Finite Group Databasebeta

← #492679

Subgroups — #492679

Order 512.

397 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 118 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#33909C211yesmaximal, characteristic, complemented
2#33905C211yesmaximal, characteristic, complemented
3#20011C212yesmaximal
4#12522C211yesmaximal, characteristic, complemented
5#12452C211yesmaximal, characteristic, complemented
6#12451C211yesmaximal, characteristic, complemented
7#12450C212yesmaximal, complemented
8#12397C212yesmaximal, complemented
9#12394C212yesmaximal, complemented
10C8×Q32C211yesmaximal, characteristic, complemented
11C16×D8C211yesmaximal, characteristic
12#3360—22no—
13#3342C2212yes—
14#3338C2212yes—
15#2254C2212yes—
16#2252C2211yescharacteristic
17#2252C2211yescharacteristic
18#2237—22no—
19#2236—22no—
20#2227C2212yes—
21#2225C2212yes—
22#2216—21no—
23#2216—21no—
24C4×Q32C2211yescharacteristic
25D4×C16C2211yescharacteristic
26D4×C16C2211yescharacteristic
27#2158—21no—
28C2×#366—21no—
29#1710C2212yes—
30#1684—22no—
31#1675C2212yes—
32#1673C2212yes—
33#1643—22no—
34#1638—22no—
35C8×Q16C2212yes—
36C8×D8C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1364C2212yes—
41#1332C2212yes—
42D4⋊C16C2211yescharacteristic
43D4⋊C16C2211yescharacteristic
44C8×C16C2211yescharacteristic
45#556—22no—
46#554—22no—
47C2×Q32D411yescharacteristic
48C22×C16—21no—
49C22×C16—21no—
50#500C2312yes—
51#500C2312yes—
52#488—42no—
53#487—22no—
54#487—22no—
55#485—42no—
56#484—42no—
57#479—42no—
58#477—22no—
59#476—22no—
60#475—22no—
61#467—22no—
62#467—22no—
63#463—42no—
64#461—22no—
65C8×Q8—22no—
66C4×Q16C2312yes—
67C4×SD16—42no—
68C4×D8D411yescharacteristic
69C8×D4C2311yescharacteristic
70C8×D4C2311yescharacteristic
71#427—22no—
72C2×#107—22no—
73#367—22no—
74#367—42no—
75#366—21no—
76#366—21no—
77#366—21no—
78#366—21no—
79#366—22no—
80#366D411yescharacteristic
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#358—22no—
84#358—22no—
85#358D412yes—
86C22⋊C16—21no—
87C22⋊C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335D411yescharacteristic
91#335C2312yes—
92#333—22no—
93#332—22no—
94Q8⋊C8—22no—
95D4⋊C8D411yescharacteristic
96D4⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×Q16C2×D412yes—
99C2×SD16—42no—
100C2×D8D811yescharacteristic
101C22×C8—21no—
102C22×C8—21no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105#126—82no—
106#125—42no—
107#124—42no—
108C22⋊Q8—42no—
109C22⋊Q8—42no—
110C22⋊Q8—82no—
111C4×Q8—22no—
112C4×Q8—42no—
113C4×D4—42no—
114C4×D4C2×D411yescharacteristic
115C4×D4C2×D411yescharacteristic
116#117—42no—
117C2×C4⋊C4—42no—
118Q32—22no—
119C2×C16—21no—
120C2×C16—21no—
121C2×C16—21no—
122C2×C16—42no—
123C2×C16—42no—
124C2×C16C2×D411yescharacteristic
125C2×C16C2×D411yescharacteristic
126#107—22no—
127#107—22no—
128#107—22no—
129#107—22no—
130#107—42no—
131#107D811yescharacteristic
132#107C2×D412yes—
133#106—42no—
134#106—42no—
135C4⋊C8—22no—
136C4⋊C8—42no—
137C4⋊C8C2×D411yescharacteristic
138C4⋊C8C2×D411yescharacteristic
139#103—22no—
140#103—42no—
141#103—42no—
142#103—42no—
143#103—42no—
144#103—42no—
145#103—82no—
146#102—42no—
147#102D811yescharacteristic
148#102D811yescharacteristic
149C22⋊C8—21no—
150C22⋊C8—21no—
151C4×C8—42no—
152C4×C8C2×D411yescharacteristic
153C4×C8C2×D411yescharacteristic
154C4×C8C2411yescharacteristic
155C2×Q8—22no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×D4C2×D811yescharacteristic
159C2×D4C2×D811yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—82no—
163Q16—22no—
164Q16—22no—
165SD16—84no—
166D8—22no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8C2×D811yescharacteristic
176C2×C8C2×D811yescharacteristic
177C2×C8C22×D411yescharacteristic
178C2×C8H(5,𝔽2)11yescharacteristic
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4C2×D811yescharacteristic
189C4⋊C4C2×D811yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C42—42no—
196C42—82no—
197C42C22×D411yescharacteristic
198C16—22no—
199C16—21no—
200C16—41no—
201C16—41no—
202C16#13512yes—
203C23—21no—
204C23—21no—
205Q8—42no—
206Q8—42no—
207Q8—82no—
208Q8—82no—
209Q8—162no—
210D4—21no—
211D4—21no—
212D4—21no—
213D4#50812yes—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—322no—
225C2×C4D4211yescharacteristic
226C2×C4C22×D811yescharacteristic
227C2×C4#57611yescharacteristic
228C8—21no—
229C8—21no—
230C8—21no—
231C8—41no—
232C8—41no—
233C8—41no—
234C8—82no—
235C8C2×#13511yescharacteristic
236C8#57811yescharacteristic
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22D4×D811yescharacteristic
244C4—21no—
245C4—41no—
246C4—41no—
247C4—82no—
248C4—82no—
249C4—162no—
250C4—162no—
251C4#328311yescharacteristic
252C4#329311yescharacteristic
253C4C2×#50811yescharacteristic
254C4#341711yescharacteristic
255C2—42no—
256C2—81no—
257C2#1997111yescharacteristic
258C2#3385111yescharacteristic
259C2#3385911yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×D8D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33909C2semidirect
2. #33905C2semidirect
4. #12522C2semidirect
5. #12452C2semidirect
6. #12451C2semidirect
7. #12450C2semidirect
8. #12397C2semidirect
9. #12394C2semidirect
10. C8×Q32C2semidirect

Maximal subgroups

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

subgroupindex
#339092
#339052
#200112
#125222
#124522
#124512
#124502
#123972
#123942
C8×Q322
C16×D82