Finite Group Databasebeta

← #11868979

Subgroups — #11868979

Order 1152.

965 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10595992C211yesmaximal, characteristic
2C3×#5519C211yesmaximal, characteristic, complemented
3C3×#4999C211yesmaximal, characteristic, complemented
4#10591130C211yesmaximal, characteristic, complemented
5#10591122C211yesmaximal, characteristic, complemented
6#10591096C211yesmaximal, characteristic
7C2×#63564C211yesmaximal, characteristic
8#10590313C212yesmaximal, complemented
9#10590308C212yesmaximal, complemented
10#10590308C212yesmaximal, complemented
11#10590300C212yesmaximal, complemented
12#83474—31nomaximal
13#79090—31nomaximal
14C22×C4⋊(C3⋊D3)C2211yescharacteristic
15C32×#117C2211yescharacteristic
16#64135C2211yescharacteristic
17#64102—21no—
18#64102—21no—
19C22⋊C4×C3⋊D3—22no—
20C4×C4⋊(C3⋊D3)C2212yes—
21C4×C4⋊(C3⋊D3)C2212yes—
22C3×#926C2211yescharacteristic
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#890—21no—
27C3×#888—21no—
28C3×#878—22no—
29C3×#870C2212yescomplemented
30C3×#869C2212yescomplemented
31C3×C4⋊(C3⋊C8)C2212yescomplemented
32C3×C4⋊(C3⋊C8)C2212yescomplemented
33#63575C2211yescharacteristic, complemented
34#63575C2212yescomplemented
35#63575C2211yescharacteristic, complemented
36#63573C2211yescharacteristic, complemented
37#63573C2212yescomplemented
38#63573C2211yescharacteristic, complemented
39#63564C2212yes—
40#63564C2212yes—
41#63564C2212yes—
42#63564C2212yes—
43C3×#544—31no—
44#5521—31no—
45#5521—31no—
46#5521—61no—
47#5519D311yescharacteristic, complemented
48C3×#424—31no—
49#5131—31no—
50#5121—31no—
51C2×#821—31no—
52#5106—31no—
53#5087—32no—
54#5084—32no—
55#5066—32no—
56#5066—32no—
57#5001—31no—
58#4999D311yescharacteristic, complemented
59#4998—31no—
60C2×#809—31no—
61#4964—31no—
62#4847—32no—
63#4847—32no—
64#4846—32no—
65#4842—32no—
66C23×C3⋊D3—21no—
67C22×C32×C4C2311yescharacteristic
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)—22no—
70C2×C4⋊(C3⋊D3)Q814yescomplemented
71C2×C4⋊(C3⋊D3)C2312yes—
72C2×C4×C3⋊D3—22no—
73C2×C4×C3⋊D3—22no—
74C22×C3×Dic3—21no—
75C2×C3×Dic6—22no—
76C32×C4⋊C4D411yescharacteristic
77C32×C4⋊C4D411yescharacteristic
78C32×C22⋊C4D412yes—
79C32×C42C2312yes—
80#3757—21no—
81#3752—22no—
82#3752—22no—
83C4⋊(C3⋊Dic3)C2311yescharacteristic
84C4⋊(C3⋊Dic3)C2311yescharacteristic
85C4×C3⋊Dic3—21no—
86C3×#217—21no—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3C2312yes—
90C3×C4⋊Dic3C2311yescharacteristic
91C3×C4⋊Dic3C2311yescharacteristic
92C3×Dic3⋊C4—22no—
93C3×Dic3⋊C4—22no—
94C3×C4×Dic3—21no—
95C3×#208—22no—
96C2×C3×C3⋊C8—22no—
97C2×C3×C3⋊C8C2311yescharacteristic
98C2×C3×C3⋊C8C2311yescharacteristic
99C2×C3×C3⋊C8C2312yes—
100#3081—91noSylow
101C22×D12—31no—
102C22×D12—31no—
103C22×D12—61no—
104C2×C3×M16—31no—
105C22×C3×C8—31no—
106C3×C4⋊Q8—32no—
107C3×C4⋊Q8—61no—
108C3×#125—32no—
109C3×#125—61no—
110C3×C22⋊Q8—62no—
111C3×#117—21no—
112C3×#117—31no—
113C3×#117C2×D311yescharacteristic
114C3×#117C2×D311yescharacteristic
115C2×C3×C4⋊C4—31no—
116#926—31no—
117#926—31no—
118#926—61no—
119#926C2×D311yescharacteristic
120C2×C4⋊Dic3C2×D311yescharacteristic
