic, complemented
3C2×#5032C214yesmaximal, complemented
4#82125C212yesmaximal, complemented
5#72139C211yesmaximal, characteristic, complemented
6#69598C214yesmaximal, complemented
7#69537C212yesmaximal, complemented
8#32899—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10#5964—216no—
11C2×#1010—22no—
12C2×#1009—22no—
13C2×D3×#41—24no—
14C2×#1007—21no—
15C2×#1004—21no—
16C2×#1004C2214yes—
17C22×#235—22no—
18C22×#235C2212yes—
19C2×C3×#120C2212yescomplemented
20C22×C3×C22⋊C4C2211yescharacteristic, complemented
21C2×#953C2214yescomplemented
22C3×#393—21no—
23C3×#392—21no—
24C3×#386—22no—
25C3×#386C2212yescomplemented
26C3×#381C2211yescharacteristic, complemented
27C2×#834C2214yescomplemented
28#5213C2214yescomplemented
29#5211C2212yescomplemented
30#5032—216no—
31#4960—22no—
32#4956—24no—
33C2×#806C2218yescomplemented
34#4947—22no—
35C2×#585—31no—
36C2×B(4,𝔽2)—34no—
37#2404—32no—
38#2404D311yescharacteristic, complemented
39#1790—34no—
40#1782—32no—
41#1782—31no—
42C25×C3C2311yescharacteristic
43C2×C3×#41—28no—
44C2×C3×#41—22no—
45C2×C3×#41—22no—
46C22×C3×Q8D411yescharacteristic, complemented
47C22×C3×D4—22no—
48C22×C3×D4D411yescharacteristic, complemented
49C22×C3×D4C2312yes—
50C23×C3×C4D412yescomplemented
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C22×#67—24no—
54C22×#67—24no—
55C22×#67C2314yes—
56#1010—232no—
57#1009—232no—
58D3×#41—232no—
59D3×#41—232no—
60#1007—216no—
61C2×#239—24no—
62C2×D3×Q8—24no—
63#1004—216no—
64#1004—28no—
65#1004—216no—
66#1004D4116yescomplemented
67C2×#237—24no—
68C2×#237—28no—
69C2×D3×D4—24no—
70C2×D3×D4—28no—
71C2×#235—24no—
72C2×#235—24no—
73C2×#235—216no—
74C2×#235—216no—
75C2×#235—22no—
76C2×#235—24no—
77C2×#235—22no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—216no—
81C22×D12—22no—
82C22×D12D412yescomplemented
83C22×C4×D3—24no—
84C22×C4×D3D414yescomplemented
85C22×Dic6—22no—
86C22×Dic6D412yescomplemented
87C3×#120—28no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4C2312yes—
95#953—216no—
96C2×#217—24no—
97C2×#217C2314yes—
98C2×(C2×D3)⋊C4—24no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—22no—
102#834—28no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—24no—
106#806—216no—
107#585—616no—
108C2×2−1+4—62no—
109C2×2−1+4—61no—
110C2×H(5,𝔽2)—34no—
111C2×H(5,𝔽2)—62no—
112C2×H(5,𝔽2)—61no—
113C22×#41—32no—
114C22×#41—64no—
115C22×#41—62no—
116C22×#41C2×D311yescharacteristic
117C2×#120—34no—
118C2×#120C2×D312yescomplemented
119C22×C22⋊C4C2×D311yescharacteristic, complemented
120B(4,𝔽2)—616no—
121C2×#99—34no—
122C2×#99—38no—
123#393—21no—
124#393—62no—
125#392—21no—
126#392—62no—
127#386—22no—
128#386—34no—
129#386—64no—
130#386C2×D312yescomplemented
131#381—32no—
132#381C2×D311yescharacteristic, complemented
133C24×C3—22no—
134C24×C3—21no—
135C24×C3—24no—
136C24×C3—42no—
137C24×C3—81no—
138C24×C3C2×D412yes—
139C24×C3C2411yescharacteristic
140C23×D3—24no—
141C23×D3C2×D414yes—
142C3×#41—216no—
143C3×#41—216no—
144C2×C3×Q8—22no—
145C2×C3×Q8—24no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—22no—
150C2×C3×D4—44no—
151C2×C3×D4—44no—
152C2×C3×D4—82no—
153C2×C3×D4C2×D418yes—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—81no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C2×#67—28no—
169C2×#67—216no—
170C2×#67—216no—
171C2×#67—24no—
172C2×#67—24no—
173C2×#67—28no—
174C2×#67—48no—
175C2×#67—48no—
176C2×#67—84no—
177C2×#67C2×D4116yes—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3C2×D414yes—
181#239—232no—
182D3×Q8—232no—
183#237—232no—
184#237—232no—
185#237—232no—
186D3×D4—232no—
187D3×D4—232no—
188D3×D4—232no—
189#235—216no—
190#235—216no—
191#235—216no—
192#235—232no—
193#235—216no—
194#235—232no—
195#235—216no—
196#235—216no—
197C2×D12—24no—
