Finite Group Databasebeta

← #125290

Subgroups — #125290

Order 512.

377 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 82 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#33136C211yesmaximal, characteristic
2C2×#1377C211yesmaximal, characteristic
3#7861C211yesmaximal, characteristic
4#7126C212yesmaximal, complemented
5#7111C212yesmaximal, complemented
6C22×M32—21no—
7C2×#467C411yescharacteristic
8#2920—21no—
9C22×#97C2211yescharacteristic
10#2855—21no—
11#1971C411yescharacteristic
12#1971C2211yescharacteristic
13#1944C411yescharacteristic
14#1944C2211yescharacteristic
15#1935—21no—
16#1935—21no—
17#1892—21no—
18#1807C411yescharacteristic
19#1377—21no—
20#1377C2212yes—
21#1275—21no—
22#1275C2212yes—
23C22×Q16C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C22⋊Q8—21no—
27C22×C42D411yescharacteristic
28C2×M32—24no—
29C2×M32—22no—
30C2×M32—41no—
31C2×M32—42no—
32C22×C16—41no—
33#467—41no—
34#467—41no—
35#432—41no—
36#432—41no—
37#431—22no—
38#431—22no—
39C2×#107C2×C411yescharacteristic
40C2×#106C2×C411yescharacteristic
41C2×#106C2×C411yescharacteristic
42#423—41no—
43C2×#103—21no—
44C2×#103—21no—
45C2×#103—21no—
46#407—41no—
47C4×M16—22no—
48C2×#97—41no—
49C2×#97—41no—
50C2×#97C2×C411yescharacteristic
51C2×#97D411yescharacteristic
52C2×#97D411yescharacteristic
53C2×#97C2311yescharacteristic
54#391—21no—
55#391—21no—
56#389—21no—
57#389—21no—
58#387—41no—
59#387—41no—
60#385—41no—
61#381—41no—
62#380—21no—
63#340—21no—
64#340—21no—
65#340—21no—
66C22×Q8—21no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4D811yescharacteristic
69C23×C4SD1611yescharacteristic
70C2×Q16—22no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×M16—42no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22×C411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22⋊Q8—41no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42D811yescharacteristic
105C2×C42SD1611yescharacteristic
106C2×C42C2×D411yescharacteristic
107M32—24no—
108M32—44no—
109M32—42no—
110C2×C16—44no—
111C2×C16—42no—
112#107—41no—
113#106—22no—
114#106—41no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117#103—42no—
118#103—42no—
119#103—41no—
120#103—41no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123#97—41no—
124#97—41no—
125#97—41no—
126#97—41no—
127#97—42no—
128C4×C8—24no—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136C24#10211yescharacteristic
137C2×Q8—41no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—81no—
157C22×C4#9911yescharacteristic
158C22×C4#9911yescharacteristic
159C22×C4#10211yescharacteristic
160C22×C4#10211yescharacteristic
161C22×C4#10211yescharacteristic
162C22×C4#10411yescharacteristic
163C22×C4#10411yescharacteristic
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C2×D811yescharacteristic
166C22×C4C2×SD1611yescharacteristic
167Q16—82no—
168M16—84no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—44no—
185C2×C8#10412yes—
186C2×C8#13112yes—
187C2×C8AGL(1,ℤ/8)11yescharacteristic
188C2×C8#13711yescharacteristic
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C22⋊C4—82no—
200C22⋊C4—81no—
201C22⋊C4—81no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C16—44no—
211C23—21no—
212C23—22no—
213C23—21no—
214C23—41no—
215C23#32711yescharacteristic
216C23#32711yescharacteristic
217C23C2×#10211yescharacteristic
218Q8—81no—
219Q8—81no—
220Q8—81no—
221Q8—81no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4#32711yescharacteristic
257C2×C4#32711yescharacteristic
258C2×C4#40811yescharacteristic
259C2×C4C2×#9911yescharacteristic
260C2×C4#41311yescharacteristic
261C2×C4#41711yescharacteristic
262C2×C4C2×#10411yescharacteristic
263C2×C4C2×#10411yescharacteristic
264C8—22no—
265C8—21no—
266C8—22no—
267C8—21no—
268C8—41no—
269C8—84no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22#134012yes—
280C22#149211yescharacteristic
281C22C2×#32711yescharacteristic
282C22#154711yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—161no—
290C4—161no—
291C4—161no—
292C4—161no—
293C4#146811yescharacteristic
294C4#148311yescharacteristic
295C4#150411yescharacteristic
296C4#152411yescharacteristic
297C2—21no—
298C2—21no—
299C2—22no—
300C2—41no—
301C2#767811yescharacteristic
302C2C2×#134011yescharacteristic
303C2#1168811yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7126C2semidirect
5. #7111C2semidirect

Maximal subgroups

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

subgroupindex
#331362
C2×#13772
#78612
#71262
#71112