Finite Group Databasebeta

← #11865382

Subgroups — #11865382

Order 1152.

1711 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic, complemented
2#10596301C211yesmaximal, characteristic, complemented
3#10595045C212yesmaximal, complemented
4C3×#5240C211yesmaximal, characteristic, complemented
5#10591531C218yesmaximal, complemented
6#10591231C212yesmaximal, complemented
7#84126—31nomaximal
8#78691—31nomaximal
9C2×C32×#41C2211yescharacteristic
10C22×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
11C3×#1010—22no—
12C3×#1009—22no—
13C3×D3×#41—24no—
14C3×#1007—21no—
15C3×#1004—21no—
16C3×#1004C2214yes—
17C2×C3×#235—22no—
18C2×C3×#235C2212yes—
19#64156—21no—
20#64149—21no—
21#64146C2214yescomplemented
22#64139—22no—
23#64135C2211yescharacteristic, complemented
24C3×#926C2212yescomplemented
25#63981C2218yescomplemented
26#63911—22no—
27#63909—24no—
28#63864—22no—
29C3×#834C2214yescomplemented
30#63580C2218yescomplemented
31C3×#585—31no—
32#5964D311yescharacteristic, complemented
33#5830—31no—
34#5830—31no—
35#5830—32no—
36#5830—61no—
37#5525—32no—
38C3×#410—31no—
39#5244—38no—
40#5240—32no—
41#5240D311yescharacteristic, complemented
42#5006—32no—
43#4741—38no—
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×#638—24no—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)—21no—
55C2×C4⋊(C3⋊D3)—22no—
56C2×C4⋊(C3⋊D3)—22no—
57C2×C3×#67—24no—
58C2×C3×#67C2314yes—
59C3×#239—24no—
60C3×D3×Q8—24no—
61C3×#237—24no—
62C3×#237—28no—
63C3×D3×D4—24no—
64C3×D3×D4—28no—
65C3×#235—24no—
66C3×#235—24no—
67C3×#235—22no—
68C3×#235—24no—
69C3×#235—22no—
70C3×#235—22no—
71C3×#235—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×#626—28no—
79#3757C2312yes—
80#3752—24no—
81C4⋊(C3⋊Dic3)—21no—
82C4×C3⋊Dic3—21no—
83C3×#217C2314yes—
84C3×C4⋊Dic3—22no—
85C3×C4×Dic3—22no—
86#3722—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
99#1010—22no—
100#1009—22no—
101D3×#41—24no—
102#1007—21no—
103#1004—21no—
104#1004C2×D314yes—
105C2×#235—22no—
106C2×#235C2×D312yes—
107C22×D12—31no—
108C22×D12—31no—
109C22×D12—61no—
110C3×#117—32no—
111#947—62no—
112#947—62no—
113#947—61no—
114#947—61no—
115#940—62no—
116#940—62no—
117#940—61no—
118#940—61no—
119#937—34no—
120#937—34no—
121#937—38no—
122#937—64no—
123#930—64no—
124#930—64no—
125#930—62no—
126#930—62no—
127#926—31no—
128#926—31no—
129#926—61no—
130#926C2×D312yescomplemented
131#895—64no—
132#882—38no—
133#875—62no—
134#874—62no—
135C3×#99—34no—
136#834—38no—
137#834C2×D314yescomplemented
138#806—38no—
139C23×C32C2×D412yes—
140C23×C32C2411yescharacteristic
141C22×C3⋊D3—22no—
142C22×C3⋊D3—24no—
143C22×C3⋊D3—82no—
144C22×C3×D3—24no—
145C22×C3×D3C2×D414yes—
146C32×Q8—22no—
147C32×D4—24no—
148C32×D4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4C2×D412yes—
155C2×C32×C4C2×D412yes—
156#638—84no—
157C2×C3⋊Dic3—22no—
158C4⋊(C3⋊D3)—41no—
159C4⋊(C3⋊D3)—42no—
160C4⋊(C3⋊D3)—41no—
161C3×#67—28no—
