Finite Group Databasebeta

← #345691

Subgroups — #345691

Order 512.

509 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 152 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#26512C212yesmaximal
2#26445C212yesmaximal
3#10593C212yesmaximal
4#10593C212yesmaximal
5#10581C212yesmaximal
6#9835C211yesmaximal, characteristic
7#9598C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic
9#9066C212yesmaximal
10C2×#494C2211yescharacteristic
11#2552C2212yes—
12#2369C2212yes—
13#2100—22no—
14#2087—22no—
15#2086—22no—
16#2059—22no—
17#1990C2212yes—
18#1990C2212yes—
19#1989—22no—
20#1987C2212yes—
21#1987C2212yes—
22#1947—22no—
23#1947C2212yes—
24#1947C2212yes—
25#1939C2212yes—
26#1939C2212yes—
27#1921—22no—
28#1920—22no—
29#1834C2211yescharacteristic
30C4×#107C2211yescharacteristic
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#333C2212yes—
35C2×#322C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C2×C4⋊Q8C2312yes—
41C2×#125—22no—
42#494—41no—
43#494Q814yes—
44C2×#107—22no—
45C2×#107—22no—
46C2×#107D412yes—
47C2×#106D411yescharacteristic
48C2×#106D411yescharacteristic
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#398—42no—
57#391—42no—
58#389—22no—
59#389—22no—
60#389—22no—
61#389—22no—
62#384—22no—
63#384C2312yes—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4C2312yes—
67C4×C4⋊C4C2312yes—
68#340—22no—
69#340—22no—
70#340—22no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—41no—
79#340D412yes—
80#340D411yescharacteristic
81#340D411yescharacteristic
82#336—22no—
83#336—22no—
84#335—41no—
85#335—41no—
86#333—42no—
87#322—41no—
88C22×Q8—22no—
89C2×Q16—44no—
90C2×Q16—84no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8C2×Q814yes—
102#125—44no—
103#125—82no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—42no—
120C2×C42C2411yescharacteristic
121#107—42no—
122#107—44no—
123#107—82no—
124#106—41no—
125#106Q1614yescomplemented
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—44no—
131C4⋊C8—82no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8C2×Q812yes—
138#95—22no—
139#95—22no—
140#95—42no—
141#95—42no—
142#95—42no—
143C2×Q8—24no—
144C2×Q8—42no—
145C2×Q8—82no—
146C2×Q8—82no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4C22×D411yescharacteristic
154C22×C4C22×D411yescharacteristic
155C22×C4H(5,𝔽2)11yescharacteristic
156Q16—88no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8—21no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8C2×Q1612yes—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—82no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C4⋊C4—162no—
194C4⋊C4—162no—
195C4⋊C4C2×Q1614yes—
196C42—42no—
197C42—42no—
198C42—42no—
199C42—42no—
200C42—162no—
201C42C22×Q811yescharacteristic
202C42H(5,𝔽2)11yescharacteristic
203C422−1+411yescharacteristic
204C422−1+411yescharacteristic
205C23D4211yescharacteristic
206Q8—84no—
207Q8—162no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4#53711yescharacteristic
228C2×C4#53711yescharacteristic
229C2×C4#54311yescharacteristic
230C2×C4D4×Q811yescharacteristic
231C2×C4D4×Q811yescharacteristic
232C2×C4#56311yescharacteristic
233C2×C4C22×Q1611yescharacteristic
234C2×C4#57511yescharacteristic
235C2×C4#57512yes—
236C2×C4#57611yescharacteristic
237C2×C4#57812yes—
238C2×C4#57811yescharacteristic
239C8—22no—
240C8—42no—
241C8—41no—
242C8—41no—
243C8—41no—
244C8—82no—
245C8—82no—
246C22#268311yescharacteristic
247C22#326912yes—
248C22#327311yescharacteristic
249C22D4×Q1611yescharacteristic
250C22#328812yes—
251C4—21no—
252C4—21no—
253C4—44no—
254C4—82no—
255C4—82no—
256C4—82no—
257C4—84no—
258C4—162no—
259C4#324911yescharacteristic
260C4#325611yescharacteristic
261C4#326111yescharacteristic
262C4#326211yescharacteristic
263C4Q8×Q1611yescharacteristic
264C4#339011yescharacteristic
265C4#339012yes—
266C2#1981612yes—
267C2#2022612yes—
268C2#3123911yescharacteristic
269C2#3124911yescharacteristic
270C2#3125811yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
125. #106Q16semidirect

Maximal subgroups

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

subgroupindex
#265122
#264452
#105932
#105812
#98352
#95982
#95962
#90662