Finite Group Databasebeta

← #134910

Subgroups — #134910

Order 512.

299 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 74 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#21718C211yesmaximal, characteristic
2#11831C212yesmaximal
3#8621C211yesmaximal, characteristic
4#8617C211yesmaximal, characteristic
5#7323C212yesmaximal
6C2×#446C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2399C2211yescharacteristic
9#1985C411yescharacteristic
10#1985C2211yescharacteristic
11#1939—21no—
12#1939—21no—
13#1843C411yescharacteristic
14#1801C2212yes—
15#1771—21no—
16#1771—21no—
17#1726C411yescharacteristic
18#1568—21no—
19#1567—21no—
20#1467C2212yes—
21#1354—22no—
22#1346—22no—
23#1292—21no—
24#1290—21no—
25C2×C4×Q8C2×C411yescharacteristic
26C2×C4×Q8C2311yescharacteristic
27#446—41no—
28#446—41no—
29C8×Q8—41no—
30C8×Q8—41no—
31#432—22no—
32#423—22no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8D411yescharacteristic
39C2×C4⋊C8D411yescharacteristic
40C2×#101—22no—
41C2×#97—21no—
42C2×#97—21no—
43C2×#97C2×C411yescharacteristic
44C2×C4×C8—21no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#398—21no—
49#398D411yescharacteristic
50#398Q811yescharacteristic
51#395—21no—
52#395D411yescharacteristic
53#395Q811yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4—21no—
56#376C2×C411yescharacteristic
57#343—22no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#335—41no—
63#335—41no—
64#335—41no—
65#334—41no—
66#324—22no—
67#324—22no—
68C22×Q8C4⋊C411yescharacteristic
69C22×Q8#4111yescharacteristic
70C2×M16—42no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8—41no—
80C4×Q8—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4C4⋊C411yescharacteristic
87C2×C4⋊C4C2×D411yescharacteristic
88C2×C4⋊C4C2×Q811yescharacteristic
89C2×C4⋊C4#4111yescharacteristic
90C2×C42—21no—
91C2×C42—21no—
92C2×C42C22×C411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C4⋊C8—42no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C4⋊C8—42no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105#101—82no—
106C22⋊C8—42no—
107#97—24no—
108#97—41no—
109#97—42no—
110#97—41no—
111#97—41no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—42no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95C4⋊C411yescharacteristic
125#95C4⋊C411yescharacteristic
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—81no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4C2×C22⋊C411yescharacteristic
137C22×C4C2×C4⋊C411yescharacteristic
138C22×C4#11711yescharacteristic
139C22×C4#12111yescharacteristic
140C22×C4#12111yescharacteristic
141C22×C4C22⋊Q811yescharacteristic
142C22×C4C22⋊Q811yescharacteristic
143M16—82no—
144M16—82no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—82no—
159C2×C8—82no—
160C2×C8#9911yescharacteristic
161C2×C8#9911yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C42—22no—
173C42—22no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—41no—
181C42—41no—
182C23#38011yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—82no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4C2×#9911yescharacteristic
215C2×C4#41011yescharacteristic
216C2×C4#47211yescharacteristic
217C2×C4#47311yescharacteristic
218C8—21no—
219C8—42no—
220C8—41no—
221C8—41no—
222C8—41no—
223C8—81no—
224C8—162no—
225C8—162no—
226C8#35612yes—
227C22—21no—
228C22—21no—
229C22#179911yescharacteristic
230C22#181411yescharacteristic
231C22#181611yescharacteristic
232C4—21no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—41no—
241C4—41no—
242C4—81no—
243C4—81no—
244C4C2×#35611yescharacteristic
245C4#213211yescharacteristic
246C2—21no—
247C2—21no—
248C2#862811yescharacteristic
249C2#1280211yescharacteristic
250C2#1280611yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC23#380
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#217182
#118312
#86212
#86172
#73232