Finite Group Databasebeta

← #490615

Subgroups — #490615

Order 512.

408 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2254C211yesmaximal, characteristic
2#33474C211yesmaximal, characteristic
3#19209C211yesmaximal, characteristic
4#19208C211yesmaximal, characteristic
5#12462C214yesmaximal, complemented
6#12459C214yesmaximal, complemented
7#12008C211yesmaximal, characteristic
8#11941C211yesmaximal, characteristic
9#11940C211yesmaximal, characteristic
10#3230—21no—
11#3209—21no—
12C2×#500C2211yescharacteristic
13C2×#500C2211yescharacteristic
14#2960C2211yescharacteristic
15#2254—21no—
16#2254—21no—
17#2254C2212yes—
18#2254C2212yes—
19#2183—21no—
20#2183C2212yes—
21Q8×C16—21no—
22Q8×C16C2212yes—
23C2×#367—21no—
24C2×#367—21no—
25#2151—21no—
26#2150C2211yescharacteristic
27#2108—21no—
28C2×C4×C16C2211yescharacteristic
29#1696—22no—
30#1694—22no—
31#1676C2214yes—
32#1674C2214yes—
33#1564C2211yescharacteristic
34#1564C2211yescharacteristic
35#1549—21no—
36#1548—21no—
37#1372C2212yes—
38#1371C2212yes—
39#1365C2214yes—
40#1364C2214yes—
41#533D411yescharacteristic
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×M32—21no—
45C22×C16D411yescharacteristic
46C22×C16D411yescharacteristic
47#500—21no—
48#500—21no—
49#500—21no—
50#500—21no—
51#500—21no—
52#500—21no—
53#500—41no—
54#500C2312yes—
55#500C2312yes—
56#494—41no—
57#487—41no—
58#487—41no—
59#484—42no—
60#477—24no—
61#462—24no—
62#451—42no—
63#446—21no—
64#446D412yes—
65C8×Q8—21no—
66C8×Q8C2312yes—
67#431—21no—
68#431—21no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—21no—
73#424—21no—
74#423D411yescharacteristic
75#405—21no—
76C2×C4×C8C2311yescharacteristic
77#367—22no—
78#367—22no—
79#367—22no—
80#367—22no—
81C4⋊C16—22no—
82C4⋊C16—22no—
83C4⋊C16—21no—
84C4⋊C16C2312yes—
85#346—22no—
86C4×C16—22no—
87C4×C16D411yescharacteristic
88C4×C16D411yescharacteristic
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335C2312yes—
92#335C2312yes—
93#334—21no—
94#334—21no—
95#333—24no—
96#332—24no—
97C22×Q8—41no—
98C22×Q8—41no—
99C2×Q16—44no—
100C2×M16—21no—
101C22×C8C2×D411yescharacteristic
102C22×C8#4111yescharacteristic
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—41no—
106C4⋊Q8—41no—
107C4⋊Q8—41no—
108C4⋊Q8—41no—
109C4⋊Q8—41no—
110C4⋊Q8—41no—
111C4⋊Q8D811yescharacteristic
112C4⋊Q8D811yescharacteristic
113#125—21no—
114C22⋊Q8—21no—
115C22⋊Q8D812yes—
116C4×Q8—21no—
117C4×Q8C2×D412yes—
118#117—21no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C42C2×D411yescharacteristic
124M32—42no—
125C2×C16—22no—
126C2×C16—22no—
127C2×C16Q1614yescomplemented
128C2×C16C2×D411yescharacteristic
129C2×C16C2×D411yescharacteristic
130C2×C16C2×D411yescharacteristic
131C2×C16C2×D411yescharacteristic
132#107—22no—
133#107—22no—
134#107—22no—
135#107—22no—
136#107—41no—
137#107—41no—
138#107—41no—
139#107—41no—
140#107—44no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—21no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8C2×D412yes—
147#103—44no—
148#103—44no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151#97—22no—
152C4×C8—22no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8C2×D411yescharacteristic
156C4×C8C2×D411yescharacteristic
157C4×C8#4111yescharacteristic
158C4×C8C2411yescharacteristic
159C2×Q8—21no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—81no—
163C2×Q8—81no—
164C2×Q8—162no—
165C2×Q8—162no—
166C2×Q8C2×D812yes—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4#12111yescharacteristic
170C22×C4C2×D811yescharacteristic
171C22×C4#13511yescharacteristic
172Q16—164no—
173M16—42no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8#12111yescharacteristic
179C2×C8#12111yescharacteristic
180C2×C8C2×Q1612yes—
181C2×C8#13712yes—
182C2×C8C22×D411yescharacteristic
183C2×C8C2×#4111yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4C2×D812yes—
198C22⋊C4—22no—
199C42—22no—
200C42#12111yescharacteristic
201C42C2×D811yescharacteristic
202C42#13511yescharacteristic
203C42C22×D411yescharacteristic
204C16—21no—
205C16—21no—
206C16—42no—
207C16C2×Q1612yes—
208C16C2×Q1612yes—
209C23#46611yescharacteristic
210Q8—24no—
211Q8—42no—
212Q8—84no—
213Q8—162no—
214Q8—162no—
215Q8—162no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—162no—
223C2×C4—162no—
224C2×C4#46212yes—
225C2×C4#46611yescharacteristic
226C2×C4#46611yescharacteristic
227C2×C4#46611yescharacteristic
228C2×C4#47312yes—
229C2×C4C2×#12111yescharacteristic
230C2×C4C22×D811yescharacteristic
231C2×C4C2×#13511yescharacteristic
232C8—21no—
233C8—21no—
234C8—42no—
235C8—82no—
236C8—82no—
237C8#46212yes—
238C8C22×Q1611yescharacteristic
239C8C2×#13711yescharacteristic
240C22—21no—
241C22—21no—
242C22#167412yes—
243C22C2×#46611yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—42no—
248C4—162no—
249C4—162no—
250C4#167412yes—
251C4#222312yes—
252C4C2×#46211yescharacteristic
253C4C2×#47311yescharacteristic
254C4#341612yes—
255C2—22no—
256C2C2×#167411yescharacteristic
257C2#3360911yescharacteristic
258C2C2×#222311yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radicalC23#466
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #12462C2semidirect
6. #12459C2semidirect
127. C2×C16Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#22542
#334742
#192092
#192082
#124622
#124592
#120082
#119412
#119402