Finite Group Databasebeta

← #49124187

Subgroups — #49124187

Order 36224.

660 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D1132C211yesmaximal, characteristic
2C2×#22583568C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D283C211yesmaximal, characteristic, complemented
4#31826354C212yesmaximal, complemented
5#31826348C212yesmaximal, complemented
6#31826343C212yesmaximal
7#31826322C211yesmaximal, characteristic, complemented
8#31826322C211yesmaximal, characteristic, complemented
9C4×C4⋊Dic283C211yesmaximal, characteristic, complemented
10#31826312C212yesmaximal, complemented
11#382×C283C211yesmaximal, characteristic, complemented
12C22×D1132C2211yescharacteristic
13C22×C4×D283C2212yes—
14C22×C4×D283C2211yescharacteristic
15C22×C4×D283C2211yescharacteristic
16C4×D1132C418yescomplemented
17#22583568—24no—
18C2×C4⋊Dic283C2211yescharacteristic
19C2×C4⋊Dic283C2211yescharacteristic
20C2×C4⋊Dic283C2211yescharacteristic
21C2×Dic283⋊C4C2212yes—
22C4⋊C4×D283—24no—
23C2×(C2×D283)⋊C4C2212yes—
24C2×(C2×D283)⋊C4C2212yes—
25C2×(C2×D283)⋊C4C2212yes—
26C2×(C2×D283)⋊C4C2211yescharacteristic
27C2×(C2×D283)⋊C4C2211yescharacteristic
28C2×C4×Dic283C2212yes—
29C2×C4×Dic283C2212yes—
30C283⋊#95C2212yes—
31C283⋊#95C2212yes—
32C283⋊#95C2212yes—
33C2×C4⋊C4×C283C2211yescharacteristic
34C2×C4⋊C4×C283C2211yescharacteristic
35C2×C42×C283C2211yescharacteristic
36#95×C283C2212yes—
37#95×C283C2212yes—
38C23×D283C2311yescharacteristic
39C23×D283C2311yescharacteristic
40C2×D1132—22no—
41C2×D1132—22no—
42C2×D1132C2×C414yes—
43C22×Dic283C2312yes—
44C22×Dic283C2311yescharacteristic
45C22×Dic283C2311yescharacteristic
46C22×Dic283C2312yes—
47C2×C4×D283—24no—
48C2×C4×D283—24no—
49C2×C4×D283—22no—
50C2×C4×D283—22no—
51C2×C4×D283—22no—
52C2×C4×D283—22no—
53C2×C4×D283—42no—
54C2×C4×D283C2×C414yes—
55C2×C4×D283C2×C414yescomplemented
56C4⋊Dic283—22no—
57C4⋊Dic283C2×C414yescomplemented
58C4⋊Dic283D414yescomplemented
59Dic283⋊C4—24no—
60(C2×D283)⋊C4—24no—
61(C2×D283)⋊C4—24no—
62(C2×D283)⋊C4—42no—
63(C2×D283)⋊C4C2×C414yes—
64(C2×D283)⋊C4C2×C414yescomplemented
65C4×Dic283—24no—
66C4×Dic283—42no—
67C22×C4×C283C2312yes—
68C22×C4×C283C2312yes—
69C22×C4×C283C2311yescharacteristic
70C22×C4×C283C2311yescharacteristic
71C22×C4×C283C2311yescharacteristic
72C4⋊C4×C283—22no—
73C4⋊C4×C283—22no—
74C42×C283C2×C414yescomplemented
75C22×D283—21no—
76C22×D283—21no—
77C22×D283—21no—
78C22×D283—21no—
79C22×D283—21no—
80C22×D283—21no—
81C22×D283—21no—
82C22×D283—21no—
83C22×D283—21no—
84C22×D283—21no—
85C22×D283C22×C412yes—
86C22×D283C22×C412yes—
87C22×D283#4112yes—
88C22×D283#4112yes—
89D1132—28no—
90C2×Dic283—21no—
91C2×Dic283—21no—
92C2×Dic283—22no—
93C2×Dic283—21no—
94C2×Dic283—21no—
95C2×Dic283—22no—
96C2×Dic283—22no—
