Finite Group Databasebeta

← #11889191

Subgroups — #11889191

Order 1152.

1206 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#546C211yesmaximal, characteristic
2#10596013C211yesmaximal, characteristic, complemented
3#10592938C211yesmaximal, characteristic, complemented
4#10592932C211yesmaximal, characteristic, complemented
5#10592917C211yesmaximal, characteristic, complemented
6#10592914C211yesmaximal, characteristic, complemented
7#10592754C211yesmaximal, characteristic, complemented
8#10592753C211yesmaximal, characteristic, complemented
9#10592751C211yesmaximal, characteristic, complemented
10#10592744C211yesmaximal, characteristic, complemented
11#10592740C211yesmaximal, characteristic, complemented
12#10592734C211yesmaximal, characteristic, complemented
13#10592717C211yesmaximal, characteristic, complemented
14#10592710C211yesmaximal, characteristic, complemented
15#10592707C211yesmaximal, characteristic, complemented
16#83497—34nomaximal
17C2×C32×#41C2211yescharacteristic
18C22×C32×D4C2211yescharacteristic
19C32×#124C2211yescharacteristic
20C32×#123—21no—
21C32×C22⋊Q8C2211yescharacteristic
22C32×#121—21no—
23C32×#121C2211yescharacteristic
24C32×#120—21no—
25C32×C4×D4C2211yescharacteristic
26C32×C4×D4C2211yescharacteristic
27C2×C32×C22⋊C4—21no—
28C2×#3756C2211yescharacteristic, complemented
29#64145—21no—
30C2×#3754C2211yescharacteristic
31C2×#3754C2211yescharacteristic
32#64135C2211yescharacteristic
33#64135C2211yescharacteristic, complemented
34#64133C2211yescharacteristic
35#64133C2211yescharacteristic, complemented
36C2×#3748C2211yescharacteristic
37C22×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
38#64096—21no—
39#64088—21no—
40#64088—21no—
41#64078C2211yescharacteristic, complemented
42C4×#634C2211yescharacteristic, complemented
43#63662C2211yescharacteristic, complemented
44#63662C2211yescharacteristic, complemented
45#63662C2211yescharacteristic, complemented
46#63659C2211yescharacteristic
47#63659C2211yescharacteristic
48#63659C2211yescharacteristic
49#63659C2211yescharacteristic
50#63659C2211yescharacteristic
51#63659C2211yescharacteristic
52#63657C2211yescharacteristic
53#63657C2211yescharacteristic, complemented
54#63637C2211yescharacteristic, complemented
55#63635C2211yescharacteristic, complemented
56#63635C2211yescharacteristic, complemented
57#63634C2211yescharacteristic, complemented
58C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
59#63630C2211yescharacteristic, complemented
60C3×#546D314yes—
61#5542—34no—
62#5239—34no—
63#5233—34no—
64#5218—34no—
65#5215—34no—
66#5055—34no—
67#5054—34no—
68#5052—34no—
69#5045—34no—
70#5041—34no—
71#5035—34no—
72#5018—34no—
73#5011—34no—
74#5008—34no—
75C24×C32—21no—
76C32×#41—22no—
77C32×#41—41no—
78C2×C32×Q8C2311yescharacteristic
79C2×C32×D4—22no—
80C2×C32×D4—21no—
81C2×C32×D4—21no—
82C2×C32×D4—21no—
83C2×C32×D4—41no—
84C2×C32×D4D411yescharacteristic
85C2×C32×D4C2311yescharacteristic
86C2×C32×D4C2311yescharacteristic
87C22×C32×C4—21no—
88C22×C32×C4—21no—
89C22×C32×C4C2311yescharacteristic
90C22×C32×C4C2311yescharacteristic
91C2×#634—21no—
92C2×#634C2311yescharacteristic
93C32×C4⋊C4—21no—
