Finite Group Databasebeta

← #10943972

Subgroups — #10943972

Order 768.

3706 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 3706 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
2C3×#2406C211yesmaximal, characteristic, complemented
3C2×#5053C214yesmaximal, complemented
4#82192C212yesmaximal, complemented
5#72139C211yesmaximal, characteristic, complemented
6#69598C214yesmaximal, complemented
7#69537C212yesmaximal, complemented
8#32901—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×#123C2212yescomplemented
20C22×C3×C4⋊C4C2211yescharacteristic, complemented
21C2×#929C2214yescomplemented
22C3×#395—21no—
23C3×#394—21no—
24C3×#386C2212yescomplemented
25C3×#385—22no—
26C3×#381C2211yescharacteristic, complemented
27C2×#834C2214yescomplemented
28#5213C2214yescomplemented
29#5211C2212yescomplemented
30#5053—216no—
31#4991—22no—
32#4989—24no—
33#4979—22no—
34C2×#806C2218yescomplemented
35C2×#585—31no—
36C2×#458—34no—
37#2406—32no—
38#2406D311yescharacteristic, complemented
39#1790—34no—
40#1782—32no—
41#1782—31no—
42C2×C3×#41—28no—
43C2×C3×#41—22no—
44C2×C3×#41—22no—
45C22×C3×Q8D411yescharacteristic
46C22×C3×D4D411yescharacteristic
47C22×C3×D4C2312yes—
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C22×#67—24no—
53C22×#67C2314yes—
54#1010—232no—
55#1009—232no—
56D3×#41—232no—
57D3×#41—232no—
58#1007—216no—
59C2×#239—24no—
60C2×D3×Q8—24no—
61#1004—216no—
62#1004—28no—
63#1004—216no—
64#1004D4116yes—
65C2×#237—24no—
66C2×#237—28no—
67C2×D3×D4—24no—
68C2×D3×D4—28no—
69C2×#235—24no—
70C2×#235—24no—
71C2×#235—216no—
72C2×#235—216no—
73C2×#235—22no—
74C2×#235—24no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235—22no—
78C2×#235—216no—
79C22×D12—22no—
80C22×D12D412yes—
81C22×C4×D3—24no—
82C22×C4×D3D414yes—
83C22×Dic6—22no—
84C22×Dic6D412yes—
85C3×#123—28no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4C2312yes—
94C2×#217C2314yes—
95#929—216no—
96C2×(C2×D3)⋊C4—24no—
97C2×(C2×D3)⋊C4—24no—
98C2×Dic3⋊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×#123—34no—
118C2×#123C2×D312yescomplemented
119C22×C4⋊C4C2×D311yescharacteristic, complemented
120#458—616no—
121C2×#99—34no—
122C2×#99—38no—
123#395—21no—
124#395—62no—
125#394—21no—
126#394—62no—
127#386—34no—
128#386C2×D312yescomplemented
129#385—22no—
130#385—64no—
131#381—32no—
132#381C2×D311yescharacteristic, complemented
133C24×C3C2×D412yes—
134C24×C3C2411yescharacteristic
135C23×D3—24no—
136C23×D3C2×D414yes—
137C3×#41—216no—
138C3×#41—216no—
139C2×C3×Q8—22no—
140C2×C3×Q8—24no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4C2×D418yes—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—24no—
156C22×C3×C4—21no—
157C22×C3×C4—42no—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C2×#67—28no—
164C2×#67—216no—
165C2×#67—216no—
166C2×#67—24no—
167C2×#67—24no—
168C2×#67—28no—
169C2×#67C2×D4116yes—
170C22×Dic3—24no—
171C22×Dic3—24no—
172C22×Dic3C2×D414yes—
173#239—232no—
174D3×Q8—232no—
175#237—232no—
176#237—232no—
177#237—232no—
178D3×D4—232no—
179D3×D4—232no—
180D3×D4—232no—
181#235—216no—
182#235—216no—
183#235—216no—
184#235—232no—
185#235—216no—
186#235—232no—
187#235—216no—
188#235—216no—
189C2×D12—24no—
190C2×D12—216no—
191C2×D12—28no—
