Finite Group Databasebeta

← #11895978

Subgroups — #11895978

Order 1152.

818 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5721C211yesmaximal, characteristic, complemented
2#10593900C211yesmaximal, characteristic
3#10592786C211yesmaximal, characteristic, complemented
4C3×#5088C211yesmaximal, characteristic, complemented
5#10590550C212yesmaximal, complemented
6#10590549C212yesmaximal, complemented
7#10590467C212yesmaximal, complemented
8#10590454C212yesmaximal, complemented
9#10590364C211yesmaximal, characteristic, complemented
10C4×#3717C211yesmaximal, characteristic, complemented
11#10590258C211yesmaximal, characteristic, complemented
12#83690—31nomaximal
13#81590—31nomaximal
14C2×#3796—21no—
15C32×C4⋊Q8C2211yescharacteristic
16C4×C4⋊(C3⋊D3)C2211yescharacteristic
17C3×Q8×Dic3—21no—
18C3×#944—21no—
19C3×#890C2212yes—
20C3×#889—22no—
21C3×#887C2212yes—
22C3×#874C2211yescharacteristic
23C3×C4×Dic6C2211yescharacteristic
24C3×C4×Dic6C2211yescharacteristic
25#63916—21no—
26#63911C2212yes—
27Dic3×D12C2212yes—
28#63886—22no—
29C2×#3717C2211yescharacteristic
30C2×#3715C2212yes—
31#63636C2212yes—
32#63634C2212yes—
33C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
34C3×#809C2212yes—
35C3×#809C2212yes—
36C3×C4×C3⋊C8C2211yescharacteristic
37#63573C2211yescharacteristic
38Dic3⋊(C3⋊C8)C2212yes—
39#63566C2211yescharacteristic
40#63564C2211yescharacteristic
41#63562C2212yes—
42#63554C2212yes—
43C3×#557—31no—
44#5721D311yescharacteristic, complemented
45#5718—31no—
46#5577—31no—
47C3×#486—31no—
48#5088D311yescharacteristic, complemented
49#5087—31no—
50#5087—31no—
51#5087—61no—
52#5075—31no—
53#5067—32no—
54#5066—32no—
55#5027—31no—
56#5027—32no—
57#5026—32no—
58C4×#215—31no—
59#5024—31no—
60#4764—32no—
61#4763—32no—
62#4762—32no—
63#4757—32no—
64C4×#204—31no—
65#4679—31no—
66C2×C32×Q8—21no—
67C2×C4⋊(C3⋊D3)C2311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12C2312yes—
72C2×C3×Dic6D411yescharacteristic
73C2×C3×Dic6C2311yescharacteristic
74C2×#626—22no—
75C2×#626—22no—
76C2×D3×Dic3—22no—
77#3796—44no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4C2312yes—
80C32×C42C2311yescharacteristic
81#3752—21no—
82C4⋊(C3⋊Dic3)C2311yescharacteristic
83C2×C3⋊(C3⋊C8)C2312yes—
84C3×C4⋊Dic3—22no—
85C3×C4⋊Dic3D411yescharacteristic
86C3×C4⋊Dic3C2311yescharacteristic
87C3×Dic3⋊C4—21no—
88C3×Dic3⋊C4—22no—
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4D412yes—
91C3×C4×Dic3—21no—
92C3×C4×Dic3—22no—
93C3×C4×Dic3C2312yes—
94C2×C3×C3⋊C8C2311yescharacteristic
95C2×C3×C3⋊C8C2311yescharacteristic
96(C2×D3)⋊Dic3—22no—
97#3717—22no—
98#3715—24no—
99#3368—91noSylow
100C2×#239—61no—
101C2×#237—61no—
102C3×C4⋊Q8—21no—
103C3×C4⋊Q8—61no—
104C3×C4⋊Q8C2×D311yescharacteristic
105C3×C4⋊Q8C2×D311yescharacteristic
106C3×#127—31no—
107C3×#125—32no—
108C3×#125—62no—
109C3×C4×Q8—31no—
110C3×C4×Q8—32no—
111C3×C4×Q8—31no—
