Finite Group Databasebeta

← #11865384

Subgroups — #11865384

Order 1152.

1623 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic, complemented
2C3×#5830C211yesmaximal, characteristic, complemented
3#10595040C212yesmaximal, complemented
4#10592939C211yesmaximal, characteristic, complemented
5#10591530C218yesmaximal, complemented
6#10591231C212yesmaximal, complemented
7#84126—31nomaximal
8#83396—31nomaximal
9C2×C32×#41C2211yescharacteristic
10C3×#1010—22no—
11C3×#1009—22no—
12C3×D3×#41—24no—
13C3×#1007—21no—
14C3×#1004—21no—
15C3×#1004C2214yes—
16C2×C3×#235—22no—
17C2×C3×#235C2212yes—
18C22×C3×D12C2211yescharacteristic, complemented
19#64135C2212yescomplemented
20C3×#947—21no—
21C3×#940—21no—
22C3×#937C2214yescomplemented
23C3×#930—22no—
24C3×#926C2211yescharacteristic, complemented
25#63976C2218yescomplemented
26#63914—22no—
27#63908—24no—
28#63863—22no—
29#63661C2214yescomplemented
30#63580C2218yescomplemented
31C3×#585—31no—
32#5964D311yescharacteristic, complemented
33C3×#535—31no—
34#5830—32no—
35#5830D311yescharacteristic, complemented
36#5805—32no—
37#5244—38no—
38#5240—31no—
39#5240—31no—
40#5240—32no—
41#5240—61no—
42#5032—38no—
43#5006—32no—
44C23×C3×D3C2312yes—
45C32×#41—22no—
46C32×#41—22no—
47C2×C32×Q8D411yescharacteristic, complemented
48C2×C32×D4D411yescharacteristic, complemented
49C2×C32×D4C2312yes—
50C22×C32×C4D412yescomplemented
51C22×C32×C4C2311yescharacteristic
52C2×C3×#67—24no—
53C2×C3×#67—24no—
54C2×C3×#67C2314yes—
55C3×#239—24no—
56C3×D3×Q8—24no—
57C3×#237—24no—
58C3×#237—28no—
59C3×D3×D4—24no—
60C3×D3×D4—28no—
61C3×#235—24no—
62C3×#235—24no—
63C3×#235—22no—
64C3×#235—24no—
65C3×#235—22no—
66C3×#235—22no—
67C3×#235—22no—
68C2×C3×D12—22no—
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12—22no—
72C2×C3×D12—22no—
73C2×C3×D12D412yescomplemented
74C2×C3×C4×D3—24no—
75C2×C3×C4×D3D414yescomplemented
76C2×C3×Dic6—22no—
77C2×C3×Dic6D412yescomplemented
78C2×#625—28no—
79#3757C2314yes—
80C4⋊(C3⋊Dic3)—22no—
81C4×C3⋊Dic3—22no—
82C3×#217C2312yes—
83C3×(C2×D3)⋊C4—24no—
84C3×C4⋊Dic3—21no—
85C3×C4×Dic3—21no—
86(C2×D3)⋊Dic3—28no—
87#3027—91noSylow
88C3×2−1+4—62no—
89C3×2−1+4—61no—
90C3×H(5,𝔽2)—34no—
91C3×H(5,𝔽2)—62no—
92C3×H(5,𝔽2)—61no—
93C2×C3×#41—21no—
94C2×C3×#41—32no—
95C2×C3×#41—64no—
96C2×C3×#41—62no—
97C2×C3×#41C2×D311yescharacteristic
98C2×C3×#41C2×D311yescharacteristic
99C22×C3×D4—31no—
100#1010—22no—
101#1009—22no—
102D3×#41—24no—
103#1007—21no—
104#1004—21no—
105#1004C2×D314yes—
106C2×#235—22no—
107C2×#235C2×D312yes—
108C22×D12C2×D311yescharacteristic, complemented
109C3×#127—61no—
110C3×#124—61no—
111C3×#121—62no—
112C3×#120—34no—
113C3×#117—31no—
114#953—38no—
115#947—21no—
116#947—62no—
117#940—21no—
118#940—62no—
119#940—62no—
120#938—64no—
121#937—38no—
122#937C2×D314yescomplemented
123#936—62no—
124#930—22no—
125#930—64no—
126#926—32no—
127#926—32no—
128#926—62no—
129#926C2×D311yescharacteristic, complemented
130#834—34no—
131#834—34no—
132#834—38no—
133#834—64no—
134#806—38no—
135C23×C32C2×D412yes—
136C23×C32C2411yescharacteristic
137C22×C3×D3—24no—
138C22×C3×D3—24no—
139C22×C3×D3—22no—
