Finite Group Databasebeta

← #101264

Subgroups — #101264

Order 512.

545 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 90 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#23507C211yesmaximal, characteristic
2#21921C211yesmaximal, characteristic
3#8340C212yesmaximal
4#7598C211yesmaximal, characteristic
5#7045C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C2×#419—21no—
8C2×#419—21no—
9#2296C2211yescharacteristic
10#2043—22no—
11#2043C414yes—
12#1917—22no—
13#1916—22no—
14#1818—21no—
15#1797C2212yes—
16#1797C2212yes—
17#1750—21no—
18#1728C2211yescharacteristic
19C2×#332—22no—
20C2×#332—22no—
21#1271—22no—
22C23×Q8—21no—
23C22×M16—21no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—21no—
26C2×C4⋊Q8—42no—
27C2×C4⋊Q8—42no—
28C2×C4⋊Q8C2×C412yes—
29C2×#117D411yescharacteristic
30C2×#117D411yescharacteristic
31C22×C4⋊C4—22no—
32C22×C42C2311yescharacteristic
33C2×C4⋊C8—22no—
34C2×C4⋊C8D412yes—
35C2×C4⋊C8D412yes—
36#419—44no—
37#419—42no—
38#419—42no—
39C2×#103—24no—
40C2×#103—24no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×#97—21no—
44C2×C4×C8—21no—
45#389C2×C412yes—
46#389D412yes—
47#385—22no—
48C4×C4⋊C4—22no—
49#336—21no—
50#336—22no—
51#336—21no—
52#336—22no—
53#336—22no—
54#336—21no—
55#336—22no—
56#336C2×C412yes—
57#332—44no—
58#332—44no—
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—41no—
63C22×Q8—41no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—41no—
67C22×Q8—42no—
68C22×Q8—42no—
69C23×C4—41no—
70C23×C4C2×D411yescharacteristic
71C23×C4#4112yes—
72C2×M16—42no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—42no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—42no—
80C4⋊Q8—24no—
81C4⋊Q8—24no—
82C4⋊Q8—42no—
83C4⋊Q8—42no—
84C4⋊Q8—44no—
85C4⋊Q8—44no—
86#117—22no—
87#117—22no—
88#117—41no—
89#117—41no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—48no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4C22⋊C412yes—
96C2×C4⋊C4C22⋊C412yes—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—41no—
102C2×C42—41no—
103C2×C42C22×C411yescharacteristic
104C2×C42C2×D412yes—
105C2×C42C2×D411yescharacteristic
106C4⋊C8—42no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110#103—48no—
111#103—48no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—82no—
115#97—81no—
116C4×C8—81no—
117#95—22no—
118#95—22no—
119C24#12311yescharacteristic
120C2×Q8—24no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—44no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—82no—
133C2×Q8—82no—
134C2×Q8—82no—
135C2×Q8—82no—
136C2×Q8—82no—
137C2×Q8—82no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—82no—
151C22×C4—82no—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C4×D412yes—
154C22×C4#12011yescharacteristic
155C22×C4#12112yes—
156C22×C4#13512yes—
157C22×C4AGL(1,ℤ/8)12yes—
158M16—82no—
159M16—82no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—82no—
170C2×C8—82no—
171C2×C8—82no—
172C4⋊C4—28no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—82no—
182C22⋊C4—42no—
183C22⋊C4—81no—
184C42—22no—
185C42—22no—
186C42—41no—
187C42—41no—
188C42—42no—
189C42—41no—
190C42—41no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23—22no—
195C23#38611yescharacteristic
196C23#48212yes—
197Q8—44no—
198Q8—44no—
199Q8—82no—
200Q8—82no—
201Q8—84no—
202Q8—84no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—84no—
224C2×C4—84no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4#41611yescharacteristic
228C2×C4#41811yescharacteristic
229C2×C4#44912yes—
230C2×C4#45412yes—
231C2×C4B(4,𝔽2)11yescharacteristic
232C2×C4B(4,𝔽2)11yescharacteristic
233C2×C4#47112yes—
234C2×C4#47312yes—
235C8—82no—
236C8—82no—
237C8—82no—
238C8—162no—
239C22—22no—
240C22—24no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22#187511yescharacteristic
245C22#188811yescharacteristic
246C22#188911yescharacteristic
247C22#190212yes—
248C22#190812yes—
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—82no—
255C4—84no—
256C4—82no—
257C4#159712yes—
258C4#159712yes—
259C2—42no—
260C2#901811yescharacteristic
261C2#917212yes—
262C2#942211yescharacteristic
263C2#943611yescharacteristic
264C2#951912yes—
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×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
#235072
#219212
#83402
#75982
#70452