Finite Group Databasebeta

← #141966

Subgroups — #141966

Order 512.

338 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 96 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#21025C211yesmaximal, characteristic
2#12786C211yesmaximal, characteristic
3#12602C211yesmaximal, characteristic
4#7194C212yesmaximal
5#7193C212yesmaximal
6C2×#424C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2285C411yescharacteristic
9#2150—21no—
10C2×#349C2212yes—
11C2×#349C2212yes—
12C4×M32—21no—
13#1917—21no—
14#1917—21no—
15#1916—21no—
16#1916—21no—
17#1729C411yescharacteristic
18#1729C2211yescharacteristic
19#1728C2211yescharacteristic
20C2×#336C411yescharacteristic
21#1377—21no—
22#1377—21no—
23C22×M16C2×C411yescharacteristic
24C22×M16C2311yescharacteristic
25C23×C8C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C2×#117C2×C411yescharacteristic
28C2×#117C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C2×M32—22no—
31C2×M32—22no—
32#424—41no—
33#424—41no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36#405—41no—
37#405—41no—
38C2×#97D411yescharacteristic
39C2×#97D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#384C811yescharacteristic
43#384C811yescharacteristic
44#382C811yescharacteristic
45#382C811yescharacteristic
46#380—21no—
47#380—21no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4—21no—
50C4⋊C16—41no—
51C4⋊C16—41no—
52#349—22no—
53#349—24no—
54#349Q814yes—
55#346—41no—
56C4×C16—41no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65C23×C4—21no—
66C23×C4C22×C411yescharacteristic
67C2×M16—22no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16C2×Q812yes—
73C2×M16#4112yes—
74C22×C8—22no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87#117—41no—
88#117—41no—
89#117—41no—
90#117—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4C2×C811yescharacteristic
96C2×C4⋊C4C2×C811yescharacteristic
97C2×C4⋊C4C2×C811yescharacteristic
98C2×C4⋊C4C2×C811yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4M1611yescharacteristic
101C2×C22⋊C4M1611yescharacteristic
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C22⋊C411yescharacteristic
108C2×C42C2×C811yescharacteristic
109C2×C42C2×C811yescharacteristic
110M32—24no—
111M32—42no—
112M32—42no—
113M32—42no—
114C2×C16—42no—
115C2×C16—42no—
116C4⋊C8—42no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#97—41no—
120#97—41no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129C24C2×M1611yescharacteristic
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—81no—
141C22×C4C22⋊C811yescharacteristic
142C22×C4C22⋊C811yescharacteristic
143C22×C4C22⋊C811yescharacteristic
144C22×C4C22⋊C811yescharacteristic
145C22×C4C2×C22⋊C411yescharacteristic
146C22×C4C4×Q812yes—
147C22×C4C22×C811yescharacteristic
148C22×C4#13112yes—
149M16—42no—
150M16—42no—
151M16—82no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—82no—
170C2×C8#9911yescharacteristic
171C2×C8#9911yescharacteristic
172C2×C8C22⋊Q812yes—
173C2×C8#12411yescharacteristic
174C2×C8C4⋊Q811yescharacteristic
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—41no—
191C42—41no—
192C42—41no—
193C42—41no—
194C42—41no—
195C42—41no—
196C16—44no—
197C16—82no—
198C23—21no—
199C23—21no—
200C23—41no—
201C23—41no—
202C23C2×C22⋊C811yescharacteristic
203C23#44612yes—
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—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—84no—
231C2×C4#39111yescharacteristic
232C2×C4#40811yescharacteristic
233C2×C4C2×#9911yescharacteristic
234C2×C4#41011yescharacteristic
235C2×C4#41311yescharacteristic
236C2×C4#41311yescharacteristic
237C2×C4C8×Q812yes—
238C8—21no—
239C8—21no—
240C8—22no—
241C8—21no—
242C8—21no—
243C8—41no—
244C8—81no—
245C8—81no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—41no—
252C22—41no—
253C22#145811yescharacteristic
254C22#145911yescharacteristic
255C22#198411yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4#146811yescharacteristic
266C4#147311yescharacteristic
267C4#199611yescharacteristic
268C4#199811yescharacteristic
269C2—21no—
270C2—21no—
271C2—42no—
272C2#988011yescharacteristic
273C2#1148511yescharacteristic
274C2#1148611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22#1459
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#210252
#127862
#126022
#71942
#71932