Finite Group Databasebeta

← #49119779

Subgroups — #49119779

Order 36224.

294 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C22⋊Q8×D283C211yesmaximal, characteristic
2#31826427C211yesmaximal, characteristic
3#31826312C211yesmaximal, characteristic
4#31826203C211yesmaximal, characteristic, complemented
5#31826202C211yesmaximal, characteristic, complemented
6#31826201C211yesmaximal, characteristic, complemented
7#328×C283C211yesmaximal, characteristic, complemented
8C2×Q8×D283C411yescharacteristic, complemented
9C22×C4×D283C2211yescharacteristic
10C2×#17985375—21no—
11#22583590C411yescharacteristic, complemented
12#22583589C2211yescharacteristic
13#22583588—21no—
14#22583587—21no—
15#22583586C411yescharacteristic, complemented
16C4⋊C4×D283—21no—
17C4⋊C4×D283C411yescharacteristic, complemented
18C2×(C2×D283)⋊C4—21no—
19C22⋊C4×D283—21no—
20C2×C4×Dic283—21no—
21C22⋊(C283⋊C8)C2211yescharacteristic
22(C2×D283)⋊C8—21no—
23C283⋊#95C2211yescharacteristic
24C22⋊Q8×C283C2211yescharacteristic
25C22⋊C8×C283C2211yescharacteristic
26#95×C283C2211yescharacteristic
27C23×D283D411yescharacteristic
28C2×Dic566—21no—
29C2×Dic566C2×C411yescharacteristic
30Q8×D283—42no—
31C22×Dic283—21no—
32C22×Dic283D411yescharacteristic
33C2×C4×D283—21no—
34C2×C4×D283—21no—
35C2×C4×D283—42no—
36C2×C4×D283C2×C411yescharacteristic
37C2×C4×D283C2×C411yescharacteristic
38C2×C4×D283D411yescharacteristic
39C2×C4×D283D411yescharacteristic
40#17985375—42no—
41C2×C283⋊C8—21no—
42C4⋊Dic283—21no—
43C4⋊Dic283C2×C411yescharacteristic
44Dic283⋊C4—21no—
45Dic283⋊C4—21no—
46Dic283⋊C4—21no—
47C22⋊Dic283—21no—
48(C2×D283)⋊C4—24no—
49(C2×D283)⋊C4—21no—
50(C2×D283)⋊C4—21no—
51C4×Dic283—42no—
52C2×Q8×C283C2×C411yescharacteristic
53C22×C4×C283—21no—
54C22×C4×C283C2311yescharacteristic
55C2×C8×C283—21no—
56C4⋊C4×C283—21no—
57C4⋊C4×C283C2×C411yescharacteristic
58C22⋊C4×C283—21no—
59C22×D283—21no—
60C22×D283—21no—
61C22×D283—22no—
62C22×D283—41no—
63C22×D283C22⋊C411yescharacteristic
64C22×D283C22⋊C411yescharacteristic
65Dic566—42no—
66Dic566—41no—
67C2×Dic283—22no—
68C2×Dic283—22no—
69C2×Dic283—21no—
70C2×Dic283—21no—
71C2×Dic283—41no—
72C2×Dic283—41no—
73C2×Dic283C22⋊C411yescharacteristic
74C2×Dic283C22⋊C411yescharacteristic
75C4×D283—22no—
76C4×D283—22no—
77C4×D283—41no—
78C4×D283—41no—
79C4×D283—42no—
80C4×D283—42no—
81C283⋊C8—41no—
82C23×C283C2×D411yescharacteristic
83Q8×C283—41no—
84C2×C4×C283—22no—
85C2×C4×C283—21no—
86C2×C4×C283—21no—
87C2×C4×C283—41no—
88C2×C4×C283—41no—
89C2×C4×C283—41no—
90C2×C4×C283C22×C411yescharacteristic
91C2×C4×C283C2×D411yescharacteristic
92C8×C283—41no—
93C22×C283—21no—
94C22×C283—21no—
95C22×C283C2×C22⋊C411yescharacteristic
96C22×C283#13711yescharacteristic
97C22×C283#13711yescharacteristic
98C2×D283—21no—
99C2×D283—21no—
100C2×D283—21no—
101C2×D283—21no—
102C2×D283—21no—
103C2×D283—42no—
104C2×D283—42no—
105C2×D283#10412yes—
106C4×C283—21no—
107C4×C283—21no—
108C4×C283—42no—
109C4×C283—41no—
110C4×C283—41no—
111Dic283—21no—
112Dic283—42no—
113Dic283—41no—
114Dic283—41no—
115Dic283#9912yes—
116C2×C283—21no—
117C2×C283—21no—
118C2×C283C2×#9911yescharacteristic
