Finite Group Databasebeta

← #345334

Subgroups — #345334

Order 512.

583 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 136 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#26299C212yesmaximal, complemented
2#24677C212yesmaximal
3#10593C212yesmaximal, complemented
4#10481C212yesmaximal, complemented
5#10056C212yesmaximal, complemented
6#9835C211yesmaximal, characteristic, complemented
7#9663C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic, complemented
9#8897C212yesmaximal, complemented
10C2×#494C2211yescharacteristic
11C2×#476C2212yes—
12C2×C4×D8C2211yescharacteristic
13#2563C2212yes—
14#2354C2212yes—
15#2077—22no—
16#2075—22no—
17#2073—22no—
18#2054—22no—
19#1990C2212yes—
20#1990C2212yes—
21#1987C2212yes—
22#1954C2212yes—
23#1949—22no—
24#1947C2212yes—
25#1939C2212yes—
26#1926—22no—
27#1854—22no—
28#1834C2211yescharacteristic
29#1825—22no—
30#1802C2212yes—
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#333C2212yes—
35C2×D4⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C22×D8D411yescharacteristic
41C2×C4⋊Q8C2312yes—
42C2×C4×D4C2312yes—
43C2×#117—22no—
44#494—22no—
45#494—41no—
46#476—42no—
47#476—42no—
48C4×D8—41no—
49C4×D8—41no—
50C2×#107—22no—
51C2×#107D411yescharacteristic
52C2×#106—22no—
53C2×#106D411yescharacteristic
54C2×#106D411yescharacteristic
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×#102—22no—
59C2×#102—22no—
60C2×#102D412yes—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#393—42no—
67#390—22no—
68#390—22no—
69#388—42no—
70#384C2312yes—
71#382—22no—
72#380—22no—
73#380—22no—
74#380—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2312yes—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—41no—
86#340D412yes—
87#336—22no—
88#335—41no—
89#335—41no—
90#333—42no—
91D4⋊C8—24no—
92#322—41no—
93C22×Q8—22no—
94C22×D4C2×D412yes—
95C23×C4—22no—
96C2×Q16—44no—
97C2×Q16—84no—
98C2×D8—42no—
99C2×D8—41no—
100C22×C8—21no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4×D4—22no—
112C4×D4—22no—
113C4×D4—22no—
114C4×D4—22no—
115#117—44no—
116#117—82no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D412yes—
124C2×C4⋊C4C2×D412yes—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C42—22no—
131C2×C42—42no—
132C2×C42C2411yescharacteristic
133#107—41no—
134#107—82no—
135#106—22no—
136#106—41no—
137#106—44no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—44no—
143#102—42no—
144#102—44no—
145#102—44no—
146C22⋊C8—44no—
147C22⋊C8—82no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8#4112yes—
154#95—22no—
155#95—42no—
156#95—42no—
157#95—42no—
158C24—22no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—82no—
162C2×Q8—82no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D411yescharacteristic
182C22×C4H(5,𝔽2)11yescharacteristic
183Q16—88no—
184D8—44no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—21no—
193C2×C8—22no—
194C2×C8—21no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8#13512yes—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—82no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—84no—
221C22⋊C4—162no—
222C22⋊C4—162no—
223C22⋊C4—162no—
224C42—42no—
225C42—42no—
226C42—84no—
227C42C2×#4111yescharacteristic
228C42H(5,𝔽2)11yescharacteristic
229C42H(5,𝔽2)11yescharacteristic
230C422−1+411yescharacteristic
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23D4211yescharacteristic
239Q8—84no—
240Q8—162no—
241D4—42no—
242D4—42no—
243D4—42no—
244D4—42no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4#53411yescharacteristic
266C2×C4#53711yescharacteristic
267C2×C4#53811yescharacteristic
268C2×C4#54711yescharacteristic
269C2×C4#55011yescharacteristic
270C2×C4#56211yescharacteristic
271C2×C4C2×#13511yescharacteristic
272C2×C4#57512yes—
273C2×C4#57511yescharacteristic
274C2×C4#57611yescharacteristic
275C2×C4#57812yes—
276C2×C4#57811yescharacteristic
277C8—22no—
278C8—42no—
279C8—41no—
280C8—41no—
281C8—41no—
282C8—82no—
283C8—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22#268211yescharacteristic
292C22#326912yes—
293C22#327311yescharacteristic
294C22#328311yescharacteristic
295C22#328812yes—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—162no—
302C4—162no—
303C4#319711yescharacteristic
304C4#320211yescharacteristic
305C4#325011yescharacteristic
306C4#326211yescharacteristic
307C4#334912yes—
308C4#339611yescharacteristic
309C4#340211yescharacteristic
310C2—162no—
311C2#1986412yes—
312C2#2022612yes—
313C2#3120511yescharacteristic
314C2#3121011yescharacteristic
315C2#3123111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26299C2semidirect
3. #10593C2semidirect
4. #10481C2semidirect
5. #10056C2semidirect
6. #9835C2semidirect
8. #9596C2semidirect
9. #8897C2semidirect

Maximal subgroups

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

subgroupindex
#262992
#246772
#105932
#104812
#100562
#98352
#96632
#95962
#88972