Finite Group Databasebeta

← #134832

Subgroups — #134832

Order 512.

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

normal subgroups only

This group has 301 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#21714C211yesmaximal, characteristic
2#11532C212yesmaximal
3#8623C211yesmaximal, characteristic
4#8621C211yesmaximal, characteristic
5#7248C212yesmaximal
6C2×#446C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8C4×C4⋊Q8C2211yescharacteristic
9#1984C411yescharacteristic
10#1984C2211yescharacteristic
11#1940—21no—
12#1940—21no—
13#1842C411yescharacteristic
14#1771—21no—
15#1771—21no—
16C4×#97C411yescharacteristic
17#1568—21no—
18#1568—21no—
19#1467C2212yes—
20#1467C2212yes—
21#1325—22no—
22#1319—22no—
23#1290—21no—
24#1290—21no—
25C2×C4⋊Q8Q811yescharacteristic
26C2×C4×Q8C2×C411yescharacteristic
27C2×C4×Q8C2311yescharacteristic
28#446—41no—
29#446—41no—
30C8×Q8—41no—
31C8×Q8—41no—
32#432—22no—
33#432—22no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8D411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×#97—21no—
42C2×#97—21no—
43C2×#97—21no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#391D411yescharacteristic
49#391Q811yescharacteristic
50#389—21no—
51#389—21no—
52#384D411yescharacteristic
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55C43C2×C411yescharacteristic
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#335—41no—
61#335—41no—
62#334—41no—
63#334—41no—
64#324—22no—
65#324—22no—
66#324—22no—
67#324—22no—
68C22×Q8C4⋊C411yescharacteristic
69C22×Q8C2×Q811yescharacteristic
70C22×C8—21no—
71C22×C8—21no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C4⋊Q8—42no—
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⋊C4#4111yescharacteristic
89C2×C4⋊C4#4111yescharacteristic
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C4⋊C411yescharacteristic
94C2×C42C4⋊C411yescharacteristic
95C2×C42C22×C411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42C2×D411yescharacteristic
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—42no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109#97—24no—
110#97—42no—
111#97—42no—
112#97—42no—
113#97—41no—
114#97—41no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
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
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—82no—
157C2×C8—82no—
158C2×C8#9911yescharacteristic
159C2×C8#9911yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C42—22no—
170C42—22no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—22no—
178C42—41no—
179C42—41no—
180C42—41no—
181C42—81no—
182C23#38011yescharacteristic
183Q8—44no—
184Q8—82no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
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—41no—
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×C4C2×#9911yescharacteristic
214C2×C4#41011yescharacteristic
215C2×C4#43012yes—
216C8—21no—
217C8—42no—
218C8—41no—
219C8—41no—
220C8—41no—
221C8—81no—
222C8—162no—
223C8—162no—
224C8#35612yes—
225C22—21no—
226C22—21no—
227C22#179911yescharacteristic
228C22#181111yescharacteristic
229C22#181311yescharacteristic
230C4—21no—
231C4—22no—
232C4—42no—
233C4—42no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—81no—
240C4—81no—
241C4C2×#35611yescharacteristic
242C4#213211yescharacteristic
243C2—21no—
244C2—21no—
245C2#861311yescharacteristic
246C2#1280211yescharacteristic
247C2#1280811yescharacteristic
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
#217142
#115322
#86232
#86212
#72482