Finite Group Databasebeta

← #49131268

Subgroups — #49131268

Order 36224.

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

normal subgroups only

This group has 448 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
1#31827020C211yesmaximal, characteristic, complemented
2C4×D4×D283C211yesmaximal, characteristic
3#31826794C212yesmaximal, complemented
4#31826756C211yesmaximal, characteristic, complemented
5#31826755C211yesmaximal, characteristic, complemented
6#31826667C211yesmaximal, characteristic, complemented
7#31826614C212yesmaximal
8SD16×Dic283C212yesmaximal, complemented
9#31826550C212yesmaximal, complemented
10#31826517C211yesmaximal, characteristic, complemented
11#461×C283C211yesmaximal, characteristic, complemented
12C2×D4×D283C2211yescharacteristic
13C22×C4×D283—21no—
14C2×#17985379C2212yes—
15#22583610C2211yescharacteristic, complemented
16#22583601—21no—
17#22583590—21no—
18#22583586C2212yes—
19Q8×Dic283C2212yescomplemented
20#22583575—21no—
21C4×D1132C2211yescharacteristic
22D4×Dic283C2211yescharacteristic
23#22583572C2211yescharacteristic
24C4×C22⋊D283—21no—
25(C22⋊D283)⋊C4—21no—
26#22583568—21no—
27C4⋊C4×D283C2211yescharacteristic
28C22⋊C4×D283—21no—
29C42×D283C2211yescharacteristic
30#22583542C2212yescomplemented
31#22583541C2212yescomplemented
32C4⋊(C283⋊C8)C2211yescharacteristic, complemented
33#22583534C2212yescomplemented
34#22583532C2212yes—
35#22583531C2212yes—
36(C2×D283)⋊C8C2212yes—
37C8×Dic283C2212yescomplemented
38C2×SD16×C283C2212yes—
39C4⋊Q8×C283C2211yescharacteristic, complemented
40C4×D4×C283C2211yescharacteristic
41C4⋊C8×C283C2211yescharacteristic, complemented
42#103×C283C2212yescomplemented
43C23×D283—21no—
44C2×Dic566—21no—
45C2×D1132C2311yescharacteristic
46C2×C22⋊D283—21no—
47D4×D283—22no—
48D4×D283—22no—
49C22×Dic283—21no—
50C2×C4×D283—21no—
51C2×C4×D283—21no—
52C2×C4×D283—21no—
53C2×C4×D283—44no—
54C2×C4×D283D411yescharacteristic
55C2×C4×D283D411yescharacteristic
56C2×C4×D283C2311yescharacteristic
57#17985379—24no—
58C2×C283⋊C8C2312yes—
59C4⋊Dic283—22no—
60C4⋊Dic283—21no—
61C4⋊Dic283C2311yescharacteristic
62C4⋊Dic283C2312yes—
63Dic283⋊C4—21no—
64Dic283⋊C4—22no—
65C22⋊Dic283—21no—
66(C2×D283)⋊C4—22no—
67(C2×D283)⋊C4—21no—
68(C2×D283)⋊C4—21no—
69C4×Dic283—22no—
70C4×Dic283D411yescharacteristic
71C4×Dic283D411yescharacteristic
72C4×Dic283C2311yescharacteristic
73C2×Q8×C283C2312yes—
74C2×D4×C283C2311yescharacteristic
75C22×C4×C283—21no—
76SD16×C283—24no—
77C2×C8×C283C2312yes—
78C4⋊C4×C283—21no—
79C4⋊C4×C283—21no—
80C4⋊C4×C283C2311yescharacteristic
81C22⋊C4×C283—21no—
82C42×C283C2311yescharacteristic
83C22×D283—24no—
84C22×D283—21no—
85C22×D283—42no—
86C22×D283—42no—
87C22×D283C2×D411yescharacteristic
88Dic566—42no—
89D1132—21no—
90D1132#4112yes—
91C22⋊D283—24no—
92C2×Dic283—22no—
93C2×Dic283—21no—
94C2×Dic283—42no—
95C2×Dic283C2×D411yescharacteristic
96C2×Dic283C2×D411yescharacteristic
97C2×Dic283C2×D411yescharacteristic
98C4×D283—21no—
99C4×D283—21no—
100C4×D283—41no—
101C4×D283—42no—
102C4×D283—42no—
103C4×D283—42no—
104C4×D283SD1614yescomplemented
105C283⋊C8—22no—
106C23×C283—21no—
107Q8×C283—22no—
108Q8×C283—22no—
109D4×C283—21no—
110D4×C283#4112yes—
111C2×C4×C283—21no—
112C2×C4×C283—22no—
113C2×C4×C283—42no—
114C2×C4×C283C2×D411yescharacteristic
115C2×C4×C283C2×D411yescharacteristic
116C2×C4×C283C2411yescharacteristic
117C8×C283—22no—
118C22×C283—22no—
119C22×C283—41no—
120C22×C283—41no—
121C22×C283C22×D411yescharacteristic
122C2×D283—21no—
123C2×D283—21no—
124C2×D283—22no—
125C2×D283—48no—
126C2×D283—41no—
127C2×D283—41no—
128C2×D283C2×SD1612yes—
129C4×C283—21no—
130C4×C283—42no—
131C4×C283—41no—
132C4×C283C2×SD1612yes—
133C4×C283C2×#4111yescharacteristic
134C4×C283C2×#4111yescharacteristic
135Dic283—21no—
136Dic283—21no—
137Dic283—41no—
138Dic283—42no—
139Dic283C2×SD1612yes—
140C2×C283—42no—
141C2×C283#52511yescharacteristic
142C2×C283C22×SD1611yescharacteristic
