Finite Group Databasebeta

← #99023

Subgroups — #99023

Order 512.

318 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 94 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#21673C211yesmaximal, characteristic
2#8274C211yesmaximal, characteristic
3#8273C211yesmaximal, characteristic
4#7636C212yesmaximal
5#7042C212yesmaximal
6C4×#125C2211yescharacteristic
7#1939C411yescharacteristic
8#1939C411yescharacteristic
9#1916—21no—
10#1916—21no—
11#1831C2211yescharacteristic
12#1771—21no—
13#1771—21no—
14#1770C411yescharacteristic
15#1770C411yescharacteristic
16#1767C2212yes—
17#1767C2212yes—
18C42×C8C2211yescharacteristic
19C2×#335—21no—
20C2×#334—21no—
21#1277—21no—
22#1277—21no—
23#1271—22no—
24#1271—22no—
25C2×#125Q811yescharacteristic
26C2×C4⋊C8—22no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×#97—21no—
33C2×#97—21no—
34C2×#97—22no—
35C2×#97—22no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#389D411yescharacteristic
43#389D411yescharacteristic
44#387—21no—
45#387—21no—
46#382Q811yescharacteristic
47C4×C4⋊C4—21no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4C2×C411yescharacteristic
50C4×C4⋊C4C2×C411yescharacteristic
51C43C2311yescharacteristic
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—22no—
59#336—22no—
60#335—24no—
61#334—24no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—42no—
65C22×C8—42no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70#125—42no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C4⋊C411yescharacteristic
76C2×C4⋊C4C4⋊C411yescharacteristic
77C2×C4⋊C4C4⋊C411yescharacteristic
78C2×C4⋊C4C4⋊C411yescharacteristic
79C2×C42—21no—
80C2×C42—21no—
81C2×C42C22×C411yescharacteristic
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C42C2×Q811yescharacteristic
86C2×C42#4111yescharacteristic
87C2×C42#4111yescharacteristic
88C4⋊C8—24no—
89C4⋊C8—24no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96#97—42no—
97#97—42no—
98#97—82no—
99#97—82no—
100C4×C8—22no—
101C4×C8—22no—
102C4×C8—22no—
103C4×C8—41no—
104C4×C8—41no—
105C4×C8—41no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4C2×C22⋊C411yescharacteristic
117C22×C4C2×C4⋊C411yescharacteristic
118C22×C4#11711yescharacteristic
119C22×C4#12111yescharacteristic
120C22×C4#12111yescharacteristic
121C22×C4C22⋊Q811yescharacteristic
122C22×C4C22⋊Q811yescharacteristic
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—44no—
133C2×C8—44no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—82no—
141C2×C8—82no—
142C2×C8—82no—
143C2×C8—82no—
144C2×C8#10112yes—
145C2×C8#10412yes—
146C2×C8#10412yes—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—22no—
163C42—22no—
164C42—21no—
165C42—21no—
166C42—41no—
167C42—41no—
168C42—41no—
169C42—41no—
170C42—41no—
171C23#38011yescharacteristic
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
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—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4C2×#10111yescharacteristic
200C2×C4#41311yescharacteristic
201C2×C4C2×#10411yescharacteristic
202C2×C4C2×#10411yescharacteristic
203C2×C4#42111yescharacteristic
204C2×C4#42111yescharacteristic
205C2×C4#43012yes—
206C2×C4#47112yes—
207C2×C4#47211yescharacteristic
208C2×C4#47311yescharacteristic
209C8—22no—
210C8—22no—
211C8—22no—
212C8—41no—
213C8—84no—
214C8—84no—
215C8—81no—
216C8—81no—
217C22#180111yescharacteristic
218C22#180611yescharacteristic
219C22#180611yescharacteristic
220C22#181111yescharacteristic
221C22#181211yescharacteristic
222C22#181511yescharacteristic
223C22#181611yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—81no—
233C4—81no—
234C4—81no—
235C4#152611yescharacteristic
236C4#153211yescharacteristic
237C4#153211yescharacteristic
238C4#153411yescharacteristic
239C2#858411yescharacteristic
240C2#862511yescharacteristic
241C2#864111yescharacteristic
242C2#867211yescharacteristic
243C2#867311yescharacteristic
244C2#869411yescharacteristic
245C2#869511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#216732
#82742
#82732
#76362
#70422