121C2×#208C2×D311yescharacteristic
122C22×C3⋊C8C2×D311yescharacteristic
123#893—62no—
124#893—62no—
125#893—61no—
126#893—61no—
127#893—61no—
128#893—61no—
129#890—21no—
130#888—21no—
131D3×C22⋊C4—64no—
132D3×C22⋊C4—62no—
133D3×C22⋊C4—62no—
134#878—22no—
135C4×D12—32no—
136C4×D12—32no—
137C4×D12—32no—
138C4×D12—32no—
139C4×D12—62no—
140C4×D12—62no—
141#870C2×D312yescomplemented
142#869C2×D312yescomplemented
143C3×C4⋊C8—32no—
144C3×C4⋊C8—32no—
145#821—32no—
146#821—32no—
147#821—32no—
148#821—32no—
149#818—32no—
150#818—31no—
151#818—31no—
152#817—32no—
153#817—31no—
154#817—31no—
155#809—32no—
156#809—32no—
157#809—32no—
158#809—32no—
159#808—32no—
160#808—31no—
161#808—31no—
162#807—32no—
163#807—31no—
164#807—31no—
165C4⋊(C3⋊C8)C2×D312yescomplemented
166C4⋊(C3⋊C8)C2×D312yescomplemented
167C23×C32C2×D411yescharacteristic
168C22×C3⋊D3—22no—
169C22×C3⋊D3—24no—
170C22×C3⋊D3—81no—
171C22×C3⋊D3—81no—
172C2×C32×C4C2×D412yes—
173C2×C32×C4C2×D412yes—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×Q812yes—
176C2×C32×C4#4112yes—
177C2×C32×C4C2411yescharacteristic
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C4⋊(C3⋊D3)—24no—
181C4⋊(C3⋊D3)—22no—
182C4⋊(C3⋊D3)C2×Q814yes—
183C4×C3⋊D3—82no—
184C4×C3⋊D3—82no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—81no—
189C3×Dic6—82no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—22no—
193#544—31no—
194#518—91no—
195#478—92no—
196#478—92no—
197#476—92no—
198#474—92no—
199#428—91no—
200#427—91no—
201#424—31no—
202#418—91no—
203C2×#102—91no—
204C23×D3—62no—
205C23×D3—61no—
206C23×D3—61no—
207C2×C3×Q8—62no—
208C22×C3×C4—21no—
209C22×C3×C4—61no—
210C22×C3×C4C22×D311yescharacteristic
211C22×C3×C4C22×D311yescharacteristic
212C22×Dic3—21no—
213C2×D12—34no—
214C2×D12—34no—
215C2×D12—32no—
216C2×D12—32no—
217C2×D12—62no—
218C2×D12—64no—
219C2×D12—62no—
220C2×D12—64no—
221C2×D12—61no—
222C2×D12—62no—
223C2×D12—62no—
224C2×D12—61no—
225C2×C4×D3—64no—
226C2×C4×D3—64no—
227C2×C4×D3—62no—
228C2×C4×D3—62no—
229C2×C4×D3—62no—
230C2×C4×D3—62no—
231C2×Dic6—22no—
232C3×M16—62no—
233C2×C3×C8—31no—
234C2×C3×C8—32no—
235C2×C3×C8—31no—
236C2×C3×C8—62no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—32no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—62no—
243C3×C4⋊C4—62no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4—62no—
246C3×C4⋊C4D1211yescharacteristic
247C3×C4⋊C4D1211yescharacteristic
248C3×C4⋊C4#6711yescharacteristic
249C3×C4⋊C4#6711yescharacteristic
250C3×C22⋊C4—22no—
251C3×C22⋊C4—61no—
252C3×C22⋊C4D1212yes—
253C3×C22⋊C4#6712yes—
254C3×C42—22no—
255C3×C42—61no—
256C3×C42C22×D312yes—
257C3×C42C22×D312yes—
258#217—21no—
259#217—62no—
260#217—61no—
261#217—61no—
262(C2×D3)⋊C4—64no—
263(C2×D3)⋊C4—64no—
264(C2×D3)⋊C4—62no—
265(C2×D3)⋊C4—62no—
266(C2×D3)⋊C4—62no—
267(C2×D3)⋊C4—62no—
268C4⋊Dic3—22no—
269C4⋊Dic3—21no—
270C4⋊Dic3—31no—
271C4⋊Dic3—31no—
272C4⋊Dic3—31no—
273C4⋊Dic3—31no—
274C4⋊Dic3—61no—
275C4⋊Dic3—61no—
276C4⋊Dic3C22×D311yescharacteristic
277C4⋊Dic3C22×D312yes—
278C4⋊Dic3C22×D311yescharacteristic
279Dic3⋊C4—22no—
280Dic3⋊C4—22no—
281C4×Dic3—21no—
282C4×Dic3—62no—
283C4×Dic3—61no—
284C4×Dic3—61no—
