Finite Group Databasebeta

← #49134750

Subgroups — #49134750

Order 36224.

1648 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#17985393C211yesmaximal, characteristic, complemented
2C2×#41×D283C211yesmaximal, characteristic, complemented
3#31827134C211yesmaximal, characteristic, complemented
4Q8×C22⋊D283C211yesmaximal, characteristic, complemented
5#31827127C211yesmaximal, characteristic, complemented
6#31827117C213yesmaximal, complemented
7#31827111C213yesmaximal, complemented
8#31827108C213yesmaximal, complemented
9#31827099C213yesmaximal, complemented
10#31827014C213yesmaximal, complemented
11#31826978C213yesmaximal, complemented
12#31826975C213yesmaximal, complemented
13C4×#17985393C211yesmaximal, characteristic, complemented
14C4×D4×D283C213yesmaximal, complemented
15#548×C283C211yesmaximal, characteristic, complemented
16C2×#17985393—28no—
17C2×#17985393C2211yescharacteristic
18C2×#17985393C2213yescomplemented
19C2×#17985393C2213yescomplemented
20C2×#17985393C2211yescharacteristic, complemented
21C2×#17985392C2213yescomplemented
22C2×#17985391C2213yescomplemented
23#41×D283—28no—
24C2×Q8×D283C2211yescharacteristic, complemented
25C22×D1132C2213yescomplemented
26C2×D4×D283C2213yes—
27C2×D4×D283C2213yescomplemented
28C22×C4×D283C2213yescomplemented
29C22×C4×D283C2213yescomplemented
30#22583608C2213yescomplemented
31#22583606C2213yescomplemented
32#22583598C2213yescomplemented
33#22583595C2216yescomplemented
34#22583592C2216yescomplemented
35#22583590C2213yescomplemented
36#22583589C2213yescomplemented
37#22583586C2213yescomplemented
38#22583585C2213yescomplemented
39#22583584C2213yescomplemented
40#22583583C2216yescomplemented
41#22583583C2216yescomplemented
42#22583581C2213yescomplemented
43#22583580C2216yescomplemented
44#22583578C2216yescomplemented
45Q8×Dic283C2211yescharacteristic, complemented
46C4×D1132C2213yescomplemented
47C4×D1132C2213yescomplemented
48D4×Dic283C2213yescomplemented
49#22583572C2213yescomplemented
50#22583572C2213yescomplemented
51C4×C22⋊D283C2213yes—
52C4×C22⋊D283C2213yescomplemented
53(C22⋊D283)⋊C4C2211yescharacteristic
54(C22⋊D283)⋊C4C2216yescomplemented
55(C22⋊D283)⋊C4C2213yescomplemented
56#22583568C2213yescomplemented
57C4⋊C4×D283C2213yescomplemented
58C22⋊C4×D283C2213yescomplemented
59C22⋊C4×D283C2213yescomplemented
60C42×D283C2213yescomplemented
61C2×#41×C283C2211yescharacteristic, complemented
62C22×Q8×C283C2211yescharacteristic, complemented
63#124×C283C2213yescomplemented
64C22⋊Q8×C283C2213yescomplemented
65#121×C283C2213yescomplemented
66C4×Q8×C283C2211yescharacteristic, complemented
67C4×D4×C283C2213yescomplemented
68C23×D283C2313yes—
69C23×D283C2313yes—
70#17985393—212no—
71#17985393—212no—
72#17985393—24no—
73#17985393D418yescomplemented
74#17985392—212no—
75#17985391—212no—
76C2×Dic566C2313yes—
77Q8×D283—24no—
78C2×D1132—212no—
79C2×D1132C2313yes—
80C2×D1132C2316yes—
81C2×D1132C2313yes—
82C2×D1132C2313yes—
83C2×C22⋊D283C2313yes—
84C2×C22⋊D283C2316yes—
85C2×C22⋊D283C2316yes—
86C2×C22⋊D283C2311yescharacteristic
87D4×D283—212no—
88D4×D283—212no—
89C22×Dic283C2311yescharacteristic
90C22×Dic283C2313yes—
91C2×C4×D283—212no—
92C2×C4×D283—212no—