162C3×#67—24no—
163C3×#67—24no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—24no—
166C2×C3×Dic3C2×D414yes—
167C3×D12—24no—
168C3×D12—22no—
169C3×D12—22no—
170C3×C4×D3—28no—
171C3×C4×D3—24no—
172C3×C4×D3—24no—
173C3×Dic6—24no—
174C3×Dic6—22no—
175C3×Dic6—22no—
176#626—88no—
177#585—31no—
178#535—92no—
179#535—91no—
180B(4,𝔽2)—98no—
181#410—31no—
182#410—92no—
183C23×D3—32no—
184C23×D3—32no—
185C23×D3—62no—
186C3×#41—22no—
187C3×#41—22no—
188C3×#41—22no—
189C3×#41—22no—
190C3×#41—42no—
191C3×#41—42no—
192C3×#41—64no—
193C3×#41—64no—
194C3×#41—68no—
195C3×#41—64no—
196C3×#41—64no—
197C3×#41—62no—
198C3×#41—62no—
199C3×#41—64no—
200C3×#41—62no—
201C3×#41—62no—
202C2×C3×Q8—21no—
203C2×C3×Q8—32no—
204C2×C3×Q8—62no—
205C2×C3×Q8—64no—
206C2×C3×Q8D1211yescharacteristic, complemented
207C2×C3×Q8#6711yescharacteristic, complemented
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—34no—
211C2×C3×D4—32no—
212C2×C3×D4—62no—
213C2×C3×D4—68no—
214C2×C3×D4—64no—
215C2×C3×D4—64no—
216C2×C3×D4D1211yescharacteristic, 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×C4D1212yescomplemented
225C22×C3×C4#6712yescomplemented
226C22×C3×C4C22×D311yescharacteristic
227C22×C3×C4C22×D311yescharacteristic
228C2×#67—24no—
229C2×#67—68no—
230C2×#67—68no—
231C2×#67—64no—
232C2×#67—64no—
233C2×#67C22×D314yes—
234#239—24no—
235D3×Q8—24no—
236#237—24no—
237#237—28no—
238D3×D4—24no—
239D3×D4—28no—
240#235—24no—
241#235—24no—
242#235—22no—
243#235—24no—
244#235—22no—
245#235—22no—
246#235—22no—
247C2×D12—22no—
248C2×D12—62no—
249C2×D12—62no—
250C2×D12—61no—
251C2×D12—61no—
252C2×D12—62no—
253C2×D12—62no—
254C2×D12—62no—
255C2×D12—62no—
256C2×D12—61no—
257C2×D12—61no—
258C2×D12—68no—
259C2×D12—122no—
260C2×D12—122no—
261C2×D12#6712yescomplemented
262C2×C4×D3—24no—
263C2×C4×D3#6714yescomplemented
264C2×Dic6—22no—
265C2×Dic6#6712yescomplemented
266C3×C4⋊C4—62no—
267C3×C22⋊C4—34no—
268C3×C42—62no—
269#217—32no—
270#217—32no—
271#217—62no—
272#217C22×D314yes—
273(C2×D3)⋊C4—68no—
274(C2×D3)⋊C4—68no—
275(C2×D3)⋊C4—64no—
276(C2×D3)⋊C4—64no—
277(C2×D3)⋊C4—68no—
278C4⋊Dic3—22no—
279C4⋊Dic3—62no—
280C4⋊Dic3—61no—
281C4⋊Dic3—61no—
282C4×Dic3—22no—
283C4×Dic3—62no—
284C4×Dic3—61no—
285C4×Dic3—61no—
286C22×C32—22no—
287C22×C32—41no—
288C22×C32—42no—
289C22×C32C22×D411yescharacteristic
290C22×C32C22×D412yes—
291C2×C3⋊D3—44no—
292C2×C3⋊D3—42no—
293C2×C3⋊D3—48no—
294C2×C3×D3—24no—
295C2×C3×D3—44no—
296C2×C3×D3—84no—
297C32×C4—22no—
298C32×C4#12011yescharacteristic
299C32×C4#12012yes—
300C32×C4#12011yescharacteristic
301C3⋊Dic3—42no—
302C3×Dic3—24no—
303C3×Dic3—44no—
3042−1+4—62no—
3052−1+4—61no—
306H(5,𝔽2)—34no—
307H(5,𝔽2)—62no—
308H(5,𝔽2)—61no—
309C2×#41—32no—
310C2×#41—64no—
311C2×#41—62no—
312C2×#41D3211yescharacteristic
313C22×D4—91no—
314#127—182no—
315#127—181no—
316#124—182no—
317#124—181no—
318#121—184no—
319#121—182no—
320#120—98no—
321#120—94no—
322#117—32no—