97C2×Dic283—22no—
98C2×Dic283—22no—
99C2×Dic283C22×C412yes—
100C2×Dic283C22×C412yes—
101C2×Dic283C2×D414yes—
102C4×D283—28no—
103C4×D283—28no—
104C4×D283—48no—
105C23×C283C2411yescharacteristic
106C2×C4×C283—21no—
107C2×C4×C283—21no—
108C2×C4×C283—21no—
109C2×C4×C283—21no—
110C2×C4×C283—22no—
111C2×C4×C283—22no—
112C2×C4×C283—22no—
113C2×C4×C283—22no—
114C2×C4×C283—22no—
115C2×C4×C283—22no—
116C2×C4×C283C22×C412yes—
117C2×C4×C283C22×C412yes—
118C2×C4×C283C22×C412yes—
119C2×C4×C283C2×D412yes—
120C2×C4×C283#4112yes—
121C22×C283C23×C411yescharacteristic
122C22×C283C22×D411yescharacteristic
123C22×C283C2×#4111yescharacteristic
124C22×C283C2×#4111yescharacteristic
125C22×C283C2×#4111yescharacteristic
126C22×C283H(5,𝔽2)11yescharacteristic
127C22×C2832−1+411yescharacteristic
128C2×D283—22no—
129C2×D283—22no—
130C2×D283—22no—
131C2×D283—22no—
132C2×D283—22no—
133C2×D283—22no—
134C2×D283—22no—
135C2×D283—22no—
136C2×D283—22no—
137C2×D283—41no—
138C2×D283—41no—
139C2×D283—41no—
140C2×D283—41no—
141C2×D283#11714yescomplemented
142C4×C283—22no—
143C4×C283—22no—
144C4×C283—42no—
145C4×C283—42no—
146C4×C283C4×D414yes—
147Dic283—22no—
148Dic283—22no—
149Dic283—24no—
150Dic283—42no—
151C2×C283C2×#11711yescharacteristic
152C2×C283C2×C4×D411yescharacteristic
153C2×C283#52011yescharacteristic
154C2×C283#54611yescharacteristic
155C2×C283#54811yescharacteristic
156C2×C283#55211yescharacteristic
157C2×C283#55311yescharacteristic
158D283—24no—
159D283—42no—
160C283#235211yescharacteristic, complemented, Sylow
161#2352—2831nomaximal, Sylow
162C2×C4×D4—2831no—
163C2×#117—2831no—
164C22×C4⋊C4—2831no—
165#388—2832no—
166#387—2832no—
167#386—2832no—
168#382D28311yescharacteristic, complemented
169#380—2831no—
170#380—2831no—
171C4×C4⋊C4—2831no—
172C4×C22⋊C4—2832no—
173C22×D4—2831no—
174C23×C4—2831no—
175C23×C4—2831no—
176C23×C4—2832no—
177C4×D4—2838no—
178#117—5664no—
179C2×C4⋊C4—2831no—
180C2×C4⋊C4—2831no—
181C2×C4⋊C4—2832no—
182C2×C4⋊C4—2831no—
183C2×C4⋊C4—5664no—
184C2×C4⋊C4C2×D28311yescharacteristic
185C2×C4⋊C4C2×D28311yescharacteristic
186C2×C22⋊C4—2831no—
187C2×C22⋊C4—2831no—
188C2×C22⋊C4—2832no—
189C2×C22⋊C4—2832no—
190C2×C22⋊C4—2832no—
191C2×C42—2832no—
192C2×C42—2832no—
193C2×C42C2×D28311yescharacteristic
194#95—2832no—
195#95—2832no—
196#95—2832no—
197#95C2×D28312yes—
198#95C2×D28312yes—
199C24—2831no—
200C24—2831no—
201C2×D4—2834no—
202C2×D4—5662no—
203C2×D4—5662no—
204C22×C4—2831no—
205C22×C4—2831no—
206C22×C4—2832no—
207C22×C4—2834no—
208C22×C4—2834no—
209C22×C4—2832no—
210C22×C4—5664no—
211C22×C4—5664no—
212C22×C4—5662no—
213C22×C4—5662no—
214C22×C4—5662no—
215C22×C4—5662no—
216C22×C4—11322no—
217C22×C4C22×D28312yes—
218C22×C4C22×D28312yes—
219C22×C4C22×D28311yescharacteristic