192C2×D12—22no—
193C2×D12—22no—
194C2×C4×D3—28no—
195C2×C4×D3—232no—
196C2×C4×D3—28no—
197C2×C4×D3—24no—
198C2×C4×D3—24no—
199C2×C4×D3—28no—
200C2×Dic6—24no—
201C2×Dic6—216no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—28no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4—44no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—44no—
211C3×C22⋊C4—44no—
212#217—28no—
213(C2×D3)⋊C4—48no—
214(C2×D3)⋊C4—48no—
215Dic3⋊C4—48no—
2162−1+4—616no—
2172−1+4—632no—
218H(5,𝔽2)—316no—
219H(5,𝔽2)—616no—
220H(5,𝔽2)—616no—
221H(5,𝔽2)—632no—
222H(5,𝔽2)—68no—
223C2×#41—28no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—616no—
227C2×#41—64no—
228C2×#41—64no—
229C2×#41—616no—
230C2×#41—616no—
231C2×#41—68no—
232C2×#41—64no—
233C2×#41—64no—
234C2×#41—632no—
235C2×#41—632no—
236C2×#41—62no—
237C2×#41—62no—
238C2×#41—62no—
239C2×#41—62no—
240C2×#41—64no—
241C22×Q8—32no—
242C22×Q8—62no—
243C22×Q8—64no—
244C22×Q8#6711yescharacteristic
245C22×D4—34no—
246C22×D4—32no—
247C22×D4—62no—
248C22×D4—68no—
249C22×D4—64no—
250C22×D4—64no—
251C22×D4#6711yescharacteristic
252C22×D4C22×D312yes—
253C23×C4—34no—
254C23×C4—64no—
255C23×C4#6712yes—
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259#123—28no—
260#123—616no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—64no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—34no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4C22×D312yes—
273#99—616no—
274#99—68no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95—64no—
279#95—62no—
280#95—62no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3#12014yes—
288C23×C3C22×D412yes—
289C23×C3C22×D414yes—
290C23×C3C22×D411yescharacteristic
291C22×D3—24no—
292C22×D3—216no—
293C22×D3—24no—
294C22×D3—44no—
295C22×D3—44no—
296C22×D3—84no—
297C22×D3—84no—
298C22×D3#12018yes—
299C3×Q8—28no—
300C3×D4—28no—
301C3×D4—216no—
302C2×C3×C4—28no—
303C2×C3×C4—28no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4#12011yescharacteristic
322C2×C3×C4#12012yes—
323C2×C3×C4#12011yescharacteristic
324C2×C3×C4#12014yes—
325#67—232no—
326#67—216no—
327#67—216no—
328C2×Dic3—216no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—44no—
332C2×Dic3—44no—
333C2×Dic3—44no—
334C2×Dic3#12018yes—
335D12—28no—
336D12—216no—
337D12—28no—
338C4×D3—216no—
339C4×D3—232no—
340C4×D3—216no—
341Dic6—28no—
342Dic6—216no—
343Dic6—28no—
344C24—34no—
345C24—64no—
346C24C2×#6712yes—
347C24C23×D311yescharacteristic
348#41—216no—
349#41—216no—
350#41—632no—
351#41—616no—
352#41—616no—
353#41—632no—
354#41—616no—
355#41—616no—
356#41—616no—
357#41—632no—
358#41—632no—
359#41—632no—
360#41—632no—
361#41—616no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—68no—
365C2×Q8—64no—
366C2×Q8—616no—
367C2×Q8—632no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—316no—
375C2×D4—68no—
376C2×D4—64no—
377C2×D4—616no—
378C2×D4—632no—
379C2×D4—632no—
380C2×D4—632no—
381C2×D4—68no—
382C2×D4—616no—
383C2×D4—616no—
384C2×D4—64no—
385C2×D4—68no—
386C2×D4—64no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4D3×D418yes—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—34no—
402C22×C4—42no—
403C22×C4—68no—
404C22×C4—68no—
405C22×C4—68no—
