Finite Group Databasebeta

← #345662

Subgroups — #345662

Order 512.

516 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 146 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#26507C212yesmaximal
2#26446C212yesmaximal
3#10597C212yesmaximal
4#10589C212yesmaximal
5#10577C211yesmaximal, characteristic
6#10574C211yesmaximal, characteristic
7#9839C211yesmaximal, characteristic
8#9599C212yesmaximal
9#9066C212yesmaximal
10C2×#494C2211yescharacteristic
11#2571C2211yescharacteristic
12#2551C2211yescharacteristic
13#2369C2212yes—
14#2101—22no—
15#2087—22no—
16#2086—22no—
17#2057—22no—
18#1990C2212yes—
19#1990C2212yes—
20#1987—22no—
21#1987C2212yes—
22#1987C2212yes—
23#1949—22no—
24#1948C2212yes—
25#1947C2212yes—
26#1939C2212yes—
27#1939C2212yes—
28#1921—22no—
29#1920—22no—
30#1844C2211yescharacteristic
31C4×#107C2212yes—
32C2×#334C2212yes—
33C2×#333C2212yes—
34C2×#322C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2312yes—
42#494—41no—
43#494Q814yescomplemented
44C2×#107—22no—
45C2×#107D412yes—
46C2×#107D412yes—
47C2×#106—22no—
48C2×#106—41no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×C4×C8C2312yes—
54C2×C4×C8C2311yescharacteristic
55#400—41no—
56#395—41no—
57#391—42no—
58#389—22no—
59#389—22no—
60#389—22no—
61#389—22no—
62#384C2311yescharacteristic
63#384C2311yescharacteristic
64#382—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4C2312yes—
68C4×C4⋊C4C2312yes—
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—22no—
79#340D412yes—
80#340D412yes—
81#336—22no—
82#336—22no—
83#334—42no—
84#333—42no—
85#322—41no—
86C22×Q8—22no—
87C22×Q8—41no—
88C2×Q16—44no—
89C2×Q16—82no—
90C2×Q16—82no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8C2×D412yes—
95C22×C8C2×D412yes—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—44no—
100C4⋊Q8—82no—
101C4⋊Q8C2×Q814yes—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D412yes—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—42no—
118C2×C42C2411yescharacteristic
119#107—24no—
120#107—42no—
121#107—82no—
122#106—44no—
123#106—82no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—44no—
129C4⋊C8—82no—
130C4×C8—22no—
131C4×C8—21no—
132C4×C8—22no—
133C4×C8C2×Q812yes—
134#95—22no—
135#95—22no—
136#95—42no—
137#95—42no—
138#95—41no—
139#95—42no—
140#95—41no—
141C2×Q8—24no—
142C2×Q8—42no—
143C2×Q8—82no—
144C2×Q8—82no—
145C2×Q8—84no—
146C2×Q8—161no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4C22×D412yes—
155C22×C4H(5,𝔽2)11yescharacteristic
156Q16—88no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—84no—
190C4⋊C4—161no—
191C4⋊C4—162no—
192C4⋊C4#13714yes—
193C42—42no—
194C42—42no—
195C42—42no—
196C42—42no—
197C42—162no—
198C42C22×Q811yescharacteristic
199C42H(5,𝔽2)11yescharacteristic
200C422−1+412yes—
201C23D4211yescharacteristic
202Q8—84no—
203Q8—162no—
204Q8—164no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4#53712yes—
227C2×C4#54311yescharacteristic
228C2×C4D4×Q812yes—
229C2×C4#56311yescharacteristic
230C2×C4C2×#13712yes—
231C2×C4#57512yes—
232C2×C4#57712yes—
233C2×C4#57812yes—
234C8—24no—
235C8—42no—
236C8—42no—
237C8—82no—
238C8—82no—
239C22#268311yescharacteristic
240C22#326712yes—
241C22#327012yes—
242C22#329111yescharacteristic
243C22#329211yescharacteristic
244C4—21no—
245C4—21no—
246C4—44no—
247C4—84no—
248C4—82no—
249C4—82no—
250C4—82no—
251C4—161no—
252C4—161no—
253C4#325212yes—
254C4#325812yes—
255C4#338712yes—
256C4#339012yes—
257C2#2009212yes—
258C2#2011711yescharacteristic
259C2#2013111yescharacteristic
260C2#3125312yes—
261C2#3125411yescharacteristic
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
43. #494Q8semidirect

Maximal subgroups

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

subgroupindex
#265072
#264462
#105972
#105892
#105772
#105742
#98392
#95992
#90662