Finite Group Databasebeta

← #49130844

Subgroups — #49130844

Order 36224.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22583633C211yesmaximal, characteristic, complemented
2C22×C8×D283C211yesmaximal, characteristic, complemented
3#31826978C212yesmaximal, complemented
4#31826703C211yesmaximal, characteristic, complemented
5#31826702C211yesmaximal, characteristic, complemented
6#31826701C211yesmaximal, characteristic, complemented
7#31826697C212yesmaximal, complemented
8#31826696C212yesmaximal, complemented
9#31826502C211yesmaximal, characteristic, complemented
10#31826460C212yesmaximal, complemented
11#467×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2212yes—
13C22×C4×D283C2211yescharacteristic
14#22583633—24no—
15C2×#17985379C2212yescomplemented
16C2×D2264C2211yescharacteristic, complemented
17C22×C283⋊C8C2211yescharacteristic
18C2×C8×D283—24no—
19C2×C8×D283C2211yescharacteristic
20C2×C8×D283C2211yescharacteristic
21C2×C8×D283C2211yescharacteristic
22C2×C8×D283C2211yescharacteristic
23#22583595—22no—
24#22583586C2212yes—
25#22583585C2212yes—
26#22583584C2212yes—
27#22583580—22no—
28#22583572—22no—
29(C22⋊D283)⋊C4—22no—
30#22583568C2212yes—
31#22583544C2211yescharacteristic, complemented
32#22583541C2212yescomplemented
33#22583534C2212yescomplemented
34#22583532C2212yescomplemented
35#22583530C2212yescomplemented
36C2×Q16×C283C2211yescharacteristic, complemented
37C22×C8×C283C2211yescharacteristic
38C22⋊Q8×C283C2212yes—
39#107×C283C2211yescharacteristic, complemented
40#103×C283C2212yescomplemented
41C23×D283D411yescharacteristic
42#17985393—24no—
43#17985393—24no—
44C2×D1132—22no—
45C2×D1132—22no—
46C2×D1132C2312yes—
47C2×C22⋊D283—22no—
48C22×Dic283D411yescharacteristic
49C2×C4×D283—22no—
50C2×C4×D283—22no—
51C2×C4×D283—22no—
52C2×C4×D283D411yescharacteristic
53C2×C4×D283D411yescharacteristic
54C2×C4×D283C2311yescharacteristic
55C2×C4×D283C2311yescharacteristic
56#17985379—24no—
57D2264—22no—
58C2×C283⋊C8—22no—
59C2×C283⋊C8C2311yescharacteristic
60C2×C283⋊C8C2311yescharacteristic
61C8×D283—22no—
62C8×D283—22no—
63C8×D283—22no—
64C8×D283D414yescomplemented
65C4⋊Dic283C2312yes—
66Dic283⋊C4—22no—
67(C2×D283)⋊C4—22no—
68(C2×D283)⋊C4—22no—
69(C2×D283)⋊C4—22no—
70(C2×D283)⋊C4—22no—
71C4×Dic283—22no—
72C2×Q8×C283C2312yes—
73C22×C4×C283C2311yescharacteristic
74Q16×C283—22no—
75C2×C8×C283—22no—
76C2×C8×C283C2311yescharacteristic
77C2×C8×C283C2311yescharacteristic
78C4⋊C4×C283—22no—
79C4⋊C4×C283C2312yes—
80C22⋊C4×C283—22no—
81C22×D283—22no—
82C22×D283—22no—
83C22×D283—22no—
84C22×D283—22no—
85C22×D283C2×D411yescharacteristic
86C22×D283C2×D411yescharacteristic
87D1132—22no—
88D1132—22no—
89D1132—44no—
90D1132—44no—
91C22⋊D283—44no—
92C2×Dic283—22no—
93C2×Dic283—22no—
94C2×Dic283C2×D411yescharacteristic
95C2×Dic283C2×D411yescharacteristic
96C4×D283—22no—
97C4×D283—21no—
98C4×D283—21no—
99C4×D283—22no—
100C4×D283—44no—
101C4×D283C2×D412yes—
102C4×D283#4112yes—
103C283⋊C8—21no—
104C283⋊C8C2×D412yes—
105C23×C283C2×D411yescharacteristic
106Q8×C283—22no—
107Q8×C283—22no—
108C2×C4×C283—21no—
109C2×C4×C283—21no—
110C2×C4×C283—22no—
111C2×C4×C283—22no—
112C2×C4×C283C2×D411yescharacteristic
113C2×C4×C283C2411yescharacteristic
114C8×C283—21no—
115C8×C283C2×D412yes—
116C22×C283—21no—
117C22×C283—21no—
118C22×C283#13512yescomplemented
119C22×C283C22×D411yescharacteristic
120C2×D283—28no—
121C2×D283—21no—
122C2×D283—21no—
123C2×D283—21no—
124C2×D283—21no—
125C2×D283—42no—
126C2×D283—42no—
127C2×D283—42no—
128C2×D283#12112yes—
129C2×D283#13512yescomplemented
130C4×C283—21no—
131C4×C283—42no—
132C4×C283—42no—
133C4×C283C22×D411yescharacteristic
134C4×C283C2×#4111yescharacteristic
135Dic283—21no—
136Dic283—42no—
137Dic283#12112yes—
138C2×C283—22no—
139C2×C283C2×#12111yescharacteristic
140C2×C283C2×#13511yescharacteristic
141C2×C283C2×#13511yescharacteristic
142D283—22no—
143D283—22no—
144D283—82no—
145C283#305011yescharacteristic, complemented, Sylow