140C22×C3×D3—82no—
141C22×C3×D3C2×D414yes—
142C32×Q8—22no—
143C32×D4—24no—
144C32×D4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4C2×D412yes—
148C2×C32×C4C2×D412yes—
149C2×C32×C4C2×D412yes—
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D411yescharacteristic
152C2×C3⋊Dic3—24no—
153C3×#67—28no—
154C3×#67—24no—
155C3×#67—24no—
156C3×#67—84no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3C2×D414yes—
160C3×D12—24no—
161C3×D12—22no—
162C3×D12—22no—
163C3×D12—41no—
164C3×D12—42no—
165C3×D12—41no—
166C3×C4×D3—28no—
167C3×C4×D3—24no—
168C3×C4×D3—24no—
169C3×Dic6—24no—
170C3×Dic6—22no—
171C3×Dic6—22no—
172#625—88no—
173#585—31no—
174#535—31no—
175#535—92no—
176B(4,𝔽2)—98no—
177#410—92no—
178#410—91no—
179C24×C3—32no—
180C23×D3C22×D312yes—
181C3×#41—22no—
182C3×#41—22no—
183C3×#41—22no—
184C3×#41—22no—
185C3×#41—42no—
186C3×#41—42no—
187C3×#41—64no—
188C3×#41—64no—
189C3×#41—68no—
190C3×#41—64no—
191C3×#41—64no—
192C3×#41—62no—
193C3×#41—62no—
194C3×#41—62no—
195C3×#41—62no—
196C3×#41—64no—
197C2×C3×Q8—21no—
198C2×C3×Q8—32no—
199C2×C3×Q8—62no—
200C2×C3×Q8—64no—
201C2×C3×Q8#6711yescharacteristic, complemented
202C2×C3×Q8#6711yescharacteristic, complemented
203C2×C3×D4—21no—
204C2×C3×D4—22no—
205C2×C3×D4—34no—
206C2×C3×D4—32no—
207C2×C3×D4—62no—
208C2×C3×D4—68no—
209C2×C3×D4—64no—
210C2×C3×D4—64no—
211C2×C3×D4—64no—
212C2×C3×D4—62no—
213C2×C3×D4—62no—
214C2×C3×D4—61no—
215C2×C3×D4—61no—
216C2×C3×D4#6711yescharacteristic, complemented
217C2×C3×D4#6711yescharacteristic, complemented
218C2×C3×D4C22×D312yes—
219C2×C3×D4C22×D312yes—
220C22×C3×C4—22no—
221C22×C3×C4—21no—
222C22×C3×C4—34no—
223C22×C3×C4—64no—
224C22×C3×C4#6712yescomplemented
225C22×C3×C4#6712yescomplemented
226C22×C3×C4C22×D311yescharacteristic
227C22×C3×C4C22×D311yescharacteristic
228C2×#67—24no—
229C2×#67—24no—
230C2×#67—68no—
231C2×#67—68no—
232C2×#67C22×D314yes—
233#239—24no—
234D3×Q8—24no—
235#237—24no—
236#237—28no—
237D3×D4—24no—
238D3×D4—28no—
239#235—24no—
240#235—24no—
241#235—22no—
242#235—24no—
243#235—22no—
244#235—22no—
245#235—22no—
246C2×D12—22no—
247C2×D12—22no—
248C2×D12—22no—
249C2×D12—21no—
250C2×D12—21no—
251C2×D12#6712yescomplemented
252C2×C4×D3—24no—
253C2×C4×D3#6714yescomplemented
254C2×Dic6—22no—
255C2×Dic6#6712yescomplemented
256C3×C4⋊C4—61no—
257C3×C22⋊C4—32no—
258C3×C22⋊C4—64no—
259C3×C42—61no—
260#217—34no—
261#217—34no—
262#217—64no—
263#217—68no—
264#217C22×D312yes—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—68no—
267C4⋊Dic3—21no—
268C4⋊Dic3—62no—
269C4⋊Dic3—62no—
270C4⋊Dic3—122no—
271C4×Dic3—21no—
272C4×Dic3—62no—
273C4×Dic3—62no—
274C4×Dic3—122no—
275C22×C32—22no—
276C22×C32—41no—
277C22×C32—42no—
278C22×C32C22×D412yes—
279C22×C32C22×D411yescharacteristic
280C2×C3×D3—24no—
281C2×C3×D3—44no—
282C2×C3×D3—48no—
283C2×C3×D3—44no—
284C2×C3×D3—42no—
285C2×C3×D3—84no—
286C32×C4—22no—
287C32×C4#12011yescharacteristic
288C32×C4#12012yes—
289C32×C4#12011yescharacteristic
290C3⋊Dic3—44no—
291C3×Dic3—24no—
292C3×Dic3—42no—
2932−1+4—62no—
2942−1+4—61no—
295H(5,𝔽2)—34no—
296H(5,𝔽2)—62no—
297H(5,𝔽2)—61no—
