Finite Group Databasebeta

← #49131176

Subgroups — #49131176

Order 36224.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31826976C211yesmaximal, characteristic
2C2×#22583569C211yesmaximal, characteristic, complemented
3#31826807C212yesmaximal, complemented
4#31826801C211yesmaximal, characteristic, complemented
5#31826796C211yesmaximal, characteristic, complemented
6#31826790C212yesmaximal, complemented
7#31826784C211yesmaximal, characteristic, complemented
8#31826779C211yesmaximal, characteristic, complemented
9#31826503C212yesmaximal, complemented
10#31826462C212yesmaximal, complemented
11C2×C22⋊C8×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13#22583593—21no—
14#22583589C2211yescharacteristic
15#22583589C2211yescharacteristic
16#22583584C2211yescharacteristic
17#22583584C2211yescharacteristic
18#22583579—21no—
19C4×C22⋊D283—21no—
20C4×C22⋊D283—21no—
21#22583569—21no—
22#22583569—21no—
23#22583569C2212yes—
24#22583569C2211yescharacteristic
25#22583569C2212yes—
26C2×C4⋊Dic283C2211yescharacteristic
27C2×C4⋊Dic283C2211yescharacteristic
28C2×C22⋊Dic283—21no—
29C2×C4×Dic283—21no—
30#22583544C2212yescomplemented
31#22583544C2212yescomplemented
32#22583542C2212yescomplemented
33#22583542C2212yescomplemented
34#22583535C2212yescomplemented
35#22583535C2212yescomplemented
36#22583532C2212yescomplemented
37#22583532C2212yescomplemented
38C23×C4×C283C2211yescharacteristic
39C22×C8×C283C2212yes—
40C22⋊C8×C283C2211yescharacteristic
41C22⋊C8×C283C2211yescharacteristic
42C22⋊C8×C283C2212yes—
43#17985392—24no—
44C2×Dic566C2311yescharacteristic
45C2×D1132C2311yescharacteristic
46C2×C22⋊D283—21no—
47C2×C22⋊D283—21no—
48C2×C22⋊D283—21no—
49C22×Dic283—21no—
50C22×Dic283—21no—
51C2×C4×D283—21no—
52C4⋊Dic283—21no—
53C4⋊Dic283—21no—
54C4⋊Dic283C2312yes—
55C4⋊Dic283C2312yes—
56C4⋊Dic283C2311yescharacteristic
57C4⋊Dic283C2311yescharacteristic
58Dic283⋊C4—21no—
59Dic283⋊C4—21no—
60C22⋊Dic283—22no—
61C22⋊Dic283—22no—
62C22⋊Dic283—21no—
63C22⋊Dic283—21no—
64C22⋊Dic283—21no—
65(C2×D283)⋊C4—21no—
66(C2×D283)⋊C4—21no—
67C4×Dic283—22no—
68C4×Dic283—22no—
69C4×Dic283—21no—
70C24×C283D411yescharacteristic
71C22×C4×C283—24no—
72C22×C4×C283D411yescharacteristic
73C22×C4×C283D411yescharacteristic
74C22×C4×C283D411yescharacteristic
75C22×C4×C283C2311yescharacteristic
76C22×C4×C283C2311yescharacteristic
77C22×C4×C283C2311yescharacteristic
78C2×C8×C283—24no—
79C2×C8×C283C2312yes—
80C2×C8×C283C2312yes—
81C22×D283—21no—
82Dic566—22no—
83D1132—22no—
84C22⋊D283—42no—
85C22⋊D283—42no—
86C22⋊D283—42no—
87C2×Dic283—22no—
88C2×Dic283—22no—
89C2×Dic283—21no—
90C2×Dic283—21no—
91C2×Dic283—21no—
92C2×Dic283—42no—
93C2×Dic283—42no—
94C4×D283—42no—
95C23×C283—22no—
96C23×C283—22no—
97C23×C283—41no—
98C23×C283C2×D411yescharacteristic
99C23×C283C2×D411yescharacteristic
100C23×C283C2×D411yescharacteristic
101C2×C4×C283—22no—
102C2×C4×C283—22no—
103C2×C4×C283—22no—
104C2×C4×C283—22no—
105C2×C4×C283—22no—
106C2×C4×C283C2×D411yescharacteristic
107C2×C4×C283C2×D411yescharacteristic
108C2×C4×C283C2×D411yescharacteristic
109C2×C4×C283#4114yes—
110C2×C4×C283C2411yescharacteristic
111C8×C283—22no—
112C8×C283—22no—
113C22×C283—21no—
114C22×C283—21no—
115C22×C283—21no—
116C22×C283—21no—
117C22×C283—21no—
118C22×C283—21no—
119C22×C283—21no—
120C22×C283—44no—
121C22×C283#12312yes—
122C22×C283#13512yes—
123C22×C283C22×D411yescharacteristic
124C2×D283—41no—
125C2×D283—41no—
126C2×D283—41no—
127C4×C283—21no—
128C4×C283—21no—
129C4×C283#12312yes—
130C4×C283C2×#4112yes—
131Dic283—41no—
132Dic283—42no—
133Dic283—41no—
134Dic283—42no—
135Dic283—41no—
136C2×C283—22no—
137C2×C283—22no—
138C2×C283—41no—
139C2×C283C2×#12311yescharacteristic
140C2×C283C2×#13511yescharacteristic
141C2×C283#57511yescharacteristic
142D283—81no—