285#208—22no—
286C2×C3⋊C8—22no—
287C2×C3⋊C8C22×D311yescharacteristic
288C2×C3⋊C8C22×D312yes—
289C2×C3⋊C8C22×D311yescharacteristic
290C22×C32—21no—
291C22×C32—21no—
292C22×C32C22⋊Q812yes—
293C22×C32C22×D411yescharacteristic
294C2×C3⋊D3—24no—
295C2×C3⋊D3—42no—
296C2×C3⋊D3—42no—
297C2×C3⋊D3—84no—
298C32×C4—22no—
299C32×C4—22no—
300C32×C4C22⋊Q812yes—
301C32×C4C22×Q811yescharacteristic
302C32×C4C2×#4111yescharacteristic
303C3⋊Dic3—41no—
304C3⋊Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—42no—
307C3×Dic3—41no—
308C22×D4—91no—
309C2×M16—31no—
310C22×C8—31no—
311C4⋊Q8—32no—
312C4⋊Q8—61no—
313#125—32no—
314#125—61no—
315C22⋊Q8—62no—
316C4×D4—92no—
317C4×D4—92no—
318C4×D4—181no—
319C4×D4—181no—
320#117—31no—
321#117—91no—
322#117D3211yescharacteristic
323C2×C4⋊C4—31no—
324C2×C22⋊C4—182no—
325#107—91no—
326#107—91no—
327#107—92no—
328#106—91no—
329#106—91no—
330#106—92no—
331C4⋊C8—32no—
332C4⋊C8—32no—
333#102—92no—
334#102—92no—
335#102—92no—
336#102—92no—
337C23×C3—21no—
338C23×C3C2×D1211yescharacteristic
339C23×C3C2×#6711yescharacteristic
340C22×D3—64no—
341C22×D3—62no—
342C22×D3—62no—
343C22×D3—68no—
344C22×D3—64no—
345C22×D3—64no—
346C22×D3—241no—
347C22×D3—241no—
348C22×D3—241no—
349C22×D3—241no—
350C22×D3—242no—
351C22×D3—242no—
352C3×Q8—242no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—62no—
362C2×C3×C4—241no—
363C2×C3×C4C2×D1212yes—
364C2×C3×C4C2×D1212yes—
365C2×C3×C4C2×D1211yescharacteristic
366C2×C3×C4#23712yes—
367C2×C3×C4D3×Q812yes—
368C2×C3×C4D3×Q812yes—
369C2×C3×C4#23912yes—
370C2×C3×C4C2×#6712yes—
371C2×C3×C4C2×#6712yes—
372C2×C3×C4C2×#6711yescharacteristic
373C2×C3×C4C23×D311yescharacteristic
374C2×C3×C4C23×D311yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—62no—
379C2×Dic3—62no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—81no—
385D12—34no—
386D12—34no—
387D12—64no—
388D12—64no—
389D12—68no—
390D12—64no—
391D12—64no—
392D12—62no—
393D12—62no—
394C4×D3—242no—
395C4×D3—242no—
396C4×D3—242no—
397C4×D3—242no—
398C4×D3—244no—
399C4×D3—244no—
400Dic6—82no—
401C3×C8—61no—
402C3×C8—61no—
403C3×C8—62no—
404C3⋊C8—21no—
405C3⋊C8—21no—
406C3⋊C8—22no—
407C2×C32—22no—
408C2×C32C2×C22⋊Q811yescharacteristic
409C2×C32#57611yescharacteristic
410C2×C32#57711yescharacteristic
411C3⋊D3—44no—
412C24—181no—
413C2×Q8—62no—
414C2×D4—92no—
415C2×D4—94no—
416C2×D4—182no—
417C2×D4—181no—
418C22×C4—61no—
419C22×C4—182no—
420C22×C4—182no—
421C22×C4C2×D3211yescharacteristic
422M16—62no—
423C2×C8—31no—
424C2×C8—32no—
425C2×C8—31no—
426C2×C8—62no—
427C4⋊C4—32no—
428C4⋊C4—31no—
429C4⋊C4—31no—
430C4⋊C4—61no—
431C4⋊C4—62no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4—91no—
435C4⋊C4—91no—
436C4⋊C4#62611yescharacteristic
437C4⋊C4#62611yescharacteristic
438C22⋊C4—61no—
439C22⋊C4—181no—
440C22⋊C4—182no—
441C22⋊C4—182no—
442C22⋊C4#62612yes—
443C42—61no—
444C42—181no—
445C42C2×D3212yes—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—22no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3#89712yes—
455C22×C3#94612yes—
456C22×C3C22×D1211yescharacteristic
457C22×C3C22×#6711yescharacteristic
458C2×D3—68no—
459C2×D3—64no—
