Finite Group Databasebeta

← #139421

Subgroups — #139421

Order 512.

383 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 84 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#33497C211yesmaximal, characteristic, complemented
2#28736C211yesmaximal, characteristic
3#11964C212yesmaximal
4#8623C211yesmaximal, characteristic, complemented
5#7301C212yesmaximal, complemented
6C2×#493C2211yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2548—21no—
9#2183—24no—
10#2163—21no—
11#2163C2212yes—
12#2162—21no—
13#2054C412yes—
14#2054C412yes—
15#1982—21no—
16#1970—21no—
17#1970—21no—
18C4×#107C2211yescharacteristic
19#1568C2212yes—
20#1467—21no—
21#1361—24no—
22#1290—21no—
23C22×D8—21no—
24C22×D8C2×C412yes—
25C2×#127—21no—
26C2×#124—41no—
27C2×C4×Q8D411yescharacteristic
28#493—22no—
29#493—22no—
30C8×Q8—22no—
31C8×Q8—22no—
32#432—22no—
33C2×#107—21no—
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8D411yescharacteristic
37C2×#102—22no—
38C2×#102—22no—
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8C2311yescharacteristic
42#394—41no—
43#394—41no—
44#390—22no—
45#390—22no—
46#390—41no—
47C4×C4⋊C4—21no—
48C4⋊C16—22no—
49C4⋊C16—22no—
50C4⋊C16D414yes—
51C22⋊C16—22no—
52#346—22no—
53#346—22no—
54C4×C16—22no—
55#340C2×C412yes—
56#335—22no—
57#334—22no—
58#324—41no—
59#324—41no—
60C22×Q8SD1611yescharacteristic
61C22×D4—22no—
62C22×D4—41no—
63C22×D4—41no—
64C2×D8—24no—
65C2×D8—42no—
66C2×D8—42no—
67C2×D8—41no—
68C2×D8—41no—
69C2×D8—82no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74#127—42no—
75#127—42no—
76#124—84no—
77C4×Q8—22no—
78C4×Q8—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4D811yescharacteristic
81C2×C4⋊C4D811yescharacteristic
82C2×C4⋊C4SD1611yescharacteristic
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C42—41no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90C2×C42C2×D411yescharacteristic
91C2×C16—24no—
92#107—42no—
93C4⋊C8—22no—
94C4⋊C8—22no—
95C4⋊C8—22no—
96C4⋊C8—42no—
97#102—82no—
98#102—82no—
99C22⋊C8—42no—
100#97—42no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8C2×D412yes—
106C4×C8#4112yes—
107#95—41no—
108C24—42no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—41no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—82no—
119C2×D4—82no—
120C2×D4—82no—
121C2×D4—82no—
122C2×D4—162no—
123C22×C4—42no—
124C22×C4#10211yescharacteristic
125C22×C4#10211yescharacteristic
126C22×C4#10211yescharacteristic
127C22×C4#10211yescharacteristic
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C2×D811yescharacteristic
130C22×C4C2×SD1611yescharacteristic
131D8—84no—
132D8—82no—
133D8—82no—
134C2×C8—22no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—82no—
141C2×C8C4×D412yes—
142C2×C8#12011yescharacteristic
143C2×C8#12311yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C22⋊C4—84no—
150C22⋊C4—84no—
151C22⋊C4—162no—
152C22⋊C4—162no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—82no—
158C42#12112yes—
159C42#13512yes—
160C16—44no—
161C23—82no—
162C23—82no—
163C23—82no—
164C23—82no—
165C23—82no—
166C23—82no—
167C23—82no—
168C23C2×#10211yescharacteristic
169Q8—44no—
170D4—82no—
171D4—82no—
172D4—82no—
173D4—82no—
174D4—164no—
175D4—164no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4#36112yes—
194C2×C4#38611yescharacteristic
195C2×C4#41611yescharacteristic
196C2×C4#46512yes—
197C2×C4#46612yes—
198C8—21no—
199C8—22no—
200C8—41no—
201C8—41no—
202C8—162no—
203C8#45412yes—
204C22—21no—
205C22—21no—
206C22—162no—
207C22—162no—
208C22—162no—
209C22—162no—
210C22—162no—
211C22—162no—
212C22—162no—
213C22#189311yescharacteristic
214C22C2×#36111yescharacteristic
215C22#214811yescharacteristic
216C4—22no—
217C4—22no—
218C4—22no—
219C4—21no—
220C4—41no—
221C4—41no—
222C4—41no—
223C4—162no—
224C4#188811yescharacteristic
225C4#191411yescharacteristic
226C2—21no—
227C2—21no—
228C2—322no—
229C2#939811yescharacteristic
230C2#1306711yescharacteristic
231C2#1307411yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#493C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33497C2semidirect
4. #8623C2semidirect
5. #7301C2semidirect

Maximal subgroups

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

subgroupindex
#334972
#287362
#119642
#86232
#73012