93C2×C4×D283C2313yes—
94C2×C4×D283C2313yes—
95C2×C4×D283C2313yes—
96C2×C4×D283C2311yescharacteristic
97C2×C4×D283C2313yes—
98C2×C4×D283C2313yes—
99C2×C4×D283C2313yes—
100C2×C4×D283C2313yes—
101C2×C4×D283C2313yes—
102C4⋊Dic283C2313yes—
103C4⋊Dic283C2313yes—
104Dic283⋊C4C2311yescharacteristic
105Dic283⋊C4C2313yes—
106Dic283⋊C4C2316yes—
107C22⋊Dic283C2313yes—
108C22⋊Dic283C2313yes—
109(C2×D283)⋊C4C2311yescharacteristic
110(C2×D283)⋊C4C2313yes—
111(C2×D283)⋊C4C2313yes—
112(C2×D283)⋊C4C2313yes—
113(C2×D283)⋊C4C2316yes—
114(C2×D283)⋊C4C2316yes—
115C4×Dic283C2311yescharacteristic
116C4×Dic283C2313yes—
117C4×Dic283C2313yes—
118#41×C283—24no—
119C2×Q8×C283—24no—
120C2×Q8×C283C2313yes—
121C2×Q8×C283C2311yescharacteristic
122C2×D4×C283C2313yes—
123C2×D4×C283C2313yes—
124C22×C4×C283C2313yes—
125C22×C4×C283C2313yes—
126C4⋊C4×C283C2313yes—
127C4⋊C4×C283C2313yes—
128C22⋊C4×C283C2311yescharacteristic
129C22⋊C4×C283C2313yes—
130C22⋊C4×C283C2316yes—
131C42×C283C2313yes—
132C22×D283—26no—
133C22×D283—26no—
134C22×D283—26no—
135C22×D283—26no—
136C22×D283—43no—
137C22×D283—43no—
138C22×D283C2413yes—
139C22×D283C2413yes—
140C22×D283C2411yescharacteristic
141Dic566—26no—
142D1132—212no—
143D1132—26no—
144D1132—26no—
145D1132C2×D4112yescomplemented
146C22⋊D283—212no—
147C22⋊D283—26no—
148C22⋊D283—46no—
149C22⋊D283#4114yescomplemented
150C2×Dic283—22no—
151C2×Dic283—26no—
152C2×Dic283C2413yes—
153C2×Dic283C2413yes—
154C2×Dic283C2411yescharacteristic
155C2×Dic283C2411yescharacteristic
156C2×Dic283C2411yescharacteristic
157C4×D283—26no—
158C4×D283—26no—
159C4×D283—26no—
160C4×D283—26no—
161C4×D283—22no—
162C4×D283—26no—
163C4×D283—26no—
164C4×D283—43no—
165C4×D283C2×D4112yescomplemented
166C23×C283C2413yes—
167C23×C283C2411yescharacteristic
168Q8×C283—26no—
169Q8×C283C2×D414yescomplemented
170D4×C283—26no—
171D4×C283—26no—
172C2×C4×C283—26no—
173C2×C4×C283—26no—
174C2×C4×C283C2411yescharacteristic
175C2×C4×C283C2413yes—
176C2×C4×C283C2413yes—
177C2×C4×C283C2411yescharacteristic
178C2×C4×C283C2413yes—
179C22×C283—21no—
180C22×C283—21no—
181C22×C283—23no—
182C22×C283—23no—
183C22×C283—23no—
184C22×C283C2×#4112yescomplemented
185C22×C283C2511yescharacteristic
186C2×D283—21no—
187C2×D283—23no—
188C2×D283—23no—
189C2×D283—21no—
190C2×D283—23no—
191C2×D283—23no—
192C2×D283—23no—
193C2×D283—412no—
194C2×D283—412no—
195C2×D283C22×D416yes—
196C2×D283C2×#4112yescomplemented
197C4×C283—21no—
198C4×C283—23no—
199C4×C283—23no—
200C4×C283—21no—
201C4×C283C22×D416yes—
202Dic283—21no—
203Dic283—23no—
204Dic283—23no—
205Dic283C22×D412yes—
206Dic283C2×#4112yescomplemented
207C2×C283—22no—
208C2×C283—43no—
209C2×C283C23×D411yescharacteristic
210C2×C283C22×#4111yescharacteristic
211C2×C283#58511yescharacteristic
212D283—22no—
213D283—26no—
214D283—43no—
215C283D4×#4111yescharacteristic, complemented, Sylow
216D4×#41—2831nomaximal, Sylow
217C22×#41—2831no—
218C22×#41—2831no—
219#550—2831no—
220D4×Q8—2831no—
221#548—2831no—
222#548D28311yescharacteristic, complemented
223#547—2833no—
224#546—2833no—
225#546—2833no—
226D42—2833no—
227#532—2833no—
228#525—2833no—
229#525—2833no—
230C4×#41—2831no—
231C2×C4×D4—2833no—
232C2×#41—2831no—
233C2×#41—2833no—
