Finite Group Databasebeta

← #102092

Subgroups — #102092

Order 512.

472 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C2×#1942C211yesmaximal, characteristic
2#9879C212yesmaximal
3#8595C211yesmaximal, characteristic
4#8145C211yesmaximal, characteristic
5#7576C212yesmaximal
6C22×#107C411yescharacteristic
7C22×#107C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C2×#389C2212yes—
10#1942—21no—
11#1942—21no—
12#1942—21no—
13#1942—21no—
14#1797C2212yes—
15C4×C22⋊C8C2211yescharacteristic
16C4×C22⋊C8C2211yescharacteristic
17C2×#340C412yes—
18C2×#334—21no—
19C2×#322—21no—
20#1280—24no—
21#1277—21no—
22#1273—21no—
23#1271—24no—
24C23×C8C2×C411yescharacteristic
25C23×C8C2×C411yescharacteristic
26C22×C4⋊C4—22no—
27C22×C4⋊C4C2×C412yes—
28C22×C4⋊C4C2×C412yes—
29C22×C42C2311yescharacteristic
30C2×#107—21no—
31C2×#107—21no—
32C2×#107—24no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—42no—
36C2×C4⋊C8—22no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8D412yes—
40C2×C4×C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8—42no—
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#389—22no—
48#389—22no—
49#389—22no—
50#389—22no—
51C2×#95—22no—
52#340—22no—
53#340—22no—
54#336—22no—
55#336—21no—
56#336D412yes—
57#334—42no—
58#322—42no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4C22×C411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C23×C4#4111yescharacteristic
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8—42no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—84no—
89C2×C4⋊C4M1614yes—
90C2×C42—41no—
91C2×C42—41no—
92C2×C42C2×D411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42#4111yescharacteristic
96#107—24no—
97#107—24no—
98#107—42no—
99#107—42no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C22⋊C8—24no—
103C22⋊C8—24no—
104C22⋊C8—44no—
105C4×C8—22no—
106C4×C8—22no—
107C4×C8—41no—
108C4×C8—41no—
109C4×C8—41no—
110C4×C8—41no—
111C4×C8—41no—
112C4×C8—41no—
113#95—42no—
114#95—42no—
115C24C4×D411yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—44no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4—82no—
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4C4×D411yescharacteristic
136C22×C4#12111yescharacteristic
137C22×C4#12111yescharacteristic
138C22×C4#12411yescharacteristic
139C22×C4#12711yescharacteristic
140C22×C4C2×M1612yes—
141C22×C4#13112yes—
142C22×C4#13714yes—
143C2×C8—28no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—24no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—82no—
161C2×C8—82no—
162C2×C8—82no—
163C2×C8#9912yes—
164C2×C8#10112yes—
165C2×C8#10112yes—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C4⋊C4—84no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—41no—
180C42—41no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23#39011yescharacteristic
186C23#43512yes—
187C23#44112yes—
188C23#44312yes—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—84no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—84no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—164no—
215C2×C4#40711yescharacteristic
216C2×C4#40811yescharacteristic
217C2×C4C2×#9911yescharacteristic
218C2×C4#41011yescharacteristic
219C2×C4C2×#10111yescharacteristic
220C2×C4C2×#10111yescharacteristic
221C2×C4#43512yes—
222C2×C4#47012yes—
223C2×C4#47212yes—
224C2×C4#49712yes—
225C8—22no—
226C8—22no—
227C8—22no—
228C8—44no—
229C8—41no—
230C8—82no—
231C8—162no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22#158714yes—
239C22#195511yescharacteristic
240C22#196411yescharacteristic
241C22#196611yescharacteristic
242C22#197111yescharacteristic
243C22#197411yescharacteristic
244C22#197611yescharacteristic
245C22#197811yescharacteristic
246C4—22no—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—41no—
251C4—84no—
252C4—162no—
253C4#146311yescharacteristic
254C4#146711yescharacteristic
255C4#147011yescharacteristic
256C4#147211yescharacteristic
257C2—24no—
258C2#965411yescharacteristic
259C2#966411yescharacteristic
260C2#967011yescharacteristic
261C2#978411yescharacteristic
262C2#978811yescharacteristic
263C2#981011yescharacteristic
264C2#981511yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19422
#98792
#85952
#81452
#75762