198C2×D12—216no—
199C2×D12—28no—
200C2×D12—22no—
201C2×D12—22no—
202C2×C4×D3—28no—
203C2×C4×D3—232no—
204C2×C4×D3—28no—
205C2×C4×D3—24no—
206C2×C4×D3—24no—
207C2×C4×D3—28no—
208C2×Dic6—24no—
209C2×Dic6—216no—
210C2×Dic6—22no—
211C2×Dic6—22no—
212C2×Dic6—28no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—41no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—41no—
217C3×C22⋊C4—44no—
218C3×C22⋊C4—82no—
219#217—28no—
220#217—48no—
221(C2×D3)⋊C4—84no—
222C25C22×D311yescharacteristic
2232−1+4—616no—
2242−1+4—632no—
225H(5,𝔽2)—316no—
226H(5,𝔽2)—616no—
227H(5,𝔽2)—616no—
228H(5,𝔽2)—632no—
229H(5,𝔽2)—68no—
230C2×#41—28no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—616no—
234C2×#41—64no—
235C2×#41—64no—
236C2×#41—616no—
237C2×#41—616no—
238C2×#41—68no—
239C2×#41—64no—
240C2×#41—64no—
241C2×#41—632no—
242C2×#41—632no—
243C2×#41—62no—
244C2×#41—62no—
245C2×#41—62no—
246C2×#41—62no—
247C2×#41—64no—
248C22×Q8—32no—
249C22×Q8—62no—
250C22×Q8—64no—
251C22×Q8#6711yescharacteristic, complemented
252C22×D4—22no—
253C22×D4—34no—
254C22×D4—32no—
255C22×D4—62no—
256C22×D4—68no—
257C22×D4—64no—
258C22×D4—64no—
259C22×D4—64no—
260C22×D4#6711yescharacteristic, complemented
261C22×D4C22×D312yes—
262C23×C4—34no—
263C23×C4—64no—
264C23×C4#6712yescomplemented
265C23×C4C22×D311yescharacteristic
266C23×C4C22×D311yescharacteristic
267#120—28no—
268#120—616no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—34no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—64no—
278C2×C22⋊C4C22×D312yes—
279#99—616no—
280#99—68no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—64no—
285#95—62no—
286#95—62no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—24no—
290C23×C3—22no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—44no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—44no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—88no—
303C23×C3#12014yes—
304C23×C3C22×D412yes—
305C23×C3C22×D414yes—
306C23×C3C22×D411yescharacteristic
307C22×D3—24no—
308C22×D3—216no—
309C22×D3—24no—
310C22×D3—44no—
311C22×D3—44no—
312C22×D3—84no—
313C22×D3—84no—
314C22×D3#12018yes—
315C3×Q8—28no—
316C3×D4—28no—
317C3×D4—216no—
318C3×D4—88no—
319C2×C3×C4—28no—
320C2×C3×C4—28no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—44no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12012yes—
335C2×C3×C4#12011yescharacteristic
336C2×C3×C4#12014yes—
337#67—232no—
338#67—216no—
339#67—216no—
340#67—816no—
341C2×Dic3—216no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—44no—
345C2×Dic3—44no—
346C2×Dic3—44no—
347C2×Dic3#12018yes—
348D12—28no—
349D12—216no—
350D12—28no—
351C4×D3—216no—
352C4×D3—232no—
353C4×D3—216no—
354Dic6—28no—
355Dic6—216no—
356Dic6—28no—
357C24—22no—
358C24—21no—
359C24—24no—
360C24—34no—
361C24—42no—
362C24—64no—
363C24—81no—
364C24C2×#6712yes—
365C24C23×D311yescharacteristic
366#41—216no—
367#41—216no—
368#41—632no—
369#41—616no—
370#41—616no—
371#41—632no—
372#41—616no—
373#41—616no—
374#41—616no—
375#41—632no—
376#41—632no—
377#41—632no—
378#41—632no—
379#41—616no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—68no—
383C2×Q8—64no—
384C2×Q8—616no—
385C2×Q8—632no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—316no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—68no—
396C2×D4—64no—
397C2×D4—616no—
398C2×D4—632no—
399C2×D4—632no—
400C2×D4—632no—
401C2×D4—68no—
402C2×D4—616no—
403C2×D4—616no—
404C2×D4—64no—
405C2×D4—68no—
406C2×D4—64no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—82no—
410C2×D4—128no—
411C2×D4—128no—
412C2×D4—244no—
413C2×D4D3×D418yes—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—34no—
423C22×C4—68no—
424C22×C4—68no—
425C22×C4—68no—
426C22×C4—632no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4—81no—
