Finite Group Databasebeta

← #345627

Subgroups — #345627

Order 512.

648 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 146 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#26530C212yesmaximal, complemented
2#26463C212yesmaximal, complemented
3#10599C212yesmaximal, complemented
4#10591C212yesmaximal, complemented
5#10580C212yesmaximal, complemented
6#9837C211yesmaximal, characteristic
7#9600C211yesmaximal, characteristic, complemented
8#9600C211yesmaximal, characteristic, complemented
9#9061C212yesmaximal
10C2×#493C2211yescharacteristic
11C2×#478—22no—
12#2552C2212yes—
13#2366C2212yes—
14#2094—22no—
15#2065—22no—
16#2065—22no—
17#2025—22no—
18#1982C2212yes—
19#1982C2212yes—
20#1957C2212yes—
21#1957C2212yes—
22#1948C2212yes—
23#1947C2212yes—
24#1939C2212yes—
25#1939C2212yes—
26#1864—22no—
27#1862—22no—
28#1844C2211yescharacteristic
29#1802—22no—
30C4×#106C2211yescharacteristic
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#331C2212yes—
35C2×#322C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1279C2212yes—
39C22×D8—41no—
40C2×#127C2312yes—
41C2×#125—22no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44#493—41no—
45#493Q814yes—
46#478—48no—
47C2×#107—22no—
48C2×#106—22no—
49C2×#106—41no—
50C2×#106D412yes—
51C2×#106D412yes—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×#102—22no—
57C2×#102—22no—
58C2×#102—22no—
59C2×#102—22no—
60C2×#102—22no—
61C2×#102—22no—
62C2×#102—22no—
63C2×#102—22no—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#398—42no—
68#390—22no—
69#390—22no—
70#389—22no—
71#389—22no—
72#386—42no—
73#384—22no—
74#384C2312yes—
75C4×C4⋊C4C2312yes—
76C4×C4⋊C4C2312yes—
77#340—22no—
78#340—22no—
79#340D411yescharacteristic
80#340D411yescharacteristic
81#340D411yescharacteristic
82#340D411yescharacteristic
83#336—22no—
84#336—22no—
85#335—41no—
86#335—41no—
87#331—42no—
88#322—41no—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4—22no—
92C23×C4—42no—
93C2×D8—44no—
94C2×D8—82no—
95C2×D8—82no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103#127—22no—
104#127—22no—
105#127—22no—
106#127C2×Q814yes—
107#125—44no—
108#125—82no—
109C4×D4—44no—
110C4×D4—44no—
111C4×D4—82no—
112C4×D4—82no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D412yes—
122C2×C4⋊C4C2×D412yes—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42C2411yescharacteristic
129#107—44no—
130#106—24no—
131#106—42no—
132#106—82no—
133#106—82no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—44no—
139C4⋊C8—82no—
140#102—44no—
141#102—44no—
142#102—44no—
143#102—44no—
144#102—82no—
145#102—82no—
146#102—82no—
147#102—82no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8C2×Q812yes—
154#95—22no—
155#95—22no—
156#95—42no—
157C24—42no—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—82no—
173C2×D4—82no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—84no—
180C22×C4—84no—
181C22×C4—162no—
182C22×C4C22×D411yescharacteristic
183C22×C4C22×D411yescharacteristic
184C22×C4H(5,𝔽2)11yescharacteristic
185D8—88no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—162no—
218C4⋊C4AGL(1,ℤ/8)14yes—
219C22⋊C4—84no—
220C22⋊C4—84no—
221C22⋊C4—162no—
222C42—42no—
223C42—42no—
224C42—42no—
225C42—42no—
226C42C22×Q811yescharacteristic
227C42H(5,𝔽2)11yescharacteristic
228C422−1+411yescharacteristic
229C422−1+411yescharacteristic
230C23—44no—
231C23—82no—
232C23—82no—
233C23—82no—
234C23—82no—
235C23—82no—
236C23—82no—
237C23D4211yescharacteristic
238D4—44no—
239D4—44no—
240D4—44no—
241D4—44no—
242D4—84no—
243D4—82no—
244D4—82no—
245D4—82no—
246D4—82no—
247D4—162no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—168no—
266C2×C4#53711yescharacteristic
267C2×C4#53711yescharacteristic
268C2×C4#54311yescharacteristic
269C2×C4D4×Q811yescharacteristic
270C2×C4D4×Q811yescharacteristic
271C2×C4#56311yescharacteristic
272C2×C4C2×AGL(1,ℤ/8)12yes—
273C2×C4#57512yes—
274C2×C4#57612yes—
275C2×C4#57712yes—
276C8—24no—
277C8—41no—
278C8—41no—
279C8—41no—
280C8—41no—
281C8—82no—
282C8—82no—
283C22—84no—
284C22—84no—
285C22—84no—
286C22—162no—
287C22—162no—
288C22—162no—
289C22—162no—
290C22#268311yescharacteristic
291C22#326512yes—
292C22#326812yes—
293C22#329111yescharacteristic
294C22#329211yescharacteristic
295C4—21no—
296C4—21no—
297C4—44no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—162no—
302C4#325311yescharacteristic
303C4#325611yescharacteristic
304C4#325911yescharacteristic
305C4#326311yescharacteristic
306C4#338412yes—
307C4#338912yes—
308C2—164no—
309C2#2007612yes—
310C2#2008212yes—
311C2#3124511yescharacteristic
312C2#3124711yescharacteristic
313C2#3125411yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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. #26530C2semidirect
2. #26463C2semidirect
3. #10599C2semidirect
4. #10591C2semidirect
5. #10580C2semidirect
7. #9600C2semidirect
8. #9600C2semidirect

Maximal subgroups

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

subgroupindex
#265302
#264632
#105992
#105912
#105802
#98372
#96002
#90612