Finite Group Databasebeta

← #116359

Subgroups — #116359

Order 512.

712 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 130 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#21935C211yesmaximal, characteristic
2#20694C212yesmaximal
3#20688C211yesmaximal, characteristic
4#8112C211yesmaximal, characteristic
5#8104C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2212yes—
9#2280C2211yescharacteristic
10#2280C2211yescharacteristic
11#1851—24no—
12#1822—24no—
13#1821—22no—
14#1821—22no—
15#1820—22no—
16#1819C414yes—
17#1798—22no—
18#1742—24no—
19#1740—22no—
20#1306—22no—
21#1305—24no—
22C24×C4C2311yescharacteristic
23C22×M16—22no—
24C22×M16D414yes—
25C2×#117—21no—
26C2×#117—22no—
27C2×#117—21no—
28C2×#117—22no—
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—21no—
34C2×#117D412yes—
35C22×C4⋊C4D412yes—
36C22×C22⋊C4C2×C412yes—
37C22×C42Q812yes—
38C2×C4⋊C8—24no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407D414yes—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51#381—21no—
52#381D412yes—
53#380—22no—
54#380—22no—
55#379C2×C412yes—
56#379C2×C412yes—
57C4×C22⋊C4—22no—
58C4×C22⋊C4—22no—
59#343—28no—
60#343—28no—
61#342—28no—
62#336—24no—
63C25C22×C411yescharacteristic
64C23×C4—44no—
65C23×C4C4⋊C414yes—
66C23×C4C2×D412yes—
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D412yes—
70C23×C4C2×D411yescharacteristic
71C23×C4C2×Q811yescharacteristic
72C23×C4#4111yescharacteristic
73C23×C4#4112yes—
74C23×C4#4111yescharacteristic
75C23×C4#4112yes—
76C2×M16—24no—
77C2×M16—24no—
78C2×M16—24no—
79C2×M16—42no—
80C2×M16—48no—
81C2×M16—48no—
82C22×C8—24no—
83C22×C8—42no—
84#117—41no—
85#117—42no—
86#117—42no—
87#117—42no—
88#117—41no—
89#117—42no—
90#117—42no—
91#117—41no—
92#117—42no—
93#117—41no—
94#117—44no—
95#117—44no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—41no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4C22⋊C414yes—
112C2×C42—22no—
113C2×C42—24no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—24no—
117C2×C42—41no—
118C4⋊C8—84no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127#95—24no—
128#95—42no—
129C24—42no—
130C24—82no—
131C24C2×C22⋊C411yescharacteristic
132C24C2×C4⋊C411yescharacteristic
133C24#11711yescharacteristic
134C24C4×D412yes—
135C24C4×D412yes—
136C22×C4—24no—
137C22×C4—28no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—44no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—84no—
152C22×C4#12012yes—
153C22×C4#12011yescharacteristic
154C22×C4#12112yes—
155C22×C4#12111yescharacteristic
156C22×C4#12112yes—
157C22×C4#12111yescharacteristic
158C22×C4#12112yes—
159C22×C4C22⋊Q812yes—
160C22×C4C22⋊Q811yescharacteristic
161C22×C4C22⋊Q811yescharacteristic
162C22×C4C22⋊Q812yes—
163C22×C4#12311yescharacteristic
164C22×C4#12312yes—
165C22×C4#12312yes—
166C22×C4#12412yes—
167C22×C4#12411yescharacteristic
168C22×C4#12612yes—
169C22×C4#12711yescharacteristic
170M16—48no—
171M16—48no—
172M16—84no—
173M16—84no—
174C2×C8—44no—
175C2×C8—44no—
176C2×C8—48no—
177C2×C8—84no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C22⋊C4—42no—
185C22⋊C4—44no—
186C22⋊C4—42no—
187C22⋊C4—44no—
188C22⋊C4—44no—
189C22⋊C4—84no—
190C22⋊C4—84no—
191C42—42no—
192C42—44no—
193C42—42no—
194C42—44no—
195C42—82no—
196C23—22no—
197C23—24no—
198C23—41no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—41no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—88no—
209C23#38011yescharacteristic
210C23#38512yes—
211C23#38611yescharacteristic
212C23#38812yes—
213C23#39011yescharacteristic
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—48no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—84no—
235C2×C4—84no—
236C2×C4—84no—
237C2×C4—82no—
238C2×C4—84no—
239C2×C4#39211yescharacteristic
240C2×C4#39311yescharacteristic
241C2×C4#39412yes—
242C2×C4#39612yes—
243C2×C4#39711yescharacteristic
244C2×C4#39911yescharacteristic
245C2×C4#45814yes—
246C8—84no—
247C8—84no—
248C22—22no—
249C22—21no—
250C22—21no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—44no—
260C22—82no—
261C22—84no—
262C22—84no—
263C22—82no—
264C22#143911yescharacteristic
265C22#189012yes—
266C4—22no—
267C4—22no—
268C4—21no—
269C4—21no—
270C4—84no—
271C4—84no—
272C4—162no—
273C4#202212yes—
274C4#203012yes—
275C2—22no—
276C2—41no—
277C2—41no—
278C2—42no—
279C2—42no—
280C2#1107011yescharacteristic
281C2#1109012yes—
subgroupG/H
centerC2×C4#393
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
socleC22#1439
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#219352
#206942
#206882
#81122
#81042