Finite Group Databasebeta

← #329591

Subgroups — #329591

Order 512.

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

normal subgroups only

This group has 830 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#32903C212yesmaximal, complemented
2#29806C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic, complemented
4#10623C212yesmaximal, complemented
5#10474C214yesmaximal, complemented
6#10262C212yesmaximal, complemented
7#9134C212yesmaximal, complemented
8#8599C211yesmaximal, characteristic, complemented
9C22×C4×Q8C2211yescharacteristic
10C2×#493C2211yescharacteristic
11C2×#460C2214yes—
12C2×#448—22no—
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2856C2212yes—
16#2054—22no—
17#1999—22no—
18#1984—22no—
19#1984C2212yes—
20#1984C2212yes—
21#1957C2212yes—
22#1880—24no—
23#1874—22no—
24#1834C2211yescharacteristic
25#1817—22no—
26#1803C2214yes—
27#1797C2212yes—
28C2×#335C2212yes—
29C2×#331C2214yes—
30#1296C2214yes—
31#1277C2212yes—
32C23×Q8D411yescharacteristic
33C22×SD16—24no—
34C22×D8—22no—
35C2×#127C2312yes—
36C2×#121—22no—
37C2×#121—22no—
38C2×#121—22no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—42no—
42C2×C4×Q8—42no—
43C2×C4×Q8D412yes—
44C2×C4×Q8C2312yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D411yescharacteristic
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#493—42no—
50#460—44no—
51#460—44no—
52#448—48no—
53#431—42no—
54#431—42no—
55C2×C4⋊C8D412yes—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2314yes—
58C2×#103—24no—
59C2×#102—24no—
60C2×#102—24no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8D412yes—
64C2×C4×C8—21no—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#390—22no—
68#390—22no—
69#390—22no—
70#384C2312yes—
71#340—22no—
72#340—24no—
73#336—24no—
74#336—24no—
75#336—22no—
76#335—42no—
77#331—44no—
78C22×Q8—21no—
79C22×Q8—22no—
80C22×Q8—42no—
81C22×Q8—42no—
82C22×Q8D814yescomplemented
83C22×Q8C2×D412yes—
84C22×D4—22no—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—42no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4#4111yescharacteristic
93C2×SD16—48no—
94C2×SD16—48no—
95C2×SD16—48no—
96C2×D8—44no—
97C2×D8—84no—
98C22×C8—24no—
99C22×C8—22no—
100C22×C8C2×D412yes—
101C22×C8C2×D412yes—
102#127—24no—
103#127—24no—
104#121—44no—
105#121—44no—
106#121—84no—
107#121—84no—
108C4×Q8—24no—
109C4×Q8—24no—
110C4×Q8—44no—
111C4×Q8—42no—
112C4×Q8—42no—
113C4×Q8—44no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4C2×D412yes—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42#4111yescharacteristic
135C2×C42C2411yescharacteristic
136C4⋊C8—24no—
137C4⋊C8—24no—
138C4⋊C8—24no—
139C4⋊C8—42no—
140#103—48no—
141#102—48no—
142#102—84no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145C22⋊C8—44no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—42no—
149C24—42no—
150C24#12111yescharacteristic
151C2×Q8—22no—
152C2×Q8—24no—
153C2×Q8—24no—
154C2×Q8—24no—
155C2×Q8—48no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—48no—
159C2×Q8C2×D814yes—
160C2×D4—44no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—44no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—82no—
170C2×D4—82no—
171C2×D4—84no—
172C2×D4—84no—
173C2×D4—84no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—82no—
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4C2×D812yes—
193C22×C4#13512yes—
194C22×C4C22×D411yescharacteristic
195C22×C4C22×D411yescharacteristic
196C22×C4C2×#4111yescharacteristic
197SD16—816no—
198D8—88no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—44no—
204C2×C8—42no—
205C2×C8—44no—
206C2×C8—44no—
207C2×C8—44no—
208C4⋊C4—24no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—82no—
216C22⋊C4—84no—
217C22⋊C4—84no—
218C22⋊C4—84no—
219C42—21no—
220C42—22no—
221C42—21no—
222C42—22no—
223C42—44no—
224C42—42no—
225C42—42no—
226C42H(5,𝔽2)12yes—
227C42H(5,𝔽2)12yes—
228C23—21no—
229C23—22no—
230C23—21no—
231C23—22no—
232C23—82no—
233C23—82no—
234C23—82no—
235C23—82no—
236C23—82no—
237C23—82no—
238C23—82no—
239C23#46612yes—
240C23C2×#12111yescharacteristic
241Q8—28no—
242Q8—48no—
243Q8—44no—
244D4—84no—
245D4—84no—
246D4—84no—
247D4—84no—
248D4—164no—
249D4—168no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4#46612yes—
273C2×C4#53412yes—
274C2×C4#53411yescharacteristic
275C2×C4#53511yescharacteristic
276C2×C4#53812yes—
277C2×C4C22×D811yescharacteristic
278C2×C4C2×#13511yescharacteristic
279C2×C4#57512yes—
280C2×C4#57714yes—
281C8—42no—
282C8—42no—
283C8—84no—
284C8—82no—
285C22—22no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—162no—
291C22—162no—
292C22—162no—
293C22—162no—
294C22—162no—
295C22—162no—
296C22—162no—
297C22#261711yescharacteristic
298C22#304312yes—
299C22C2×#46611yescharacteristic
300C22#305411yescharacteristic
301C22#306812yes—
302C4—22no—
303C4—22no—
304C4—41no—
305C4—44no—
306C4—41no—
307C4—82no—
308C4—162no—
309C4#318612yes—
310C4#319112yes—
311C4#320012yes—
312C4#322112yes—
313C2—42no—
314C2—322no—
315C2#1939812yes—
316C2#1943712yes—
317C2#2972211yescharacteristic
318C2#2973011yescharacteristic
319C2#2975111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#29806C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #32903C2semidirect
3. #21818C2semidirect
4. #10623C2semidirect
5. #10474C2semidirect
6. #10262C2semidirect
7. #9134C2semidirect
8. #8599C2semidirect
82. C22×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#329032
#298062
#218182
#106232
#104742
#102622
#91342
#85992