Finite Group Databasebeta

← #108570

Subgroups — #108570

Order 512.

453 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 150 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
1#20545C211yesmaximal, characteristic
2C2×#1292C211yesmaximal, characteristic
3#7988C211yesmaximal, characteristic
4#7962C212yesmaximal
5#7934C212yesmaximal
6#2920C2212yes—
7C2×#431C2211yescharacteristic
8C2×#423C2212yes—
9C22×C4⋊C8C2211yescharacteristic
10#2283C2211yescharacteristic
11#1770—21no—
12C8×C4⋊C4—21no—
13#1768—22no—
14#1752C412yes—
15C8×C22⋊C4C412yes—
16#1292—21no—
17#1292—21no—
18#1290—22no—
19#1286—22no—
20#1286—22no—
21#1283C418yescomplemented
22C22×M16—22no—
23C23×C8C2×C412yes—
24C22×C42C2311yescharacteristic
25#432—22no—
26#432—22no—
27#431—22no—
28#431—22no—
29#431—22no—
30#431C2×C414yes—
31#423—22no—
32#423—22no—
33#423—22no—
34#423—22no—
35#423—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8C2×C414yes—
41C2×C4⋊C8D412yes—
42C2×C4⋊C8D411yescharacteristic
43C2×C4⋊C8D411yescharacteristic
44#407—22no—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8C2×C411yescharacteristic
47C4×M16C2×C414yes—
48C2×#97—22no—
49C2×C4×C8—21no—
50C2×C4×C8—22no—
51#383Q812yes—
52#382D411yescharacteristic
53#382D411yescharacteristic
54#381—21no—
55#380—21no—
56C4×C4⋊C4—21no—
57C4×C22⋊C4C2×C412yes—
58#376—21no—
59#336—22no—
60#336—22no—
61C23×C4C22×C411yescharacteristic
62C23×C4#4111yescharacteristic
63C23×C4#4111yescharacteristic
64C2×M16—24no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—82no—
68C2×M16—82no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—24no—
74C22×C8—82no—
75C22×C8C22⋊C412yes—
76C22×C8C22⋊C412yes—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C42—22no—
81C2×C42C22×C412yes—
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C42C2×Q811yescharacteristic
86C2×C42#4112yes—
87C2×C42#4112yes—
88C2×C42#4112yes—
89C4⋊C8—24no—
90C4⋊C8—24no—
91C4⋊C8—24no—
92C4⋊C8—24no—
93C4⋊C8—22no—
94C4⋊C8—22no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C22⋊C8—21no—
100C22⋊C8—21no—
101C22⋊C8—42no—
102C22⋊C8—42no—
103C22⋊C8C4212yes—
104C22⋊C8C4212yes—
105#97—24no—
106#97—82no—
107C4×C8—24no—
108C4×C8—24no—
109C4×C8—82no—
110#95—21no—
111#95—21no—
112#95—22no—
113#95—22no—
114#95C4⋊C412yes—
115#95C4⋊C412yes—
116C24#11711yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—42no—
125C22×C4C2×C4211yescharacteristic
126C22×C4C2×C22⋊C411yescharacteristic
127C22×C4C2×C4⋊C411yescharacteristic
128C22×C4#11711yescharacteristic
129C22×C4#11711yescharacteristic
130C22×C4#11712yes—
131C22×C4#11712yes—
132C22×C4#11711yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4C22⋊Q811yescharacteristic
136C22×C4C22⋊Q811yescharacteristic
137C22×C4#12612yes—
138C22×C4#12612yes—
139M16—48no—
140M16—84no—
141M16—82no—
142M16—82no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—44no—
154C2×C8—44no—
155C2×C8—82no—
156C2×C8—82no—
157C2×C8—84no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C22⋊C4—44no—
162C42—42no—
163C42—42no—
164C42C4×D412yes—
165C42C4×D412yes—
166C42C4×D412yes—
167C42C4×Q812yes—
168C42#12312yes—
169C42#12312yes—
170C42#12412yes—
171C42#12512yes—
172C23—21no—
173C23—21no—
174C23—22no—
175C23—41no—
176C23#37611yescharacteristic
177C23#38011yescharacteristic
178C23#38311yescharacteristic
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—24no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4C4×C22⋊C411yescharacteristic
205C2×C4C4×C4⋊C411yescharacteristic
206C2×C4#38111yescharacteristic
207C2×C4#38211yescharacteristic
208C2×C4#38712yes—
209C2×C4#38712yes—
210C2×C4#38812yes—
211C2×C4#38812yes—
212C2×C4#47112yes—
213C2×C4#47211yescharacteristic
214C2×C4#47311yescharacteristic
215C8—42no—
216C8—42no—
217C8—42no—
218C8—42no—
219C8—84no—
220C8—84no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—42no—
229C22—42no—
230C22#143411yescharacteristic
231C22#174712yes—
232C22#181511yescharacteristic
233C22#181611yescharacteristic
234C4—22no—
235C4—22no—
236C4—22no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—82no—
241C4—82no—
242C4#182911yescharacteristic
243C4#183011yescharacteristic
244C4#193012yes—
245C4#193112yes—
246C2—21no—
247C2—21no—
248C2—22no—
249C2—41no—
250C2#1080011yescharacteristic
251C2#1082711yescharacteristic
252C2#1083011yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1283C4semidirect

Maximal subgroups

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

subgroupindex
#205452
C2×#12922
#79882
#79622
#79342