Finite Group Databasebeta

← #49131188

Subgroups — #49131188

Order 36224.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#31826978C211yesmaximal, characteristic
2C2×C4⋊C4×D283C211yesmaximal, characteristic, complemented
3#31826798C211yesmaximal, characteristic, complemented
4#31826794C212yesmaximal, complemented
5#31826784C211yesmaximal, characteristic, complemented
6#31826783C211yesmaximal, characteristic, complemented
7#31826780C212yesmaximal, complemented
8#31826506C212yesmaximal, complemented
9#31826465C212yesmaximal, complemented
10#31826427C211yesmaximal, characteristic, complemented
11#484×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2211yescharacteristic
13C22×C4×D283—21no—
14C22×C4×D283C2211yescharacteristic
15C2×#17985375C2212yes—
16#22583595—21no—
17#22583586C2211yescharacteristic
18#22583585C2211yescharacteristic
19#22583584C2211yescharacteristic
20#22583580—21no—
21#22583572—21no—
22(C22⋊D283)⋊C4—21no—
23#22583568C2211yescharacteristic
24C2×C4⋊Dic283C2211yescharacteristic
25C2×Dic283⋊C4—21no—
26C4⋊C4×D283—21no—
27C4⋊C4×D283—21no—
28C4⋊C4×D283C2212yes—
29C4⋊C4×D283C2212yes—
30#22583543C2212yescomplemented
31#22583542C2212yescomplemented
32#22583541C2212yescomplemented
33#22583534C2212yescomplemented
34#22583532C2212yescomplemented
35#22583530C2212yescomplemented
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
44#17985393—24no—
45C2×D1132—21no—
46C2×D1132—21no—
47C2×D1132C2311yescharacteristic
48C2×C22⋊D283—21no—
49C22×Dic283—21no—
50C22×Dic283D411yescharacteristic
51C2×C4×D283—24no—
52C2×C4×D283—22no—
53C2×C4×D283—22no—
54C2×C4×D283—21no—
55C2×C4×D283—44no—
56C2×C4×D283D411yescharacteristic
57C2×C4×D283D411yescharacteristic
58C2×C4×D283C2311yescharacteristic
59C2×C4×D283C2311yescharacteristic
60#17985375—24no—
61C2×C283⋊C8C2312yes—
62C4⋊Dic283—21no—
63C4⋊Dic283C2312yes—
64C4⋊Dic283C2311yescharacteristic
65Dic283⋊C4—22no—
66Dic283⋊C4—22no—
67Dic283⋊C4—21no—
68(C2×D283)⋊C4—21no—
69(C2×D283)⋊C4—21no—
70(C2×D283)⋊C4—21no—
71(C2×D283)⋊C4—21no—
72C4×Dic283—21no—
73C2×Q8×C283C2311yescharacteristic
74C22×C4×C283—21no—
75C22×C4×C283C2311yescharacteristic
76C2×C8×C283C2312yes—
77C4⋊C4×C283—21no—
78C4⋊C4×C283—21no—
79C4⋊C4×C283C2312yes—
80C4⋊C4×C283C2311yescharacteristic
81C22⋊C4×C283—21no—
82C22×D283—22no—
83C22×D283—22no—
84C22×D283—22no—
85C22×D283—21no—
86C22×D283C2×D411yescharacteristic
87C22×D283C2×D411yescharacteristic
88D1132—22no—
89D1132—42no—
90D1132—42no—
91C22⋊D283—42no—
92C2×Dic283—22no—
93C2×Dic283—21no—
94C2×Dic283—22no—
95C2×Dic283—42no—
96C2×Dic283C2×D411yescharacteristic
97C2×Dic283C2×D411yescharacteristic
98C4×D283—22no—
99C4×D283—22no—
100C4×D283—22no—
101C4×D283—44no—
102C4×D283—44no—
103C4×D283—42no—
104C4×D283#4114yes—
105C283⋊C8—22no—
106C23×C283C2×D411yescharacteristic
107Q8×C283—22no—
108C2×C4×C283—22no—
109C2×C4×C283—21no—
110C2×C4×C283—21no—
111C2×C4×C283—22no—
112C2×C4×C283—42no—
113C2×C4×C283C2×D411yescharacteristic
114C2×C4×C283C2411yescharacteristic
115C8×C283—22no—
116C22×C283—21no—
117C22×C283—21no—
118C22×C283#13712yes—
119C22×C283C22×D411yescharacteristic
120C2×D283—28no—
121C2×D283—21no—
122C2×D283—21no—
123C2×D283—21no—
124C2×D283—21no—
125C2×D283—41no—
126C2×D283—41no—
127C2×D283—41no—
128C2×D283#12312yes—
129C2×D283AGL(1,ℤ/8)12yes—
130C4×C283—21no—
131C4×C283—41no—
132C4×C283—42no—
133C4×C283—41no—
134C4×C283C2×#4112yes—
135Dic283—21no—
136Dic283—42no—
137Dic283—41no—
138Dic283#12312yes—
139C2×C283—22no—
140C2×C283C2×#12311yescharacteristic
141C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C283C2×#13711yescharacteristic
143D283—22no—
144D283—22no—
145D283—81no—
146C283#309211yescharacteristic, complemented, Sylow
