Finite Group Databasebeta

← #122834

Subgroups — #122834

Order 512.

311 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 66 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#28185C211yesmaximal, characteristic
2#11494C211yesmaximal, characteristic, complemented
3#8615C211yesmaximal, characteristic, complemented
4#7091C212yesmaximal, complemented
5#7088C212yesmaximal
6C2×C8×Q8C2211yescharacteristic
7C2×C4×D8C411yescharacteristic
8#2538—21no—
9#2163—21no—
10#2086C411yescharacteristic
11#2086C2211yescharacteristic
12#2054C411yescharacteristic
13#2054C2211yescharacteristic
14#1934—21no—
15#1934—21no—
16#1834C411yescharacteristic
17#1802—21no—
18#1801—21no—
19#1566—21no—
20#1322—21no—
21#1322C2212yes—
22#1292C2212yes—
23C22×D8C2×C411yescharacteristic
24C2×C22⋊Q8—41no—
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4—21no—
27C8×Q8—41no—
28C8×Q8—41no—
29C4×D8—41no—
30C4×D8—41no—
31C2×#107C2×C411yescharacteristic
32#423—22no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8D411yescharacteristic
35C2×C4⋊C8C2311yescharacteristic
36C2×#102—21no—
37C2×#102—21no—
38C2×#102—21no—
39C2×#101—41no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#396—41no—
44#396—41no—
45#390—21no—
46#390—21no—
47#389—21no—
48#389—21no—
49#384—21no—
50#380—41no—
51C4⋊C16—41no—
52C22⋊C16—22no—
53#346—41no—
54#343—41no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340C2×C411yescharacteristic
59#340C2×C411yescharacteristic
60#334—41no—
61#334—41no—
62C22×Q8D811yescharacteristic
63C22×D4—21no—
64C23×C4—41no—
65C2×D8—41no—
66C2×D8—42no—
67C2×M16—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C22⋊Q8—84no—
73C4×Q8—41no—
74C4×Q8—41no—
75C4×D4—42no—
76C4×D4—42no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
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×C42C22⋊C411yescharacteristic
93C2×C42C22⋊C411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C16—42no—
96#107—41no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102#102—42no—
103#102—42no—
104#102—41no—
105#102—41no—
106#101—82no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—21no—
113#95—41no—
114#95—41no—
115C24—41no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—41no—
119C2×D4—22no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—82no—
129C22×C4—82no—
130C22×C4—82no—
131C22×C4#10211yescharacteristic
132C22×C4#10211yescharacteristic
133C22×C4#10211yescharacteristic
134C22×C4#10211yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C2×D811yescharacteristic
137C22×C4C2×SD1611yescharacteristic
138D8—42no—
139D8—42no—
140M16—82no—
141M16—82no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—82no—
150C2×C8#9911yescharacteristic
151C2×C8#9911yescharacteristic
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C4⋊C4—161no—
165C22⋊C4—82no—
166C22⋊C4—82no—
167C22⋊C4—82no—
168C22⋊C4—161no—
169C22⋊C4—161no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—21no—
176C16—82no—
177C23—42no—
178C23—81no—
179C23—81no—
180C23—81no—
181C23—81no—
182C23—81no—
183C23—81no—
184C23C2×#10211yescharacteristic
185Q8—44no—
186D4—42no—
187D4—42no—
188D4—81no—
189D4—81no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—164no—
215C2×C4#36012yes—
216C2×C4C2×#9911yescharacteristic
217C2×C4#42111yescharacteristic
218C2×C4#50912yes—
219C8—21no—
220C8—21no—
221C8—41no—
222C8—41no—
223C8—41no—
224C8—162no—
225C22—21no—
226C22—21no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22—161no—
231C22—161no—
232C22—161no—
233C22—161no—
234C22#151911yescharacteristic
235C22#214011yescharacteristic
236C22C2×#36011yescharacteristic
237C4—21no—
238C4—22no—
239C4—22no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—161no—
245C4—161no—
246C4—161no—
247C4#212011yescharacteristic
248C4#213311yescharacteristic
249C2—21no—
250C2—21no—
251C2—162no—
252C2#1187211yescharacteristic
253C2#1187911yescharacteristic
254C2#1188811yescharacteristic
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. #11494C2semidirect
3. #8615C2semidirect
4. #7091C2semidirect

Maximal subgroups

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

subgroupindex
#281852
#114942
#86152
#70912
#70882