Finite Group Databasebeta

← #49131180

Subgroups — #49131180

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#31826975C211yesmaximal, characteristic
2C2×C4⋊C4×D283C211yesmaximal, characteristic, complemented
3#31826802C212yesmaximal, complemented
4#31826796C211yesmaximal, characteristic, complemented
5#31826795C211yesmaximal, characteristic, complemented
6#31826782C212yesmaximal, complemented
7#31826776C211yesmaximal, characteristic, complemented
8#31826506C212yesmaximal, complemented
9#31826463C212yesmaximal, complemented
10#31826427C211yesmaximal, characteristic, complemented
11#480×C283C211yesmaximal, characteristic, complemented
12C2×#17985391C2211yescharacteristic
13C22×C4×D283—21no—
14C22×C4×D283C2211yescharacteristic
15C2×#17985375C2212yes—
16#22583592—21no—
17#22583590C2211yescharacteristic
18#22583589C2211yescharacteristic
19#22583581C2211yescharacteristic
20#22583578—21no—
21D4×Dic283—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#22583535C2212yescomplemented
34#22583533C2212yescomplemented
35#22583531C2212yescomplemented
36C22⋊(C283⋊C8)C2211yescharacteristic
37(C2×D283)⋊C8C2212yes—
38#121×C283C2211yescharacteristic
39C2×C4⋊C4×C283C2211yescharacteristic
40#107×C283C2212yescomplemented
41#102×C283C2212yescomplemented
42C22⋊C8×C283C2211yescharacteristic
43C23×D283D411yescharacteristic
44#17985391—24no—
45C2×Dic566C2311yescharacteristic
46C2×C22⋊D283—21no—
47C2×C22⋊D283—21no—
48C22×Dic283—21no—
49C22×Dic283—21no—
50C22×Dic283D411yescharacteristic
51C2×C4×D283—24no—
52C2×C4×D283—22no—
53C2×C4×D283—22no—
54C2×C4×D283—44no—
55C2×C4×D283D411yescharacteristic
56C2×C4×D283D411yescharacteristic
57C2×C4×D283C2311yescharacteristic
58C2×C4×D283C2311yescharacteristic
59#17985375—24no—
60C2×C283⋊C8C2312yes—
61C4⋊Dic283—21no—
62C4⋊Dic283—21no—
63C4⋊Dic283C2312yes—
64C4⋊Dic283C2311yescharacteristic
65Dic283⋊C4—22no—
66Dic283⋊C4—22no—
67Dic283⋊C4—21no—
68C22⋊Dic283—21no—
69C22⋊Dic283—21no—
70(C2×D283)⋊C4—21no—
71(C2×D283)⋊C4—21no—
72C4×Dic283—21no—
73C2×D4×C283—21no—
74C2×D4×C283C2311yescharacteristic
75C22×C4×C283—21no—
76C22×C4×C283C2311yescharacteristic
77C2×C8×C283C2312yes—
78C4⋊C4×C283—21no—
79C4⋊C4×C283C2312yes—
80C4⋊C4×C283C2311yescharacteristic
81C22⋊C4×C283—21no—
82C22×D283—22no—
83C22×D283—22no—
84C22×D283—22no—
85C22×D283C2×D411yescharacteristic
86C22×D283C2×D411yescharacteristic
87Dic566—22no—
88C22⋊D283—42no—
89C22⋊D283—42no—
90C2×Dic283—22no—
91C2×Dic283—21no—
92C2×Dic283—21no—
93C2×Dic283—22no—
94C2×Dic283—42no—
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#4114yes—
104C283⋊C8—22no—
105C23×C283—21no—
106C23×C283C2×D411yescharacteristic
107D4×C283—22no—
108D4×C283—42no—
109C2×C4×C283—22no—
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—41no—
119C22×C283—41no—
120C22×C283—41no—
121C22×C283AGL(1,ℤ/8)12yes—
122C22×C283C22×D411yescharacteristic
123C2×D283—28no—
124C2×D283—21no—
125C2×D283—21no—
126C2×D283—21no—
127C2×D283—21no—
128C2×D283#12312yes—
129C2×D283#13712yes—
130C4×C283—21no—
131C4×C283—42no—
132C4×C283—41no—
133C4×C283C2×#4112yes—
134Dic283—21no—
135Dic283—41no—
136Dic283—42no—
137Dic283—41no—
138Dic283#12312yes—
139C2×C283—22no—
140C2×C283—81no—
141C2×C283C2×#12311yescharacteristic
