Finite Group Databasebeta

← #134932

Subgroups — #134932

Order 512.

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

normal subgroups only

This group has 336 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#21567C211yesmaximal, characteristic
2#11830C212yesmaximal
3#8614C211yesmaximal, characteristic
4#8612C211yesmaximal, characteristic
5#7321C212yesmaximal
6C2×C8×D4C2211yescharacteristic
7C2×#405C411yescharacteristic
8#2395C2211yescharacteristic
9#1923—21no—
10#1916—21no—
11#1870C2211yescharacteristic
12#1851C411yescharacteristic
13#1818C411yescharacteristic
14#1810—21no—
15#1800C2212yes—
16#1771—21no—
17#1752—21no—
18#1752C411yescharacteristic
19#1566—21no—
20#1465C2212yes—
21#1353—22no—
22#1346—22no—
23#1293—21no—
24#1292—21no—
25C22×M16C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C4×D4C2311yescharacteristic
29C2×#117C2×C411yescharacteristic
30C8×D4—41no—
31C8×D4—41no—
32C2×#108—41no—
33#423—22no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8D411yescharacteristic
36C2×#100—22no—
37#407—22no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8C2×C411yescharacteristic
40C2×C22⋊C8C2×C411yescharacteristic
41C2×C22⋊C8D411yescharacteristic
42#405—41no—
43#405—41no—
44C2×#97—21no—
45C2×#97—21no—
46C2×C4×C8—21no—
47C2×C4×C8D411yescharacteristic
48#400D411yescharacteristic
49#399Q811yescharacteristic
50#398D411yescharacteristic
51#397—21no—
52#397Q811yescharacteristic
53#394—21no—
54C4×C4⋊C4—21no—
55C4×C22⋊C4—21no—
56C4×C22⋊C4C2×C411yescharacteristic
57#343—22no—
58#341—41no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#334—41no—
65#334—41no—
66#324—22no—
67#323—22no—
68C22×D4#4111yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×M16—42no—
72C2×M16—41no—
73C2×M16—42no—
74C2×M16—42no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22⋊C411yescharacteristic
87C4×D4—41no—
88C4×D4—41no—
89#117—41no—
90#117—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C4⋊C411yescharacteristic
94C2×C4⋊C4C4⋊C411yescharacteristic
95C2×C4⋊C4C2×D411yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4C2×Q811yescharacteristic
101C2×C22⋊C4#4111yescharacteristic
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2×D411yescharacteristic
105#108—82no—
106C4⋊C8—41no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110#100—82no—
111C22⋊C8—41no—
112C22⋊C8—41no—
113C22⋊C8—42no—
114C22⋊C8—41no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117#97—42no—
118#97—41no—
119#97—41no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—42no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95C4⋊C411yescharacteristic
128#95C4⋊C411yescharacteristic
129C24#11711yescharacteristic
130C24C22⋊Q811yescharacteristic
131C2×D4—22no—
132C2×D4—41no—
133C2×D4—41no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C2×C4⋊C411yescharacteristic
148C22×C4#12111yescharacteristic
149C22×C4#12111yescharacteristic
150C22×C4C22⋊Q811yescharacteristic
151M16—42no—
152M16—42no—
153M16—82no—
154M16—82no—
155M16—82no—
156M16—82no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—42no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—81no—
174C2×C8—82no—
175C2×C8—82no—
176C2×C8#9911yescharacteristic
177C2×C8#9911yescharacteristic
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—81no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C42—21no—
190C42—21no—
191C42—41no—
192C42—41no—
193C42—41no—
194C42—41no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23#38011yescharacteristic
206D4—44no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
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—41no—
233C2×C4—41no—
234C2×C4—81no—
235C2×C4C2×#9911yescharacteristic
236C2×C4#41011yescharacteristic
237C2×C4#47112yes—
238C8—21no—
239C8—41no—
240C8—41no—
241C8—41no—
242C8—42no—
243C8—81no—
244C8—162no—
245C8—162no—
246C8#35212yes—
247C22—21no—
248C22—21no—
249C22—42no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22#179911yescharacteristic
260C22#181411yescharacteristic
261C22#181511yescharacteristic
262C4—22no—
263C4—21no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4C2×#35211yescharacteristic
272C4#212011yescharacteristic
273C2—21no—
274C2—21no—
275C2—81no—
276C2—81no—
277C2#862911yescharacteristic
278C2#1279411yescharacteristic
279C2#1281011yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4#41
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#215672
#118302
#86142
#86122
#73212