Finite Group Databasebeta

← #24907758

Subgroups — #24907758

Order 10624.

408 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 109 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×#13050574C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D83C211yesmaximal, characteristic
3C2×(C2×D83)⋊C8C212yesmaximal, complemented
4#19127455C212yesmaximal, complemented
5C2×C22⋊C8×C83C211yesmaximal, characteristic, complemented
6C22×C22⋊D83C411yescharacteristic
7C22×C4×D83C2211yescharacteristic
8C2×#13050574—24no—
9C2×#13050574—22no—
10C22×C83⋊C8C2212yes—
11C4×C22⋊D83—22no—
12C4×C22⋊D83—22no—
13C2×Dic83⋊C4C411yescharacteristic
14C2×C22⋊Dic83C411yescharacteristic
15C2×(C2×D83)⋊C4C411yescharacteristic
16C2×C4×Dic83C2211yescharacteristic
17(C2×D83)⋊C8—24no—
18C23×C4×C83C2211yescharacteristic
19C22×C8×C83C2212yes—
20C22⋊C8×C83—22no—
21C23×D83C2×C411yescharacteristic
22C2×C22⋊D83—22no—
23C2×C22⋊D83—22no—
24C2×C22⋊D83—22no—
25C22×Dic83C2×C411yescharacteristic
26C22×Dic83C2×C411yescharacteristic
27C22×Dic83C2×C411yescharacteristic
28C2×C4×D83—22no—
29C2×C4×D83—21no—
30C2×C4×D83—21no—
31C2×C4×D83—22no—
32#13050574—28no—
33C2×C83⋊C8—22no—
34C2×C83⋊C8—22no—
35C2×C83⋊C8D414yescomplemented
36Dic83⋊C4—22no—
37C22⋊Dic83—22no—
38(C2×D83)⋊C4—22no—
39C4×Dic83—21no—
40C4×Dic83—21no—
41C24×C83C2×C411yescharacteristic
42C22×C4×C83—22no—
43C22×C4×C83—22no—
44C22×C4×C83C2×C411yescharacteristic
45C22×C4×C83D412yes—
46C22×C4×C83D412yes—
47C22×C4×C83C2311yescharacteristic
48C2×C8×C83—22no—
49C2×C8×C83—22no—
50C2×C8×C83—22no—
51C23×C83—22no—
52C23×C83—21no—
53C23×C83—21no—
54C23×C83—22no—
55C23×C83C22⋊C412yes—
56C23×C83C22×C411yescharacteristic
57C22×D83—21no—
58C22×D83—22no—
59C22×D83—21no—
60C22×D83—22no—
61C22×D83M1612yescomplemented
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—24no—
65C2×C4×C83—22no—
66C2×C4×C83C22⋊C412yes—
67C2×C4×C83C2×D412yes—
68C2×C4×C83C2×D411yescharacteristic
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83#4112yes—
71C22⋊D83—28no—
72C2×Dic83—21no—
73C2×Dic83—21no—
74C2×Dic83—21no—
75C2×Dic83—21no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83M1612yescomplemented
79C4×D83—41no—
80C4×D83—42no—
81C4×D83—41no—
82C8×C83—42no—
83C83⋊C8—24no—
84C22×C83—22no—
85C22×C83—22no—
86C22×C83—22no—
87C22×C83—42no—
88C22×C83—42no—
89C22×C83C2×C22⋊C411yescharacteristic
90C22×C83C4×D412yes—
91C22×C83C2×M1611yescharacteristic
92C22×C83C2×M1611yescharacteristic
93C22×C83#13112yes—
94C2×D83—22no—
95C2×D83—24no—
96C2×D83—41no—
97C2×D83—41no—
98C2×D83—42no—
99C4×C83—22no—
100C4×C83#12011yescharacteristic
101C4×C83#12112yes—
102C4×C83#12311yescharacteristic
103Dic83—22no—
104Dic83—41no—
105Dic83—41no—
106C2×C83—42no—
107C2×C83#38611yescharacteristic
108C2×C83#40711yescharacteristic
109C2×C83#40811yescharacteristic
110C2×C83#43512yes—
111C2×C83#43612yes—
112D83—42no—
