Finite Group Databasebeta

← #49131207

Subgroups — #49131207

Order 36224.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#22583586C211yesmaximal, characteristic
2#117×D283C211yesmaximal, characteristic
3#31826805C212yesmaximal, complemented
4#31826804C212yesmaximal, complemented
5#31826799C212yesmaximal, complemented
6#31826794C212yesmaximal, complemented
7#31826508C211yesmaximal, characteristic, complemented
8#31826507C211yesmaximal, characteristic, complemented
9#31826474C211yesmaximal, characteristic, complemented
10#31826428C211yesmaximal, characteristic
11#419×C283C211yesmaximal, characteristic, complemented
12C22×C4×D283C2211yescharacteristic
13C2×#17985376C2211yescharacteristic
14#22583586—21no—
15#22583586—21no—
16#22583586C2212yes—
17#22583586C2212yes—
18#22583569C2211yescharacteristic
19#22583568C2211yescharacteristic
20#22583568C2211yescharacteristic
21#22583567—21no—
22#22583566—21no—
23C2×C4⋊Dic283C2211yescharacteristic
24C2×Dic283⋊C4—21no—
25C4⋊C4×D283C2211yescharacteristic
26C4⋊C4×D283C2211yescharacteristic
27C2×(C2×D283)⋊C4—21no—
28C22⋊C4×D283—21no—
29C42×D283—21no—
30#22583544C2212yescomplemented
31#22583543C2212yescomplemented
32#22583542C2212yescomplemented
33#22583541C2212yescomplemented
34#22583534C2212yescomplemented
35#22583534C2212yescomplemented
36(C2×D283)⋊C8C2212yes—
37(C2×D283)⋊C8C2212yes—
38C22×Q8×C283C2211yescharacteristic
39C2×M16×C283C2211yescharacteristic
40#117×C283C2211yescharacteristic
41#103×C283C2212yescomplemented
42#103×C283C2212yescomplemented
43C23×D283D411yescharacteristic
44C22×Dic283—21no—
45C22×Dic283D411yescharacteristic
46C2×C4×D283—21no—
47C2×C4×D283—21no—
48C2×C4×D283—22no—
49C2×C4×D283—22no—
50C2×C4×D283D411yescharacteristic
51C2×C4×D283D411yescharacteristic
52C2×C4×D283C2311yescharacteristic
53C2×C4×D283C2311yescharacteristic
54#17985376—22no—
55C2×C283⋊C8C2312yes—
56C4⋊Dic283—21no—
57C4⋊Dic283C2311yescharacteristic
58C4⋊Dic283C2311yescharacteristic
59C4⋊Dic283C2312yes—
60Dic283⋊C4—21no—
61Dic283⋊C4—21no—
62Dic283⋊C4—22no—
63Dic283⋊C4—22no—
64C22⋊Dic283—21no—
65(C2×D283)⋊C4—21no—
66(C2×D283)⋊C4—21no—
67(C2×D283)⋊C4—22no—
68(C2×D283)⋊C4—22no—
69C4×Dic283—21no—
70C4×Dic283—21no—
71C4×Dic283—21no—
72C2×Q8×C283—21no—
73C2×Q8×C283—22no—
74C2×Q8×C283—22no—
75C2×Q8×C283C2312yes—
76C22×C4×C283—21no—
77C22×C4×C283C2311yescharacteristic
78M16×C283—22no—
79C2×C8×C283C2312yes—
80C4⋊C4×C283C2311yescharacteristic
81C4⋊C4×C283C2311yescharacteristic
82C22⋊C4×C283—21no—
83C42×C283—21no—
84C22×D283—22no—
85C22×D283—22no—
86C22×D283—41no—
87C22×D283C2×D411yescharacteristic
88C22×D283C2×D411yescharacteristic
89C2×Dic283—22no—
90C2×Dic283—21no—
91C2×Dic283—21no—
92C2×Dic283—22no—
93C2×Dic283—42no—
94C2×Dic283C2×D411yescharacteristic
95C2×Dic283C2×D411yescharacteristic
96C4×D283—22no—
97C4×D283—22no—
98C4×D283—21no—
99C4×D283—21no—
100C4×D283—21no—
101C4×D283—21no—
102C4×D283—42no—
103C4×D283—42no—
104C283⋊C8—22no—
105C23×C283C2×D411yescharacteristic
106Q8×C283—22no—
107Q8×C283—22no—
108Q8×C283—42no—
109C2×C4×C283—21no—
110C2×C4×C283—21no—
111C2×C4×C283—22no—
112C2×C4×C283—42no—
113C2×C4×C283C2×D411yescharacteristic
114C2×C4×C283#4112yes—
115C2×C4×C283#4112yes—
116C2×C4×C283C2411yescharacteristic
117C8×C283—22no—
118C22×C283—21no—
119C22×C283—21no—
120C22×C283#12312yes—
121C22×C283C22×D411yescharacteristic
122C2×D283—21no—
123C2×D283—21no—
124C2×D283—21no—
125C2×D283—21no—
126C2×D283—21no—
127C2×D283—44no—
128C2×D283#13712yes—
129C4×C283—42no—
130C4×C283—41no—
131C4×C283—41no—
132C4×C283#12312yes—
133C4×C283C2×#4111yescharacteristic
134C4×C283C2×#4111yescharacteristic
135Dic283—21no—
136Dic283—21no—
137Dic283—41no—
138Dic283—41no—
139Dic283—42no—
140C2×C283—22no—
141C2×C283C2×#12311yescharacteristic