234C2×#41—2833no—
235C2×#41—2831no—
236C2×#41—2833no—
237C2×#41—2833no—
238C2×#41—5668no—
239C2×#41—5668no—
240C2×#41C2×D28311yescharacteristic, complemented
241C22×Q8—2831no—
242C22×Q8C2×D28311yescharacteristic, complemented
243C22×D4—2833no—
244C22×D4—2833no—
245C22×D4—2833no—
246C23×C4—2833no—
247C23×C4—2833no—
248C4⋊Q8—2833no—
249#127—2833no—
250#124—2833no—
251#124C2×D28313yescomplemented
252#123—2836no—
253#123—2836no—
254C22⋊Q8—2833no—
255C22⋊Q8—2833no—
256C22⋊Q8—2833no—
257C22⋊Q8C2×D28313yescomplemented
258#121—2833no—
259#121—2833no—
260#121—2833no—
261#121—2836no—
262#121—2836no—
263#121C2×D28313yescomplemented
264#120—2836no—
265#120—2836no—
266C4×Q8—2831no—
267C4×Q8C2×D28311yescharacteristic, complemented
268C4×D4—2833no—
269C4×D4—2833no—
270C4×D4—2833no—
271C4×D4—2833no—
272C4×D4—2833no—
273C4×D4—2831no—
274C4×D4—2833no—
275C4×D4—2836no—
276C4×D4—2833no—
277C4×D4—2833no—
278C4×D4C2×D28313yescomplemented
279#117—2833no—
280C2×C4⋊C4—2833no—
281C2×C22⋊C4—2833no—
282C2×C22⋊C4—2833no—
283C2×C42—2833no—
284C24—2833no—
285C24—2833no—
286#41—24no—
287#41—2838no—
288#41—56612no—
289#41—56612no—
290#41—56612no—
291#41—56612no—
292#41—5664no—
293C2×Q8—24no—
294C2×Q8—2833no—
295C2×Q8—5664no—
296C2×Q8C22×D28313yes—
297C2×Q8C22×D28311yescharacteristic
298C2×D4—2831no—
299C2×D4—2833no—
300C2×D4—2836no—
301C2×D4—2836no—
302C2×D4—2833no—
303C2×D4—2833no—
304C2×D4—2836no—
305C2×D4—2833no—
306C2×D4—56612no—
307C2×D4—56612no—
308C2×D4—56612no—
309C2×D4C22×D28313yes—
310C2×D4C22×D28313yes—
311C22×C4—2833no—
312C22×C4—2831no—
313C22×C4—2833no—
314C22×C4—2833no—
315C22×C4—2833no—
316C22×C4—2833no—
317C22×C4—2831no—
318C22×C4—2833no—
319C22×C4—2833no—
320C22×C4—2833no—
321C22×C4—2833no—
322C22×C4—56612no—
323C22×C4—56612no—
324C22×C4C22×D28313yes—
325C22×C4C22×D28313yes—
326C4⋊C4—2831no—
327C4⋊C4—2833no—
328C4⋊C4—2836no—
329C4⋊C4—2833no—
330C4⋊C4—2833no—
331C4⋊C4C22×D28313yes—
332C4⋊C4C22×D28313yes—
333C22⋊C4—2833no—
334C22⋊C4—2833no—
335C22⋊C4—2831no—
336C22⋊C4—2833no—
337C22⋊C4—2833no—
338C22⋊C4—2833no—
339C22⋊C4—2836no—
340C22⋊C4—2836no—
341C22⋊C4C22×D28311yescharacteristic
342C22⋊C4C22×D28313yes—
343C22⋊C4C22×D28316yes—
344C42—2831no—
345C42—2833no—
346C42—2833no—
347C42C22×D28313yes—
348C23—2831no—
349C23—2833no—
350C23—2833no—
351C23—5666no—
352C23—5666no—
353C23—5666no—
354C23—5666no—
355C23—11323no—
356C23—11323no—
357C23C23×D28313yes—
358C23C23×D28311yescharacteristic
359Q8—26no—
360Q8—5666no—
361Q8D4×D28314yescomplemented
362D4—26no—
363D4—26no—
364D4—2834no—
365D4—28312no—
366D4—56612no—
367D4—5666no—
368D4—5666no—
369D4—56612no—
370D4—5666no—
371D4—11326no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—2831no—
375C2×C4—2831no—
376C2×C4—28312no—
377C2×C4—2833no—
378C2×C4—2831no—
379C2×C4—2833no—
380C2×C4—5662no—
381C2×C4—5666no—
382C2×C4—5666no—
383C2×C4—5666no—
384C2×C4—5666no—
385C2×C4—5662no—
386C2×C4—5666no—
387C2×C4—5666no—
388C2×C4—5666no—
389C2×C4—11323no—
390C2×C4C23×D28311yescharacteristic
391C2×C4C23×D28313yes—
392C2×C4C23×D28311yescharacteristic
393C2×C4C23×D28313yes—
394C2×C4C23×D28313yes—
395C22—21no—
396C22—21no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—2836no—
