Finite Group Databasebeta

← #122605

Subgroups — #122605

Order 512.

329 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 76 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#29771C211yesmaximal, characteristic
2C2×#1322C211yesmaximal, characteristic
3C2×#1290C211yesmaximal, characteristic
4#7082C212yesmaximal, complemented
5#7081C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C2×#432—21no—
8C22×C4⋊C8C2211yescharacteristic
9#2855—21no—
10C2×C22⋊C16—21no—
11#2087C411yescharacteristic
12#2087C2211yescharacteristic
13#2068—21no—
14#2068—21no—
15#2056C411yescharacteristic
16#2056C2211yescharacteristic
17#1892—21no—
18#1808C411yescharacteristic
19#1322—21no—
20#1322C2212yes—
21#1290—21no—
22#1290C2212yes—
23C22×Q16C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C22⋊Q8—21no—
27C22×C42D411yescharacteristic
28C22×C16—41no—
29#467—41no—
30#467—41no—
31#432—22no—
32#432—22no—
33#432—41no—
34#432—41no—
35C2×#107—21no—
36C2×#107C2×C411yescharacteristic
37C2×C4⋊C8—41no—
38C2×C4⋊C8—41no—
39C2×C4⋊C8—41no—
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×C4⋊C8D411yescharacteristic
43C2×C4⋊C8C2311yescharacteristic
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×C22⋊C8—41no—
48C2×#97—22no—
49#391—21no—
50#391—21no—
51#389—21no—
52#389—21no—
53#387—41no—
54#387—41no—
55#385—41no—
56#381—41no—
57#380—21no—
58C22⋊C16—22no—
59C22⋊C16—41no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C22×Q8—21no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4SD1611yescharacteristic
68C2×Q16—41no—
69C2×Q16—81no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—42no—
75C22×C8—81no—
76C22×C8—81no—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42D811yescharacteristic
99C2×C42SD1611yescharacteristic
100C2×C42C2×D411yescharacteristic
101C2×C16—42no—
102C2×C16—82no—
103#107—41no—
104#107—81no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111#103—42no—
112#103—81no—
113#103—81no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116#97—24no—
117#97—42no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125C24#10211yescharacteristic
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—81no—
145C22×C4#9911yescharacteristic
146C22×C4#9911yescharacteristic
147C22×C4#10211yescharacteristic
148C22×C4#10211yescharacteristic
149C22×C4#10211yescharacteristic
150C22×C4#10411yescharacteristic
151C22×C4#10411yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×D811yescharacteristic
154C22×C4C2×SD1611yescharacteristic
155Q16—82no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—42no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—44no—
166C2×C8—81no—
167C2×C8—81no—
168C2×C8—81no—
169C2×C8—84no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—81no—
181C22⋊C4—81no—
182C22⋊C4—82no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C16—82no—
192C23—41no—
193C23—41no—
194C23#32711yescharacteristic
195C23#32711yescharacteristic
196C23#35912yes—
197C23#36212yes—
198C23C2×#10211yescharacteristic
199Q8—81no—
200Q8—81no—
201Q8—81no—
202Q8—81no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4#32711yescharacteristic
239C2×C4#32711yescharacteristic
240C2×C4C2×#9911yescharacteristic
241C2×C4C2×#10411yescharacteristic
242C8—41no—
243C8—41no—
244C8—41no—
245C8—41no—
246C8—84no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—42no—
254C22—42no—
255C22#135012yes—
256C22#135012yes—
257C22C2×#32711yescharacteristic
258C22C2×#35911yescharacteristic
259C22C2×#36211yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—161no—
268C4—161no—
269C4—161no—
270C4—161no—
271C4#212111yescharacteristic
272C4#213311yescharacteristic
273C2—21no—
274C2—22no—
275C2—22no—
276C2—21no—
277C2#1168511yescharacteristic
278C2#1169811yescharacteristic
279C2C2×#135011yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7082C2semidirect
5. #7081C2semidirect

Maximal subgroups

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

subgroupindex
#297712
C2×#13222
C2×#12902
#70822
#70812