432C22×C4D3×D412yes—
433C22×C4D3×D412yes—
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6712yes—
436C22×C4C2×#6711yescharacteristic
437C22⋊C4—24no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—68no—
443C22⋊C4—82no—
444C22⋊C4—128no—
445C22⋊C4—244no—
446C22×C3—22no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—24no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—48no—
454C22×C3—44no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—82no—
459C22×C3—84no—
460C22×C3—82no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3C2×#12012yes—
464C22×C3C2×#12011yescharacteristic
465C22×C3C2×#12014yes—
466C2×D3—28no—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—48no—
470C2×D3—84no—
471C2×D3—816no—
472C3×C4—21no—
473C3×C4—24no—
474C3×C4—22no—
475C3×C4—21no—
476C3×C4—82no—
477C3×C4—81no—
478Dic3—28no—
479Dic3—84no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—38no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—44no—
496C23—616no—
497C23—64no—
498C23—64no—
499C23—88no—
500C23—124no—
501C23—124no—
502C23—244no—
503C23—244no—
504C23#93714yes—
505C23C2×D3×D412yes—
506C23C2×D3×D414yes—
507C23C22×#6711yescharacteristic
508Q8—28no—
509Q8—68no—
510Q8—68no—
511Q8—616no—
512D4—28no—
513D4—216no—
514D4—68no—
515D4—68no—
516D4—616no—
517D4—632no—
518D4—616no—
519D4—616no—
520D4—88no—
521D4—2416no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—38no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—616no—
538C2×C4—616no—
539C2×C4—632no—
540C2×C4—616no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—124no—
544C2×C4—124no—
545C2×C4—124no—
546C2×C4#93714yes—
547C2×C4#93711yescharacteristic
548C2×C4#93711yescharacteristic
549C2×C4#93712yes—
550C2×C3—24no—
551C2×C3—22no—
552C2×C3—44no—
553C2×C3—82no—
554C2×C3#284611yescharacteristic
555C2×C3#302511yescharacteristic
556C2×C3#302711yescharacteristic
557D3—48no—
558C22—21no—
559C22—22no—
560C22—21no—
561C22—22no—
562C22—24no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—48no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—68no—
571C22—82no—
572C22—82no—
573C22—84no—
574C22—81no—
575C22—81no—
576C22—128no—
577C22—124no—
578C22—124no—
579C22—244no—
580C22—2416no—
581C22D3×#12014yes—
582C22C2×#93712yes—
583C22C2×#93711yescharacteristic
584C4—21no—
585C4—21no—
586C4—24no—
587C4—22no—
588C4—68no—
589C4—82no—
590C4—81no—
591C4—244no—
592C3#3289911yescharacteristic, complemented, Sylow
593C2—22no—
594C2—24no—
595C2—44no—
596C2—82no—
597C2—128no—
598C2#8261211yescharacteristic
599C2#8293611yescharacteristic
600C2#8339611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2404C2
socleC22×C3C2×#120
hypercenter#2404D3
hypocenterC3#32899
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. C2×#5964C2semidirect
2. C3×#2404C2semidirect
3. C2×#5032C2semidirect
4. #82125C2semidirect
5. #72139C2semidirect
6. #69598C2semidirect
7. #69537C2semidirect
19. C2×C3×#120C22semidirect
20. C22×C3×C22⋊C4C22semidirect
21. C2×#953C22semidirect
25. C3×#386C22semidirect
26. C3×#381C22semidirect
27. C2×#834C22semidirect
28. #5213C22semidirect
29. #5211C22semidirect
33. C2×#806C22semidirect
38. #2404D3semidirect
46. C22×C3×Q8D4semidirect
48. C22×C3×D4D4semidirect
50. C23×C3×C4D4semidirect
66. #1004D4semidirect
82. C22×D12D4semidirect
84. C22×C4×D3D4semidirect
86. C22×Dic6D4semidirect
118. C2×#120C2×D3semidirect
119. C22×C22⋊C4C2×D3semidirect
130. #386C2×D3semidirect
132. #381C2×D3semidirect
251. C22×Q8#67semidirect
260. C22×D4#67semidirect
264. C23×C4#67semidirect
592. C3#32899semidirect

Maximal subgroups

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

subgroupindex
C2×#59642
C3×#24042
C2×#50322
#821252
#721392
#695982
#695372
#328993