Finite Group Databasebeta

← #24907372

Subgroups — #24907372

Order 10624.

852 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C23×C4×D83C211yesmaximal, characteristic
2C2×C22⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
3C2×(C2×D83)⋊C8C214yesmaximal, complemented
4C2×C22⋊C8×C83C211yesmaximal, characteristic, complemented
5C24×D83C411yescharacteristic
6C23×Dic83C411yescharacteristic
7C22×C4×D83—28no—
8C22×C4×D83—22no—
9C22×C4×D83—22no—
10C22×C4×D83C412yes—
11C22×C4×D83C2212yes—
12C22×C83⋊C8C2212yescomplemented
13C22⋊(C83⋊C8)—22no—
14(C2×D83)⋊C8—28no—
15C23×C4×C83C2211yescharacteristic
16C22×C8×C83C2212yescomplemented
17C22⋊C8×C83—22no—
18C23×D83—22no—
19C23×D83—22no—
20C23×D83—24no—
21C23×D83—24no—
22C23×D83C814yescomplemented
23C23×D83C2×C412yes—
24C22×Dic83—22no—
25C22×Dic83—24no—
26C22×Dic83C2×C412yes—
27C2×C4×D83—216no—
28C2×C4×D83—216no—
29C2×C4×D83—24no—
30C2×C4×D83—24no—
31C2×C4×D83—24no—
32C2×C4×D83—22no—
33C2×C4×D83—22no—
34C2×C4×D83—24no—
35C2×C4×D83D418yes—
36C2×C83⋊C8—24no—
37C2×C83⋊C8—22no—
38C24×C83C2×C411yescharacteristic
39C22×C4×C83—24no—
40C22×C4×C83C2×C411yescharacteristic
41C22×C4×C83D412yes—
42C22×C4×C83D412yes—
43C22×C4×C83C2311yescharacteristic
44C2×C8×C83—22no—
45C2×C8×C83—24no—
46C23×C83—22no—
47C23×C83—22no—
48C23×C83C22⋊C412yes—
49C23×C83C2×C812yes—
50C23×C83M1612yes—
51C23×C83C22×C411yescharacteristic
52C22×D83—28no—
53C22×D83—216no—
54C22×D83—28no—
55C22×D83—28no—
56C22×D83—22no—
57C22×D83—22no—
58C22×D83—22no—
59C22×D83—22no—
60C22×D83—216no—
61C22×D83C22⋊C414yes—
62C22×D83C2×C814yes—
63C22×D83M1614yes—
64C2×C4×C83—22no—
65C2×C4×C83—24no—
66C2×C4×C83—24no—
67C2×C4×C83C22⋊C412yes—
68C2×C4×C83C2×D414yes—
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C2×Dic83—28no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83C22⋊C414yes—
75C4×D83—216no—
76C4×D83—28no—
77C4×D83—28no—
78C8×C83—42no—
79C83⋊C8—42no—
80C22×C83—22no—
81C22×C83—24no—
82C22×C83—22no—
83C22×C83—24no—
84C22×C83C22⋊C814yescomplemented
85C22×C83C2×C22⋊C412yes—
86C22×C83C2×C22⋊C411yescharacteristic
87C22×C83C22×C811yescharacteristic
88C22×C83C2×M1612yes—
89C22×C83C2×M1611yescharacteristic
90C2×D83—232no—
91C2×D83—24no—
92C2×D83—24no—
93C2×D83—24no—
94C2×D83—24no—
95C2×D83—24no—
96C2×D83—24no—
97C2×D83C22⋊C818yescomplemented
98C4×C83—22no—
99C4×C83#12012yes—
100C4×C83#12012yes—
101Dic83—24no—
102C2×C83—24no—
103C2×C83#37911yescharacteristic
104C2×C83C2×C22⋊C812yes—
105C2×C83C2×C22⋊C811yescharacteristic
106C2×C83#40712yes—
107C2×C83#40711yescharacteristic
108D83—28no—
109#1779—831nomaximal, Sylow
110C83#177911yescharacteristic, complemented, Sylow
111C24×C4—831no—
112C2×C22⋊C8—831no—
113C2×C22⋊C8—834no—
114C2×C22⋊C8D8311yescharacteristic, complemented