112C3×C4×Q8—61no—
113Q8×Dic3—21no—
114#944—21no—
115C2×#215—32no—
116C2×#215—31no—
117#940—32no—
118#939—62no—
119#938—61no—
120D4×Dic3—32no—
121C2×#214—32no—
122C2×#204—31no—
123#895—62no—
124#895—61no—
125#890C2×D312yes—
126#889—22no—
127#887C2×D312yes—
128#874—32no—
129#874C2×D311yescharacteristic
130C4×D12—32no—
131C4×D12—31no—
132C4×D12—31no—
133C4×D12—61no—
134C4×Dic6C2×D311yescharacteristic
135C4×Dic6C2×D311yescharacteristic
136C3×#102—32no—
137C3×#102—32no—
138C3×C4×C8—31no—
139#835—31no—
140#832—32no—
141#821—31no—
142#817—31no—
143#816—31no—
144#815—32no—
145Dic3⋊C8—32no—
146#809—32no—
147#809—32no—
148#809—32no—
149#809—31no—
150#809—62no—
151#809C2×D312yes—
152#809C2×D312yes—
153#808—31no—
154#807—32no—
155#807—32no—
156#807—62no—
157C4⋊(C3⋊C8)—31no—
158C4⋊(C3⋊C8)—31no—
159C4⋊(C3⋊C8)—32no—
160C4⋊(C3⋊C8)—61no—
161#803—32no—
162C4×C3⋊C8C2×D311yescharacteristic
163C22×C3⋊D3—21no—
164C22×C3×D3—22no—
165C32×Q8—42no—
166C2×C32×C4—22no—
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4C2411yescharacteristic
170C2×C3⋊Dic3—21no—
171C4⋊(C3⋊D3)—21no—
172C4⋊(C3⋊D3)—21no—
173C4×C3⋊D3—42no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3C2×D412yes—
177C2×C3×Dic3C2×D412yes—
178C3×D12—22no—
179C3×D12—22no—
180C3×D12—42no—
181C3×D12—81no—
182C3×Dic6—22no—
183C3×Dic6—21no—
184C3×Dic6#4112yescomplemented
185#626—44no—
186#626—82no—
187D3×Dic3—44no—
188C3⋊(C3⋊C8)—22no—
189C3×C3⋊C8—21no—
190C3×C3⋊C8—21no—
191#557—31no—
192#550—91no—
193#490—92no—
194#486—31no—
195#478—92no—
196#474—91no—
197#460—92no—
198#460—91no—
199#446—91no—
200#440—92no—
201C4×SD16—91no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—31no—
205C2×C3×Q8—31no—
206C2×C3×Q8—41no—
207C2×C3×D4—32no—
208C2×C3×D4—61no—
209C2×C3×D4—61no—
210C2×#67—62no—
211C2×#67—62no—
212C22×Dic3—62no—
213#239—124no—
214#237—124no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—31no—
218C2×D12—31no—
219C2×D12—61no—
220C2×D12—62no—
221C2×D12—62no—
222C2×D12C22×D312yes—
223C2×C4×D3—62no—
224C2×C4×D3—61no—
225C2×C4×D3—61no—
226C2×C4×D3—121no—
227C2×Dic6#6711yescharacteristic
228C2×Dic6C22×D311yescharacteristic
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—32no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—62no—
239C3×C4⋊C4—62no—
240C3×C4⋊C4—62no—
241C3×C4⋊C4—61no—
242C3×C4⋊C4C22×D312yes—
243C3×C4⋊C4C22×D312yes—
244C3×C42—21no—
245C3×C42—32no—
246C3×C42—62no—
247C3×C42—61no—
248C3×C42C22×D311yescharacteristic
249C3×C42C22×D311yescharacteristic
250#217—62no—
251#215—62no—
252#215—64no—
253#214—64no—
254(C2×D3)⋊C4—62no—
255(C2×D3)⋊C4—61no—
256(C2×D3)⋊C4—61no—
257(C2×D3)⋊C4—121no—
258C4⋊Dic3—22no—
259C4⋊Dic3—31no—
260C4⋊Dic3—31no—
261C4⋊Dic3—61no—
262C4⋊Dic3#6711yescharacteristic
263C4⋊Dic3C22×D311yescharacteristic
264Dic3⋊C4—21no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267Dic3⋊C4#6712yes—
268C4×Dic3—21no—
