Finite Group Databasebeta

← #492707

Subgroups — #492707

Order 512.

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

normal subgroups only

This group has 408 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#33960C211yesmaximal, characteristic, complemented
2#33908C211yesmaximal, characteristic
3#20016C212yesmaximal, complemented
4#12504C211yesmaximal, characteristic, complemented
5#12458C211yesmaximal, characteristic, complemented
6#12431C211yesmaximal, characteristic, complemented
7#12420C212yesmaximal, complemented
8#12387C212yesmaximal, complemented
9#12347C212yesmaximal, complemented
10C16×SD16C211yesmaximal, characteristic, complemented
11C8×D16C211yesmaximal, characteristic, complemented
12#3393C2212yes—
13#3341C2212yes—
14#3340—22no—
15#2252C2211yescharacteristic
16#2249—21no—
17#2247C2211yescharacteristic
18#2241C2212yes—
19#2234—22no—
20#2226C2212yes—
21#2215—21no—
22#2211—22no—
23#2208C2212yes—
24Q8×C16C2211yescharacteristic
25C4×Q32—21no—
26C4×D16C2211yescharacteristic
27D4×C16C2211yescharacteristic
28#2158—21no—
29#1707C2212yes—
30#1676C2212yes—
31#1671—22no—
32#1669C2212yes—
33#1646—22no—
34#1638—22no—
35C8×SD16C2211yescharacteristic
36C8×D8C2212yes—
37#1371C2211yescharacteristic
38#1367C2211yescharacteristic
39#1365C2212yes—
40#1362C2212yes—
41Q8⋊C16C2211yescharacteristic
42#1331C2212yes—
43D4⋊C16C2211yescharacteristic
44C8×C16C2211yescharacteristic
45#556—22no—
46#555—22no—
47C2×Q32—21no—
48C2×Q32—21no—
49C2×SD32—22no—
50C2×D16D411yescharacteristic
51C22×C16—21no—
52#500C2312yes—
53#498—22no—
54#495—22no—
55#494C2312yes—
56#487—22no—
57#485—42no—
58#482—42no—
59#482—42no—
60#477—22no—
61#476—22no—
62#474—22no—
63#466—22no—
64#465—42no—
65#464—42no—
66#461—22no—
67C8×Q8C2311yescharacteristic
68C4×Q16—22no—
69C4×SD16D411yescharacteristic
70C4×D8C2312yes—
71C8×D4—22no—
72C8×D4C2311yescharacteristic
73#427—22no—
74#427—42no—
75#367—22no—
76#366—21no—
77#366—21no—
78#366D411yescharacteristic
79C4⋊C16—21no—
80C4⋊C16C2311yescharacteristic
81C4⋊C16C2311yescharacteristic
82#358—22no—
83#358—42no—
84#357—22no—
85#357D412yes—
86C22⋊C16—21no—
87C4×C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335C2312yes—
91#334D411yescharacteristic
92#333—22no—
93#332—22no—
94Q8⋊C8D411yescharacteristic
95D4⋊C8—22no—
96D4⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×Q16—22no—
99C2×Q16—22no—
100C2×SD16—42no—
101C2×SD16D811yescharacteristic
102C2×D8C2×D412yes—
103C22×C8—21no—
104C22×C8—42no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107#126—82no—
108#124—42no—
109#124—82no—
110C22⋊Q8—42no—
111#121—42no—
112#121—42no—
113C4×Q8—42no—
114C4×Q8C2×D411yescharacteristic
115C4×D4—22no—
116C4×D4C2×D411yescharacteristic
117#117—42no—
118#117—42no—
119#117—42no—
120Q32—42no—
121Q32—81no—
122SD32—82no—
123D16—22no—
124C2×C16—21no—
125C2×C16—21no—
126C2×C16—21no—
127C2×C16—42no—
128C2×C16C2×D411yescharacteristic
129C2×C16C2×D411yescharacteristic
130#107—22no—
131#107—22no—
132#107—42no—
133#107—42no—
134#107C2×D412yes—
135#106—42no—
136#106—42no—
137#106—42no—
138#106D811yescharacteristic
139C4⋊C8—21no—
140C4⋊C8—22no—
141C4⋊C8C2×D411yescharacteristic
142C4⋊C8C2×D411yescharacteristic
143#103—42no—
144#103—42no—
145#103—82no—
146#103D811yescharacteristic
147#102—22no—
148#102—42no—
149#102—42no—
150#102D811yescharacteristic
151C22⋊C8—21no—
152C22⋊C8—42no—
153C4×C8—21no—
154C4×C8C2×D411yescharacteristic
155C4×C8C2×D411yescharacteristic
156C4×C8C2411yescharacteristic
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8C2×D811yescharacteristic
160C2×D4—22no—
161C2×D4—82no—
162C2×D4C2×D811yescharacteristic
163C22×C4—21no—
164C22×C4—42no—
165Q16—42no—
166Q16—42no—
167Q16—82no—
168SD16—22no—
169SD16—162no—
170D8—22no—
171D8—22no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—84no—
179C2×C8C2×D811yescharacteristic
180C2×C8C2×D811yescharacteristic
181C2×C8C22×D411yescharacteristic
182C2×C8H(5,𝔽2)11yescharacteristic
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—82no—
190C4⋊C4C2×D811yescharacteristic
191C4⋊C4C2×D811yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—42no—
194C22⋊C4—82no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C22⋊C4—82no—
198C42—21no—
199C42—82no—
200C42—82no—
201C42C22×D411yescharacteristic
202C16—21no—
203C16—41no—
204C16—41no—
205C16—41no—
206C16#13512yes—
207C23—21no—
208C23—42no—
209Q8—21no—
210Q8—21no—
211Q8—82no—
212Q8—82no—
213Q8—162no—
214D4—21no—
215D4—21no—
216D4—42no—
217D4—42no—
218D4—322no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4D4211yescharacteristic
230C2×C4C22×D811yescharacteristic
231C2×C4#57611yescharacteristic
232C8—21no—
233C8—21no—
234C8—41no—
235C8—41no—
236C8—41no—
237C8—82no—
238C8#50812yes—
239C8C2×#13511yescharacteristic
240C8#57711yescharacteristic
241C22—41no—
242C22—41no—
243C22—41no—
244C22—82no—
245C22—82no—
246C22—82no—
247C22D4×D811yescharacteristic
248C4—21no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—82no—
253C4—162no—
254C4—162no—
255C4#328411yescharacteristic
256C4#329011yescharacteristic
257C4C2×#50811yescharacteristic
258C4#341711yescharacteristic
259C2—81no—
260C2—162no—
261C2#1997211yescharacteristic
262C2#3384911yescharacteristic
263C2#3385911yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radical#33908C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33960C2semidirect
3. #20016C2semidirect
4. #12504C2semidirect
5. #12458C2semidirect
6. #12431C2semidirect
7. #12420C2semidirect
8. #12387C2semidirect
9. #12347C2semidirect
10. C16×SD16C2semidirect
11. C8×D16C2semidirect

Maximal subgroups

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

subgroupindex
#339602
#339082
#200162
#125042
#124582
#124312
#124202
#123872
#123472
C16×SD162
C8×D162