Finite Group Databasebeta

← #102497

Subgroups — #102497

Order 512.

592 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 76 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#32957C212yesmaximal
2#32762C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7725C212yesmaximal
5#7044C211yesmaximal, characteristic
6C22×C4×Q8C2211yescharacteristic
7C2×#477—22no—
8#2857C2212yes—
9#2072—22no—
10#2016—22no—
11#2001—22no—
12#1863—24no—
13#1852—22no—
14#1817C2211yescharacteristic
15#1817C2211yescharacteristic
16#1803—22no—
17#1797C2212yes—
18#1787—22no—
19C2×#324—22no—
20#1271—24no—
21C23×Q8D411yescharacteristic
22C22×Q16—22no—
23C2×C4⋊Q8D412yes—
24C2×C22⋊Q8—22no—
25C2×C4×Q8—22no—
26C2×C4×Q8—22no—
27C2×C4×Q8—44no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4D411yescharacteristic
30C22×C42—21no—
31C22×C42C2311yescharacteristic
32#477—48no—
33C2×#107—24no—
34C2×C4⋊C8—22no—
35C2×#103—24no—
36C2×#103—24no—
37C2×#103—24no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8D412yes—
41#391—22no—
42#389—22no—
43#389—22no—
44#384D412yes—
45#380—22no—
46#380—22no—
47#340—24no—
48#340—22no—
49#336—22no—
50#336—22no—
51#336D414yes—
52#324—44no—
53C22×Q8—21no—
54C22×Q8—22no—
55C22×Q8—22no—
56C22×Q8—22no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8Q1612yescomplemented
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4#4111yescharacteristic
65C2×Q16—84no—
66C2×Q16—82no—
67C22×C8—22no—
68C22×C8—24no—
69C22×C8—41no—
70C4⋊Q8—44no—
71C22⋊Q8—84no—
72C4×Q8—28no—
73C4×Q8—44no—
74C4×Q8—44no—
75C4×Q8—44no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C42—42no—
90C2×C42—42no—
91C2×C42—42no—
92C2×C42—42no—
93C2×C42C2×D412yes—
94C2×C42C2×D412yes—
95#107—48no—
96C4⋊C8—44no—
97#103—48no—
98#103—48no—
99#103—84no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104#95—42no—
105#95—42no—
106C24#12111yescharacteristic
107C2×Q8—24no—
108C2×Q8—24no—
109C2×Q8—24no—
110C2×Q8—44no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—44no—
114C2×Q8—42no—
115C2×Q8—44no—
116C2×Q8—42no—
117C2×Q8—48no—
118C2×Q8—42no—
119C2×Q8—44no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—82no—
139C22×C4#12012yes—
140C22×C4#12011yescharacteristic
141C22×C4#12112yes—
142C22×C4#12711yescharacteristic
143C22×C4C2×Q1611yescharacteristic
144C22×C4#13511yescharacteristic
145C22×C4#13712yes—
146Q16—88no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8—42no—
152C2×C8—81no—
153C2×C8—82no—
154C4⋊C4—24no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C22⋊C4—84no—
165C42—24no—
166C42—42no—
167C42—48no—
168C23—22no—
169C23—21no—
170C23—22no—
171C23—21no—
172C23#39211yescharacteristic
173C23#46711yescharacteristic
174C23#47011yescharacteristic
175Q8—28no—
176Q8—44no—
177Q8—44no—
178Q8—44no—
179Q8—84no—
180Q8—84no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4#45112yes—
211C2×C4#45212yes—
212C2×C4B(4,𝔽2)12yes—
213C2×C4#47314yes—
214C8—84no—
215C8—82no—
216C8—161no—
217C22—21no—
218C22—21no—
219C22—22no—
220C22—22no—
221C22—41no—
222C22—42no—
223C22—41no—
224C22#200111yescharacteristic
225C22#200812yes—
226C22#201011yescharacteristic
227C22#202012yes—
228C22#202111yescharacteristic
229C4—24no—
230C4—22no—
231C4—22no—
232C4—22no—
233C4—44no—
234C4—41no—
235C4—82no—
236C4—162no—
237C4—162no—
238C2—41no—
239C2—41no—
240C2#998011yescharacteristic
241C2#999111yescharacteristic
242C2#1004112yes—
243C2#1018312yes—
244C2#1032411yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
59. C22×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#329572
#327622
#83032
#77252
#70442