298C2×#41—32no—
299C2×#41—64no—
300C2×#41—62no—
301C2×#41D3211yescharacteristic
302C22×D4—31no—
303#127—61no—
304#127—182no—
305#124—61no—
306#124—182no—
307#121—62no—
308#121—184no—
309#120—34no—
310#120—98no—
311#117—31no—
312#117—92no—
313#99—98no—
314#99—94no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—34no—
318C23×C3—64no—
319C23×C3—64no—
320C23×C3—62no—
321C23×C3—242no—
322C23×C3C2×#6712yes—
323C23×C3C2×#6712yes—
324C23×C3C23×D311yescharacteristic
325C23×C3C23×D311yescharacteristic
326C22×D3—24no—
327C22×D3—24no—
328C22×D3—22no—
329C22×D3—82no—
330C22×D3C2×#6714yes—
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—42no—
334C3×Q8—64no—
335C3×Q8—62no—
336C3×Q8—62no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—42no—
342C3×D4—44no—
343C3×D4—64no—
344C3×D4—68no—
345C3×D4—64no—
346C3×D4—64no—
347C3×D4—62no—
348C3×D4—62no—
349C3×D4—121no—
350C3×D4—122no—
351C3×D4—121no—
352C3×D4—244no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—34no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—68no—
366C2×C3×C4—64no—
367C2×C3×C4—64no—
368C2×C3×C4—64no—
369C2×C3×C4—62no—
370C2×C3×C4D3×D412yes—
371C2×C3×C4D3×D412yes—
372C2×C3×C4C2×#6712yes—
373C2×C3×C4C2×#6712yes—
374C2×C3×C4C2×#6711yescharacteristic
375C2×C3×C4C2×#6711yescharacteristic
376C2×C3×C4C2×#6711yescharacteristic
377C2×C3×C4C2×#6711yescharacteristic
378C2×C3×C4C2×#6712yes—
379C2×C3×C4C2×#6712yes—
380#67—28no—
381#67—24no—
382#67—24no—
383#67—84no—
384#67—248no—
385#67—248no—
386C2×Dic3—24no—
387C2×Dic3—22no—
388C2×Dic3—64no—
389C2×Dic3—64no—
390C2×Dic3—124no—
391C2×Dic3C2×#6714yes—
392D12—24no—
393D12—22no—
394D12—22no—
395D12—41no—
396D12—41no—
397D12—42no—
398C4×D3—28no—
399C4×D3—24no—
400C4×D3—24no—
401Dic6—24no—
402Dic6—22no—
403Dic6—22no—
404C2×C32—21no—
405C2×C32—22no—
406C2×C32—42no—
407C2×C32C2×#12011yescharacteristic
408C3×D3—44no—
409C3×D3—82no—
410C24—32no—
411#41—22no—
412#41—22no—
413#41—64no—
414#41—64no—
415#41—68no—
416#41—64no—
417#41—64no—
418#41—62no—
419#41—62no—
420#41—64no—
421#41—62no—
422#41—62no—
423C2×Q8—32no—
424C2×Q8—62no—
425C2×Q8—64no—
426C2×Q8#62511yescharacteristic, complemented
427C2×D4—34no—
428C2×D4—32no—
429C2×D4—62no—
430C2×D4—68no—
431C2×D4—64no—
432C2×D4—64no—
433C2×D4—64no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—62no—
437C2×D4—62no—
438C2×D4—188no—
439C2×D4#62511yescharacteristic, complemented
440C2×D4C2×D3212yes—
441C22×C4—34no—
442C22×C4—64no—
443C22×C4#62512yescomplemented
444C22×C4C2×D3211yescharacteristic
445C4⋊C4—61no—
446C4⋊C4—182no—
447C22⋊C4—32no—
448C22⋊C4—64no—
449C22⋊C4—94no—
450C22⋊C4—188no—
451C42—61no—
452C42—182no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—22no—
457C22×C3—42no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—64no—
463C22×C3—81no—
464C22×C3—82no—
465C22×C3—124no—
466C22×C3—128no—
467C22×C3—124no—
468C22×C3—122no—
469C22×C3—244no—
470C22×C3C2×D3×D412yes—
471C22×C3C22×#6711yescharacteristic
472C22×C3C22×#6711yescharacteristic
473C22×C3C22×#6712yes—
474C2×D3—24no—
475C2×D3—44no—
476C2×D3—44no—
477C2×D3—48no—
478C2×D3—42no—