269C4×Dic3—22no—
270C4×Dic3C22×D312yes—
271C2×C3⋊C8—32no—
272C2×C3⋊C8—32no—
273C2×C3⋊C8—62no—
274C2×C3⋊C8C22×D311yescharacteristic
275C2×C3⋊C8C22×D311yescharacteristic
276#204—62no—
277C22×C32C22×D411yescharacteristic
278C2×C3⋊D3—41no—
279C2×C3⋊D3—41no—
280C2×C3⋊D3—41no—
281C2×C3×D3—42no—
282C2×C3×D3—42no—
283C2×C3×D3—42no—
284C32×C4—21no—
285C32×C4—42no—
286C32×C4AGL(1,ℤ/8)12yes—
287C32×C4C2×#4111yescharacteristic
288C32×C42−1+411yescharacteristic
289C3⋊Dic3—41no—
290C3×Dic3—22no—
291C3×Dic3—22no—
292C3×Dic3—22no—
293C3×Dic3—41no—
294C2×#41—181no—
295C2×SD16—91no—
296C2×SD16—92no—
297C4⋊Q8—61no—
298C4⋊Q8D3211yescharacteristic
299#127—31no—
300#127—92no—
301#125—32no—
302#125—62no—
303#121—181no—
304#121—182no—
305C4×Q8—31no—
306C4×Q8—32no—
307C4×Q8—31no—
308C4×Q8—61no—
309C4×D4—92no—
310C4×D4—91no—
311#107—92no—
312#106—91no—
313C4⋊C8—92no—
314C4⋊C8—91no—
315#103—91no—
316#102—32no—
317#102—32no—
318#102—91no—
319#102—92no—
320#102—92no—
321#97—92no—
322C4×C8—31no—
323C23×C3—62no—
324C22×D3—22no—
325C22×D3—61no—
326C22×D3—61no—
327C22×D3—121no—
328C3×Q8—32no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—62no—
332C3×Q8—61no—
333C3×Q8—82no—
334C3×D4—62no—
335C3×D4—62no—
336C3×D4—122no—
337C3×D4—241no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—32no—
344C2×C3×C4—32no—
345C2×C3×C4—42no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4D3×D411yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351C2×C3×C4C2×#6711yescharacteristic
352C2×C3×C4C23×D311yescharacteristic
353C2×C3×C4C23×D311yescharacteristic
354#67—124no—
355#67—242no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—121no—
361C2×Dic3—124no—
362C2×Dic3C2×#6712yes—
363C2×Dic3C2×#6712yes—
364D12—22no—
365D12—22no—
366D12—42no—
367D12—62no—
368D12—61no—
369D12—61no—
370D12—61no—
371D12—61no—
372D12—81no—
373D12—121no—
374D12—124no—
375D12—242no—
376C4×D3—124no—
377C4×D3—122no—
378C4×D3—122no—
379C4×D3—242no—
380Dic6—22no—
381Dic6—21no—
382Dic6#23912yescomplemented
383C3×C8—61no—
384C3×C8—61no—
385C3⋊C8—21no—
386C3⋊C8—21no—
387C3⋊C8—62no—
388C3⋊C8—62no—
389C3⋊C8—122no—
390C2×C32#54811yescharacteristic
391C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
392C2×C32#57711yescharacteristic
393C3⋊D3—81no—
394C3×D3—82no—
395#41—364no—
396C2×Q8—21no—
397C2×Q8—31no—
398C2×Q8—31no—
399C2×D4—32no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—91no—
403C2×D4—182no—
404C2×D4—182no—
405C22×C4—182no—
406C22×C4—181no—
407SD16—182no—
408SD16—184no—
409C2×C8—31no—
410C2×C8—31no—
411C2×C8—92no—
412C4⋊C4—21no—
413C4⋊C4—31no—
414C4⋊C4—32no—
415C4⋊C4—31no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4—62no—
419C4⋊C4—61no—
420C4⋊C4—91no—
421C4⋊C4C2×D3212yes—
422C22⋊C4—182no—
423C22⋊C4—181no—
424C42—32no—
425C42—62no—
426C42—61no—
427C42C2×D3211yescharacteristic
428C22×C3—21no—
429C22×C3—122no—
430C22×C3—122no—
431C22×C3—122no—
