Finite Group Databasebeta

← #98839

Subgroups — #98839

Order 512.

434 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 126 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#20435C211yesmaximal, characteristic
2#8596C212yesmaximal
3#7598C212yesmaximal
4#7034C212yesmaximal
5C2×#424C411yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9#2296C2211yescharacteristic
10#1797C2214yes—
11#1771—22no—
12C8×C4⋊C4—22no—
13#1729C411yescharacteristic
14#1728C2211yescharacteristic
15C2×#335—22no—
16#1277—22no—
17#1271—24no—
18C22×M16C2×C411yescharacteristic
19C22×M16C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C2×#117C2×C411yescharacteristic
23C2×#117C2×C411yescharacteristic
24C22×C42C2311yescharacteristic
25#424—22no—
26#424—22no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8—24no—
29C2×C22⋊C8—24no—
30#405—22no—
31#405—42no—
32#405C814yescomplemented
33C2×#97—21no—
34C2×#97—21no—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8—42no—
38C2×C4×C8D412yes—
39C2×C4×C8D412yes—
40#389D412yes—
41#389Q812yes—
42#385—22no—
43C4×C4⋊C4—22no—
44#336—22no—
45#336—22no—
46#336—24no—
47#335—44no—
48C23×C4C22×C411yescharacteristic
49C23×C4#4112yes—
50C2×M16—21no—
51C2×M16—21no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16C2×C812yes—
57C2×M16M1612yes—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—44no—
67C22×C8C22⋊C412yes—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C2×C812yes—
71C22×C8M1612yes—
72#117—21no—
73#117—21no—
74#117—41no—
75#117—41no—
76#117C2×C812yes—
77#117M1612yes—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4C4⋊C412yes—
81C2×C4⋊C4C4⋊C412yes—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—41no—
87C2×C42—41no—
88C2×C42C2×D412yes—
89C2×C42C2×D411yescharacteristic
90C2×C42C2×Q811yescharacteristic
91C4⋊C8—28no—
92C4⋊C8—44no—
93C4⋊C8—44no—
94C22⋊C8—84no—
95#97—24no—
96#97—42no—
97C4×C8—24no—
98C4×C8—24no—
99C4×C8—42no—
100C4×C8—42no—
101#95—22no—
102#95—22no—
103C24#11711yescharacteristic
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4C2×C22⋊C411yescharacteristic
115C22×C4C2×C4⋊C411yescharacteristic
116C22×C4#12112yes—
117C22×C4C22⋊Q812yes—
118C22×C4C22×C811yescharacteristic
119C22×C4C2×M1611yescharacteristic
120C22×C4C2×M1611yescharacteristic
121C22×C4C2×M1611yescharacteristic
122M16—42no—
123M16—42no—
124M16—42no—
125M16—42no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—24no—
129C2×C8—24no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—84no—
142C2×C8—84no—
143C2×C8—84no—
144C2×C8C22⋊C812yes—
145C2×C8C22⋊C812yes—
146C2×C8#9912yes—
147C2×C8#10112yes—
148C4⋊C4—22no—
149C4⋊C4—42no—
150C4⋊C4—44no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4C4⋊C814yes—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—42no—
157C42—22no—
158C42—22no—
159C42—22no—
160C42—22no—
161C42—41no—
162C42—41no—
163C42—41no—
164C42—41no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—22no—
169C23#38011yescharacteristic
170C23#43111yescharacteristic
171C23#43211yescharacteristic
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4C2×C22⋊C811yescharacteristic
196C2×C4#40711yescharacteristic
197C2×C4C2×#9911yescharacteristic
198C2×C4#41011yescharacteristic
199C2×C4C2×#10111yescharacteristic
200C2×C4#41311yescharacteristic
201C2×C4C2×C4⋊C811yescharacteristic
202C2×C4#42311yescharacteristic
203C2×C4#47212yes—
204C2×C4#47312yes—
205C8—22no—
206C8—42no—
207C8—42no—
208C8—42no—
209C8—164no—
210C8#32414yes—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—22no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22#179711yescharacteristic
219C22#179911yescharacteristic
220C22#180111yescharacteristic
221C22#181412yes—
222C22#181612yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—42no—
229C4—82no—
230C4C2×#32411yescharacteristic
231C4#145811yescharacteristic
232C4#146511yescharacteristic
233C4#146711yescharacteristic
234C2—41no—
235C2—41no—
236C2#855811yescharacteristic
237C2#860512yes—
238C2#862712yes—
239C2#863311yescharacteristic
240C2#863911yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
32. #405C8semidirect

Maximal subgroups

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

subgroupindex
#204352
#85962
#75982
#70342