479C2×D3—84no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—22no—
485C3×C4—42no—
486C3×C4—64no—
487C3×C4—122no—
488C3×C4#93711yescharacteristic
489C3×C4#93711yescharacteristic
490C3×C4#93712yes—
491C3×C4#95311yescharacteristic
492C3×C4#95312yes—
493C3×C4#95311yescharacteristic
494Dic3—24no—
495Dic3—42no—
496Dic3—124no—
497Dic3—124no—
498Dic3—244no—
499C32#302711yescharacteristic, complemented, Sylow
500C23—34no—
501C23—64no—
502C23—62no—
503C23—64no—
504C23—242no—
505C23C2×#62512yes—
506C23C22×D3211yescharacteristic
507Q8—22no—
508Q8—64no—
509Q8—62no—
510Q8—62no—
511D4—24no—
512D4—22no—
513D4—64no—
514D4—68no—
515D4—64no—
516D4—64no—
517D4—62no—
518D4—62no—
519D4—121no—
520D4—121no—
521D4—122no—
522D4—244no—
523D4—728no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—34no—
527C2×C4—68no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—62no—
532C2×C4—184no—
533C2×C4C2×#62512yes—
534C2×C4C2×#62511yescharacteristic
535C2×C4C2×#62511yescharacteristic
536C2×C4D3×#6712yes—
537C2×C4D3×#6712yes—
538C2×C3—22no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—42no—
544C2×C3—41no—
545C2×C3—42no—
546C2×C3—42no—
547C2×C3—82no—
548C2×C3—124no—
549C2×C3—242no—
550C2×C3C2×#93711yescharacteristic
551C2×C3C2×#95311yescharacteristic
552D3—44no—
553D3—82no—
554C22—22no—
555C22—41no—
556C22—42no—
557C22—64no—
558C22—124no—
559C22—124no—
560C22—128no—
561C22—122no—
562C22—244no—
563C22C22×#62511yescharacteristic
564C22C2×D3×#6712yes—
565C4—22no—
566C4—64no—
567C4—122no—
568C4—364no—
569C4#6397611yescharacteristic
570C4#6397611yescharacteristic
571C4#6397612yes—
572C3—21no—
573C3#8339611yescharacteristic, complemented
574C3#8412611yescharacteristic, complemented
575C2—21no—
576C2—22no—
577C2—42no—
578C2—124no—
579C2—242no—
580C2C2×#6397611yescharacteristic
subgroupG/H
centerC2C2×#63976
commutator, cosocleC23×C32C24
FrattiniC23C22×D32
FittingC2×C32×#41C22
socleC2×C32C2×#120
hypercenterC2×#41D32
hypocenterC32#3027
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C3×#5964C2semidirect
2. C3×#5830C2semidirect
3. #10595040C2semidirect
4. #10592939C2semidirect
5. #10591530C2semidirect
6. #10591231C2semidirect
18. C22×C3×D12C22semidirect
19. #64135C22semidirect
22. C3×#937C22semidirect
24. C3×#926C22semidirect
25. #63976C22semidirect
29. #63661C22semidirect
30. #63580C22semidirect
32. #5964D3semidirect
35. #5830D3semidirect
47. C2×C32×Q8D4semidirect
48. C2×C32×D4D4semidirect
50. C22×C32×C4D4semidirect
73. C2×C3×D12D4semidirect
75. C2×C3×C4×D3D4semidirect
77. C2×C3×Dic6D4semidirect
108. C22×D12C2×D3semidirect
122. #937C2×D3semidirect
129. #926C2×D3semidirect
201. C2×C3×Q8#67semidirect
202. C2×C3×Q8#67semidirect
216. C2×C3×D4#67semidirect
217. C2×C3×D4#67semidirect
224. C22×C3×C4#67semidirect
225. C22×C3×C4#67semidirect
251. C2×D12#67semidirect
253. C2×C4×D3#67semidirect
255. C2×Dic6#67semidirect
426. C2×Q8#625semidirect
439. C2×D4#625semidirect
443. C22×C4#625semidirect
499. C32#3027semidirect
573. C3#83396semidirect
574. C3#84126semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C3×#58302
#105950402
#105929392
#105915302
#105912312
#841263
#833963