94C32×C4⋊C4D411yescharacteristic
95C32×C4⋊C4C2311yescharacteristic
96C32×C22⋊C4—21no—
97C32×C22⋊C4—21no—
98C32×C22⋊C4—21no—
99C32×C22⋊C4—21no—
100C32×C22⋊C4—21no—
101C32×C22⋊C4D411yescharacteristic
102C32×C22⋊C4D411yescharacteristic
103C32×C42C2311yescharacteristic
104#3757—21no—
105#3757—21no—
106#3757—21no—
107#3757—21no—
108#3756—41no—
109#3754—22no—
110#3754—41no—
111C4⋊(C3⋊Dic3)—21no—
112C4⋊(C3⋊Dic3)C2311yescharacteristic
113C4⋊(C3⋊Dic3)C2311yescharacteristic
114C4⋊(C3⋊Dic3)C2311yescharacteristic
115#3750—21no—
116#3750—21no—
117#3750—21no—
118C4×C3⋊Dic3—21no—
119C4×C3⋊Dic3—21no—
120C4×C3⋊Dic3—21no—
121#3748—41no—
122C2×C3⋊(C3⋊C8)—41no—
123C2×C3⋊(C3⋊C8)C2311yescharacteristic
124C2×C3⋊(C3⋊C8)C2311yescharacteristic
125C2×C3⋊(C3⋊C8)C2311yescharacteristic
126C2×C3⋊(C3⋊C8)C2311yescharacteristic
127#3311—91noSylow
128C2×C3×#41C2×D314yes—
129C22×C3×D4C2×D314yes—
130C3×#124C2×D314yes—
131C3×#123—24no—
132C3×C22⋊Q8C2×D314yes—
133C3×#121—24no—
134C3×#121C2×D314yes—
135C3×#120—24no—
136C3×C4×D4C2×D314yes—
137C3×C4×D4C2×D314yes—
138C2×C3×C22⋊C4—24no—
139C2×#216—34no—
140#936—64no—
141C2×#214—34no—
142C2×#214—34no—
143#926—34no—
144#926—34no—
145#924—34no—
146#924—34no—
147C2×#208—34no—
148C22×C3⋊C8—34no—
149#887—64no—
150#879—64no—
151#879—64no—
152#869—34no—
153C4×Dic6—34no—
154#835—34no—
155#835—34no—
156#835—34no—
157#832—34no—
158#832—34no—
159#832—34no—
160#832—34no—
161#832—34no—
162#832—34no—
163#830—34no—
164#830—34no—
165#810—34no—
166#808—34no—
167#808—34no—
168#807—34no—
169C4⋊(C3⋊C8)—34no—
170#803—34no—
171C23×C32—21no—
172C23×C32—21no—
173C23×C32—42no—
174C23×C32—42no—
175C23×C32—81no—
176C23×C32C2×D411yescharacteristic
177C23×C32C2×D411yescharacteristic
178C32×Q8—21no—
179C32×Q8—21no—
180C32×D4—21no—
181C32×D4—21no—
182C32×D4—21no—
183C32×D4—41no—
184C32×D4—42no—
185C32×D4—42no—
186C32×D4—81no—
187C32×D4#4112yes—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—42no—
195C2×C32×C4—81no—
196C2×C32×C4C2×D411yescharacteristic
197C2×C32×C4C2×D411yescharacteristic
198C2×C32×C4C2×D411yescharacteristic
199C2×C32×C4C2×D411yescharacteristic
200C2×C32×C4C2411yescharacteristic
201C2×C3⋊Dic3—21no—
202C2×C3⋊Dic3—21no—
203C2×C3⋊Dic3—21no—
204C2×C3⋊Dic3—21no—
205#634—21no—
206#634—21no—
207#634—42no—
208C3⋊(C3⋊C8)—21no—
209C3⋊(C3⋊C8)—21no—
210C3⋊(C3⋊C8)—21no—
211C3⋊(C3⋊C8)—21no—
212#556—91no—
213#546C3⋊D311yescharacteristic
214#489—91no—
215#485—91no—
216#482—91no—
217#476—91no—
218#470—91no—
219#465—91no—
220#464—91no—
221#452—91no—
222#450—91no—
223#440—91no—
224#435—91no—
225#418—91no—
226#416—91no—
227C24×C3—24no—
228C3×#41—28no—
229C3×#41—44no—
230C2×C3×Q8C22×D314yes—
231C2×C3×D4—28no—
232C2×C3×D4—24no—
233C2×C3×D4—24no—
234C2×C3×D4—24no—
235C2×C3×D4—44no—
236C2×C3×D4#6714yes—
237C2×C3×D4C22×D314yes—
238C2×C3×D4C22×D314yes—
239C22×C3×C4—24no—
240C22×C3×C4—24no—
241C22×C3×C4C22×D314yes—
242C22×C3×C4C22×D314yes—
243C2×Dic6—34no—
244C2×Dic6—64no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4#6714yes—
247C3×C4⋊C4C22×D314yes—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4#6714yes—
254C3×C22⋊C4#6714yes—
255C3×C42C22×D314yes—
256#217—64no—
257#217—64no—
