Finite Group Databasebeta

← #474817

Subgroups — #474817

Order 512.

698 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 186 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#23747C212yesmaximal
2C2×#1941C211yesmaximal, characteristic
3#21143C211yesmaximal, characteristic
4#11303C214yesmaximal
5#11238C212yesmaximal
6#11238C212yesmaximal
7#11204C212yesmaximal
8#10822C211yesmaximal, characteristic
9C22×#106C2211yescharacteristic
10C22×#106C2211yescharacteristic
11C2×#424C2212yes—
12C2×#423C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#389C2212yes—
15#2285C2211yescharacteristic
16#2101—24no—
17#2100—22no—
18#2099—22no—
19#1941—24no—
20#1939—24no—
21C2×#343C2214yes—
22C2×#343C2214yes—
23C2×#341C2212yes—
24C2×#341C2212yes—
25#1736C2214yes—
26#1736C2214yes—
27C2×#340C2212yes—
28#1729C2212yes—
29#1728C2212yes—
30C22×M16C2312yes—
31C22×M16C2314yes—
32C23×C8C2311yescharacteristic
33C23×C8C2311yescharacteristic
34C2×#117C2312yes—
35C22×C4⋊C4—22no—
36C22×C4⋊C4C2312yes—
37C22×C4⋊C4C2312yes—
38C22×C42C2311yescharacteristic
39C2×#106—22no—
40C2×#106—24no—
41C2×#106—22no—
42C2×#106—42no—
43#424—28no—
44#423—42no—
45C2×C4⋊C8—24no—
46C2×C4⋊C8—22no—
47C2×C4×C8—42no—
48C2×C4×C8D414yes—
49#389—22no—
50#389—22no—
51#389—22no—
52#389—22no—
53#384D411yescharacteristic
54#384D411yescharacteristic
55#382D412yes—
56#380—22no—
57C4×C4⋊C4—22no—
58C2×#95—22no—
59#343—28no—
60#343—44no—
61#341—24no—
62#341—42no—
63#340—24no—
64#340—24no—
65#340—24no—
66#340—24no—
67#340—24no—
68#336—24no—
69#336—22no—
70#336—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4#4112yes—
74C23×C4C2411yescharacteristic
75C2×M16—24no—
76C2×M16—24no—
77C2×M16—24no—
78C2×M16—24no—
79C2×M16—24no—
80C2×M16—24no—
81C2×M16—24no—
82C2×M16—24no—
83C2×M16—24no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—24no—
90C22×C8—22no—
91C22×C8C2×D412yes—
92C22×C8C2×D412yes—
93C22×C8#4114yes—
94#117—24no—
95#117—24no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—84no—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4#4114yes—
113C2×C22⋊C4—22no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D412yes—
120#106—28no—
121#106—44no—
122C4⋊C8—44no—
123C4⋊C8—48no—
124C4×C8—44no—
125#95—24no—
126#95—42no—
127#95—42no—
128C24C2×#4111yescharacteristic
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—44no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4#12112yes—
144C22×C4#12112yes—
145C22×C4C22×D411yescharacteristic
146C22×C4C22×D411yescharacteristic
147C22×C4C2×#4112yes—
148C22×C4C2×#4112yes—
149C22×C4H(5,𝔽2)12yes—
150C22×C42−1+412yes—
151C22×C42−1+412yes—
152C22×C42−1+412yes—
153M16—44no—
154M16—44no—
155M16—48no—
156M16—44no—
157M16—44no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—28no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—44no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—44no—
169C2×C8—44no—
170C2×C8—44no—
171C2×C8#12414yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—84no—
184C4⋊C4—84no—
185C22⋊C4—44no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—42no—
191C42—42no—
192C23—22no—
193C23—22no—
194C23C2×#12111yescharacteristic
195C23#53912yes—
196C23#55212yes—
197C23#55612yes—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—48no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—84no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—84no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—164no—
217C2×C4#47114yes—
218C2×C4#52511yescharacteristic
219C2×C4C2×#12411yescharacteristic
220C2×C4#53411yescharacteristic
221C2×C4#53611yescharacteristic
222C2×C4#53611yescharacteristic
223C2×C4#53611yescharacteristic
224C2×C4#53712yes—
225C2×C4#54114yes—
226C2×C4#54114yes—
227C2×C4#54712yes—
228C2×C4#54812yes—
229C2×C4#55314yes—
230C2×C4#56314yes—
231C8—24no—
232C8—42no—
233C8—82no—
234C8—84no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—24no—
240C22#253611yescharacteristic
241C22#254711yescharacteristic
242C22#261811yescharacteristic
243C22#277614yes—
244C22C2×#47112yes—
245C22#306712yes—
246C4—24no—
247C4—42no—
248C4—84no—
249C4—82no—
250C4—82no—
251C4—162no—
252C4#259111yescharacteristic
253C4#263411yescharacteristic
254C4#264211yescharacteristic
255C4#272514yes—
256C4#281711yescharacteristic
257C2—24no—
258C2#1701011yescharacteristic
259C2#2821812yes—
260C2#2873211yescharacteristic
261C2#2873311yescharacteristic
262C2#2979012yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#237472
C2×#19412
#211432
#113032
#112382
#112042
#108222