Finite Group Databasebeta

← #326952

Subgroups — #326952

Order 512.

816 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 136 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#37641C211yesmaximal, characteristic
2#27980C211yesmaximal, characteristic
3#25818C212yesmaximal
4#10428C214yesmaximal, complemented
5#10087C214yesmaximal, complemented
6#8722C211yesmaximal, characteristic
7#8338C212yesmaximal
8C2×#533C2212yes—
9C2×#501C2214yes—
10#2857C2211yescharacteristic
11#2822C2214yes—
12#2625—24no—
13#2621—22no—
14#2582—22no—
15#2568—22no—
16#2546—22no—
17#2285C2211yescharacteristic
18#2072—22no—
19#2058—22no—
20#2057—22no—
21#1992C2214yes—
22#1960C2212yes—
23#1873—22no—
24#1854—24no—
25#1836C2212yes—
26#1749C2212yes—
27#1749C2211yescharacteristic
28C2×#330C2218yes—
29#1295C2214yes—
30#533—48no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8D412yes—
33C2×C4⋊Q8D412yes—
34C2×C4⋊Q8C2311yescharacteristic
35C2×#125C2314yes—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—21no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×#117D414yes—
42C2×#117D412yes—
43C22×C42C2311yescharacteristic
44#501—48no—
45C2×#107—24no—
46C2×#106—24no—
47C2×#103—24no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—21no—
50C2×#97C2314yes—
51C2×#97C2312yes—
52#400—28no—
53#398—28no—
54#398—28no—
55#397—28no—
56#391—24no—
57#391—24no—
58#391—24no—
59#391—21no—
60#389—24no—
61#389—24no—
62#389—22no—
63#384D412yes—
64#384C2311yescharacteristic
65#382—22no—
66#382C2312yes—
67#380—22no—
68#380—22no—
69C4×C4⋊C4—22no—
70#340—24no—
71#340—24no—
72#340—24no—
73#336—22no—
74#336—21no—
75#330—48no—
76C22×Q8—22no—
77C22×Q8—22no—
78C22×Q8—21no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D412yes—
81C22×C8—24no—
82C22×C8—22no—
83C4⋊Q8—24no—
84C4⋊Q8—44no—
85C4⋊Q8—44no—
86C4⋊Q8—84no—
87#125—28no—
88#125—28no—
89C22⋊Q8—48no—
90C22⋊Q8—48no—
91C22⋊Q8—44no—
92C4×Q8—48no—
93C4×Q8—48no—
94#117—44no—
95#117—44no—
96#117—82no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C2×D414yes—
109C2×C4⋊C4C2×D414yes—
110C2×C4⋊C4C2×D414yes—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—41no—
114C2×C42—24no—
115C2×C42—24no—
116C2×C42—22no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D412yes—
119C2×C42C2411yescharacteristic
120#107—48no—
121#106—48no—
122#103—84no—
123C22⋊C8—28no—
124C22⋊C8—42no—
125#97—24no—
126#97—28no—
127#95—28no—
128#95—28no—
129#95—28no—
130#95—24no—
131#95—42no—
132C24#12011yescharacteristic
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—44no—
140C2×Q8—44no—
141C2×Q8—42no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4#12011yescharacteristic
154C22×C4#12012yes—
155C22×C4C22×D412yes—
156C22×C4C22×D411yescharacteristic
157C2×C8—28no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—48no—
174C4⋊C4—48no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—88no—
178C4⋊C4—82no—
179C22⋊C4—48no—
180C22⋊C4—82no—
181C22⋊C4—82no—
182C42—24no—
183C42—22no—
184C42—48no—
185C42—44no—
186C42H(5,𝔽2)14yes—
187C23—21no—
188C23—21no—
189C23—22no—
190C23—21no—
191C23—22no—
192C23C2×#12011yescharacteristic
193Q8—88no—
194Q8—88no—
195Q8—84no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—48no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—82no—
220C2×C4#45412yes—
221C2×C4#45614yes—
222C2×C4#53414yes—
223C2×C4#53512yes—
224C8—48no—
225C8—82no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—42no—
230C22—41no—
231C22#261311yescharacteristic
232C22C2×#45411yescharacteristic
233C22C2×#45612yes—
234C22#301911yescharacteristic
235C22#302012yes—
236C4—24no—
237C4—24no—
238C4—22no—
239C4—22no—
240C4—84no—
241C4—84no—
242C4—84no—
243C4—161no—
244C4—161no—
245C2—81no—
246C2#1906311yescharacteristic
247C2#1906612yes—
248C2#1908411yescharacteristic
249C2#2961312yes—
250C2#2961711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10428C2semidirect
5. #10087C2semidirect

Maximal subgroups

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

subgroupindex
#376412
#279802
#258182
#104282
#100872
#87222
#83382