Finite Group Databasebeta

← #24913208

Subgroups — #24913208

Order 10624.

632 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#14581776C212yesmaximal, complemented
2C2×#14581766C211yesmaximal, characteristic
3C22×Dic83⋊C4C211yesmaximal, characteristic, complemented
4#19127633C212yesmaximal, complemented
5#19127623C214yesmaximal, complemented
6#19127619C212yesmaximal, complemented
7#19127578C212yesmaximal, complemented
8#380×C83C211yesmaximal, characteristic, complemented
9C22×D332C2211yescharacteristic
10C22×C22⋊D83C2212yes—
11C23×Dic83C2211yescharacteristic
12C23×Dic83C2211yescharacteristic
13#14581776—28no—
14#14581766—24no—
15C2×C4⋊Dic83C2214yes—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4—24no—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2214yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2214yes—
23C83⋊#95C2214yes—
24C23×C4×C83C2211yescharacteristic
25C2×C4⋊C4×C83C2212yes—
26C2×C22⋊C4×C83C2212yes—
27#95×C83C2212yes—
28C23×D83C2312yes—
29C2×D332—22no—
30C2×D332—42no—
31C2×C22⋊D83—24no—
32C2×C22⋊D83—24no—
33C2×C22⋊D83—24no—
34C22×Dic83—22no—
35C22×Dic83—22no—
36C22×Dic83—22no—
37C22×Dic83—42no—
38C22×Dic83D414yescomplemented
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—22no—
43C4⋊Dic83—28no—
44Dic83⋊C4—24no—
45Dic83⋊C4—24no—
46(C2×D83)⋊C4—24no—
47(C2×D83)⋊C4—28no—
48(C2×D83)⋊C4—44no—
49C24×C83C2311yescharacteristic
50C22×C4×C83—42no—
51C22×C4×C83D414yes—
52C22×C4×C83C2314yes—
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—42no—
56C22⋊C4×C83—24no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83C2×D412yes—
60C23×C83C2×D412yes—
61C23×C83#4112yes—
62C23×C83C2411yescharacteristic
63C22×D83—22no—
64C22×D83—22no—
65C22×D83—22no—
66C22×D83—24no—
67C22×D83—24no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—24no—
71C2×C4×C83—24no—
72C2×C4×C83—24no—
73C2×C4×C83—44no—
74C2×C4×C83C2×D412yes—
75C2×C4×C83C2×D412yes—
76C22⋊D83—48no—
77C2×Dic83—28no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—24no—
85C2×Dic83—44no—
86C2×Dic83C2×D414yes—
87D332—44no—
88C22×C83—22no—
89C22×C83—22no—
90C22×C83—24no—
91C22×C83—24no—
92C22×C83#12114yes—
93C22×C83C22×D411yescharacteristic
94C22×C83C22×D411yescharacteristic
95C22×C83C2×#4111yescharacteristic
96C22×C83H(5,𝔽2)12yes—
97C22×C832−1+412yes—
98C2×D83—42no—
99C2×D83—42no—
100C2×D83—42no—
101C2×D83—44no—
102C2×D83—44no—
103C4×C83—22no—
104C4×C83—22no—
105C4×C83—44no—
106Dic83—24no—
107Dic83—42no—
108Dic83—42no—
109C2×C83—24no—
110C2×C83C2×#12111yescharacteristic
111C2×C83#53411yescharacteristic
112C2×C83#53611yescharacteristic
113C2×C83#53712yes—
114C2×C83#53912yes—
115D83—82no—
116#2618—831nomaximal, Sylow
117C83#261811yescharacteristic, complemented, Sylow
118C2×#123—832no—
119C2×#121—831no—
120C22×C4⋊C4—831no—
121#398—832no—
122#396—834no—
123#394—832no—
124#386—832no—
125#380D8311yescharacteristic, complemented
126C22×D4—832no—
127C22×D4—831no—
128C23×C4—831no—
129C23×C4—831no—
130C23×C4C2×D8311yescharacteristic
131#123—1668no—
132#121—1664no—
133C2×C4⋊C4—832no—
134C2×C4⋊C4—832no—
135C2×C4⋊C4—831no—
136C2×C4⋊C4—834no—
137C2×C4⋊C4—1664no—
138C2×C4⋊C4C2×D8312yes—
139C2×C22⋊C4—834no—
140C2×C22⋊C4—832no—
141C2×C22⋊C4—834no—
142C2×C22⋊C4C2×D8312yes—
143#95—834no—
144#95C2×D8312yes—
145C24—832no—
146C24C22×D8311yescharacteristic
147C2×D4—3322no—
148C2×D4—1664no—
149C2×D4—1664no—
150C2×D4—1664no—
151C2×D4—1662no—
152C22×C4—42no—
153C22×C4—3322no—
154C22×C4—834no—
155C22×C4—832no—
156C22×C4—832no—
157C22×C4—832no—
158C22×C4—1662no—
159C22×C4—1662no—
160C22×C4—1662no—
161C22×C4D33214yes—
162C22×C4C22×D8314yes—
163C22×C4C22×D8311yescharacteristic
164C22×C4C22×D8311yescharacteristic
165C4⋊C4—42no—
166C4⋊C4—1668no—
167C4⋊C4—1664no—
168C4⋊C4—1664no—
169C4⋊C4—1662no—
170C22⋊C4—24no—
171C22⋊C4—3324no—
172C22⋊C4—1668no—
173C22⋊C4—1664no—
174C23—22no—
175C23—22no—
176C23—1662no—
177C23—1662no—
178C23—1662no—
179C23—1664no—
180C23—1664no—
181C23D4×D8312yes—
182C23C2×D33212yes—
183C23#1305059112yes—
184C23C23×D8311yescharacteristic
185D4—3324no—
186D4—3328no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—44no—
193C2×C4—3324no—
194C2×C4—834no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1664no—
200C2×C4—1668no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4C2×D33212yes—
204C2×C4C2×D33212yes—
205C22—22no—
206C22—22no—
207C22—24no—
208C22—24no—
209C22—3322no—
210C22—3322no—
211C22—3322no—
212C22—3324no—
213C22—3324no—
214C22#1458176714yes—
215C22C2×D4×D8311yescharacteristic
216C22C22×D33211yescharacteristic
217C22C2×#1305059111yescharacteristic
218C22#1458183712yes—
219C22#1458183912yes—
220C4—22no—
221C4—22no—
222C4—44no—
223C4—3322no—
224C4—3322no—
225C4—1664no—
226C2—24no—
227C2—6642no—
228C2C2×#1458176711yescharacteristic
229C2#1912826311yescharacteristic
230C2#1912827711yescharacteristic
231C2#1912827812yes—
232C2#1912830012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#380×C83C2
prime radicalC2×#14581766C2
hypercenter#380D83
hypocenterC83#2618
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×#14581776C2semidirect
3. C22×Dic83⋊C4C2semidirect
4. #19127633C2semidirect
5. #19127623C2semidirect
6. #19127619C2semidirect
7. #19127578C2semidirect
8. #380×C83C2semidirect
38. C22×Dic83D4semidirect
117. C83#2618semidirect
125. #380D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817762
C2×#145817662
C22×Dic83⋊C42
#191276332
#191276232
#191276192
#191275782
#380×C832
#261883