Finite Group Databasebeta

← #102462

Subgroups — #102462

Order 512.

624 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 98 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#30291C211yesmaximal, characteristic
3#11380C211yesmaximal, characteristic
4#7667C212yesmaximal
5#7020C211yesmaximal, characteristic
6C22×C4×Q8C2211yescharacteristic
7C2×#477—22no—
8#2857C2212yes—
9#2083—22no—
10#2057—22no—
11#1863—24no—
12#1852—22no—
13#1808—22no—
14#1803—22no—
15#1797C2212yes—
16#1787—22no—
17C2×#336—22no—
18C2×#336C2211yescharacteristic
19C2×#336C2211yescharacteristic
20#1272—24no—
21C23×Q8D411yescharacteristic
22C23×C8—21no—
23C23×C8D412yes—
24C2×C4⋊Q8D412yes—
25C2×C22⋊Q8—22no—
26C2×C4×Q8—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—44no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4D411yescharacteristic
31C22×C42—21no—
32C22×C42C2311yescharacteristic
33#477—48no—
34C2×#107—24no—
35C2×#107—22no—
36C2×#107—22no—
37C2×C4⋊C8—22no—
38C2×#103—24no—
39C2×#103—24no—
40C2×C22⋊C8—22no—
41#391—22no—
42#389—22no—
43#389—22no—
44#384D412yes—
45#380—22no—
46#380—22no—
47#340—24no—
48#340—24no—
49#336—22no—
50#336—22no—
51#336—44no—
52#336D414yes—
53C22×Q8—21no—
54C22×Q8—22no—
55C22×Q8—22no—
56C22×Q8—22no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8Q1612yes—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4#4111yescharacteristic
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—24no—
69C22×C8—42no—
70C22×C8—42no—
71C22×C8—41no—
72C22×C8—42no—
73C22×C8—41no—
74C22×C8—42no—
75C22×C8Q1614yes—
76C4⋊Q8—44no—
77C22⋊Q8—84no—
78C4×Q8—28no—
79C4×Q8—44no—
80C4×Q8—44no—
81C4×Q8—44no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C42—24no—
95C2×C42—42no—
96C2×C42—42no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42C2×D412yes—
100C2×C42C2×D412yes—
101#107—48no—
102#107—82no—
103#107—82no—
104C4⋊C8—44no—
105#103—48no—
106#103—48no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109#95—42no—
110#95—42no—
111C24#12111yescharacteristic
112C2×Q8—24no—
113C2×Q8—24no—
114C2×Q8—24no—
115C2×Q8—44no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—44no—
119C2×Q8—44no—
120C2×Q8—48no—
121C2×Q8—42no—
122C2×Q8—82no—
123C2×Q8—82no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4#12012yes—
147C22×C4#12011yescharacteristic
148C22×C4#12112yes—
149C22×C4#12711yescharacteristic
150C22×C4C2×Q1612yes—
151C22×C4C2×Q1611yescharacteristic
152C22×C4#13511yescharacteristic
153C22×C4#13712yes—
154C22×C4#13712yes—
155C2×C8—24no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—42no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8—82no—
165C2×C8—81no—
166C2×C8—82no—
167C2×C8—81no—
168C4⋊C4—24no—
169C4⋊C4—42no—
170C4⋊C4—48no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—82no—
177C4⋊C4—84no—
178C22⋊C4—84no—
179C42—24no—
180C42—42no—
181C42—44no—
182C42—44no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23#39211yescharacteristic
188C23#46212yes—
189C23#46711yescharacteristic
190C23#47011yescharacteristic
191C23#47212yes—
192Q8—28no—
193Q8—44no—
194Q8—44no—
195Q8—44no—
196Q8—164no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4#45112yes—
231C2×C4#45112yes—
232C2×C4#45212yes—
233C2×C4#45511yescharacteristic
234C2×C4#45611yescharacteristic
235C2×C4B(4,𝔽2)12yes—
236C2×C4#49411yescharacteristic
237C2×C4#49711yescharacteristic
238C8—44no—
239C8—84no—
240C8—82no—
241C22—21no—
242C22—21no—
243C22—22no—
244C22—22no—
245C22—21no—
246C22—21no—
247C22—22no—
248C22—22no—
249C22#167612yes—
250C22#169612yes—
251C22#200111yescharacteristic
252C22#200711yescharacteristic
253C22#201011yescharacteristic
254C22#201611yescharacteristic
255C22#201711yescharacteristic
256C22#201811yescharacteristic
257C22#202111yescharacteristic
258C4—22no—
259C4—22no—
260C4—21no—
261C4—21no—
262C4—44no—
263C4—42no—
264C4—82no—
265C4—162no—
266C4—162no—
267C4#160212yes—
268C4#160512yes—
269C2—22no—
270C2—22no—
271C2#996711yescharacteristic
272C2#997411yescharacteristic
273C2#1002811yescharacteristic
274C2#1003711yescharacteristic
275C2#1025911yescharacteristic
276C2#1027911yescharacteristic
277C2#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

Maximal subgroups

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

subgroupindex
#329572
#302912
#113802
#76672
#70202