406C22×C4—632no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4D3×D412yes—
412C22×C4D3×D412yes—
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6712yes—
415C22×C4C2×#6711yescharacteristic
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—128no—
421C22⋊C4—24no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—68no—
425C22⋊C4—128no—
426C22⋊C4—128no—
427C22×C3—22no—
428C22×C3—21no—
429C22×C3—22no—
430C22×C3—21no—
431C22×C3—24no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—82no—
437C22×C3C2×#12012yes—
438C22×C3C2×#12011yescharacteristic
439C22×C3C2×#12014yes—
440C2×D3—28no—
441C2×D3—44no—
442C2×D3—44no—
443C2×D3—48no—
444C2×D3—84no—
445C2×D3—816no—
446C3×C4—21no—
447C3×C4—24no—
448C3×C4—22no—
449C3×C4—21no—
450C3×C4—42no—
451C3×C4—82no—
452C3×C4—81no—
453Dic3—28no—
454Dic3—84no—
455C23—22no—
456C23—22no—
457C23—38no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—616no—
463C23—64no—
464C23—64no—
465C23—124no—
466C23—124no—
467C23—244no—
468C23—244no—
469C23#93714yes—
470C23C2×D3×D412yes—
471C23C2×D3×D414yes—
472C23C22×#6711yescharacteristic
473Q8—28no—
474Q8—68no—
475Q8—68no—
476Q8—616no—
477D4—28no—
478D4—216no—
479D4—68no—
480D4—68no—
481D4—616no—
482D4—632no—
483D4—616no—
484D4—616no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—38no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—616no—
506C2×C4—616no—
507C2×C4—632no—
508C2×C4—616no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—124no—
512C2×C4—124no—
513C2×C4—124no—
514C2×C4#93714yes—
515C2×C4#93711yescharacteristic
516C2×C4#93711yescharacteristic
517C2×C4#93712yes—
518C2×C3—24no—
519C2×C3—22no—
520C2×C3—44no—
521C2×C3#284611yescharacteristic
522C2×C3#302511yescharacteristic
523C2×C3#302811yescharacteristic
524D3—48no—
525C22—21no—
526C22—22no—
527C22—21no—
528C22—22no—
529C22—24no—
530C22—44no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—68no—
535C22—82no—
536C22—128no—
537C22—124no—
538C22—124no—
539C22—244no—
540C22—2416no—
541C22D3×#12014yes—
542C22C2×#93712yes—
543C22C2×#93711yescharacteristic
544C4—21no—
545C4—21no—
546C4—24no—
547C4—22no—
548C4—42no—
549C4—68no—
550C4—82no—
551C4—81no—
552C4—244no—
553C3#3290111yescharacteristic, complemented, Sylow
554C2—22no—
555C2—24no—
556C2—44no—
557C2—128no—
558C2#8261211yescharacteristic
559C2#8293611yescharacteristic
560C2#8341711yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2406C2
socleC22×C3C2×#120
prime radicalC2×#5964C2
hypercenter#2406D3
hypocenterC3#32901
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. C3×#2406C2semidirect
3. C2×#5053C2semidirect
4. #82192C2semidirect
5. #72139C2semidirect
6. #69598C2semidirect
7. #69537C2semidirect
19. C2×C3×#123C22semidirect
20. C22×C3×C4⋊C4C22semidirect
21. C2×#929C22semidirect
24. C3×#386C22semidirect
26. C3×#381C22semidirect
27. C2×#834C22semidirect
28. #5213C22semidirect
29. #5211C22semidirect
34. C2×#806C22semidirect
38. #2406D3semidirect
118. C2×#123C2×D3semidirect
119. C22×C4⋊C4C2×D3semidirect
128. #386C2×D3semidirect
132. #381C2×D3semidirect
553. C3#32901semidirect

Maximal subgroups

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

subgroupindex
C2×#59642
C3×#24062
C2×#50532
#821922
#721392
#695982
#695372
#329013