115C25—831no—
116C23×C4—832no—
117C23×C4—831no—
118C23×C4—832no—
119C23×C4—1668no—
120C23×C4—1662no—
121C23×C4—1662no—
122C23×C4C2×D8311yescharacteristic
123C22×C8—832no—
124C22×C8C2×D8312yescomplemented
125C22⋊C8—22no—
126C22⋊C8—1668no—
127C22⋊C8—1662no—
128C24—832no—
129C24—834no—
130C24—1664no—
131C24—1664no—
132C24—1662no—
133C24—1662no—
134C24C4×D8311yescharacteristic
135C22×C4—24no—
136C22×C4—832no—
137C22×C4—838no—
138C22×C4—1664no—
139C22×C4—1664no—
140C22×C4—16616no—
141C22×C4—16616no—
142C22×C4—1664no—
143C22×C4—1662no—
144C22×C4—1664no—
145C22×C4—1664no—
146C22×C4—1662no—
147C22×C4—1662no—
148C22×C4C4×D8311yescharacteristic
149C22×C4D33212yes—
150C22×C4C22⋊D8312yes—
151C22×C4C22×D8311yescharacteristic
152C2×C8—24no—
153C2×C8—22no—
154C2×C8—1664no—
155C2×C8—1662no—
156C23—22no—
157C23—22no—
158C23—834no—
159C23—834no—
160C23—834no—
161C23—16616no—
162C23—1668no—
163C23—16616no—
164C23—1668no—
165C23—1662no—
166C23—1662no—
167C23—1668no—
168C23—1662no—
169C23—1662no—
170C23(C2×D83)⋊C412yes—
171C23C8×D8312yes—
172C23#1305057312yes—
173C23C2×C4×D8311yescharacteristic
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—834no—
178C2×C4—16616no—
179C2×C4—1668no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4—1668no—
183C2×C4—1668no—
184C2×C4(C2×D83)⋊C412yes—
185C2×C4D4×D8314yes—
186C2×C4C2×C22⋊D8311yescharacteristic
187C2×C4C2×D33211yescharacteristic
188C8—42no—
189C8—3322no—
190C22—22no—
191C22—24no—
192C22—24no—
193C22—22no—
194C22—838no—
195C22—16632no—
196C22—1664no—
197C22—1664no—
198C22—1664no—
199C22—1664no—
200C22—1664no—
201C22—1664no—
202C22(C2×D83)⋊C814yescomplemented
203C22C22⋊C4×D8312yes—
204C22C2×(C2×D83)⋊C411yescharacteristic
205C22C2×C8×D8311yescharacteristic
206C22M16×D8312yes—
207C22C2×#1305057311yescharacteristic
208C4—22no—
209C4—1664no—
210C4#1458176012yes—
211C4#1458176212yes—
212C2—24no—
213C2—1668no—
214C2#1912755611yescharacteristic
215C2C22⋊C8×D8312yes—
216C2C2×(C2×D83)⋊C811yescharacteristic
217C2#1912766312yes—
218C2#1912766411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C22×D83
commutatorC22×C83C22×C8
FittingC2×C22⋊C8×C83C2
socleC23×C83C22×C4
prime radicalC24×D83C4
cosocleC22×C4×C83C23
hypercenterC2×C22⋊C8D83
hypocenterC83#1779
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊(C83⋊C8)C2semidirect
3. C2×(C2×D83)⋊C8C2semidirect
4. C2×C22⋊C8×C83C2semidirect
12. C22×C83⋊C8C22semidirect
16. C22×C8×C83C22semidirect
22. C23×D83C8semidirect
84. C22×C83C22⋊C8semidirect
97. C2×D83C22⋊C8semidirect
110. C83#1779semidirect
114. C2×C22⋊C8D83semidirect
124. C22×C8C2×D83semidirect
202. C22(C2×D83)⋊C8semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D832
C2×C22⋊(C83⋊C8)2
C2×(C2×D83)⋊C82
C2×C22⋊C8×C832
#177983