Finite Group Databasebeta

← #141995

Subgroups — #141995

Order 512.

354 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 88 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#21147C211yesmaximal, characteristic
2#12790C211yesmaximal, characteristic
3#12608C211yesmaximal, characteristic
4#7194C212yesmaximal
5#7191C212yesmaximal
6C2×#424C411yescharacteristic
7C2×#424C411yescharacteristic
8C22×C4⋊C8C411yescharacteristic
9C2×#405C2211yescharacteristic
10#2296C411yescharacteristic
11C2×#364—21no—
12C2×#349C2212yes—
13C2×#349C2212yes—
14C2×#347—21no—
15#1940—21no—
16#1940—21no—
17#1939—21no—
18#1939—21no—
19#1729C2211yescharacteristic
20#1728C2211yescharacteristic
21#1376—21no—
22#1375—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—42no—
33#424—42no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—42no—
41#405—42no—
42C2×#97D411yescharacteristic
43C2×#97D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#389—21no—
47#389—21no—
48#385—21no—
49#385—21no—
50C4×C4⋊C4—21no—
51C4×C4⋊C4—21no—
52#364—42no—
53#349—22no—
54#349—24no—
55#349Q814yes—
56#347—42no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61C23×C4M1611yescharacteristic
62C23×C4M1611yescharacteristic
63C23×C4C22×C411yescharacteristic
64C2×M16—22no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16C2×Q812yes—
70C2×M16#4112yes—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—22no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83#117—22no—
84#117—22no—
85#117—41no—
86#117—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4M1611yescharacteristic
93C2×C4⋊C4M1611yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42C22⋊C411yescharacteristic
102C2×C42C22⋊C411yescharacteristic
103M32—24no—
104M32—42no—
105M32—42no—
106M32—42no—
107C2×C16—42no—
108C2×C16—42no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115#97—22no—
116#97—41no—
117C4×C8—22no—
118C4×C8—41no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123C24C2×M1611yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—81no—
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C4×Q812yes—
140C22×C4C2×M1611yescharacteristic
141C22×C4#13112yes—
142M16—42no—
143M16—42no—
144M16—82no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—44no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—82no—
162C2×C8#10012yes—
163C2×C8C22⋊Q812yes—
164C2×C8#12411yescharacteristic
165C2×C8C4⋊Q811yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C22⋊C4—24no—
174C22⋊C4—42no—
175C42—21no—
176C42—21no—
177C42—42no—
178C42—41no—
179C42—41no—
180C42—41no—
181C42—41no—
182C42#9912yes—
183C42#10112yes—
184C16—44no—
185C16—82no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—41no—
190C23#40711yescharacteristic
191C23#44612yes—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—82no—
219C2×C4#39111yescharacteristic
220C2×C4#40811yescharacteristic
221C2×C4C2×#9911yescharacteristic
222C2×C4#41011yescharacteristic
223C2×C4C2×#10011yescharacteristic
224C2×C4C2×#10111yescharacteristic
225C2×C4#44612yes—
226C8—22no—
227C8—22no—
228C8—21no—
229C8—21no—
230C8—41no—
231C8—81no—
232C8—81no—
233C22—21no—
234C22—21no—
235C22—22no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22#146311yescharacteristic
241C22#146611yescharacteristic
242C22#198511yescharacteristic
243C4—22no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—81no—
248C4—81no—
249C4—81no—
250C4#147011yescharacteristic
251C4#147111yescharacteristic
252C4#199611yescharacteristic
253C4#199711yescharacteristic
254C2—21no—
255C2—21no—
256C2—42no—
257C2#988311yescharacteristic
258C2#1153711yescharacteristic
259C2#1153811yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC22×M16C23
socleC22#1466
prime radicalC24C2×M16
LCSG ▷ C23×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
#211472
#127902
#126082
#71942
#71912