143C283#309111yescharacteristic, complemented, Sylow
144#3091—2831nomaximal, Sylow
145#525—2831no—
146C2×#117—2831no—
147#485—2832no—
148#484—2831no—
149#483—2831no—
150#482—2832no—
151#481—2831no—
152#480—2831no—
153#427—2832no—
154#416—2832no—
155C2×C22⋊C8D28311yescharacteristic, complemented
156C2×#41—2831no—
157C23×C4C2×D28311yescharacteristic
158C22×C8C2×D28312yes—
159#123—5661no—
160C22⋊Q8—2831no—
161C22⋊Q8—2831no—
162#121—2831no—
163#121—2831no—
164#120—5661no—
165C4×D4—5661no—
166C4×D4—5661no—
167#117—2832no—
168#117—2832no—
169#117—2831no—
170#117—5661no—
171#117—5661no—
172C2×C4⋊C4—2831no—
173C2×C4⋊C4—2831no—
174C2×C22⋊C4—5661no—
175C2×C42—5661no—
176#107—2832no—
177#107—2832no—
178#106—2832no—
179#106—2832no—
180#103—2832no—
181#103—2832no—
182#102—2832no—
183#102—2832no—
184C22⋊C8C2×D28311yescharacteristic
185C22⋊C8C2×D28311yescharacteristic
186C22⋊C8C2×D28312yes—
187C24D113211yescharacteristic
188#41—5664no—
189C2×Q8—2831no—
190C2×D4—2831no—
191C2×D4—5661no—
192C2×D4—5661no—
193C2×D4—5661no—
194C22×C4—24no—
195C22×C4—5661no—
196C22×C4—5661no—
197C22×C4—5661no—
198C22×C4D113211yescharacteristic
199C22×C4D113211yescharacteristic
200C22×C4D113211yescharacteristic
201C22×C4C22×D28311yescharacteristic
202C22×C4C22×D28311yescharacteristic
203C22×C4C22×D28311yescharacteristic
204C2×C8—24no—
205C2×C8C22×D28312yes—
206C2×C8C22×D28312yes—
207C4⋊C4—2832no—
208C4⋊C4—2832no—
209C4⋊C4—2831no—
210C4⋊C4—2831no—
211C4⋊C4—5661no—
212C4⋊C4—5661no—
213C4⋊C4—5661no—
214C4⋊C4—5661no—
215C22⋊C4—5662no—
216C22⋊C4—5662no—
217C22⋊C4—5661no—
218C22⋊C4—5661no—
219C22⋊C4—5661no—
220C22⋊C4—5661no—
221C22⋊C4—5661no—
222C42—5662no—
223C42—5662no—
224C42—5661no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—5661no—
229C23C2×D113211yescharacteristic
230C23C2×D113211yescharacteristic
231C23C2×D113211yescharacteristic
232Q8—5662no—
233D4—5662no—
234D4—11322no—
235D4—11322no—
236D4—11322no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—5662no—
243C2×C4—5662no—
244C2×C4—5661no—
245C2×C4—5661no—
246C2×C4—5661no—
247C2×C4—11322no—
248C2×C4—11322no—
249C2×C4—11322no—
250C2×C4C2×D113211yescharacteristic
251C2×C4C2×D113211yescharacteristic
252C2×C4C2×D113211yescharacteristic
253C2×C4#1798539114yes—
254C2×C4C23×D28311yescharacteristic
255C8—22no—
256C8—22no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—11321no—
266C22—11321no—
267C22—11321no—
268C22#2258359412yes—
269C22#2258363212yes—
270C22C22×D113211yescharacteristic
271C4—21no—
272C4—21no—
273C4—11321no—
274C4—11322no—
275C4—11321no—
276C4—11322no—
277C4—11321no—
278C4#2258359412yes—
279C4C2×#1798539112yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—22641no—
284C2C2×#2258359411yescharacteristic
285C2C2×#2258363211yescharacteristic
286C2#3182726511yescharacteristic
subgroupG/H
centerC2×C4C2×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×C22⋊C8×C283C2
socleC22×C283C22×D4
prime radical#31826976C2
hypercenterC2×C22⋊C8D283
hypocenterC283#3091
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#22583569C2semidirect
3. #31826807C2semidirect
4. #31826801C2semidirect
5. #31826796C2semidirect
6. #31826790C2semidirect
7. #31826784C2semidirect
8. #31826779C2semidirect
9. #31826503C2semidirect
10. #31826462C2semidirect
11. C2×C22⋊C8×C283C2semidirect
30. #22583544C22semidirect
31. #22583544C22semidirect
32. #22583542C22semidirect
33. #22583542C22semidirect
34. #22583535C22semidirect
35. #22583535C22semidirect
36. #22583532C22semidirect
37. #22583532C22semidirect
143. C283#3091semidirect
155. C2×C22⋊C8D283semidirect

Maximal subgroups

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

subgroupindex
#318269762
C2×#225835692
#318268072
#318268012
#318267962
#318267902
#318267842
#318267792
#318265032
#318264622
C2×C22⋊C8×C2832
#3091283