143C2×C283#57511yescharacteristic
144D283—22no—
145D283—42no—
146C283#310111yescharacteristic, complemented, Sylow
147#3101—2831nomaximal, Sylow
148#533—2831no—
149C2×C4×D4—2831no—
150#483—2832no—
151#476—2831no—
152#476—2831no—
153#461—2831no—
154#461D28311yescharacteristic, complemented
155#450—2832no—
156C4×SD16—2832no—
157C4×SD16—2832no—
158#431—2831no—
159C22×D4—2831no—
160C23×C4—5661no—
161C2×SD16—2832no—
162C2×SD16C2×D28312yes—
163C4⋊Q8—2831no—
164C4⋊Q8C2×D28311yescharacteristic, complemented
165#125—5661no—
166C22⋊Q8—2832no—
167C22⋊Q8—5661no—
168C4×Q8—2832no—
169C4×Q8—5661no—
170C4×D4—2831no—
171C4×D4—2831no—
172C4×D4—2831no—
173C4×D4—5661no—
174C4×D4—5661no—
175C4×D4C2×D28311yescharacteristic
176#117—5661no—
177C2×C4⋊C4—2831no—
178C2×C22⋊C4—5661no—
179C2×C42—2831no—
180#106—2832no—
181#106—2832no—
182C4⋊C8—2831no—
183C4⋊C8C2×D28311yescharacteristic, complemented
184#103—2832no—
185#103C2×D28312yescomplemented
186#102—2832no—
187#102—2832no—
188C22⋊C8—2832no—
189C4×C8—2832no—
190C24—5661no—
191C2×Q8—5661no—
192C2×Q8C22×D28312yes—
193C2×D4—2831no—
194C2×D4—5661no—
195C2×D4—5662no—
196C2×D4—5662no—
197C2×D4C22×D28311yescharacteristic
198C22×C4—21no—
199C22×C4—2831no—
200C22×C4—2831no—
201C22×C4—2831no—
202C22×C4—5661no—
203C22×C4—5661no—
204C22×C4—5661no—
205C22×C4—5661no—
206C22×C4—11324no—
207SD16—24no—
208SD16—5664no—
209C2×C8—2832no—
210C2×C8C22×D28312yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—2832no—
214C4⋊C4—2831no—
215C4⋊C4—5661no—
216C4⋊C4—5662no—
217C4⋊C4—5662no—
218C4⋊C4—5661no—
219C4⋊C4C22×D28311yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—5661no—
222C22⋊C4—5662no—
223C22⋊C4—5661no—
224C22⋊C4—5661no—
225C42—2831no—
226C42—2831no—
227C42—2831no—
228C42—5662no—
229C42C22×D28311yescharacteristic
230C23—21no—
231C23—2831no—
232C23—5664no—
233C23—5661no—
234C23—11322no—
235C23—11322no—
236Q8—22no—
237Q8—22no—
238Q8—11322no—
239D4—21no—
240D4—2832no—
241D4—5664no—
242D4—5661no—
243D4#1798539112yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—2831no—
248C2×C4—2834no—
249C2×C4—2831no—
250C2×C4—2831no—
251C2×C4—5661no—
252C2×C4—5661no—
253C2×C4—5662no—
254C2×C4—5661no—
255C2×C4—11321no—
256C2×C4—11322no—
257C2×C4—11322no—
258C2×C4—11322no—
259C2×C4—11322no—
260C2×C4D4×D28311yescharacteristic
261C2×C4D4×D28311yescharacteristic
262C2×C4C23×D28311yescharacteristic
263C8—22no—
264C8—5662no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—2832no—
269C22—5661no—
270C22—5661no—
271C22—5662no—
272C22—11328no—
273C22—11321no—
274C22—11321no—
275C22C2×D4×D28311yescharacteristic
276C4—21no—
277C4—42no—
278C4—41no—
279C4—2832no—
280C4—5661no—
281C4—5661no—
282C4—11321no—
283C4—11322no—
284C4SD16×D28312yes—
285C4#41×D28311yescharacteristic
286C4C2×#1798539111yescharacteristic
287C2—42no—
288C2—5662no—
289C2—11322no—
290C2#3182697511yescharacteristic
291C2C2×SD16×D28311yescharacteristic
292C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#461×C283C2
socleC22×C283C22×D4
prime radicalC2×D4×D283C22
hypercenter#461D283
hypocenterC283#3101
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #31827020C2semidirect
3. #31826794C2semidirect
4. #31826756C2semidirect
5. #31826755C2semidirect
6. #31826667C2semidirect
8. SD16×Dic283C2semidirect
9. #31826550C2semidirect
10. #31826517C2semidirect
11. #461×C283C2semidirect
15. #22583610C22semidirect
19. Q8×Dic283C22semidirect
30. #22583542C22semidirect
31. #22583541C22semidirect
32. C4⋊(C283⋊C8)C22semidirect
33. #22583534C22semidirect
37. C8×Dic283C22semidirect
39. C4⋊Q8×C283C22semidirect
41. C4⋊C8×C283C22semidirect
42. #103×C283C22semidirect
104. C4×D283SD16semidirect
146. C283#3101semidirect
154. #461D283semidirect
164. C4⋊Q8C2×D283semidirect
183. C4⋊C8C2×D283semidirect
185. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270202
C4×D4×D2832
#318267942
#318267562
#318267552
#318266672
#318266142
SD16×Dic2832
#318265502
#318265172
#461×C2832
#3101283