142C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C283C2×#13711yescharacteristic
144D283—22no—
145D283—22no—
146C283#309211yescharacteristic, complemented, Sylow
147#3092—2831nomaximal, Sylow
148#525—2831no—
149C22×C4⋊C4—2831no—
150#484—2832no—
151#483—2831no—
152#483—2831no—
153#481—2832no—
154#480—2831no—
155#480D28311yescharacteristic, complemented
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⋊Q8—2831no—
166#121—2831no—
167#121C2×D28311yescharacteristic
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#103—2832no—
185#102—2832no—
186#102C2×D28312yescomplemented
187C22⋊C8—2831no—
188C22⋊C8—2832no—
189C22⋊C8C2×D28311yescharacteristic
190C24—2831no—
191#41—5664no—
192C2×Q8—2831no—
193C2×D4—21no—
194C2×D4—5661no—
195C2×D4—5661no—
196C2×D4C22×D28311yescharacteristic
197C22×C4—21no—
198C22×C4—2831no—
199C22×C4—2831no—
200C22×C4—2831no—
201C22×C4—2831no—
202C22×C4—2831no—
203C22×C4—5664no—
204C22×C4—5662no—
205C22×C4—5662no—
206C22×C4—5661no—
207C22×C4—5661no—
208C22×C4—11324no—
209C22×C4C22×D28311yescharacteristic
210M16—5664no—
211C2×C8—2832no—
212C2×C8C22×D28312yes—
213C4⋊C4—21no—
214C4⋊C4—2832no—
215C4⋊C4—2831no—
216C4⋊C4—5662no—
217C4⋊C4—5662no—
218C4⋊C4—5661no—
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—21no—
230C23—2831no—
231C23—2831no—
232C23—5662no—
233C23—5662no—
234C23—5662no—
235C23D4×D28311yescharacteristic
236Q8—5662no—
237D4—22no—
238D4—42no—
239D4—11322no—
240D4—11322no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—2831no—
246C2×C4—2834no—
247C2×C4—2831no—
248C2×C4—5662no—
249C2×C4—5662no—
250C2×C4—5662no—
251C2×C4—5662no—
252C2×C4—5661no—
253C2×C4—5661no—
254C2×C4—5662no—
255C2×C4—11324no—
256C2×C4—11324no—
257C2×C4—11322no—
258C2×C4—11322no—
259C2×C4D4×D28311yescharacteristic
260C2×C4C23×D28311yescharacteristic
261C8—22no—
262C8—5662no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—2832no—
269C22—2832no—
270C22—5668no—
271C22—5661no—
272C22—5661no—
273C22—5661no—
274C22—5661no—
275C22#2258363412yes—
276C22C2×D4×D28311yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—2832no—
281C4—5661no—
282C4—11321no—
283C4—11322no—
284C4—11321no—
285C4#41×D28312yes—
286C2—22no—
287C2—81no—
288C2—5662no—
289C2—5662no—
290C2#123×D28311yescharacteristic
291C2C2×#2258363411yescharacteristic
292C2#137×D28311yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#480×C283C2
socleC22×C283C22×D4
prime radical#31826975C2
hypercenter#480D283
hypocenterC283#3092
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D283C2semidirect
3. #31826802C2semidirect
4. #31826796C2semidirect
5. #31826795C2semidirect
6. #31826782C2semidirect
7. #31826776C2semidirect
8. #31826506C2semidirect
9. #31826463C2semidirect
10. #31826427C2semidirect
11. #480×C283C2semidirect
30. #22583543C22semidirect
31. #22583542C22semidirect
32. #22583541C22semidirect
33. #22583535C22semidirect
34. #22583533C22semidirect
35. #22583531C22semidirect
40. #107×C283C22semidirect
41. #102×C283C22semidirect
146. C283#3092semidirect
155. #480D283semidirect
180. #107C2×D283semidirect
186. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318269752
C2×C4⋊C4×D2832
#318268022
#318267962
#318267952
#318267822
#318267762
#318265062
#318264632
#318264272
#480×C2832
#3092283