Finite Group Databasebeta

← #122811

Subgroups — #122811

Order 512.

314 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 64 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#28554C211yesmaximal, characteristic
2#11487C211yesmaximal, characteristic
3#8611C211yesmaximal, characteristic
4#7087C212yesmaximal, complemented
5#7086C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2415—21no—
9#2162—21no—
10#2086C411yescharacteristic
11#2075C2211yescharacteristic
12#2056C411yescharacteristic
13#2001C2211yescharacteristic
14#1933—21no—
15#1892—21no—
16#1873—21no—
17#1810—21no—
18#1735C411yescharacteristic
19#1465—21no—
20#1314—21no—
21#1314C2212yes—
22#1293C2212yes—
23C22×Q16C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C22⋊Q8—21no—
27C2×C22⋊Q8—21no—
28C2×C4×D4D411yescharacteristic
29C2×#117—21no—
30#467—41no—
31#467—41no—
32C8×D4—41no—
33C8×D4—41no—
34C2×#108—41no—
35C2×#107C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40#407—22no—
41C2×C22⋊C8D411yescharacteristic
42C2×C22⋊C8C2311yescharacteristic
43C2×C4×C8D411yescharacteristic
44#397—41no—
45#394—41no—
46#391—21no—
47#389—21no—
48#389—41no—
49#388—41no—
50#380—21no—
51C4⋊C16—41no—
52C22⋊C16—22no—
53C4×C16—41no—
54#341—41no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340C2×C411yescharacteristic
59#340C2×C411yescharacteristic
60#324—41no—
61#323—41no—
62C22×Q8—21no—
63C22×D4SD1611yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C2×Q16—41no—
67C2×Q16—81no—
68C2×M16—42no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—81no—
74C22×C8—81no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C22⋊Q8—42no—
80C22⋊Q8—42no—
81C22⋊Q8—42no—
82C22⋊Q8—81no—
83C4×D4—41no—
84C4×D4—41no—
85#117—42no—
86#117—81no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4D811yescharacteristic
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4D811yescharacteristic
99C2×C22⋊C4SD1611yescharacteristic
100C2×C42—41no—
101C2×C42C22⋊C411yescharacteristic
102C2×C16—42no—
103#108—82no—
104#107—41no—
105C4⋊C8—41no—
106#103—42no—
107#103—41no—
108#103—41no—
109#103—81no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C4×C8—21no—
115C4×C8—21no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120C24#10211yescharacteristic
121C24C2×SD1611yescharacteristic
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×D4—22no—
129C2×D4—41no—
130C2×D4—41no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4#10211yescharacteristic
144C22×C4#10211yescharacteristic
145C22×C4#10211yescharacteristic
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C2×D811yescharacteristic
148Q16—82no—
149M16—82no—
150M16—82no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—41no—
155C2×C8—42no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—81no—
160C2×C8—81no—
161C2×C8—81no—
162C2×C8—82no—
163C2×C8#9911yescharacteristic
164C2×C8#9911yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C22⋊C4—82no—
180C22⋊C4—82no—
181C42—21no—
182C42—21no—
183C42—82no—
184C16—82no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23C2×#10211yescharacteristic
196Q8—81no—
197Q8—81no—
198Q8—81no—
199Q8—81no—
200D4—44no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4#35912yes—
230C2×C4C2×#9911yescharacteristic
231C2×C4#42111yescharacteristic
232C8—21no—
233C8—21no—
234C8—41no—
235C8—41no—
236C8—41no—
237C8—162no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22#151911yescharacteristic
251C22C2×#35911yescharacteristic
252C22#214811yescharacteristic
253C4—21no—
254C4—22no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—82no—
259C4—161no—
260C4—161no—
261C4—161no—
262C4#212111yescharacteristic
263C4#212611yescharacteristic
264C2—21no—
265C2—21no—
266C2—81no—
267C2—81no—
268C2#1187111yescharacteristic
269C2#1187611yescharacteristic
270C2#1189311yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C22⋊C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7087C2semidirect
5. #7086C2semidirect

Maximal subgroups

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

subgroupindex
#285542
#114872
#86112
#70872
#70862