Finite Group Databasebeta

← #49131206

Subgroups — #49131206

Order 36224.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D283C211yesmaximal, characteristic
2C2×#22583568C211yesmaximal, characteristic
3#31826808C212yesmaximal, complemented
4#31826804C212yesmaximal, complemented
5#31826800C211yesmaximal, characteristic, complemented
6#31826796C211yesmaximal, characteristic, complemented
7#31826793C211yesmaximal, characteristic, complemented
8#31826506C212yesmaximal, complemented
9#31826472C212yesmaximal, complemented
10#31826427C211yesmaximal, characteristic
11#484×C283C211yesmaximal, characteristic, complemented
12C2×Q8×D283C2211yescharacteristic
13C22×C4×D283C2211yescharacteristic
14C2×#17985375C2212yes—
15#22583590C2211yescharacteristic
16#22583589C2211yescharacteristic
17#22583588—21no—
18#22583587—21no—
19#22583586C2211yescharacteristic
20#22583568—21no—
21#22583568—21no—
22#22583568C2212yes—
23#22583568C2212yes—
24C2×C4⋊Dic283C2211yescharacteristic
25C4⋊C4×D283—21no—
26C4⋊C4×D283C2211yescharacteristic
27C2×(C2×D283)⋊C4—21no—
28C22⋊C4×D283—21no—
29C2×C4×Dic283—21no—
30#22583543C2212yescomplemented
31#22583542C2212yescomplemented
32#22583541C2212yescomplemented
33#22583535C2212yescomplemented
34#22583534C2212yescomplemented
35#22583533C2212yescomplemented
36C22⋊(C283⋊C8)C2211yescharacteristic
37(C2×D283)⋊C8C2212yes—
38C22⋊Q8×C283C2211yescharacteristic
39C2×C4⋊C4×C283C2211yescharacteristic
40#107×C283C2212yescomplemented
41#103×C283C2212yescomplemented
42C22⋊C8×C283C2211yescharacteristic
43C23×D283D411yescharacteristic
44C2×Dic566—21no—
45C2×Dic566C2311yescharacteristic
46Q8×D283—24no—
47C22×Dic283—21no—
48C22×Dic283D411yescharacteristic
49C2×C4×D283—24no—
50C2×C4×D283—21no—
51C2×C4×D283—21no—
52C2×C4×D283D411yescharacteristic
53C2×C4×D283D411yescharacteristic
54C2×C4×D283C2311yescharacteristic
55C2×C4×D283C2311yescharacteristic
56#17985375—24no—
57C2×C283⋊C8C2312yes—
58C4⋊Dic283—21no—
59C4⋊Dic283—21no—
60C4⋊Dic283C2311yescharacteristic
61C4⋊Dic283C2312yes—
62Dic283⋊C4—21no—
63Dic283⋊C4—21no—
64Dic283⋊C4—21no—
65C22⋊Dic283—21no—
66(C2×D283)⋊C4—21no—
67(C2×D283)⋊C4—21no—
68(C2×D283)⋊C4—22no—
69(C2×D283)⋊C4—22no—
70C4×Dic283—22no—
71C4×Dic283—22no—
72C2×Q8×C283C2311yescharacteristic
73C22×C4×C283—21no—
74C22×C4×C283C2311yescharacteristic
75C2×C8×C283C2312yes—
76C4⋊C4×C283—21no—
77C4⋊C4×C283—21no—
78C4⋊C4×C283C2311yescharacteristic
79C4⋊C4×C283C2312yes—
80C22⋊C4×C283—21no—
81C22×D283—22no—
82C22×D283—22no—
83C22×D283—41no—
84C22×D283C2×D411yescharacteristic
85C22×D283C2×D411yescharacteristic
86Dic566—22no—
87Dic566—42no—
88C2×Dic283—22no—
89C2×Dic283—22no—
90C2×Dic283—21no—
91C2×Dic283—21no—
92C2×Dic283—42no—
93C2×Dic283C2×D411yescharacteristic
94C2×Dic283C2×D411yescharacteristic
95C4×D283—22no—
96C4×D283—22no—
97C4×D283—22no—
98C4×D283—42no—
99C4×D283—42no—
100C4×D283#4114yes—
101C283⋊C8—22no—
102C23×C283C2×D411yescharacteristic
103Q8×C283—22no—
104C2×C4×C283—22no—
105C2×C4×C283—21no—
106C2×C4×C283—21no—
107C2×C4×C283—22no—
108C2×C4×C283—42no—
109C2×C4×C283C2×D411yescharacteristic
110C2×C4×C283C2411yescharacteristic
111C8×C283—22no—
112C22×C283—21no—
113C22×C283—21no—
114C22×C283#13712yes—
115C22×C283C22×D411yescharacteristic
116C2×D283—21no—
117C2×D283—21no—
118C2×D283—21no—
119C2×D283—21no—
120C2×D283—21no—
121C2×D283—44no—
122C2×D283#12312yes—
123C4×C283—21no—
124C4×C283—41no—
125C4×C283—42no—
126C4×C283—41no—
127C4×C283C2×#4112yes—
128Dic283—21no—
129Dic283—41no—
130Dic283—42no—
131Dic283—41no—
132Dic283#12312yes—
133C2×C283—22no—
134C2×C283C2×#12311yescharacteristic
135C2×C283C2×#13711yescharacteristic