460C2×D3—64no—
461C2×D3—124no—
462C2×D3—124no—
463C2×D3—122no—
464C2×D3—122no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—244no—
468C2×D3—244no—
469C2×D3—248no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—22no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—121no—
480C3×C4—122no—
481C3×C4—121no—
482C3×C4#89712yes—
483C3×C4#94612yes—
484C3×C4C2×#23711yescharacteristic
485C3×C4C2×D3×Q811yescharacteristic
486C3×C4C2×D3×Q811yescharacteristic
487C3×C4C2×#23911yescharacteristic
488Dic3—41no—
489Dic3—42no—
490Dic3—41no—
491Dic3—122no—
492Dic3—122no—
493Dic3—121no—
494Dic3—121no—
495Dic3—121no—
496Dic3—121no—
497C32#308111yescharacteristic, complemented, Sylow
498C23—182no—
499C23—184no—
500C23—721no—
501C23—721no—
502C23C2×#62611yescharacteristic
503Q8—242no—
504D4—94no—
505D4—182no—
506D4—184no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—62no—
510C2×C4—181no—
511C2×C4—181no—
512C2×C4—241no—
513C2×C4—722no—
514C2×C4—722no—
515C2×C4#379612yes—
516C2×C4#379712yes—
517C2×C4C2×#62612yes—
518C2×C4C2×#62612yes—
519C2×C4C2×#62611yescharacteristic
520C2×C4C22×D3211yescharacteristic
521C8—61no—
522C8—61no—
523C8—62no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—42no—
530C2×C3C2×#89711yescharacteristic
531C2×C3#575111yescharacteristic
532C2×C3#575211yescharacteristic
533C2×C3C2×#94611yescharacteristic
534C2×C3#583511yescharacteristic
535C2×C3#583711yescharacteristic
536D3—128no—
537D3—124no—
538D3—124no—
539C22—21no—
540C22—21no—
541C22—184no—
542C22—362no—
543C22—362no—
544C22—724no—
545C22#6387112yes—
546C22C22×#62611yescharacteristic
547C4—22no—
548C4—22no—
549C4—121no—
550C4—122no—
551C4—121no—
552C4—361no—
553C4—361no—
554C4#6387112yes—
555C4C2×#379611yescharacteristic
556C4C2×#379711yescharacteristic
557C3—21no—
558C3#7909011yescharacteristic, complemented
559C3#8347411yescharacteristic, complemented
560C2—22no—
561C2—364no—
562C2C2×#6387111yescharacteristic
563C2#1059544811yescharacteristic
564C2#1059544911yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#117C22
socleC22×C32C22×D4
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenter#117D32
hypocenterC32#3081
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C3×#5519C2semidirect
3. C3×#4999C2semidirect
4. #10591130C2semidirect
5. #10591122C2semidirect
8. #10590313C2semidirect
9. #10590308C2semidirect
10. #10590308C2semidirect
11. #10590300C2semidirect
29. C3×#870C22semidirect
30. C3×#869C22semidirect
31. C3×C4⋊(C3⋊C8)C22semidirect
32. C3×C4⋊(C3⋊C8)C22semidirect
33. #63575C22semidirect
34. #63575C22semidirect
35. #63575C22semidirect
36. #63573C22semidirect
37. #63573C22semidirect
38. #63573C22semidirect
47. #5519D3semidirect
58. #4999D3semidirect
70. C2×C4⋊(C3⋊D3)Q8semidirect
141. #870C2×D3semidirect
142. #869C2×D3semidirect
165. C4⋊(C3⋊C8)C2×D3semidirect
166. C4⋊(C3⋊C8)C2×D3semidirect
497. C32#3081semidirect
558. C3#79090semidirect
559. C3#83474semidirect

Maximal subgroups

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

subgroupindex
#105959922
C3×#55192
C3×#49992
#105911302
#105911222
#105910962
C2×#635642
#105903132
#105903082
#105903002
#834743
#790903