Finite Group Databasebeta

← #24905994

Subgroups — #24905994

Order 10624.

274 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 63 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×D83C211yesmaximal, characteristic
2#19127663C211yesmaximal, characteristic
3#19127563C211yesmaximal, characteristic
4#19127439C211yesmaximal, characteristic, complemented
5#19127438C211yesmaximal, characteristic, complemented
6#19127437C211yesmaximal, characteristic, complemented
7#328×C83C211yesmaximal, characteristic, complemented
8C2×Q8×D83C411yescharacteristic, complemented
9C22×C4×D83C2211yescharacteristic
10C2×#13050573—21no—
11#14581772C411yescharacteristic, complemented
12#14581771C2211yescharacteristic
13#14581770—21no—
14#14581769—21no—
15#14581768C411yescharacteristic, complemented
16C4⋊C4×D83—21no—
17C4⋊C4×D83C411yescharacteristic, complemented
18C2×(C2×D83)⋊C4—21no—
19C22⋊C4×D83—21no—
20C22⋊(C83⋊C8)C2211yescharacteristic
21(C2×D83)⋊C8—21no—
22C83⋊#95—21no—
23C83⋊#95C2211yescharacteristic
24C22⋊Q8×C83C2211yescharacteristic
25C22⋊C8×C83C2211yescharacteristic
26#95×C83C2211yescharacteristic
27C23×D83D411yescharacteristic
28C2×Dic166—21no—
29C2×Dic166C2×C411yescharacteristic
30Q8×D83—42no—
31C22×Dic83—21no—
32C22×Dic83D411yescharacteristic
33C2×C4×D83—21no—
34C2×C4×D83—21no—
35C2×C4×D83—42no—
36C2×C4×D83C2×C411yescharacteristic
37C2×C4×D83C2×C411yescharacteristic
38C2×C4×D83D411yescharacteristic
39C2×C4×D83D411yescharacteristic
40#13050573—42no—
41C2×C83⋊C8—21no—
42C4⋊Dic83—21no—
43C4⋊Dic83C2×C411yescharacteristic
44Dic83⋊C4—21no—
45Dic83⋊C4—21no—
46Dic83⋊C4—21no—
47C22⋊Dic83—21no—
48(C2×D83)⋊C4—21no—
49(C2×D83)⋊C4—21no—
50(C2×D83)⋊C4—42no—
51C2×Q8×C83C2×C411yescharacteristic
52C22×C4×C83—21no—
53C22×C4×C83C2311yescharacteristic
54C2×C8×C83—21no—
55C4⋊C4×C83—21no—
56C4⋊C4×C83C2×C411yescharacteristic
57C22⋊C4×C83—21no—
58C23×C83C2×D411yescharacteristic
59C22×D83—21no—
60C22×D83—21no—
61C22×D83—22no—
62C22×D83—41no—
63C22×D83C22⋊C411yescharacteristic
64C22×D83C22⋊C411yescharacteristic
65Q8×C83—41no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—41no—
69C2×C4×C83—41no—
70C2×C4×C83—41no—
71C2×C4×C83—41no—
72C2×C4×C83C22×C411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×Dic83—21no—
75C2×Dic83—21no—
76C2×Dic83—41no—
77C2×Dic83—41no—
78C2×Dic83—41no—
79C2×Dic83—41no—
80C2×Dic83C22⋊C411yescharacteristic
81C2×Dic83C22⋊C411yescharacteristic
82C4×D83—22no—
83C4×D83—22no—
84C4×D83—41no—
85C4×D83—41no—
86C4×D83—42no—
87C4×D83—42no—
88Dic166—42no—
89Dic166—41no—
90C8×C83—41no—
91C83⋊C8—41no—
92C22×C83—21no—
93C22×C83—21no—
94C22×C83C2×C22⋊C411yescharacteristic
95C22×C83#13711yescharacteristic
96C22×C83#13711yescharacteristic
97C2×D83—21no—
98C2×D83—21no—
99C2×D83—21no—
100C2×D83—21no—
101C2×D83—21no—
102C2×D83—42no—
103C2×D83—42no—
104C2×D83#9912yes—
105C4×C83—21no—
106C4×C83—21no—
107C4×C83—41no—
108C4×C83—41no—
109C4×C83—81no—
110Dic83—21no—