147#3092—2831nomaximal, Sylow
148#525—2831no—
149C22×C4⋊C4—2831no—
150#484—2831no—
151#484D28311yescharacteristic, complemented
152#483—2832no—
153#481—2831no—
154#481—2831no—
155#480—2832no—
156#428—2832no—
157#417—2832no—
158#407—2831no—
159C2×#41—2831no—
160C23×C4—2831no—
161C23×C4—5661no—
162C2×M16—2832no—
163#123—5661no—
164C22⋊Q8—2831no—
165C22⋊Q8C2×D28311yescharacteristic
166#121—2831no—
167#121—2831no—
168#120—5661no—
169C4×D4—5661no—
170C4×D4—5661no—
171#117—2831no—
172C2×C4⋊C4—2831no—
173C2×C4⋊C4—2832no—
174C2×C4⋊C4—2832no—
175C2×C4⋊C4—5661no—
176C2×C4⋊C4—5661no—
177C2×C4⋊C4—5661no—
178C2×C4⋊C4C2×D28311yescharacteristic
179#107—2832no—
180#107C2×D28312yescomplemented
181#106—2832no—
182#106—2832no—
183#103—2832no—
184#103C2×D28312yescomplemented
185#102—2832no—
186#102—2832no—
187C22⋊C8—2831no—
188C22⋊C8—2832no—
189C22⋊C8C2×D28311yescharacteristic
190C24—2831no—
191#41—5664no—
192C2×Q8C22×D28311yescharacteristic
193C2×D4—2831no—
194C2×D4—5661no—
195C2×D4—5661no—
196C2×D4—5661no—
197C22×C4—21no—
198C22×C4—2831no—
199C22×C4—2831no—
200C22×C4—2831no—
201C22×C4—2831no—
202C22×C4—2831no—
203C22×C4—5664no—
204C22×C4—5661no—
205C22×C4—5662no—
206C22×C4—5662no—
207C22×C4—5661no—
208C22×C4—11324no—
209C22×C4C22×D28311yescharacteristic
210M16—5664no—
211C2×C8—2832no—
212C2×C8C22×D28312yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—2832no—
216C4⋊C4—2831no—
217C4⋊C4—5662no—
218C4⋊C4—5662no—
219C4⋊C4—5661no—
220C4⋊C4—5661no—
221C4⋊C4C22×D28312yes—
222C4⋊C4C22×D28311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—5661no—
225C22⋊C4—5661no—
226C22⋊C4—5661no—
227C22⋊C4—5661no—
228C42—5661no—
229C23—2831no—
230C23—2831no—
231C23—5662no—
232C23—5662no—
233C23—5662no—
234C23—5661no—
235C23D4×D28311yescharacteristic
236Q8—22no—
237D4—5662no—
238D4—11322no—
239D4—11322no—
240D4—11322no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—2831no—
247C2×C4—2834no—
248C2×C4—2831no—
249C2×C4—5662no—
250C2×C4—5662no—
251C2×C4—5662no—
252C2×C4—5662no—
253C2×C4—5661no—
254C2×C4—5662no—
255C2×C4—11322no—
256C2×C4—11324no—
257C2×C4—11324no—
258C2×C4—11322no—
259C2×C4D4×D28311yescharacteristic
260C2×C4C23×D28311yescharacteristic
261C8—22no—
262C8—5662no—
263C22—21no—
264C22—21no—
265C22—2832no—
266C22—2832no—
267C22—5668no—
268C22—5661no—
269C22—5661no—
270C22—5661no—
271C22—5661no—
272C22—11321no—
273C22—11321no—
274C22—11321no—
275C22#2258364212yes—
276C22C2×D4×D28311yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—2832no—
282C4—5661no—
283C4—11322no—
284C4—11321no—
285C4#41×D28312yes—
286C2—22no—
287C2—5662no—
288C2—5662no—
289C2—22641no—
290C2#123×D28311yescharacteristic
291C2AGL(1,ℤ/8)×D28311yescharacteristic
292C2C2×#2258364211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#484×C283C2
socleC22×C283C22×D4
prime radical#31826978C2
hypercenter#484D283
hypocenterC283#3092
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
2. C2×C4⋊C4×D283C2semidirect
3. #31826798C2semidirect
4. #31826794C2semidirect
5. #31826784C2semidirect
6. #31826783C2semidirect
7. #31826780C2semidirect
8. #31826506C2semidirect
9. #31826465C2semidirect
10. #31826427C2semidirect
11. #484×C283C2semidirect
30. #22583543C22semidirect
31. #22583542C22semidirect
32. #22583541C22semidirect
33. #22583534C22semidirect
34. #22583532C22semidirect
35. #22583530C22semidirect
40. #107×C283C22semidirect
41. #103×C283C22semidirect
146. C283#3092semidirect
151. #484D283semidirect
180. #107C2×D283semidirect
184. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318269782
C2×C4⋊C4×D2832
#318267982
#318267942
#318267842
#318267832
#318267802
#318265062
#318264652
#318264272
#484×C2832
#3092283