Finite Group Databasebeta

← #115790

Subgroups — #115790

Order 512.

712 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 178 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#20753C211yesmaximal, characteristic
2C2×#1549C212yesmaximal
3#14282C211yesmaximal, characteristic
4C2×#1286C212yesmaximal
5#7906C211yesmaximal, characteristic
6C22×C4⋊Q8C2211yescharacteristic
7C2×#431C2212yes—
8C2×#423C2212yes—
9C2×#382C2211yescharacteristic
10C2×#376C2211yescharacteristic
11#1832—24no—
12#1728—22no—
13#1549—24no—
14#1549—24no—
15C2×#332—24no—
16#1446—22no—
17#1445C414yes—
18C4⋊#95—22no—
19#1286—24no—
20#1280—22no—
21C23×Q8—21no—
22C22×M16—22no—
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—21no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8D412yes—
30C2×C4⋊Q8Q812yes—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C412yes—
34C22×C42D412yes—
35C22×C42C2311yescharacteristic
36#431—22no—
37#431—22no—
38#431—22no—
39#431D414yes—
40#423—22no—
41#423—22no—
42#423—22no—
43#423—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8D414yes—
47C2×C22⋊C8—22no—
48C2×C4×C8—22no—
49#382—22no—
50#382—22no—
51#376—22no—
52#376—22no—
53C2×#95—21no—
54C2×#95—28no—
55C2×#95C2×C412yes—
56C2×#95C2×C412yes—
57C2×#95D412yes—
58#336—28no—
59#332—216no—
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—84no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4C22⋊C414yes—
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—44no—
74C2×M16—44no—
75C22×C8—24no—
76C22×C8—42no—
77C4⋊Q8—24no—
78C4⋊Q8—24no—
79C4⋊Q8—24no—
80C4⋊Q8—24no—
81C4⋊Q8—42no—
82C4⋊Q8—42no—
83C4⋊Q8—42no—
84C4⋊Q8—42no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4C4⋊C414yes—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—24no—
101C2×C42—24no—
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D412yes—
104C2×C42C2×D412yes—
105C2×C42C2×Q811yescharacteristic
106C2×C42#4111yescharacteristic
107C2×C42#4112yes—
108C2×C42#4112yes—
109C2×C42#4111yescharacteristic
110C4⋊C8—24no—
111C4⋊C8—24no—
112C4⋊C8—24no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C22⋊C8—24no—
117C22⋊C8—42no—
118C4×C8—24no—
119C4×C8—42no—
120#95—24no—
121#95—24no—
122#95—24no—
123#95—42no—
124#95—88no—
125C24C2×C22⋊C411yescharacteristic
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—88no—
129C2×Q8—84no—
130C2×Q8—84no—
131C2×Q8—88no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—48no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4C2×C4⋊C411yescharacteristic
148C22×C4#11711yescharacteristic
149C22×C4C4×D412yes—
150C22×C4C4×D412yes—
151C22×C4#12011yescharacteristic
152C22×C4#12112yes—
153C22×C4#12111yescharacteristic
154C22×C4#12112yes—
155C22×C4#12111yescharacteristic
156C22×C4C22⋊Q811yescharacteristic
157C22×C4C22⋊Q811yescharacteristic
158C22×C4#12311yescharacteristic
159C22×C4#12411yescharacteristic
160C22×C4#12711yescharacteristic
161C22×C4#13514yes—
162C22×C4#13714yes—
163M16—84no—
164M16—82no—
165M16—82no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8—48no—
169C2×C8—84no—
170C4⋊C4—28no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C42—44no—
179C42—44no—
180C42#12012yes—
181C42#12112yes—
182C42C22⋊Q812yes—
183C42C22⋊Q812yes—
184C42#12312yes—
185C42#12312yes—
186C42#12412yes—
187C42#12612yes—
188C23—22no—
189C23—22no—
190C23#38011yescharacteristic
191C23#38611yescharacteristic
192C23#39011yescharacteristic
193C23#41612yes—
194C23#41912yes—
195Q8—88no—
196Q8—88no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—48no—
206C2×C4—48no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—84no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—88no—
217C2×C4—82no—
218C2×C4#38512yes—
219C2×C4#38812yes—
220C2×C4#39211yescharacteristic
221C2×C4#39311yescharacteristic
222C2×C4#39412yes—
223C2×C4#39612yes—
224C2×C4#39711yescharacteristic
225C2×C4#39911yescharacteristic
226C2×C4#45214yes—
227C2×C4#46414yes—
228C2×C4#47914yes—
229C2×C4#48514yes—
230C2×C4#48812yes—
231C2×C4#49112yes—
232C2×C4#49512yes—
233C2×C4#49712yes—
234C8—84no—
235C8—84no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—24no—
241C22—24no—
242C22#143911yescharacteristic
243C22#180512yes—
244C22#187312yes—
245C22#191012yes—
246C22#191512yes—
247C22#195811yescharacteristic
248C22#196011yescharacteristic
249C4—22no—
250C4—22no—
251C4—22no—
252C4—22no—
253C4—84no—
254C4—84no—
255C4—162no—
256C4#201512yes—
257C4#202612yes—
258C4#207412yes—
259C4#209712yes—
260C2—21no—
261C2—22no—
262C2—21no—
263C2#919414yes—
264C2#1104811yescharacteristic
265C2#1105211yescharacteristic
266C2#1107311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#207532
C2×#15492
#142822
C2×#12862
#79062