Finite Group Databasebeta

← #140625

Subgroups — #140625

Order 512.

387 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×#1945C211yesmaximal, characteristic
2#12788C211yesmaximal, characteristic
3C4×C22⋊C16C211yesmaximal, characteristic
4#11444C212yesmaximal
5#11441C212yesmaximal
6C22×#108C411yescharacteristic
7C2×#427C411yescharacteristic
8C2×#424C2212yes—
9C22×C4×C8C2211yescharacteristic
10C2×#364—21no—
11#2112C2212yes—
12C2×C22⋊C16C2211yescharacteristic
13C2×C22⋊C16C2211yescharacteristic
14C2×C4×C16—21no—
15#1945—21no—
16#1945—21no—
17#1945—21no—
18#1945—21no—
19C2×#339C412yes—
20#1376—21no—
21#1374—21no—
22#1323—24no—
23#1322—24no—
24C22×M16C2×C412yes—
25C23×C8C2×C411yescharacteristic
26C23×C8C2311yescharacteristic
27C2×#117C2×C412yes—
28C22×C42C2×C411yescharacteristic
29C2×M32—22no—
30C22×C16—21no—
31C22×C16—21no—
32C2×#108—21no—
33C2×#108—21no—
34C2×#108—42no—
35#427—21no—
36#427—21no—
37#427—21no—
38#427—21no—
39C2×#107C811yescharacteristic
40C2×#107C811yescharacteristic
41C2×#106C812yes—
42#424—22no—
43#424—22no—
44#424—22no—
45#424—22no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8—22no—
48C2×C4×C8—42no—
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53#364—42no—
54C22⋊C16—21no—
55C22⋊C16—21no—
56C22⋊C16—22no—
57C22⋊C16—22no—
58C22⋊C16D412yes—
59C22⋊C16D412yes—
60C4×C16—42no—
61#339—22no—
62#339—22no—
63C23×C4—21no—
64C23×C4C22×C411yescharacteristic
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—42no—
68C2×M16—42no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C2×C811yescharacteristic
77C22×C8C2×C811yescharacteristic
78C22×C8M1611yescharacteristic
79C22×C8M1611yescharacteristic
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8#4111yescharacteristic
85C22×C8#4111yescharacteristic
86#117—22no—
87#117—22no—
88#117—22no—
89#117—22no—
90C2×C4⋊C4C2×C812yes—
91C2×C4⋊C4C2×C812yes—
92C2×C22⋊C4—22no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42—41no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99M32—42no—
100M32—42no—
101C2×C16—22no—
102C2×C16—22no—
103C2×C16—41no—
104C2×C16—41no—
105C2×C16—41no—
106C2×C16—41no—
107C2×C16—42no—
108#108—42no—
109#108—42no—
110#107—21no—
111#107—21no—
112#107—21no—
113#107—21no—
114#106—22no—
115#106—22no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4×C8—22no—
121C4×C8—22no—
122C4×C8—41no—
123C4×C8—41no—
124C24#13111yescharacteristic
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4C2×C22⋊C411yescharacteristic
137C22×C4C4×D411yescharacteristic
138C22×C4C4×D411yescharacteristic
139C22×C4C22×C811yescharacteristic
140C22×C4C2×M1611yescharacteristic
141C22×C4#13111yescharacteristic
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—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—42no—
160C2×C8—82no—
161C2×C8C22⋊C811yescharacteristic
162C2×C8C22⋊C811yescharacteristic
163C2×C8C22⋊C811yescharacteristic
164C2×C8C22⋊C811yescharacteristic
165C2×C8#9912yes—
166C2×C8#10112yes—
167C2×C8#12111yescharacteristic
168C2×C8#12111yescharacteristic
169C2×C8#12411yescharacteristic
170C2×C8#12711yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—42no—
181C42—42no—
182C42—42no—
183C42#10112yes—
184C16—42no—
185C16—42no—
186C16—82no—
187C23—22no—
188C23—21no—
189C23—21no—
190C23—41no—
191C23#40811yescharacteristic
192C23C8×D411yescharacteristic
193C23#43611yescharacteristic
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—48no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—82no—
215C2×C4#39011yescharacteristic
216C2×C4C2×C22⋊C811yescharacteristic
217C2×C4C2×#9911yescharacteristic
218C2×C4#41011yescharacteristic
219C2×C4C2×#10111yescharacteristic
220C2×C4C2×#10111yescharacteristic
221C2×C4C8×D411yescharacteristic
222C2×C4#43611yescharacteristic
223C8—21no—
224C8—21no—
225C8—21no—
226C8—21no—
227C8—21no—
228C8—21no—
229C8—82no—
230C8#32412yes—
231C8#32412yes—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—42no—
240C22—42no—
241C22#147011yescharacteristic
242C22#147211yescharacteristic
243C22#195411yescharacteristic
244C22#217012yes—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—82no—
250C4—82no—
251C4C2×#32411yescharacteristic
252C4#146111yescharacteristic
253C4#196411yescharacteristic
254C4#196611yescharacteristic
255C2—21no—
256C2—22no—
257C2—21no—
258C2—41no—
259C2#965011yescharacteristic
260C2#1319511yescharacteristic
261C2#1319711yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1954
prime radicalC24#131
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19452
#127882
C4×C22⋊C162
#114442
#114412