220C22×C4C22×D28311yescharacteristic
221C22×C4C22×D28311yescharacteristic
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—2834no—
225C4⋊C4—2834no—
226C4⋊C4—5664no—
227C4⋊C4—5662no—
228C22⋊C4—2834no—
229C22⋊C4—2834no—
230C22⋊C4—5664no—
231C22⋊C4—5664no—
232C22⋊C4—11322no—
233C42—5664no—
234C42—11322no—
235C42C4×D28314yescomplemented
236C23—2832no—
237C23—2832no—
238C23—2832no—
239C23—2832no—
240C23—5661no—
241C23—5661no—
242C23—5661no—
243C23—5661no—
244C23—5661no—
245C23—5661no—
246C23—5661no—
247C23—5661no—
248C23—5661no—
249C23—5661no—
250C23C23×D28311yescharacteristic
251D4—5668no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—2832no—
263C2×C4—2832no—
264C2×C4—2834no—
265C2×C4—5661no—
266C2×C4—5661no—
267C2×C4—5662no—
268C2×C4—5661no—
269C2×C4—5661no—
270C2×C4—5662no—
271C2×C4—5668no—
272C2×C4—5668no—
273C2×C4—5662no—
274C2×C4—5662no—
275C2×C4—5662no—
276C2×C4—11328no—
277C2×C4C2×C4×D28312yes—
278C2×C4C2×C4×D28312yes—
279C2×C4C2×C4×D28312yes—
280C2×C4D4×D28312yes—
281C2×C4#1798539312yes—
282C22—2834no—
283C22—5662no—
284C22—5662no—
285C22—5662no—
286C22—5662no—
287C22—5662no—
288C22—5662no—
289C22—5662no—
290C22—5662no—
291C22—5662no—
292C22—11321no—
293C22—11321no—
294C22—11321no—
295C22—11321no—
296C22C22×C4×D28311yescharacteristic
297C22C2×D4×D28311yescharacteristic
298C22#41×D28311yescharacteristic
299C22#41×D28311yescharacteristic
300C22C2×#1798539311yescharacteristic
301C22#2258365611yescharacteristic
302C22#2258365711yescharacteristic
303C4—22no—
304C4—22no—
305C4—42no—
306C4—42no—
307C4—5664no—
308C4—5662no—
309C4—5662no—
310C4—11322no—
311C4#2258357214yes—
312C2—5664no—
313C2—11322no—
314C2#117×D28311yescharacteristic
315C2C2×#2258357211yescharacteristic
316C2#3182694511yescharacteristic
317C2#3182710811yescharacteristic
318C2#3182712311yescharacteristic
319C2#3182715311yescharacteristic
320C2#3182716811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#382×C283C2
socle, cosocleC23×C283C24
prime radicalC22×D1132C22
hypercenter#382D283
hypocenterC283#2352
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×#22583568C2semidirect
3. C2×C4⋊C4×D283C2semidirect
4. #31826354C2semidirect
5. #31826348C2semidirect
7. #31826322C2semidirect
8. #31826322C2semidirect
9. C4×C4⋊Dic283C2semidirect
10. #31826312C2semidirect
11. #382×C283C2semidirect
16. C4×D1132C4semidirect
55. C2×C4×D283C2×C4semidirect
57. C4⋊Dic283C2×C4semidirect
58. C4⋊Dic283D4semidirect
64. (C2×D283)⋊C4C2×C4semidirect
74. C42×C283C2×C4semidirect
141. C2×D283#117semidirect
160. C283#2352semidirect
168. #382D283semidirect
235. C42C4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D11322
C2×#225835682
C2×C4⋊C4×D2832
#318263542
#318263482
#318263432
#318263222
C4×C4⋊Dic2832
#318263122
#382×C2832
#2352283