142C2×C283C2×#13711yescharacteristic
143C2×C283#57511yescharacteristic
144D283—22no—
145D283—41no—
146C283#309511yescharacteristic, complemented, Sylow
147#3095—2831nomaximal, Sylow
148C2×C22⋊Q8—2831no—
149C2×#117—2831no—
150#485—2832no—
151#485—2832no—
152#484—2832no—
153#483—2832no—
154#428—2831no—
155#428—2831no—
156#419—2831no—
157#419D28311yescharacteristic, complemented
158#407—2831no—
159C22×Q8C2×D28311yescharacteristic
160C23×C4—2831no—
161C2×M16—2831no—
162C2×M16C2×D28311yescharacteristic
163C22⋊Q8—2832no—
164C22⋊Q8—2832no—
165C22⋊Q8—5661no—
166C22⋊Q8—5661no—
167#117—2831no—
168#117—2831no—
169#117—2831no—
170#117—5661no—
171#117—5661no—
172#117C2×D28311yescharacteristic
173C2×C4⋊C4—2831no—
174C2×C4⋊C4—2831no—
175C2×C4⋊C4—2831no—
176C2×C4⋊C4—5661no—
177C2×C22⋊C4—5661no—
178C2×C22⋊C4—5661no—
179C2×C42—5661no—
180#107—2832no—
181#107—2832no—
182#106—2832no—
183#106—2832no—
184#103—2832no—
185#103—2832no—
186#103C2×D28312yescomplemented
187#103C2×D28312yescomplemented
188C22⋊C8—2832no—
189C22⋊C8—2832no—
190C24—2831no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D28312yes—
195C22×C4—21no—
196C22×C4—2831no—
197C22×C4—2831no—
198C22×C4—2831no—
199C22×C4—2831no—
200C22×C4—2831no—
201C22×C4—5661no—
202C22×C4—5661no—
203C22×C4—5662no—
204C22×C4—5662no—
205C22×C4—5661no—
206C22×C4C22×D28311yescharacteristic
207M16—22no—
208M16—5662no—
209C2×C8—2832no—
210C2×C8C22×D28312yes—
211C4⋊C4—2831no—
212C4⋊C4—2831no—
213C4⋊C4—2832no—
214C4⋊C4—5661no—
215C4⋊C4—5661no—
216C4⋊C4—5662no—
217C4⋊C4—5662no—
218C4⋊C4—5661no—
219C4⋊C4C22×D28311yescharacteristic
220C4⋊C4C22×D28311yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—5661no—
223C22⋊C4—5661no—
224C22⋊C4—5662no—
225C22⋊C4—5662no—
226C22⋊C4—5661no—
227C42—21no—
228C42—5661no—
229C42—5661no—
230C42—5661no—
231C23—2831no—
232C23—2831no—
233C23—5662no—
234C23—5662no—
235C23—11321no—
236C23D4×D28311yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—2831no—
245C2×C4—2831no—
246C2×C4—5662no—
247C2×C4—5662no—
248C2×C4—5662no—
249C2×C4—5661no—
250C2×C4—5661no—
251C2×C4—5661no—
252C2×C4—5661no—
253C2×C4—5661no—
254C2×C4—5661no—
255C2×C4—5662no—
256C2×C4—11322no—
257C2×C4—11322no—
258C2×C4—11322no—
259C2×C4D4×D28311yescharacteristic
260C2×C4#1798539112yes—
261C2×C4#1798539212yes—
262C2×C4C23×D28311yescharacteristic
263C8—22no—
264C8—5662no—
265C22—21no—
266C22—21no—
267C22—2832no—
268C22—5661no—
269C22—5661no—
270C22—5661no—
271C22—5661no—
272C22—5661no—
273C22—11324no—
274C22#2258359212yes—
275C22C2×D4×D28311yescharacteristic
276C4—42no—
277C4—41no—
278C4—41no—
279C4—5661no—
280C4—5661no—
281C4—11321no—
282C4—11321no—
283C4—11322no—
284C4#2258359212yes—
285C4C2×#1798539111yescharacteristic
286C4C2×#1798539211yescharacteristic
287C2—22no—
288C2—5662no—
289C2—11321no—
290C2C2×#2258359211yescharacteristic
291C2#137×D28311yescharacteristic
292C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#419×C283C2
socleC22×C283C22×D4
prime radicalC23×D283D4
hypercenter#419D283
hypocenterC283#3095
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #31826805C2semidirect
4. #31826804C2semidirect
5. #31826799C2semidirect
6. #31826794C2semidirect
7. #31826508C2semidirect
8. #31826507C2semidirect
9. #31826474C2semidirect
11. #419×C283C2semidirect
30. #22583544C22semidirect
31. #22583543C22semidirect
32. #22583542C22semidirect
33. #22583541C22semidirect
34. #22583534C22semidirect
35. #22583534C22semidirect
41. #103×C283C22semidirect
42. #103×C283C22semidirect
146. C283#3095semidirect
157. #419D283semidirect
186. #103C2×D283semidirect
187. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835862
#117×D2832
#318268052
#318268042
#318267992
#318267942
#318265082
#318265072
#318264742
#318264282
#419×C2832
#3095283