Finite Group Databasebeta

← #124287

Subgroups — #124287

Order 512.

353 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 100 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#21157C211yesmaximal, characteristic
2#12602C211yesmaximal, characteristic
3#8601C211yesmaximal, characteristic
4#7127C212yesmaximal, complemented
5#7126C212yesmaximal, complemented
6C2×#427C411yescharacteristic
7C22×#106C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2316—21no—
10C2×#349—21no—
11C4×M32—21no—
12#1944C411yescharacteristic
13#1944C411yescharacteristic
14#1944C2211yescharacteristic
15#1944C2211yescharacteristic
16#1941—21no—
17#1941—21no—
18#1736—21no—
19#1564—21no—
20#1458—21no—
21#1377C2212yes—
22#1275C2212yes—
23C22×M16D411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C2×#117—21no—
26C2×#117D411yescharacteristic
27C22×C4⋊C4—21no—
28C2×M32—22no—
29#431—22no—
30#427—41no—
31#427—41no—
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#106—21no—
35C2×#106—22no—
36C2×#106—21no—
37C2×#106C814yescomplemented
38C2×#106C2×C411yescharacteristic
39C2×#106C2×C411yescharacteristic
40#405—41no—
41#405—41no—
42C2×#97C2×C411yescharacteristic
43C2×#97C2311yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#389—21no—
47#389—21no—
48#389—21no—
49#389—21no—
50#389—41no—
51#389—41no—
52#380—41no—
53#380—41no—
54#349—24no—
55#346—41no—
56C4×C16—41no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61#335—41no—
62#334—41no—
63#324—41no—
64#323—41no—
65C23×C4—41no—
66C23×C4C22⋊C411yescharacteristic
67C2×M16—22no—
68C2×M16—22no—
69C2×M16—41no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—22no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C2×C812yes—
76C22×C8M1612yes—
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79#117—41no—
80#117—41no—
81#117—42no—
82#117—42no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4D811yescharacteristic
95C2×C4⋊C4SD1611yescharacteristic
96C2×C22⋊C4—41no—
97C2×C22⋊C4D811yescharacteristic
98C2×C22⋊C4SD1611yescharacteristic
99C2×C42—41no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42C2×D411yescharacteristic
102M32—24no—
103M32—42no—
104C2×C16—42no—
105#107—41no—
106#107—41no—
107#106—24no—
108#106—21no—
109#106—21no—
110#106—41no—
111#106—41no—
112#106C2×C812yes—
113#106C2×C812yes—
114C4⋊C8—42no—
115C22⋊C8—42no—
116#97—41no—
117#97—41no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—42no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127C24#10211yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—44no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—161no—
138C22×C4—161no—
139C22×C4C22⋊C812yes—
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4#10211yescharacteristic
143C22×C4#10412yes—
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C2×D811yescharacteristic
146C22×C4C2×SD1611yescharacteristic
147M16—44no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—82no—
160C2×C8C22⋊C812yes—
161C2×C8#9911yescharacteristic
162C2×C8#9911yescharacteristic
163C2×C8#10112yes—
164C2×C8C22×C811yescharacteristic
165C2×C8C2×M1611yescharacteristic
166C2×C8#13712yes—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—44no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—161no—
179C4⋊C4—161no—
180C4⋊C4—161no—
181C4⋊C4—161no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—82no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—82no—
190C16—44no—
191C23—21no—
192C23—21no—
193C23—41no—
194C23—41no—
195C23D4⋊C812yes—
196C23C2×#10211yescharacteristic
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—164no—
224C2×C4D4⋊C812yes—
225C2×C4C2×C22⋊C811yescharacteristic
226C2×C4C2×#9911yescharacteristic
227C2×C4C2×#10111yescharacteristic
228C2×C4#41911yescharacteristic
229C2×C4C2×#10411yescharacteristic
230C2×C4#42111yescharacteristic
231C8—21no—
232C8—21no—
233C8—22no—
234C8—41no—
235C8—162no—
236C8#32412yes—
237C8#32412yes—
238C22—21no—
239C22—21no—
240C22—22no—
241C22—42no—
242C22—42no—
243C22—41no—
244C22—41no—
245C22C2×D4⋊C811yescharacteristic
246C22#151911yescharacteristic
247C22#154611yescharacteristic
248C4—22no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—82no—
254C4—82no—
255C4—161no—
256C4—161no—
257C4C2×#32411yescharacteristic
258C4#147911yescharacteristic
259C4#150811yescharacteristic
260C4#152811yescharacteristic
261C2—21no—
262C2—21no—
263C2—42no—
264C2#757311yescharacteristic
265C2#1186711yescharacteristic
266C2#1187211yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC2×C8C22×C8
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7127C2semidirect
5. #7126C2semidirect
37. C2×#106C8semidirect

Maximal subgroups

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

subgroupindex
#211572
#126022
#86012
#71272
#71262