Finite Group Databasebeta

← #330532

Subgroups — #330532

Order 512.

551 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 154 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#33294C211yesmaximal, characteristic, complemented
2#29998C211yesmaximal, characteristic
3#29868C211yesmaximal, characteristic
4#21268C211yesmaximal, characteristic
5#10601C212yesmaximal, complemented
6#10597C212yesmaximal, complemented
7#10481C212yesmaximal
8#10480C212yesmaximal
9#9897C211yesmaximal, characteristic, complemented
10#9691C211yesmaximal, characteristic, complemented
11#8808C211yesmaximal, characteristic, complemented
12C2×#476C2211yescharacteristic
13C2×#476C2211yescharacteristic
14C2×#433—21no—
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17#2293C2211yescharacteristic
18#2088—22no—
19#2084—22no—
20#2071—22no—
21#2065—22no—
22#1992—22no—
23#1990C2212yes—
24#1990C2212yes—
25#1987C2211yescharacteristic
26#1987C2211yescharacteristic
27#1954C2212yes—
28#1948C2212yes—
29#1939C2212yes—
30#1924—22no—
31#1842C2211yescharacteristic
32#1836—21no—
33#1836—21no—
34#1831C2211yescharacteristic
35#1802C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37#1750—21no—
38C4×#97C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—21no—
49C2×C4×D4C2311yescharacteristic
50C2×#117—21no—
51#476—42no—
52#476—42no—
53#433—44no—
54C2×#106—21no—
55C2×#106—21no—
56C2×#106—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×#102—21no—
64C2×#102—21no—
65C2×#102—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×#97D411yescharacteristic
69C2×#97D412yes—
70C2×#97D411yescharacteristic
71C2×C4×C8—21no—
72C2×C4×C8C2312yes—
73#390D412yes—
74#389—22no—
75#389—22no—
76#389D412yes—
77#388—22no—
78#384—22no—
79#384—21no—
80#384—21no—
81#384C2312yes—
82#382—21no—
83C4×C4⋊C4C2311yescharacteristic
84C4×C22⋊C4—21no—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#336—22no—
95#336—22no—
96#336—21no—
97#335—42no—
98#334—42no—
99#333—42no—
100#333—42no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×D4C2×D411yescharacteristic
104C23×C4—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—84no—
116#125—44no—
117C4×D4—22no—
118C4×D4—22no—
119#117—42no—
120#117—42no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—22no—
137C2×C42—21no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42#4112yes—
142C2×C42C2411yescharacteristic
143#106—42no—
144#106—42no—
145#106—82no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—44no—
153C4⋊C8—82no—
154#102—42no—
155#102—42no—
156#102—82no—
157C22⋊C8—44no—
158C22⋊C8—44no—
159#97—22no—
160#97—42no—
161#97SD1614yes—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—42no—
165#95—22no—
166#95—21no—
167#95—42no—
168C24—21no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×Q8—82no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4#12112yes—
187C22×C4#12112yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C2×#4111yescharacteristic
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8C2×SD1614yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—44no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—84no—
223C4⋊C4—84no—
224C4⋊C4—84no—
225C22⋊C4—42no—
226C22⋊C4—44no—
227C22⋊C4—82no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—42no—
232C42—41no—
233C42—41no—
234C42C2×SD1612yes—
235C42#13712yes—
236C42H(5,𝔽2)12yes—
237C422−1+412yes—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—42no—
243C23C2×#12111yescharacteristic
244Q8—162no—
245Q8—162no—
246D4—41no—
247D4—41no—
248D4—42no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4#46114yes—
271C2×C4#53511yescharacteristic
272C2×C4#53611yescharacteristic
273C2×C4#53712yes—
274C2×C4#54212yes—
275C2×C4C22×SD1611yescharacteristic
276C2×C4C2×#13711yescharacteristic
277C2×C4#57512yes—
278C2×C4#57712yes—
279C2×C4#57812yes—
280C8—24no—
281C8—42no—
282C8—81no—
283C8—82no—
284C8—81no—
285C8—82no—
286C22—82no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—82no—
291C22#262011yescharacteristic
292C22C2×#46111yescharacteristic
293C22#304011yescharacteristic
294C22#306312yes—
295C22#306812yes—
296C4—24no—
297C4—22no—
298C4—42no—
299C4—82no—
300C4—81no—
301C4—161no—
302C4—161no—
303C4—162no—
304C4#322111yescharacteristic
305C4#322311yescharacteristic
306C4#324212yes—
307C4#324612yes—
308C4#325212yes—
309C2—161no—
310C2#1923312yes—
311C2#1938212yes—
312C2#2982811yescharacteristic
313C2#2985611yescharacteristic
314C2#2986111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33294C2semidirect
5. #10601C2semidirect
6. #10597C2semidirect
9. #9897C2semidirect
10. #9691C2semidirect
11. #8808C2semidirect

Maximal subgroups

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

subgroupindex
#332942
#299982
#298682
#212682
#106012
#105972
#104812
#104802
#98972
#96912
#88082