432C22×C3C2×D3×D411yescharacteristic
433C22×C3C22×#6711yescharacteristic
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—122no—
438C2×D3—121no—
439C2×D3—121no—
440C2×D3—121no—
441C2×D3—121no—
442C2×D3—121no—
443C2×D3—121no—
444C2×D3—241no—
445C2×D3—241no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—22no—
451C3×C4—41no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—62no—
455C3×C4—62no—
456C3×C4—62no—
457C3×C4—82no—
458C3×C4—121no—
459C3×C4#91112yes—
460C3×C4#94912yes—
461C3×C4C2×#23911yescharacteristic
462C3×C4#100711yescharacteristic
463C3×C4D3×#4111yescharacteristic
464C3×C4#101011yescharacteristic
465Dic3—22no—
466Dic3—22no—
467Dic3—22no—
468Dic3—41no—
469Dic3—121no—
470Dic3—121no—
471Dic3—241no—
472C32#336811yescharacteristic, complemented, Sylow
473C23—62no—
474C23—181no—
475Q8—32no—
476Q8—42no—
477Q8—61no—
478Q8—62no—
479D4—62no—
480D4—62no—
481D4—122no—
482D4—181no—
483D4—181no—
484D4—241no—
485D4—364no—
486D4—722no—
487C2×C4—22no—
488C2×C4—32no—
489C2×C4—32no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—181no—
493C2×C4—364no—
494C2×C4—362no—
495C2×C4D3×#6711yescharacteristic
496C2×C4D3×#6711yescharacteristic
497C2×C4C22×D3211yescharacteristic
498C8—61no—
499C8—61no—
500C8—182no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—242no—
505C2×C3#567711yescharacteristic
506C2×C3C2×#91111yescharacteristic
507C2×C3#576211yescharacteristic
508C2×C3#581711yescharacteristic
509C2×C3C2×#94911yescharacteristic
510C2×C3#583711yescharacteristic
511D3—82no—
512D3—242no—
513D3—241no—
514D3—241no—
515C22—122no—
516C22—122no—
517C22—122no—
518C22—361no—
519C22—361no—
520C22—361no—
521C22C2×D3×#6711yescharacteristic
522C4—21no—
523C4—42no—
524C4—62no—
525C4—62no—
526C4—62no—
527C4—121no—
528C4—361no—
529C4#6392512yes—
530C4#6431511yescharacteristic
531C4D3×#23911yescharacteristic
532C3—21no—
533C3#8159011yescharacteristic, complemented
534C3#8369011yescharacteristic, complemented
535C2—242no—
536C2—721no—
537C2C2×#6392511yescharacteristic
538C2#1059531811yescharacteristic
539C2#1059540111yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radical#10593900C2
hypercenterC4⋊Q8D32
hypocenterC32#3368
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5721C2semidirect
3. #10592786C2semidirect
4. C3×#5088C2semidirect
5. #10590550C2semidirect
6. #10590549C2semidirect
7. #10590467C2semidirect
8. #10590454C2semidirect
9. #10590364C2semidirect
10. C4×#3717C2semidirect
11. #10590258C2semidirect
44. #5721D3semidirect
48. #5088D3semidirect
184. C3×Dic6#41semidirect
382. Dic6#239semidirect
472. C32#3368semidirect
533. C3#81590semidirect
534. C3#83690semidirect

Maximal subgroups

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

subgroupindex
C3×#57212
#105939002
#105927862
C3×#50882
#105905502
#105905492
#105904672
#105904542
#105903642
C4×#37172
#105902582
#836903
#815903