119C2×C283#41911yescharacteristic
120C2×C283C2×#10411yescharacteristic
121D283—22no—
122D283—41no—
123C283#150811yescharacteristic, complemented, Sylow
124#1508—2831nomaximal, Sylow
125C2×C22⋊Q8—2831no—
126#407—2831no—
127C4×C22⋊C4—2831no—
128#328—2831no—
129#328—2831no—
130#328—2831no—
131#328D28311yescharacteristic, complemented
132C22×Q8—2831no—
133C23×C4—2831no—
134C2×M16—5661no—
135C22⋊Q8—2831no—
136C22⋊Q8—2831no—
137C22⋊Q8—2831no—
138C22⋊Q8—5661no—
139C22⋊Q8—5661no—
140C22⋊Q8C2×D28311yescharacteristic
141C2×C4⋊C4—2831no—
142C2×C4⋊C4—5661no—
143C2×C22⋊C4—5661no—
144C2×C22⋊C4—5661no—
145C2×C42—5661no—
146C22⋊C8—2831no—
147C22⋊C8—5661no—
148C22⋊C8C2×D28311yescharacteristic
149#95—2831no—
150#95C2×D28311yescharacteristic
151C24—2831no—
152C2×Q8—2831no—
153C2×Q8—5661no—
154C2×Q8—11322no—
155C2×Q8C4×D28311yescharacteristic
156C22×C4—21no—
157C22×C4—2831no—
158C22×C4—2831no—
159C22×C4—2831no—
160C22×C4—2831no—
161C22×C4—2831no—
162C22×C4—5661no—
163C22×C4—5661no—
164C22×C4—5661no—
165C22×C4—11322no—
166C22×C4C22×D28311yescharacteristic
167M16—11322no—
168C2×C8—21no—
169C2×C8—5661no—
170C4⋊C4—21no—
171C4⋊C4—2831no—
172C4⋊C4—5661no—
173C4⋊C4—5661no—
174C4⋊C4—5661no—
175C4⋊C4—5661no—
176C4⋊C4C4×D28311yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—5661no—
179C22⋊C4—5661no—
180C22⋊C4—5664no—
181C22⋊C4—5661no—
182C42—11322no—
183C23—2831no—
184C23—2831no—
185C23—5662no—
186C23—5661no—
187C23—5661no—
188C23—11321no—
189C23D4×D28311yescharacteristic
190Q8—41no—
191Q8—11321no—
192Q8—11322no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—2831no—
200C2×C4—2831no—
201C2×C4—5662no—
202C2×C4—5662no—
203C2×C4—5662no—
204C2×C4—5662no—
205C2×C4—5661no—
206C2×C4—5661no—
207C2×C4—11321no—
208C2×C4—11321no—
209C2×C4—11322no—
210C2×C4—11322no—
211C2×C4—11321no—
212C2×C4—11321no—
213C2×C4C2×C4×D28311yescharacteristic
214C2×C4D4×D28311yescharacteristic
215C8—41no—
216C8—11321no—
217C22—21no—
218C22—21no—
219C22—2832no—
220C22—5661no—
221C22—5661no—
222C22—5661no—
223C22—5661no—
224C22—5661no—
225C22—11322no—
226C22—11322no—
227C22C22⋊C4×D28311yescharacteristic
228C22#2258363911yescharacteristic
229C22#2258364211yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—41no—
235C4—2832no—
236C4—5661no—
237C4—11322no—
238C4—11321no—
239C4—11321no—
240C2—21no—
241C2—21no—
242C2—5662no—
243C2—11321no—
244C2#99×D28311yescharacteristic
245C2#3182647211yescharacteristic
246C2#104×D28311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D283
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
Fitting#328×C283C2
socleC22×C283C2×C22⋊C4
prime radicalC23×D283D4
cosocleC22×C4×C283C23
hypercenter#328D283
hypocenterC283#1508
LCSG ▷ C2×C4×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #31826203C2semidirect
5. #31826202C2semidirect
6. #31826201C2semidirect
7. #328×C283C2semidirect
8. C2×Q8×D283C4semidirect
11. #22583590C4semidirect
15. #22583586C4semidirect
17. C4⋊C4×D283C4semidirect
123. C283#1508semidirect
131. #328D283semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2832
#318264272
#318263122
#318262032
#318262022
#318262012
#328×C2832
#1508283