Finite Group Databasebeta

← #140634

Subgroups — #140634

Order 512.

383 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 104 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#21156C211yesmaximal, characteristic
2#11454C212yesmaximal
3#11450C212yesmaximal
4#8601C211yesmaximal, characteristic
5#8156C211yesmaximal, characteristic
6C2×#427C411yescharacteristic
7C22×#107C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2317C2212yes—
10#1944—21no—
11#1944—21no—
12#1942—21no—
13#1942—21no—
14#1736C412yes—
15#1564—21no—
16#1458C2212yes—
17#1458C2211yescharacteristic
18#1458C2211yescharacteristic
19#1450—21no—
20#1326—24no—
21#1325—24no—
22#1281—21no—
23#1275—21no—
24C22×M16C2×C411yescharacteristic
25C23×C8C2×C411yescharacteristic
26C2×#117C2×C412yes—
27C2×#117C2311yescharacteristic
28C22×C4⋊C4C2×C412yes—
29#431—21no—
30#431—21no—
31#431—22no—
32#427—22no—
33#427C814yescomplemented
34C2×#107—21no—
35C2×#107—21no—
36C2×#107—21no—
37C2×#107—41no—
38C2×#107—41no—
39C2×#106—21no—
40#405—41no—
41#405—41no—
42C2×#97D411yescharacteristic
43C2×#97D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#389—22no—
47#389—22no—
48#385—22no—
49#385—22no—
50#384—22no—
51#382—22no—
52#340—22no—
53#340—22no—
54#335—41no—
55#334—41no—
56#324—21no—
57#324—22no—
58#324D412yes—
59#323—21no—
60#323—22no—
61#323D412yes—
62#322—41no—
63C82—41no—
64C23×C4—22no—
65C23×C4C22×C411yescharacteristic
66C2×M16—21no—
67C2×M16—21no—
68C2×M16—22no—
69C2×M16—41no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—22no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C2×C812yes—
78C22×C8M1612yes—
79#117—22no—
80#117—22no—
81#117—22no—
82#117—41no—
83#117—41no—
84#117C2×C814yes—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4C2×D411yescharacteristic
93C2×C4⋊C4#4111yescharacteristic
94C2×C22⋊C4—22no—
95C2×C22⋊C4C2×D411yescharacteristic
96C2×C22⋊C4#4111yescharacteristic
97C2×C42—22no—
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D411yescharacteristic
100#107—24no—
101#107—42no—
102#107—41no—
103#107—41no—
104#106—24no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—42no—
108C22⋊C8—22no—
109C22⋊C8—22no—
110C22⋊C8—42no—
111#97—41no—
112#97—41no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—41no—
118C4×C8—41no—
119C4×C8—42no—
120#95—42no—
121#95—42no—
122C24C4×D411yescharacteristic
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—82no—
130C22×C4—162no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C4×D411yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4#12411yescharacteristic
136C22×C4#12711yescharacteristic
137C22×C4C22×C811yescharacteristic
138C22×C4C2×M1611yescharacteristic
139C22×C4#13111yescharacteristic
140C22×C4#13111yescharacteristic
141M16—42no—
142M16—42no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—82no—
157C2×C8C22⋊C812yes—
158C2×C8C22⋊C812yes—
159C2×C8#9911yescharacteristic
160C2×C8#9911yescharacteristic
161C2×C8#10112yes—
162C2×C8#10112yes—
163C4⋊C4—24no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—82no—
170C4⋊C4—162no—
171C4⋊C4—162no—
172C22⋊C4—24no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—42no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—24no—
181C42—42no—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23#39011yescharacteristic
187C23C8×D411yescharacteristic
188C23#43611yescharacteristic
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—44no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—164no—
214C2×C4C2×C22⋊C811yescharacteristic
215C2×C4#40811yescharacteristic
216C2×C4C2×#9911yescharacteristic
217C2×C4#41011yescharacteristic
218C2×C4C2×#10111yescharacteristic
219C2×C4C2×#10111yescharacteristic
220C2×C4C8×D411yescharacteristic
221C2×C4#43611yescharacteristic
222C8—22no—
223C8—22no—
224C8—41no—
225C8—82no—
226C8—82no—
227C8—162no—
228C8#32412yes—
229C8#32412yes—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22#195411yescharacteristic
240C22#196411yescharacteristic
241C22#196611yescharacteristic
242C4—21no—
243C4—21no—
244C4—41no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—84no—
249C4—162no—
250C4C2×#32411yescharacteristic
251C4#146111yescharacteristic
252C4#147011yescharacteristic
253C4#147211yescharacteristic
254C2—21no—
255C2—21no—
256C2—41no—
257C2—41no—
258C2#965011yescharacteristic
259C2#1323111yescharacteristic
260C2#1323311yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC22×C4C22×C8
Frattini, cosocleC2×#117C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
33. #427C8semidirect

Maximal subgroups

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

subgroupindex
#211562
#114542
#114502
#86012
#81562