113#1870—831nomaximal, Sylow
114C83#187011yescharacteristic, complemented, Sylow
115C22×M16—831no—
116C2×C4×D4—831no—
117C2×C22⋊C8—832no—
118C2×C22⋊C8D8311yescharacteristic, complemented
119#336—832no—
120C22×D4—831no—
121C23×C4—831no—
122C23×C4C2×D8311yescharacteristic
123C2×M16—1664no—
124C2×M16—1662no—
125C22×C8—832no—
126C22×C8C2×D8312yes—
127C4×D4—1662no—
128C4×D4—1662no—
129C2×C4⋊C4—831no—
130C2×C22⋊C4—831no—
131C2×C22⋊C4—831no—
132C2×C42—831no—
133C22⋊C8—22no—
134C22⋊C8—1664no—
135C24—831no—
136C24C4×D8311yescharacteristic
137C2×D4—1662no—
138C2×D4—1662no—
139C2×D4—1662no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—831no—
143C22×C4—831no—
144C22×C4—831no—
145C22×C4—1662no—
146C22×C4—1661no—
147C22×C4—1661no—
148C22×C4—1662no—
149C22×C4C4×D8311yescharacteristic
150C22×C4D33212yes—
151C22×C4C22⋊D8312yes—
152C22×C4C22×D8311yescharacteristic
153M16—1668no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—834no—
158C2×C8—1662no—
159C2×C8—1662no—
160C4⋊C4—1662no—
161C22⋊C4—1662no—
162C22⋊C4—1662no—
163C42—1661no—
164C42—1661no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—22no—
169C23—832no—
170C23—1661no—
171C23—1662no—
172C23—1661no—
173C23—1662no—
174C23(C2×D83)⋊C412yes—
175C23C2×C4×D8311yescharacteristic
176D4—1668no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—3321no—
182C2×C4—3322no—
183C2×C4—3321no—
184C2×C4—832no—
185C2×C4—1662no—
186C2×C4—1661no—
187C2×C4—1661no—
188C2×C4—1662no—
189C2×C4—1661no—
190C2×C4—1661no—
191C2×C4(C2×D83)⋊C412yes—
192C2×C4D4×D8312yes—
193C2×C4C2×C22⋊D8311yescharacteristic
194C2×C4C2×D33211yescharacteristic
195C2×C4#1305058912yes—
196C8—42no—
197C8—1664no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—3321no—
204C22—3321no—
205C22—3322no—
206C22—1662no—
207C22—1664no—
208C22C2×(C2×D83)⋊C411yescharacteristic
209C22(C22⋊D83)⋊C412yes—
210C22M16×D8311yescharacteristic
211C22M16×D8311yescharacteristic
212C22#1458179812yes—
213C4—22no—
214C4—3321no—
215C4—3321no—
216C4—1662no—
217C4#1458176211yescharacteristic
218C4#1458176412yes—
219C4#1458177611yescharacteristic
220C2—42no—
221C2—3322no—
222C2#1912757811yescharacteristic
223C2#1912766411yescharacteristic
224C2#1912766711yescharacteristic
225C2#1912777512yes—
226C2#1912777612yes—
subgroupG/H
center, FrattiniC22×C4C22×D83
commutator, socleC23×C83C22×C4
FittingC2×C22⋊C8×C83C2
prime radicalC22×C22⋊D83C4
cosocleC22×C4×C83C23
hypercenterC2×C22⋊C8D83
hypocenterC83#1870
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×#13050574C2semidirect
3. C2×(C2×D83)⋊C8C2semidirect
4. #19127455C2semidirect
5. C2×C22⋊C8×C83C2semidirect
35. C2×C83⋊C8D4semidirect
61. C22×D83M16semidirect
78. C2×Dic83M16semidirect
114. C83#1870semidirect
118. C2×C22⋊C8D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505742
C2×C4×C22⋊D832
C2×(C2×D83)⋊C82
#191274552
C2×C22⋊C8×C832
#187083