Finite Group Databasebeta

← #96832

Subgroups — #96832

Order 512.

426 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C8×#124C211yesmaximal, characteristic
2C4×#329C211yesmaximal, characteristic
3#8132C212yesmaximal, complemented
4#7567C211yesmaximal, characteristic
5#7030C212yesmaximal, complemented
6C2×C8×Q8C411yescharacteristic
7C2×C8×D4C411yescharacteristic
8C4×#124C2211yescharacteristic
9#1984C411yescharacteristic
10#1954C411yescharacteristic
11#1923—22no—
12#1831C2211yescharacteristic
13C8×C22⋊C4—22no—
14#1744C2212yes—
15C4×#97C2212yes—
16C42×C8C2211yescharacteristic
17C2×#329C412yes—
18#1295C412yes—
19#1273—22no—
20#1273—22no—
21#1272C412yes—
22#1270C412yes—
23C23×C8—21no—
24C2×#124C2×C411yescharacteristic
25C2×C4×Q8C2×C411yescharacteristic
26C2×C4×D4C2×C411yescharacteristic
27C8×Q8—42no—
28C8×D4—42no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C22⋊C8—21no—
33C2×C22⋊C8—22no—
34C2×#97—22no—
35C2×#97C2×C414yes—
36C2×C4×C8—21no—
37C2×C4×C8—22no—
38C2×C4×C8—22no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D412yes—
42C2×C4×C8D412yes—
43#391C2×C411yescharacteristic
44#390C2×C411yescharacteristic
45#388—22no—
46#382C2×C411yescharacteristic
47C4×C22⋊C4—22no—
48C43C2311yescharacteristic
49#336—22no—
50#336—22no—
51#336—22no—
52#336C2×C414yes—
53#329C818yescomplemented
54C22×Q8C4211yescharacteristic
55C22×D4C4211yescharacteristic
56C23×C4—21no—
57C22×C8—24no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—24no—
61C22×C8—44no—
62C22×C8—42no—
63C22×C8C22⋊C412yes—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66#124—22no—
67#124C2×C814yes—
68C4×Q8—41no—
69C4×Q8—41no—
70C4×D4—41no—
71C4×D4—41no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4C4211yescharacteristic
74C2×C4⋊C4C4211yescharacteristic
75C2×C22⋊C4—22no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C42—21no—
79C2×C42C22×C411yescharacteristic
80C2×C42C22×C411yescharacteristic
81C2×C42C22×C411yescharacteristic
82C2×C42C2×D412yes—
83C2×C42#4112yes—
84C4⋊C8—41no—
85C4⋊C8—41no—
86C4⋊C8—42no—
87C22⋊C8—44no—
88C22⋊C8—42no—
89#97—24no—
90#97—44no—
91#97C2×C818yescomplemented
92C4×C8—22no—
93C4×C8—41no—
94C4×C8—42no—
95C4×C8—42no—
96C4×C8—42no—
97C4×C8—42no—
98C4×C8—42no—
99C4×C8—42no—
100C4×C8—42no—
101#95—22no—
102#95—22no—
103C24—21no—
104C2×Q8—21no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×Q8C4×C812yes—
108C2×D4—21no—
109C2×D4—41no—
110C2×D4—41no—
111C2×D4C4×C812yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C2×C4211yescharacteristic
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4#11711yescharacteristic
122C22×C4C4×D412yes—
123C22×C4C4×D412yes—
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—28no—
131C2×C8—44no—
132C2×C8—44no—
133C2×C8—44no—
134C2×C8—44no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—44no—
140C2×C8—84no—
141C2×C8#10012yes—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—42no—
145C22⋊C4—28no—
146C22⋊C4—42no—
147C22⋊C4—44no—
148C42—21no—
149C42—21no—
150C42—22no—
151C42—22no—
152C42—24no—
153C42—41no—
154C42—41no—
155C42—41no—
156C42C22×C812yes—
157C42#13112yes—
158C23—22no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23—42no—
163C23C4×C22⋊C411yescharacteristic
164Q8—42no—
165Q8—42no—
166D4—42no—
167D4—42no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—82no—
186C2×C4C2×C4×C811yescharacteristic
187C2×C4#40511yescharacteristic
188C2×C4C2×#10011yescharacteristic
189C2×C4#41311yescharacteristic
190C2×C4#42112yes—
191C2×C4#42112yes—
192C2×C4C8×D414yes—
193C2×C4#44414yes—
194C8—22no—
195C8—48no—
196C8—41no—
197C8—42no—
198C8—84no—
199C8—81no—
200C8—81no—
201C22—42no—
202C22—44no—
203C22—81no—
204C22—81no—
205C22—82no—
206C22C8×C22⋊C411yescharacteristic
207C22C4×#10011yescharacteristic
208C22#175711yescharacteristic
209C22#175912yes—
210C22#176512yes—
211C4—24no—
212C4—21no—
213C4—21no—
214C4—24no—
215C4—41no—
216C4—42no—
217C4—81no—
218C4—81no—
219C4#153311yescharacteristic
220C4#153512yes—
221C4#153611yescharacteristic
222C2—82no—
223C2C8×#10011yescharacteristic
224C2#819112yes—
225C2#822411yescharacteristic
226C2#824311yescharacteristic
227C2#824812yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C2×C4×C8
Frattini, cosocleC43C23
socleC23C4×C22⋊C4
prime radicalC22×D4C42
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8132C2semidirect
5. #7030C2semidirect
53. #329C8semidirect
91. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
C4×#3292
#81322
#75672
#70302