Finite Group Databasebeta

← #317733

Subgroups — #317733

Order 512.

646 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 150 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#28286C212yesmaximal, complemented
2#22303C211yesmaximal, characteristic, complemented
3#22269C211yesmaximal, characteristic
4#10656C212yesmaximal, complemented
5#10536C212yesmaximal
6#9572C212yesmaximal, complemented
7#9571C212yesmaximal, complemented
8#8787C211yesmaximal, characteristic, complemented
9#8585C212yesmaximal, complemented
10C2×#493C2211yescharacteristic
11C2×#478—22no—
12C4×#127C2212yes—
13C4×#125C2211yescharacteristic
14#2067—22no—
15#2065—22no—
16#1982C2212yes—
17#1982C2212yes—
18#1957C2212yes—
19#1948C2212yes—
20#1947C2212yes—
21#1940—22no—
22#1939C2212yes—
23#1917—22no—
24#1879—22no—
25#1879—22no—
26#1842C2211yescharacteristic
27#1842C2212yes—
28#1834C2211yescharacteristic
29#1804—22no—
30#1770C2212yes—
31#1744C2211yescharacteristic
32C2×#331C2212yes—
33C2×#331C2212yes—
34#1296C2212yes—
35#1296C2212yes—
36#1277C2212yes—
37#1277C2211yescharacteristic
38#1277C2211yescharacteristic
39C22×D8—21no—
40C22×D8—21no—
41C2×#127C2312yes—
42C2×#125D411yescharacteristic
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46#493—24no—
47#478—48no—
48C2×#107—22no—
49C2×#106—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#102—22no—
58C2×#102—22no—
59C2×#102—22no—
60C2×#102—22no—
61C2×#102—22no—
62C2×#102—22no—
63C2×#97—22no—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#390—22no—
67#390—22no—
68#390—22no—
69#389D412yes—
70#387—22no—
71#384C2312yes—
72#382D411yescharacteristic
73C4×C4⋊C4—22no—
74C4×C4⋊C4C2312yes—
75C43C2311yescharacteristic
76#340—21no—
77#340—21no—
78#340—22no—
79#340—22no—
80#336—22no—
81#336—22no—
82#336D412yes—
83#336D412yes—
84#331—24no—
85#331—24no—
86C22×D4—22no—
87C22×D4—22no—
88C22×D4—22no—
89C23×C4—42no—
90C2×D8—24no—
91C2×D8—24no—
92C2×D8—82no—
93C2×D8—82no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C22×C8C2×D412yes—
100#127—22no—
101#127—22no—
102#127—22no—
103#127#4114yes—
104#125—24no—
105C4×D4—44no—
106C4×D4—44no—
107C4×D4—82no—
108C4×D4—82no—
109C4×D4—84no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C42—22no—
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42#4112yes—
125C2×C42C2411yescharacteristic
126#107—44no—
127#106—44no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—44no—
135C4⋊C8—82no—
136C4⋊C8—82no—
137C4⋊C8—82no—
138#102—44no—
139#102—44no—
140#102—82no—
141#102—82no—
142#102—82no—
143#102—82no—
144#97—82no—
145C4×C8—22no—
146C4×C8—22no—
147#95—22no—
148#95—22no—
149#95—22no—
150C24—42no—
151C2×D4—24no—
152C2×D4—24no—
153C2×D4—24no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—82no—
166C2×D4—82no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4—84no—
171C22×C4—84no—
172C22×C4—84no—
173C22×C4#12112yes—
174C22×C4#12112yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C2×#4111yescharacteristic
178D8—84no—
179D8—84no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C22⋊C4—84no—
208C22⋊C4—84no—
209C22⋊C4—84no—
210C42—21no—
211C42—22no—
212C42—22no—
213C42—21no—
214C42—42no—
215C42—42no—
216C42—42no—
217C42AGL(1,ℤ/8)14yes—
218C42C2×#4112yes—
219C422−1+412yes—
220C23—44no—
221C23—82no—
222C23—82no—
223C23—82no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23C2×#12111yescharacteristic
228D4—44no—
229D4—44no—
230D4—44no—
231D4—44no—
232D4—84no—
233D4—84no—
234D4—82no—
235D4—82no—
236D4—82no—
237D4—82no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—168no—
254C2×C4#46914yes—
255C2×C4#52511yescharacteristic
256C2×C4#53611yescharacteristic
257C2×C4#54812yes—
258C2×C4#55212yes—
259C2×C4C2×AGL(1,ℤ/8)12yes—
260C2×C4#57611yescharacteristic
261C2×C4#57612yes—
262C2×C4#57712yes—
263C2×C4#57811yescharacteristic
264C8—41no—
265C8—41no—
266C8—42no—
267C8—82no—
268C8—82no—
269C8—82no—
270C22—84no—
271C22—84no—
272C22—84no—
273C22—162no—
274C22—162no—
275C22—162no—
276C22—162no—
277C22#253611yescharacteristic
278C22#304512yes—
279C22C2×#46911yescharacteristic
280C22#306111yescharacteristic
281C22#306812yes—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—82no—
286C4—82no—
287C4—162no—
288C4#312211yescharacteristic
289C4#312311yescharacteristic
290C4#324112yes—
291C4#336612yes—
292C4#336712yes—
293C2—164no—
294C2#1944612yes—
295C2#1954212yes—
296C2#2819411yescharacteristic
297C2#2821411yescharacteristic
298C2#2822011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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. #28286C2semidirect
2. #22303C2semidirect
4. #10656C2semidirect
6. #9572C2semidirect
7. #9571C2semidirect
8. #8787C2semidirect
9. #8585C2semidirect

Maximal subgroups

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

subgroupindex
#282862
#223032
#222692
#106562
#105362
#95722
#95712
#87872
#85852