Finite Group Databasebeta

← #102357

Subgroups — #102357

Order 512.

460 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 118 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#33207C211yesmaximal, characteristic
2#10864C211yesmaximal, characteristic
3#8144C211yesmaximal, characteristic
4#7688C212yesmaximal
5#7687C212yesmaximal
6C2×#498C411yescharacteristic
7C22×#108—21no—
8C22×C4×C8C2211yescharacteristic
9#2857C2212yes—
10#1990C412yes—
11#1941—21no—
12#1941—21no—
13#1844C411yescharacteristic
14#1807—21no—
15#1807—21no—
16C4×C22⋊C8C2212yes—
17C2×#341—21no—
18#1728C2211yescharacteristic
19#1728C2211yescharacteristic
20C2×#322—21no—
21#1273—21no—
22#1272—24no—
23#1271—24no—
24C22×M16—21no—
25C22×M16—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C4⋊Q8C2×C412yes—
29C2×C22⋊Q8—22no—
30C22×C42C2311yescharacteristic
31#498—42no—
32C2×#108—24no—
33C2×#108—42no—
34C2×#108—42no—
35C2×#106—21no—
36C2×#106—21no—
37C2×#106—21no—
38C2×#106—21no—
39C2×C22⋊C8—22no—
40C2×C4×C8—22no—
41C2×C4×C8—42no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#391—22no—
47#389—22no—
48#389—22no—
49#384C2×C412yes—
50#380—22no—
51#380—22no—
52#341—42no—
53#340—22no—
54#340—22no—
55#336—22no—
56#336—22no—
57#336D414yes—
58#322—42no—
59C22×Q8—22no—
60C23×C4C2×D411yescharacteristic
61C23×C4#4112yes—
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—24no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8SD1612yes—
86C22×C8SD1612yes—
87C22×C8Q1612yes—
88C22×C8Q1612yes—
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91C22⋊Q8—84no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42C22×C411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D412yes—
104#108—48no—
105#106—42no—
106#106—42no—
107#106—42no—
108#106—42no—
109C22⋊C8—44no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—41no—
113C4×C8—41no—
114C4×C8—42no—
115C4×C8—42no—
116#95—42no—
117#95—42no—
118C24#12411yescharacteristic
119C2×Q8—44no—
120C2×Q8—82no—
121C2×Q8—82no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C4×D412yes—
133C22×C4#12112yes—
134C22×C4#12711yescharacteristic
135C22×C4C2×SD1611yescharacteristic
136C22×C4C2×SD1611yescharacteristic
137C22×C4C2×Q1611yescharacteristic
138C22×C4C2×Q1611yescharacteristic
139M16—42no—
140M16—42no—
141M16—42no—
142M16—42no—
143M16—81no—
144M16—81no—
145M16—81no—
146M16—81no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—24no—
154C2×C8—24no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—44no—
162C2×C8—81no—
163C2×C8—81no—
164C2×C8—81no—
165C2×C8—81no—
166C2×C8#9912yes—
167C2×C8#10312yes—
168C2×C8#10312yes—
169C2×C8#10312yes—
170C2×C8#10312yes—
171C2×C8#10412yes—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—82no—
178C4⋊C4—82no—
179C4⋊C4—84no—
180C22⋊C4—84no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—42no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23#39011yescharacteristic
191C23#48711yescharacteristic
192C23#48711yescharacteristic
193Q8—164no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—48no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4C2×#9911yescharacteristic
216C2×C4#41011yescharacteristic
217C2×C4C2×#10311yescharacteristic
218C2×C4C2×#10311yescharacteristic
219C2×C4C2×#10411yescharacteristic
220C2×C4#42111yescharacteristic
221C2×C4#44414yes—
222C2×C4#47114yes—
223C2×C4#49211yescharacteristic
224C2×C4#49411yescharacteristic
225C8—22no—
226C8—22no—
227C8—22no—
228C8—84no—
229C8—82no—
230C8—82no—
231C8#32814yes—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22#195611yescharacteristic
241C22#196411yescharacteristic
242C22#197512yes—
243C22#197712yes—
244C22#198011yescharacteristic
245C4—24no—
246C4—24no—
247C4—21no—
248C4—21no—
249C4—162no—
250C4—162no—
251C4C2×#32811yescharacteristic
252C4#150311yescharacteristic
253C4#151811yescharacteristic
254C4#152011yescharacteristic
255C2—41no—
256C2—41no—
257C2#967911yescharacteristic
258C2#972812yes—
259C2#976711yescharacteristic
260C2#981612yes—
261C2#986111yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
#332072
#108642
#81442
#76882
#76872