401C22—2832no—
402C22—5661no—
403C22—5663no—
404C22—5663no—
405C22—5661no—
406C22—5663no—
407C22—5663no—
408C22—5663no—
409C22—113212no—
410C22—113212no—
411C22#41×D28312yescomplemented
412C22C24×D28311yescharacteristic
413C4—21no—
414C4—23no—
415C4—23no—
416C4—21no—
417C4—2832no—
418C4—2832no—
419C4—5661no—
420C4—5663no—
421C4—5663no—
422C4C2×D4×D28316yes—
423C2—22no—
424C2—43no—
425C2—5662no—
426C2—5666no—
427C2—11323no—
428C2C22×D4×D28311yescharacteristic
429C2C2×#41×D28311yescharacteristic
430C2#3182730211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#548×C283C2
hypercenter#548D283
hypocenterC283D4×#41
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C22×#17985393C2semidirect
2. C2×#41×D283C2semidirect
3. #31827134C2semidirect
4. Q8×C22⋊D283C2semidirect
5. #31827127C2semidirect
6. #31827117C2semidirect
7. #31827111C2semidirect
8. #31827108C2semidirect
9. #31827099C2semidirect
10. #31827014C2semidirect
11. #31826978C2semidirect
12. #31826975C2semidirect
13. C4×#17985393C2semidirect
14. C4×D4×D283C2semidirect
15. #548×C283C2semidirect
18. C2×#17985393C22semidirect
19. C2×#17985393C22semidirect
20. C2×#17985393C22semidirect
21. C2×#17985392C22semidirect
22. C2×#17985391C22semidirect
24. C2×Q8×D283C22semidirect
25. C22×D1132C22semidirect
27. C2×D4×D283C22semidirect
28. C22×C4×D283C22semidirect
29. C22×C4×D283C22semidirect
30. #22583608C22semidirect
31. #22583606C22semidirect
32. #22583598C22semidirect
33. #22583595C22semidirect
34. #22583592C22semidirect
35. #22583590C22semidirect
36. #22583589C22semidirect
37. #22583586C22semidirect
38. #22583585C22semidirect
39. #22583584C22semidirect
40. #22583583C22semidirect
41. #22583583C22semidirect
42. #22583581C22semidirect
43. #22583580C22semidirect
44. #22583578C22semidirect
45. Q8×Dic283C22semidirect
46. C4×D1132C22semidirect
47. C4×D1132C22semidirect
48. D4×Dic283C22semidirect
49. #22583572C22semidirect
50. #22583572C22semidirect
52. C4×C22⋊D283C22semidirect
54. (C22⋊D283)⋊C4C22semidirect
55. (C22⋊D283)⋊C4C22semidirect
56. #22583568C22semidirect
57. C4⋊C4×D283C22semidirect
58. C22⋊C4×D283C22semidirect
59. C22⋊C4×D283C22semidirect
60. C42×D283C22semidirect
61. C2×#41×C283C22semidirect
62. C22×Q8×C283C22semidirect
63. #124×C283C22semidirect
64. C22⋊Q8×C283C22semidirect
65. #121×C283C22semidirect
66. C4×Q8×C283C22semidirect
67. C4×D4×C283C22semidirect
73. #17985393D4semidirect
145. D1132C2×D4semidirect
149. C22⋊D283#41semidirect
165. C4×D283C2×D4semidirect
169. Q8×C283C2×D4semidirect
184. C22×C283C2×#41semidirect
196. C2×D283C2×#41semidirect
206. Dic283C2×#41semidirect
215. C283D4×#41semidirect
222. #548D283semidirect
240. C2×#41C2×D283semidirect
242. C22×Q8C2×D283semidirect
251. #124C2×D283semidirect
257. C22⋊Q8C2×D283semidirect
263. #121C2×D283semidirect
267. C4×Q8C2×D283semidirect
278. C4×D4C2×D283semidirect
361. Q8D4×D283semidirect
411. C22#41×D283semidirect

Maximal subgroups

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

subgroupindex
C22×#179853932
C2×#41×D2832
#318271342
Q8×C22⋊D2832
#318271272
#318271172
#318271112
#318271082
#318270992
#318270142
#318269782
#318269752
C4×#179853932
C4×D4×D2832
#548×C2832
D4×#41283