136C2×C283#57511yescharacteristic
137D283—22no—
138D283—41no—
139C283#309511yescharacteristic, complemented, Sylow
140#3095—2831nomaximal, Sylow
141C2×C22⋊Q8—2831no—
142C2×#117—2831no—
143#485—2832no—
144#485—2832no—
145#484—2831no—
146#484D28311yescharacteristic, complemented
147#483—2831no—
148#483—2831no—
149#428—2832no—
150#419—2832no—
151#407—2831no—
152C22×Q8—2831no—
153C23×C4—2831no—
154C2×M16—2832no—
155C22⋊Q8—2831no—
156C22⋊Q8—2831no—
157C22⋊Q8—2831no—
158C22⋊Q8—5661no—
159C22⋊Q8—5661no—
160C22⋊Q8C2×D28311yescharacteristic
161#117—2832no—
162#117—2832no—
163#117—5661no—
164#117—5661no—
165C2×C4⋊C4—2831no—
166C2×C4⋊C4—2831no—
167C2×C4⋊C4—5661no—
168C2×C4⋊C4C2×D28311yescharacteristic
169C2×C22⋊C4—5661no—
170C2×C22⋊C4—5661no—
171C2×C42—5661no—
172#107—2832no—
173#107C2×D28312yescomplemented
174#106—2832no—
175#106—2832no—
176#103—2832no—
177#103—2832no—
178#103—2832no—
179#103C2×D28312yescomplemented
180C22⋊C8—2831no—
181C22⋊C8—2832no—
182C22⋊C8C2×D28311yescharacteristic
183C24—2831no—
184C2×Q8—2831no—
185C2×Q8—5661no—
186C2×Q8—5664no—
187C2×Q8C22×D28311yescharacteristic
188C22×C4—21no—
189C22×C4—2831no—
190C22×C4—2831no—
191C22×C4—2831no—
192C22×C4—2831no—
193C22×C4—2831no—
194C22×C4—5664no—
195C22×C4—5661no—
196C22×C4—5661no—
197C22×C4—5661no—
198C22×C4C22×D28311yescharacteristic
199M16—5664no—
200C2×C8—2832no—
201C2×C8C22×D28312yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—2832no—
205C4⋊C4—2831no—
206C4⋊C4—5661no—
207C4⋊C4—5661no—
208C4⋊C4—5661no—
209C4⋊C4—5661no—
210C4⋊C4—5661no—
211C4⋊C4C22×D28312yes—
212C4⋊C4C22×D28311yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—5662no—
215C22⋊C4—5662no—
216C22⋊C4—5661no—
217C22⋊C4—5661no—
218C22⋊C4—5661no—
219C42—5662no—
220C42—5662no—
221C23—2831no—
222C23—2831no—
223C23—5662no—
224C23—5662no—
225C23—11321no—
226C23D4×D28311yescharacteristic
227Q8—22no—
228Q8—5662no—
229Q8—11322no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—2831no—
236C2×C4—2834no—
237C2×C4—2831no—
238C2×C4—5662no—
239C2×C4—5662no—
240C2×C4—5662no—
241C2×C4—5662no—
242C2×C4—5662no—
243C2×C4—5661no—
244C2×C4—5661no—
245C2×C4—11322no—
246C2×C4—11322no—
247C2×C4—11322no—
248C2×C4D4×D28311yescharacteristic
249C2×C4C23×D28311yescharacteristic
250C8—22no—
251C8—5662no—
252C22—21no—
253C22—21no—
254C22—2832no—
255C22—5661no—
256C22—5661no—
257C22—5661no—
258C22—5661no—
259C22—5661no—
260C22—11324no—
261C22#2258364212yes—
262C22C2×D4×D28311yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—2832no—
268C4—5661no—
269C4—11321no—
270C4—11322no—
271C4—11321no—
272C4#41×D28312yes—
273C2—22no—
274C2—5662no—
275C2—11321no—
276C2#123×D28311yescharacteristic
277C2C2×#2258364211yescharacteristic
278C2#3182726111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#484×C283C2
socleC22×C283C22×D4
prime radicalC23×D283D4
hypercenter#484D283
hypocenterC283#3095
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #31826808C2semidirect
4. #31826804C2semidirect
5. #31826800C2semidirect
6. #31826796C2semidirect
7. #31826793C2semidirect
8. #31826506C2semidirect
9. #31826472C2semidirect
11. #484×C283C2semidirect
30. #22583543C22semidirect
31. #22583542C22semidirect
32. #22583541C22semidirect
33. #22583535C22semidirect
34. #22583534C22semidirect
35. #22583533C22semidirect
40. #107×C283C22semidirect
41. #103×C283C22semidirect
139. C283#3095semidirect
146. #484D283semidirect
173. #107C2×D283semidirect
179. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2832
C2×#225835682
#318268082
#318268042
#318268002
#318267962
#318267932
#318265062
#318264722
#318264272
#484×C2832
#3095283