Finite Group Databasebeta

← #134349

Subgroups — #134349

Order 512.

343 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 86 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#21024C211yesmaximal, characteristic
2#11452C212yesmaximal
3#8602C211yesmaximal, characteristic
4#8601C211yesmaximal, characteristic
5#7190C212yesmaximal
6C22×C4⋊C8C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2296C2211yescharacteristic
9#1917—21no—
10#1917—21no—
11#1916—21no—
12#1916—21no—
13#1729C411yescharacteristic
14#1729C411yescharacteristic
15#1729C2211yescharacteristic
16#1728C411yescharacteristic
17#1564—21no—
18#1564—21no—
19#1458C2212yes—
20#1458C2212yes—
21#1326—22no—
22#1326—22no—
23#1275—21no—
24#1275—21no—
25C22×M16C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C2×#117C2×C411yescharacteristic
30C2×#117C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32#431—22no—
33#431—22no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—41no—
37C2×C4⋊C8—41no—
38#405—41no—
39#405—41no—
40C2×#97D411yescharacteristic
41C2×#97D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#389—21no—
45#389—21no—
46#385D411yescharacteristic
47#385D411yescharacteristic
48#385Q811yescharacteristic
49#385Q811yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
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#323—22no—
67#323—22no—
68C23×C4C4⋊C411yescharacteristic
69C23×C4C4⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C22×C8—22no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90#117—22no—
91#117—22no—
92#117—41no—
93#117—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4#4111yescharacteristic
100C2×C4⋊C4#4111yescharacteristic
101C2×C22⋊C4C4⋊C411yescharacteristic
102C2×C22⋊C4C4⋊C411yescharacteristic
103C2×C22⋊C4C2×D411yescharacteristic
104C2×C22⋊C4C2×Q811yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#97—41no—
120#97—41no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—42no—
126C4×C8—42no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131C24C2×C4⋊C411yescharacteristic
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—81no—
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4#11711yescharacteristic
146C22×C4#12111yescharacteristic
147C22×C4#12111yescharacteristic
148C22×C4C22⋊Q811yescharacteristic
149C22×C4C22⋊Q811yescharacteristic
150C22×C4#13112yes—
151C22×C4#13112yes—
152M16—44no—
153M16—82no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—82no—
173C2×C8—82no—
174C2×C8—82no—
175C2×C8#9911yescharacteristic
176C2×C8#9911yescharacteristic
177C2×C8#10112yes—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—42no—
194C42—41no—
195C42—41no—
196C42—41no—
197C42—41no—
198C23—21no—
199C23—21no—
200C23—41no—
201C23—41no—
202C23#38011yescharacteristic
203C23#42412yes—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—82no—
233C2×C4#40811yescharacteristic
234C2×C4#40811yescharacteristic
235C2×C4C2×#9911yescharacteristic
236C2×C4#41011yescharacteristic
237C2×C4C2×#10111yescharacteristic
238C2×C4#41311yescharacteristic
239C2×C4#43312yes—
240C8—21no—
241C8—21no—
242C8—22no—
243C8—41no—
244C8—41no—
245C8—81no—
246C8—81no—
247C8—162no—
248C8—162no—
249C22—21no—
250C22—21no—
251C22—22no—
252C22—42no—
253C22—42no—
254C22—41no—
255C22—41no—
256C22#179811yescharacteristic
257C22#179911yescharacteristic
258C22#180111yescharacteristic
259C4—22no—
260C4—41no—
261C4—41no—
262C4—42no—
263C4—41no—
264C4—41no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4#146811yescharacteristic
269C4#147011yescharacteristic
270C4#147211yescharacteristic
271C4#147311yescharacteristic
272C2—21no—
273C2—21no—
274C2—42no—
275C2#856211yescharacteristic
276C2#1280511yescharacteristic
277C2#1280611yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC24C2×C4⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#210242
#114522
#86022
#86012
#71902