258#217—64no—
259#217—64no—
260#216—124no—
261#214—68no—
262#214—124no—
263C4⋊Dic3—34no—
264C4⋊Dic3—34no—
265C4⋊Dic3—34no—
266C4⋊Dic3—64no—
267Dic3⋊C4—64no—
268Dic3⋊C4—64no—
269Dic3⋊C4—64no—
270C4×Dic3—64no—
271C4×Dic3—64no—
272C4×Dic3—64no—
273#208—124no—
274C2×C3⋊C8—34no—
275C2×C3⋊C8—34no—
276C2×C3⋊C8—34no—
277C2×C3⋊C8—34no—
278C2×C3⋊C8—124no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—22no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—41no—
289C22×C32—41no—
290C22×C32—41no—
291C22×C32—84no—
292C22×C32C22×D411yescharacteristic
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4C2×#4111yescharacteristic
300C32×C4H(5,𝔽2)11yescharacteristic
301C3⋊Dic3—41no—
302C3⋊Dic3—41no—
303C3⋊Dic3—41no—
304C3⋊Dic3—41no—
305C2×#41C2×C3⋊D311yescharacteristic
306C22×D4C2×C3⋊D311yescharacteristic
307C2×Q16—91no—
308C2×SD16—91no—
309C2×SD16—91no—
310C2×M16—91no—
311C22×C8—91no—
312C4⋊Q8—91no—
313#126—181no—
314#126—181no—
315#124—181no—
316#124C2×C3⋊D311yescharacteristic
317#123—21no—
318C22⋊Q8—91no—
319C22⋊Q8—91no—
320C22⋊Q8C2×C3⋊D311yescharacteristic
321#121—21no—
322#121C2×C3⋊D311yescharacteristic
323#120—21no—
324C4×Q8—91no—
325C4×Q8—181no—
326C4×D4C2×C3⋊D311yescharacteristic
327C4×D4C2×C3⋊D311yescharacteristic
328#117—91no—
329#117—91no—
330C2×C22⋊C4—21no—
331#107—91no—
332#106—91no—
333#106—91no—
334C4⋊C8—91no—
335#103—91no—
336#103—91no—
337#103—91no—
338#103—91no—
339#102—91no—
340#102—91no—
341#102—91no—
342#102—91no—
343#102—91no—
344#102—91no—
345C22⋊C8—91no—
346C22⋊C8—91no—
347#97—91no—
348C23×C3—24no—
349C23×C3—24no—
350C23×C3—48no—
351C23×C3—48no—
352C23×C3—84no—
353C23×C3C2×#6714yes—
354C23×C3C2×#6714yes—
355C3×Q8—24no—
356C3×Q8—24no—
357C3×D4—24no—
358C3×D4—24no—
359C3×D4—24no—
360C3×D4—44no—
361C3×D4—48no—
362C3×D4—48no—
363C3×D4—84no—
364C3×D4#23718yes—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—48no—
372C2×C3×C4—84no—
373C2×C3×C4C2×#6714yes—
374C2×C3×C4C2×#6714yes—
375C2×C3×C4C2×#6714yes—
376C2×C3×C4C2×#6714yes—
377C2×C3×C4C23×D314yes—
378C2×Dic3—64no—
379C2×Dic3—64no—
380C2×Dic3—64no—
381C2×Dic3—64no—
382Dic6—64no—
383Dic6—64no—
384Dic6—128no—
385C3⋊C8—64no—
386C3⋊C8—64no—
387C3⋊C8—64no—
388C3⋊C8—64no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32—42no—
392C2×C32—81no—
393C2×C32#54611yescharacteristic
394C2×C32#57511yescharacteristic
395C2×C32#57711yescharacteristic
396C24—21no—
397#41—22no—
398#41—41no—
399C2×Q8—91no—
400C2×Q8—181no—
401C2×Q8C22×C3⋊D311yescharacteristic
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—41no—
407C2×D4#63811yescharacteristic
408C2×D4C22×C3⋊D311yescharacteristic
409C2×D4C22×C3⋊D311yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4C22×C3⋊D311yescharacteristic
413C22×C4C22×C3⋊D311yescharacteristic
414Q16—361no—
415SD16—182no—
416SD16—361no—
417M16—361no—
418C2×C8—91no—
419C2×C8—91no—
420C2×C8—91no—
421C2×C8—91no—
422C2×C8—361no—
423C4⋊C4—21no—
424C4⋊C4—91no—
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4—181no—
428C4⋊C4—181no—
429C4⋊C4—181no—
430C4⋊C4—181no—
431C4⋊C4#63811yescharacteristic
432C4⋊C4C22×C3⋊D311yescharacteristic
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—181no—
