Finite Group Databasebeta

← #492675

Subgroups — #492675

Order 512.

434 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 126 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#33902C212yesmaximal, complemented
2#20006C211yesmaximal, characteristic, complemented
3#19971C211yesmaximal, characteristic
4#12523C211yesmaximal, characteristic, complemented
5#12453C212yesmaximal, complemented
6#12447C211yesmaximal, characteristic, complemented
7#12446C211yesmaximal, characteristic, complemented
8#12396C212yesmaximal, complemented
9#12394C212yesmaximal, complemented
10C8×SD32C211yesmaximal, characteristic, complemented
11C16×D8C211yesmaximal, characteristic, complemented
12#3380—21no—
13#3338C2212yes—
14#3334C2212yes—
15#3283—21no—
16#2254C2212yes—
17#2254C2212yes—
18#2235—22no—
19#2233—22no—
20#2226C2212yes—
21#2225C2212yes—
22#2217—22no—
23C4×SD32C2211yescharacteristic
24D4×C16C2212yes—
25C2×#367—22no—
26#1713C2211yescharacteristic
27#1713C2211yescharacteristic
28#1676—21no—
29#1675C2212yes—
30#1673C2212yes—
31#1669—21no—
32#1647—22no—
33#1632—22no—
34C8×Q16C2211yescharacteristic
35C8×D8C2211yescharacteristic
36C8×D8C2211yescharacteristic
37#1372C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1364C2212yes—
41#1332C2211yescharacteristic
42#1331C2211yescharacteristic
43D4⋊C16C2212yes—
44C8×C16C2211yescharacteristic
45C2×#135—41no—
46#554—22no—
47#547—22no—
48C2×SD32D411yescharacteristic
49C22×C16—22no—
50#500—21no—
51#500—21no—
52#500C2312yes—
53#500C2312yes—
54#499—41no—
55#494—21no—
56#494—21no—
57#484—42no—
58#480—42no—
59#478—42no—
60#477—22no—
61#476—22no—
62#474—22no—
63#467—22no—
64#467—41no—
65#466—22no—
66#461—22no—
67#452—42no—
68C8×Q8—21no—
69C4×Q16C2311yescharacteristic
70C4×D8—41no—
71C4×D8D411yescharacteristic
72C4×D8C2311yescharacteristic
73C8×D4—21no—
74C8×D4C2312yes—
75C2×#107—22no—
76C2×#107—22no—
77#367—22no—
78#367—22no—
79#367—22no—
80#367—22no—
81#367—42no—
82#367D411yescharacteristic
83C4⋊C16C2312yes—
84#358—22no—
85#358D411yescharacteristic
86#357—22no—
87#357D411yescharacteristic
88C22⋊C16—22no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2311yescharacteristic
93#335C2311yescharacteristic
94#333—22no—
95#332—22no—
96Q8⋊C8—21no—
97D4⋊C8—21no—
98D4⋊C8D412yes—
99C82C2311yescharacteristic
100C2×#41—42no—
101#135—84no—
102C2×Q16—41no—
103C2×Q16—41no—
104C2×Q16—81no—
105C2×Q16C2×D411yescharacteristic
106C2×SD16—42no—
107C2×D8D811yescharacteristic
108C2×D8C2×D411yescharacteristic
109C22×C8—22no—
110C22×C8—41no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113#125—42no—
114#123—82no—
115C22⋊Q8—42no—
116C22⋊Q8—42no—
117C22⋊Q8—82no—
118#121—42no—
119C4×Q8—21no—
120C4×D4—21no—
121C4×D4—42no—
122C4×D4C2×D412yes—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125SD32—22no—
126C2×C16—21no—
127C2×C16—22no—
128C2×C16—44no—
129C2×C16C2×D411yescharacteristic
130C2×C16C2×D411yescharacteristic
131#107—22no—
132#107—22no—
133#107—22no—
134#107—22no—
135#107—42no—
136#107—42no—
137#107—41no—
138#107—41no—
139#107—81no—
140#107D811yescharacteristic
141#107C2×D411yescharacteristic
142#107C2×D411yescharacteristic
143#106—42no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—41no—
147C4⋊C8C2×D412yes—
148#103—21no—
149#103—42no—
150#103—42no—
151#102—21no—
152#102—42no—
153#102—42no—
154#102D812yes—
155C22⋊C8—22no—
156C22⋊C8—41no—
157C4×C8—41no—
158C4×C8C2×D411yescharacteristic
159C4×C8C2×D411yescharacteristic
160C4×C8C2411yescharacteristic
161#41—84no—
162#41—162no—
163C2×Q8—21no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×D4—21no—
167C2×D4—82no—
168C2×D4C2×D812yes—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—82no—
172C22×C4—82no—
173Q16—21no—
174Q16—21no—
175Q16—82no—
176Q16—161no—
177Q16—162no—
178SD16—84no—
179D8—21no—
180D8—21no—
181D8SD3214yescomplemented
182C2×C8—21no—
183C2×C8—22no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—82no—
189C2×C8C2×D811yescharacteristic
190C2×C8C2×D811yescharacteristic
191C2×C8C22×D411yescharacteristic
192C2×C8H(5,𝔽2)11yescharacteristic
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4C2×D812yes—
204C22⋊C4—22no—
205C22⋊C4—41no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C42—41no—
209C42C22×D411yescharacteristic
210C16—22no—
211C16—21no—
212C16—42no—
213C16#13512yescomplemented
214C23—22no—
215C23—41no—
216Q8—41no—
217Q8—41no—
218Q8—82no—
219Q8—82no—
220Q8—162no—
221D4—22no—
222D4—41no—
223D4—41no—
224D4—164no—
225D4C2×SD3214yes—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—164no—
237C2×C4—322no—
238C2×C4D4211yescharacteristic
239C2×C4C22×D811yescharacteristic
240C2×C4#57611yescharacteristic
241C8—22no—
242C8—21no—
243C8—21no—
244C8—42no—
245C8—81no—
246C8—81no—
247C8C2×SD3212yes—
248C8C2×#13511yescharacteristic
249C8#57811yescharacteristic
250C22—24no—
251C22—42no—
252C22—42no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22D4×D811yescharacteristic
257C4—21no—
258C4—42no—
259C4—81no—
260C4—81no—
261C4—81no—
262C4—162no—
263C4—162no—
264C4#328311yescharacteristic
265C4#329311yescharacteristic
266C4C22×SD3211yescharacteristic
267C4#341611yescharacteristic
268C2—44no—
269C2—161no—
270C2#1997111yescharacteristic
271C2D4×SD3211yescharacteristic
272C2#3385611yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radical#19971C2
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. #33902C2semidirect
2. #20006C2semidirect
4. #12523C2semidirect
5. #12453C2semidirect
6. #12447C2semidirect
7. #12446C2semidirect
8. #12396C2semidirect
9. #12394C2semidirect
10. C8×SD32C2semidirect
11. C16×D8C2semidirect
181. D8SD32semidirect
213. C16#135semidirect

Maximal subgroups

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

subgroupindex
#339022
#200062
#199712
#125232
#124532
#124472
#124462
#123962
#123942
C8×SD322
C16×D82