111Dic83—21no—
112Dic83—41no—
113Dic83—41no—
114Dic83—81no—
115C2×C83—21no—
116C2×C83—21no—
117C2×C83C2×#9911yescharacteristic
118C2×C83#41911yescharacteristic
119C2×C83#42111yescharacteristic
120D83—22no—
121D83—41no—
122#1511—831nomaximal, Sylow
123C83#151111yescharacteristic, complemented, Sylow
124C2×C22⋊Q8—831no—
125#407—831no—
126#381—831no—
127#328—831no—
128#328—831no—
129#328—831no—
130#328D8311yescharacteristic, complemented
131C22×Q8—831no—
132C23×C4—831no—
133C2×M16—1661no—
134C22⋊Q8—831no—
135C22⋊Q8—831no—
136C22⋊Q8—831no—
137C22⋊Q8—1661no—
138C22⋊Q8—1661no—
139C22⋊Q8C2×D8311yescharacteristic
140C2×C4⋊C4—831no—
141C2×C4⋊C4—1661no—
142C2×C22⋊C4—1661no—
143C2×C22⋊C4—1661no—
144C22⋊C8—831no—
145C22⋊C8—1661no—
146C22⋊C8C2×D8311yescharacteristic
147#95—831no—
148#95—1661no—
149#95C2×D8311yescharacteristic
150C24—831no—
151C2×Q8—3322no—
152C2×Q8—831no—
153C2×Q8—1661no—
154C2×Q8C4×D8311yescharacteristic
155C22×C4—21no—
156C22×C4—3322no—
157C22×C4—831no—
158C22×C4—831no—
159C22×C4—831no—
160C22×C4—831no—
161C22×C4—831no—
162C22×C4—1661no—
163C22×C4—1661no—
164C22×C4—1661no—
165C22×C4C22×D8311yescharacteristic
166M16—3322no—
167C2×C8—21no—
168C2×C8—1661no—
169C4⋊C4—21no—
170C4⋊C4—831no—
171C4⋊C4—1661no—
172C4⋊C4—1661no—
173C4⋊C4—1661no—
174C4⋊C4—1661no—
175C4⋊C4C4×D8311yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—3322no—
178C22⋊C4—1661no—
179C22⋊C4—1661no—
180C22⋊C4—1661no—
181C23—3321no—
182C23—831no—
183C23—831no—
184C23—1662no—
185C23—1661no—
186C23—1661no—
187C23D4×D8311yescharacteristic
188Q8—41no—
189Q8—3321no—
190Q8—3322no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—3321no—
198C2×C4—3321no—
199C2×C4—3322no—
200C2×C4—3322no—
201C2×C4—3321no—
202C2×C4—3321no—
203C2×C4—3321no—
204C2×C4—3321no—
205C2×C4—831no—
206C2×C4—831no—
207C2×C4—1662no—
208C2×C4—1662no—
209C2×C4—1661no—
210C2×C4—1661no—
211C2×C4C2×C4×D8311yescharacteristic
212C2×C4D4×D8311yescharacteristic
213C8—41no—
214C8—3321no—
215C22—21no—
216C22—21no—
217C22—3322no—
218C22—3322no—
219C22—832no—
220C22—1661no—
221C22—1661no—
222C22—1661no—
223C22—1661no—
224C22—1661no—
225C22C22⋊C4×D8311yescharacteristic
226C22#1458182111yescharacteristic
227C22#1458182411yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—3321no—
234C4—3321no—
235C4—6641no—
236C4—1661no—
237C4—1661no—
238C2—21no—
239C2—21no—
240C2—3321no—
241C2—1662no—
242C2#99×D8311yescharacteristic
243C2#1912770811yescharacteristic
244C2#1912771611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#328×C83C2
socleC22×C83C2×C22⋊C4
prime radicalC23×D83D4
cosocleC22×C4×C83C23
hypercenter#328D83
hypocenterC83#1511
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #19127439C2semidirect
5. #19127438C2semidirect
6. #19127437C2semidirect
7. #328×C83C2semidirect
8. C2×Q8×D83C4semidirect
11. #14581772C4semidirect
15. #14581768C4semidirect
17. C4⋊C4×D83C4semidirect
123. C83#1511semidirect
130. #328D83semidirect