Finite Group Databasebeta

← #134902

Subgroups — #134902

Order 512.

279 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 72 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#22241C211yesmaximal, characteristic
2#11831C212yesmaximal
3#8623C211yesmaximal, characteristic
4#8617C211yesmaximal, characteristic
5#7323C212yesmaximal
6C2×#446C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2399C2211yescharacteristic
9#1985C411yescharacteristic
10#1985C2211yescharacteristic
11#1917—21no—
12#1917—21no—
13#1916—21no—
14#1916—21no—
15#1832C411yescharacteristic
16#1801C2212yes—
17#1746C411yescharacteristic
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⋊C8C2×C411yescharacteristic
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8D411yescharacteristic
37C2×C4⋊C8D411yescharacteristic
38C2×#101—22no—
39C2×#97—21no—
40C2×#97C2×C411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#398—21no—
45#398D411yescharacteristic
46#398Q811yescharacteristic
47#395—21no—
48#395D411yescharacteristic
49#395Q811yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4—21no—
52#376C2×C411yescharacteristic
53#343—22no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
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—41no—
103C4⋊C8—41no—
104#101—82no—
105C22⋊C8—42no—
106#97—41no—
107#97—42no—
108#97—42no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95C4⋊C411yescharacteristic
121#95C4⋊C411yescharacteristic
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—81no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C2×C4⋊C411yescharacteristic
134C22×C4#11711yescharacteristic
135C22×C4#12111yescharacteristic
136C22×C4#12111yescharacteristic
137C22×C4C22⋊Q811yescharacteristic
138C22×C4C22⋊Q811yescharacteristic
139M16—82no—
140M16—82no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—42no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—82no—
154C2×C8—82no—
155C2×C8#9911yescharacteristic
156C2×C8#9911yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—41no—
174C42—41no—
175C42—41no—
176C42—41no—
177C23#38011yescharacteristic
178Q8—42no—
179Q8—42no—
180Q8—82no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
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×C4C2×#9911yescharacteristic
210C2×C4#41011yescharacteristic
211C2×C4#47211yescharacteristic
212C2×C4#47311yescharacteristic
213C8—21no—
214C8—21no—
215C8—41no—
216C8—41no—
217C8—41no—
218C8—81no—
219C8—81no—
220C8—162no—
221C8—162no—
222C22—21no—
223C22—21no—
224C22#179911yescharacteristic
225C22#181411yescharacteristic
226C22#181611yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—41no—
235C4—81no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4#213311yescharacteristic
240C4#213511yescharacteristic
241C2—21no—
242C2—21no—
243C2#862711yescharacteristic
244C2#1280411yescharacteristic
245C2#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
#222412
#118312
#86232
#86172
#73232