Finite Group Databasebeta

← #341559

Subgroups — #341559

Order 512.

556 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 166 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#28813C212yesmaximal
2#26446C212yesmaximal
3#10674C211yesmaximal, characteristic
4#10650C212yesmaximal
5#10573C212yesmaximal
6#10349C211yesmaximal, characteristic
7#9568C212yesmaximal
8#9536C212yesmaximal
9#8929C211yesmaximal, characteristic
10C2×#500—22no—
11#2551C2212yes—
12#2551C2211yescharacteristic
13#2369C2211yescharacteristic
14#2087—22no—
15#2057—22no—
16#2057—22no—
17#2057—22no—
18#1990C2212yes—
19#1987—22no—
20#1987C2212yes—
21#1956C2211yescharacteristic
22#1947C2212yes—
23#1947C2212yes—
24#1947C2212yes—
25#1942—22no—
26#1939C2212yes—
27#1921—22no—
28#1845C2212yes—
29#1845C2211yescharacteristic
30C4×#107C2212yes—
31C8×C4⋊C4C2212yes—
32C2×#333C2211yescharacteristic
33#1296C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2311yescharacteristic
43#500—48no—
44C2×#107—22no—
45C2×#107—22no—
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—22no—
49C2×#107—41no—
50C2×#107D412yes—
51C2×#107D412yes—
52C2×C4⋊C8—22no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×#103—41no—
56C2×C4×C8—22no—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2312yes—
59#395—42no—
60#395—41no—
61#391—41no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#384C2311yescharacteristic
67#384C2312yes—
68C4×C4⋊C4—22no—
69C4×C4⋊C4C2312yes—
70C4×C4⋊C4C2312yes—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340D412yes—
79#340D412yes—
80#336—22no—
81#336—22no—
82#333Q814yescomplemented
83C22×Q8—21no—
84C22×Q8—41no—
85C22×Q8—42no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8C2×D412yes—
90C22×C8C2×D412yes—
91C4⋊Q8—22no—
92C4⋊Q8—21no—
93C4⋊Q8—44no—
94C4⋊Q8—44no—
95C4⋊Q8—44no—
96C4⋊Q8—82no—
97C4⋊Q8—82no—
98C4⋊Q8—82no—
99C4⋊Q8C2×Q812yes—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—41no—
117C2×C42C2411yescharacteristic
118#107—24no—
119#107—44no—
120#107—44no—
121#107—44no—
122#107—82no—
123#107—82no—
124#107—82no—
125#107Q1618yes—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—82no—
130C4⋊C8C2×Q814yes—
131#103—82no—
132C4×C8—22no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—44no—
136#95—22no—
137#95—22no—
138#95—42no—
139#95—42no—
140#95—41no—
141C2×Q8—22no—
142C2×Q8—41no—
143C2×Q8—84no—
144C2×Q8—84no—
145C2×Q8—81no—
146C2×Q8—81no—
147C2×Q8—84no—
148C2×Q8—161no—
149C2×Q8—162no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4C22×D412yes—
158C22×C4H(5,𝔽2)11yescharacteristic
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—24no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8C2×Q1614yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—84no—
188C4⋊C4—84no—
189C4⋊C4—81no—
190C4⋊C4—82no—
191C4⋊C4—84no—
192C4⋊C4—161no—
193C4⋊C4C2×Q1614yes—
194C4⋊C4C2×Q1614yes—
195C4⋊C4AGL(1,ℤ/8)14yes—
196C42—42no—
197C42—42no—
198C42—42no—
199C42—42no—
200C42—161no—
201C42C22×Q811yescharacteristic
202C422−1+411yescharacteristic
203C422−1+412yes—
204C23D4211yescharacteristic
205Q8—82no—
206Q8—161no—
207Q8—164no—
208Q8—168no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—24no—
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—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—82no—
231C2×C4#53612yes—
232C2×C4#54411yescharacteristic
233C2×C4D4×Q812yes—
234C2×C4#56311yescharacteristic
235C2×C4C22×Q1612yes—
236C2×C4C2×AGL(1,ℤ/8)12yes—
237C2×C4#57612yes—
238C2×C4#57812yes—
239C8—24no—
240C8—42no—
241C8—44no—
242C8—42no—
243C8—82no—
244C22#267711yescharacteristic
245C22#326412yes—
246C22D4×Q1612yes—
247C22#328911yescharacteristic
248C22#329211yescharacteristic
249C4—21no—
250C4—21no—
251C4—44no—
252C4—44no—
253C4—44no—
254C4—82no—
255C4—82no—
256C4—161no—
257C4—162no—
258C4#323012yes—
259C4#324312yes—
260C4Q8×Q1612yes—
261C4#338612yes—
262C2#1981112yes—
263C2Q16211yescharacteristic
264C2#2005311yescharacteristic
265C2#3095712yes—
266C2#3097811yescharacteristic
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
82. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#288132
#264462
#106742
#106502
#105732
#103492
#95682
#95362
#89292