146#3050—2831nomaximal, Sylow
147C2×#135—2831no—
148C23×C8—2831no—
149#525—2832no—
150#467—2831no—
151#467D28311yescharacteristic, complemented
152#466—2831no—
153#466—2831no—
154#465—2832no—
155#465—2832no—
156#427—2831no—
157#416—2832no—
158C2×#41—2832no—
159C23×C4—2831no—
160#135—5664no—
161C2×Q16C2×D28311yescharacteristic, complemented
162C2×SD16—2832no—
163C2×D8—2831no—
164C22×C8—2831no—
165C22×C8—2831no—
166C22×C8—2831no—
167C22×C8—2831no—
168C22×C8—2831no—
169C22×C8—5664no—
170C22×C8C2×D28311yescharacteristic
171#123—5662no—
172C22⋊Q8—2832no—
173C22⋊Q8C2×D28312yes—
174#121—2832no—
175#121—2832no—
176#120—5662no—
177C4×D4—5662no—
178C4×D4—5662no—
179#117—2832no—
180#107—2831no—
181#107C2×D28311yescharacteristic, complemented
182#106—2832no—
183#103—2832no—
184#103C2×D28312yescomplemented
185#102—2832no—
186#102—2832no—
187C24—2831no—
188#41—5664no—
189#41—5664no—
190C2×Q8C22×D28312yes—
191C2×D4—2832no—
192C2×D4—5662no—
193C2×D4—5662no—
194C2×D4—5662no—
195C22×C4—2831no—
196C22×C4—2831no—
197C22×C4—2831no—
198C22×C4—2831no—
199C22×C4—2831no—
200C22×C4—5662no—
201C22×C4—5662no—
202C22×C4—5662no—
203C22×C4C22×D28311yescharacteristic
204Q16—22no—
205SD16—5664no—
206D8—5662no—
207C2×C8—22no—
208C2×C8—2831no—
209C2×C8—2834no—
210C2×C8—2831no—
211C2×C8—5662no—
212C2×C8—5662no—
213C2×C8—5662no—
214C2×C8—5662no—
215C2×C8C22×D28311yescharacteristic
216C2×C8C22×D28311yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—2832no—
219C4⋊C4—5662no—
220C4⋊C4C22×D28312yes—
221C22⋊C4—22no—
222C22⋊C4—5662no—
223C22⋊C4—5662no—
224C22⋊C4—5662no—
225C22⋊C4—5662no—
226C42—5662no—
227C23—2831no—
228C23—2831no—
229C23—5662no—
230C23—5662no—
231C23—5662no—
232C23—5662no—
233C23D4×D28311yescharacteristic
234Q8—22no—
235Q8—22no—
236D4—5662no—
237D4—5662no—
238D4—11324no—
239D4—11324no—
240D4—11324no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—2831no—
246C2×C4—2832no—
247C2×C4—2832no—
248C2×C4—2831no—
249C2×C4—5662no—
250C2×C4—5662no—
251C2×C4—5662no—
252C2×C4—5661no—
253C2×C4—5661no—
254C2×C4—5662no—
255C2×C4—11324no—
256C2×C4D4×D28311yescharacteristic
257C2×C4C23×D28311yescharacteristic
258C8—21no—
259C8—2832no—
260C8—5661no—
261C8D4×D28312yes—
262C22—21no—
263C22—21no—
264C22—2832no—
265C22—2832no—
266C22—5668no—
267C22—5661no—
268C22—5661no—
269C22—5661no—
270C22—5661no—
271C22—11322no—
272C22—11322no—
273C22—11322no—
274C22#2258363312yescomplemented
275C22C2×D4×D28311yescharacteristic
276C4—21no—
277C4—42no—
278C4—42no—
279C4—2832no—
280C4—5661no—
281C4—11322no—
282C4C2×D4×D28311yescharacteristic
283C4#41×D28311yescharacteristic
284C2—22no—
285C2—5662no—
286C2—5662no—
287C2—22642no—
288C2#121×D28311yescharacteristic
289C2#135×D28311yescharacteristic
290C2C2×#2258363311yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#467×C283C2
socleC22×C283C22×D4
hypercenter#467D283
hypocenterC283#3050
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#22583633C2semidirect
2. C22×C8×D283C2semidirect
3. #31826978C2semidirect
4. #31826703C2semidirect
5. #31826702C2semidirect
6. #31826701C2semidirect
7. #31826697C2semidirect
8. #31826696C2semidirect
9. #31826502C2semidirect
10. #31826460C2semidirect
11. #467×C283C2semidirect
15. C2×#17985379C22semidirect
16. C2×D2264C22semidirect
31. #22583544C22semidirect
32. #22583541C22semidirect
33. #22583534C22semidirect
34. #22583532C22semidirect
35. #22583530C22semidirect
36. C2×Q16×C283C22semidirect
39. #107×C283C22semidirect
40. #103×C283C22semidirect
64. C8×D283D4semidirect
118. C22×C283#135semidirect
129. C2×D283#135semidirect
145. C283#3050semidirect
151. #467D283semidirect
161. C2×Q16C2×D283semidirect
181. #107C2×D283semidirect
184. #103C2×D283semidirect
274. C22#22583633semidirect

Maximal subgroups

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

subgroupindex
C2×#225836332
C22×C8×D2832
#318269782
#318267032
#318267022
#318267012
#318266972
#318266962
#318265022
#318264602
#467×C2832
#3050283