Finite Group Databasebeta

← #122844

Subgroups — #122844

Order 512.

305 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 68 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#28739C211yesmaximal, characteristic
2#11493C211yesmaximal, characteristic, complemented
3#8615C211yesmaximal, characteristic, complemented
4#7090C212yesmaximal, complemented
5#7088C212yesmaximal
6C2×#495C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2592—21no—
9#2162—21no—
10#2056C411yescharacteristic
11#2056C2211yescharacteristic
12#2054C411yescharacteristic
13#2054C2211yescharacteristic
14#1983—21no—
15#1968—21no—
16#1968—21no—
17#1801—21no—
18C4×#106C411yescharacteristic
19#1566—21no—
20#1322—21no—
21#1322C2212yes—
22#1292C2212yes—
23C22×Q16C2×C411yescharacteristic
24C22×SD16—21no—
25C22×D8C2×C411yescharacteristic
26C2×#124—21no—
27C2×C4×Q8D411yescharacteristic
28#495—41no—
29#495—41no—
30C8×Q8—41no—
31C8×Q8—41no—
32C2×#106—21no—
33#423—22no—
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8D411yescharacteristic
36C2×C4⋊C8C2311yescharacteristic
37C2×#103—21no—
38C2×#103—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×#101—41no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#397—41no—
46#393—41no—
47#391—21no—
48#391—21no—
49#390—21no—
50#390—21no—
51#388—41no—
52#388—41no—
53C4×C4⋊C4—21no—
54C4⋊C16—41no—
55C22⋊C16—22no—
56C4×C16—41no—
57#343—41no—
58#340C2×C411yescharacteristic
59#340C2×C411yescharacteristic
60#334—41no—
61#334—41no—
62C22×Q8—21no—
63C22×Q8D811yescharacteristic
64C22×D4—21no—
65C2×Q16—41no—
66C2×Q16—81no—
67C2×SD16—42no—
68C2×SD16—42no—
69C2×SD16—41no—
70C2×SD16—41no—
71C2×D8—41no—
72C2×D8—81no—
73C2×M16—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78#124—42no—
79#124—42no—
80C4×Q8—41no—
81C4×Q8—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D811yescharacteristic
85C2×C4⋊C4SD1611yescharacteristic
86C2×C4⋊C4SD1611yescharacteristic
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C42—41no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C22⋊C411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C16—42no—
98#106—42no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—42no—
103C4⋊C8—42no—
104#103—81no—
105#103—81no—
106#102—81no—
107#102—81no—
108#101—82no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—21no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120C24—41no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4#10211yescharacteristic
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4#10211yescharacteristic
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4C2×SD1611yescharacteristic
146Q16—82no—
147SD16—82no—
148SD16—82no—
149D8—82no—
150M16—82no—
151M16—82no—
152C2×C8—22no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—82no—
160C2×C8#9911yescharacteristic
161C2×C8#9911yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C22⋊C4—82no—
170C22⋊C4—82no—
171C22⋊C4—82no—
172C22⋊C4—82no—
173C22⋊C4—161no—
174C22⋊C4—161no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—161no—
182C16—82no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23—81no—
187C23—81no—
188C23—81no—
189C23—81no—
190C23C2×#10211yescharacteristic
191Q8—44no—
192Q8—81no—
193Q8—81no—
194Q8—81no—
195Q8—81no—
196D4—81no—
197D4—81no—
198D4—81no—
199D4—81no—
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—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4#36012yes—
225C2×C4C2×#9911yescharacteristic
226C2×C4#42111yescharacteristic
227C2×C4#50812yes—
228C8—21no—
229C8—41no—
230C8—41no—
231C8—41no—
232C8—162no—
233C8#35512yes—
234C22—21no—
235C22—21no—
236C22—161no—
237C22—161no—
238C22—161no—
239C22—161no—
240C22—161no—
241C22—161no—
242C22—161no—
243C22#151911yescharacteristic
244C22#213811yescharacteristic
245C22C2×#36011yescharacteristic
246C4—21no—
247C4—22no—
248C4—22no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—161no—
254C4—161no—
255C4—161no—
256C4#212211yescharacteristic
257C4C2×#35511yescharacteristic
258C2—21no—
259C2—21no—
260C2—321no—
261C2#1187211yescharacteristic
262C2#1188211yescharacteristic
263C2#1188511yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11493C2semidirect
3. #8615C2semidirect
4. #7090C2semidirect

Maximal subgroups

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

subgroupindex
#287392
#114932
#86152
#70902
#70882