439C22⋊C4—181no—
440C22⋊C4—181no—
441C22⋊C4—181no—
442C22⋊C4#63811yescharacteristic
443C22⋊C4#63811yescharacteristic
444C42—181no—
445C42—181no—
446C42—181no—
447C42C22×C3⋊D311yescharacteristic
448C22×C3—24no—
449C22×C3—24no—
450C22×C3—28no—
451C22×C3—24no—
452C22×C3—24no—
453C22×C3—24no—
454C22×C3—24no—
455C22×C3—44no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—816no—
461C22×C3C22×#6714yes—
462C3×C4—24no—
463C3×C4—24no—
464C3×C4—24no—
465C3×C4—24no—
466C3×C4—44no—
467C3×C4—44no—
468C3×C4C2×#23714yes—
469C3×C4#100414yes—
470Dic3—124no—
471Dic3—124no—
472Dic3—124no—
473Dic3—124no—
474C32#331111yescharacteristic, complemented, Sylow
475C23—21no—
476C23—21no—
477C23—42no—
478C23—42no—
479C23—81no—
480C23C2×#63811yescharacteristic
481C23C2×#63811yescharacteristic
482Q8—21no—
483Q8—21no—
484Q8—181no—
485Q8—181no—
486Q8—362no—
487D4—21no—
488D4—21no—
489D4—21no—
490D4—41no—
491D4—42no—
492D4—42no—
493D4—81no—
494D4#383012yes—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—181no—
504C2×C4—181no—
505C2×C4—181no—
506C2×C4—181no—
507C2×C4C2×#63811yescharacteristic
508C2×C4C2×#63811yescharacteristic
509C2×C4C2×#63811yescharacteristic
510C2×C4C2×#63811yescharacteristic
511C2×C4C23×C3⋊D311yescharacteristic
512C8—181no—
513C8—181no—
514C8—181no—
515C8—181no—
516C2×C3—44no—
517C2×C3—44no—
518C2×C3—48no—
519C2×C3—84no—
520C2×C3#580614yes—
521C2×C3#582214yes—
522C2×C3#583614yes—
523C22—21no—
524C22—21no—
525C22—22no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—84no—
536C22C22×#63811yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—41no—
542C4—41no—
543C4—361no—
544C4—361no—
545C4—361no—
546C4—361no—
547C4C2×#383011yescharacteristic
548C4#6436111yescharacteristic
549C3#8349714yescomplemented
550C2—41no—
551C2—41no—
552C2—42no—
553C2—81no—
554C2#1059627711yescharacteristic
555C2#1059629311yescharacteristic
556C2#1059630711yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
Fitting, prime radicalC32×#546C2
socleC22×C32C22×D4
hypercenter#546C3⋊D3
hypocenterC32#3311
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
2. #10596013C2semidirect
3. #10592938C2semidirect
4. #10592932C2semidirect
5. #10592917C2semidirect
6. #10592914C2semidirect
7. #10592754C2semidirect
8. #10592753C2semidirect
9. #10592751C2semidirect
10. #10592744C2semidirect
11. #10592740C2semidirect
12. #10592734C2semidirect
13. #10592717C2semidirect
14. #10592710C2semidirect
15. #10592707C2semidirect
28. C2×#3756C22semidirect
33. #64135C22semidirect
35. #64133C22semidirect
37. C22×C3⋊(C3⋊C8)C22semidirect
41. #64078C22semidirect
42. C4×#634C22semidirect
43. #63662C22semidirect
44. #63662C22semidirect
45. #63662C22semidirect
53. #63657C22semidirect
54. #63637C22semidirect
55. #63635C22semidirect
56. #63635C22semidirect
57. #63634C22semidirect
58. C4⋊(C3⋊(C3⋊C8))C22semidirect
59. #63630C22semidirect
474. C32#3311semidirect
549. C3#83497semidirect

Maximal subgroups

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

subgroupindex
C32×#5462
#105960132
#105929382
#105929322
#105929172
#105929142
#105927542
#105927532
#105927512
#105927442
#105927402
#105927342
#105927172
#105927102
#105927072
#834973