323#117—91no—
324#99—34no—
325#99—98no—
326C23×C3—21no—
327C23×C3—22no—
328C23×C3—34no—
329C23×C3—64no—
330C23×C3C2×D1212yes—
331C23×C3C2×#6712yes—
332C23×C3C23×D311yescharacteristic
333C23×C3C23×D311yescharacteristic
334C22×D3—24no—
335C22×D3—64no—
336C22×D3—62no—
337C22×D3—64no—
338C22×D3—64no—
339C22×D3—62no—
340C22×D3—124no—
341C22×D3—242no—
342C22×D3—242no—
343C22×D3—244no—
344C22×D3C2×#6714yes—
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—42no—
348C3×Q8—64no—
349C3×Q8—62no—
350C3×Q8—62no—
351C3×D4—24no—
352C3×D4—24no—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4—42no—
356C3×D4—44no—
357C3×D4—64no—
358C3×D4—68no—
359C3×D4—64no—
360C3×D4—64no—
361C3×D4—62no—
362C3×D4—62no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—34no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—68no—
376C2×C3×C4—64no—
377C2×C3×C4—64no—
378C2×C3×C4—64no—
379C2×C3×C4—64no—
380C2×C3×C4C2×D1212yes—
381C2×C3×C4C2×D1211yescharacteristic
382C2×C3×C4C2×D1211yescharacteristic
383C2×C3×C4D3×D412yes—
384C2×C3×C4D3×D412yes—
385C2×C3×C4C2×#6712yes—
386C2×C3×C4C2×#6711yescharacteristic
387C2×C3×C4C2×#6711yescharacteristic
388C2×C3×C4C2×#6712yes—
389C2×C3×C4C2×#6712yes—
390#67—28no—
391#67—24no—
392#67—24no—
393#67—248no—
394#67—244no—
395#67—244no—
396#67—248no—
397C2×Dic3—24no—
398C2×Dic3—24no—
399C2×Dic3—64no—
400C2×Dic3—62no—
401C2×Dic3—62no—
402C2×Dic3C2×#6714yes—
403D12—24no—
404D12—22no—
405D12—22no—
406D12—122no—
407D12—122no—
408D12—124no—
409D12—121no—
410D12—121no—
411D12—122no—
412D12—121no—
413D12—122no—
414D12—121no—
415D12—248no—
416C4×D3—28no—
417C4×D3—24no—
418C4×D3—24no—
419Dic6—24no—
420Dic6—22no—
421Dic6—22no—
422C2×C32—21no—
423C2×C32—22no—
424C2×C32—42no—
425C2×C32C2×#12011yescharacteristic
426C3⋊D3—82no—
427C3×D3—44no—
428C24—92no—
429#41—22no—
430#41—22no—
431#41—64no—
432#41—64no—
433#41—68no—
434#41—64no—
435#41—64no—
436#41—62no—
437#41—62no—
438#41—64no—
439#41—62no—
440#41—62no—
441C2×Q8—32no—
442C2×Q8—62no—
443C2×Q8—64no—
444C2×Q8#62611yescharacteristic, complemented
445C2×D4—34no—
446C2×D4—32no—
447C2×D4—62no—
448C2×D4—68no—
449C2×D4—64no—
450C2×D4—64no—
451C2×D4—188no—
452C2×D4—184no—
453C2×D4—181no—
454C2×D4—181no—
455C2×D4—182no—
456C2×D4—182no—
457C2×D4#62611yescharacteristic, complemented
458C2×D4C2×D3212yes—
459C22×C4—34no—
460C22×C4—64no—
461C22×C4#62612yescomplemented
462C22×C4C2×D3211yescharacteristic
463C4⋊C4—62no—
464C4⋊C4—181no—
465C22⋊C4—34no—
466C22⋊C4—92no—
467C22⋊C4—188no—
468C22⋊C4—184no—
469C42—62no—
470C42—181no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—44no—
482C22×C3—64no—
483C22×C3—124no—
484C22×C3—244no—
485C22×C3C22×D1211yescharacteristic
486C22×C3C2×D3×D412yes—
487C22×C3C22×#6711yescharacteristic
488C22×C3C22×#6712yes—
489C2×D3—24no—
490C2×D3—44no—
491C2×D3—84no—
492C2×D3—124no—
493C2×D3—128no—
494C2×D3—128no—
495C2×D3—124no—
496C2×D3—124no—
497C2×D3—128no—
498C2×D3—122no—
499C2×D3—122no—
500C2×D3—248no—
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—22no—
506C3×C4—42no—
507C3×C4—64no—
508C3×C4—124no—
509C3×C4#88211yescharacteristic
510C3×C4#88211yescharacteristic
511C3×C4#88212yes—
512C3×C4#95311yescharacteristic
513C3×C4#95312yes—
514C3×C4#95311yescharacteristic
515Dic3—24no—
516Dic3—44no—
517Dic3—124no—
518Dic3—122no—
519Dic3—122no—
520C32#302711yescharacteristic, complemented, Sylow
521C23—34no—
522C23—64no—
523C23—182no—
524C23—184no—
525C23—722no—
526C23C2×#62612yes—
527C23C22×D3211yescharacteristic
528Q8—22no—
529Q8—64no—
530Q8—62no—
531Q8—62no—
532D4—24no—
533D4—22no—
534D4—64no—
535D4—68no—
536D4—64no—
537D4—64no—
538D4—62no—
539D4—62no—
540D4—361no—
541D4—361no—
542D4—362no—
543D4—728no—
544D4—724no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—34no—
548C2×C4—68no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—182no—
554C2×C4C2×#62612yes—
555C2×C4C2×#62611yescharacteristic
556C2×C4C2×#62611yescharacteristic
557C2×C4D3×#6712yes—
558C2×C4D3×#6712yes—
559C2×C3—22no—
560C2×C3—22no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—22no—
564C2×C3—21no—
565C2×C3—42no—
566C2×C3—42no—
567C2×C3—42no—
568C2×C3—82no—
569C2×C3—124no—
570C2×C3C2×#88211yescharacteristic
571C2×C3C2×#95311yescharacteristic
572D3—44no—
573D3—244no—
574D3—242no—
575D3—242no—
576C22—22no—
577C22—41no—
578C22—42no—
579C22—64no—
580C22—124no—
581C22—244no—
582C22—364no—
583C22—368no—
584C22—362no—
585C22C22×#62611yescharacteristic
586C22C2×D3×#6712yes—
587C4—22no—
588C4—64no—
589C4—124no—
590C4—362no—
591C4#6397711yescharacteristic
592C4#6397711yescharacteristic
593C4#6397712yes—
594C3—21no—
595C3#7869111yescharacteristic, complemented
596C3#8412611yescharacteristic, complemented
597C2—22no—
598C2—21no—
599C2—42no—
600C2—124no—
601C2—722no—
602C2C2×#6397711yescharacteristic
subgroupG/H
centerC2C2×#63977
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. #10596301C2semidirect
3. #10595045C2semidirect
4. C3×#5240C2semidirect
5. #10591531C2semidirect
6. #10591231C2semidirect
10. C22×C4⋊(C3⋊D3)C22semidirect
21. #64146C22semidirect
23. #64135C22semidirect
24. C3×#926C22semidirect
25. #63981C22semidirect
29. C3×#834C22semidirect
30. #63580C22semidirect
32. #5964D3semidirect
41. #5240D3semidirect
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
130. #926C2×D3semidirect
137. #834C2×D3semidirect
206. C2×C3×Q8D12semidirect
207. C2×C3×Q8#67semidirect
216. C2×C3×D4D12semidirect
217. C2×C3×D4#67semidirect
224. C22×C3×C4D12semidirect
225. C22×C3×C4#67semidirect
261. C2×D12#67semidirect
263. C2×C4×D3#67semidirect
265. C2×Dic6#67semidirect
444. C2×Q8#626semidirect
457. C2×D4#626semidirect
461. C22×C4#626semidirect
520. C32#3027semidirect
595. C3#78691semidirect
596. C3#84126semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
#105963012
#105950452
C3×#52402
#105915312
#105912312
#841263
#786913