Finite Group Databasebeta

← #149934

Subgroups — #149934

Order 512.

265 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 54 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#27985C211yesmaximal, characteristic
2#11514C211yesmaximal, characteristic
3#11450C211yesmaximal, characteristic
4#7367C212yesmaximal, complemented
5#7355C212yesmaximal, complemented
6#2621C411yescharacteristic
7#2317C2211yescharacteristic
8#2074—21no—
9#2072—21no—
10#1960C411yescharacteristic
11#1956C2211yescharacteristic
12#1873—21no—
13#1872—21no—
14#1836C2211yescharacteristic
15#1835C411yescharacteristic
16#1750C411yescharacteristic
17#1463—21no—
18#1458—21no—
19#1325—21no—
20#1325C2212yes—
21#1325C2212yes—
22#1324—21no—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×C22⋊Q8—21no—
25C2×#117C2×C411yescharacteristic
26C2×#117D411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28#431—41no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#103—21no—
32C2×#103—21no—
33#407—41no—
34C2×C22⋊C8—21no—
35C2×#97C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37#398—41no—
38#397—41no—
39#391—21no—
40#389—41no—
41#385—41no—
42#384D411yescharacteristic
43#384C2311yescharacteristic
44#382C2×C411yescharacteristic
45#382C2×C411yescharacteristic
46#382D411yescharacteristic
47#340—21no—
48#340—21no—
49#340—21no—
50#340—21no—
51#336—21no—
52#324—22no—
53#324—22no—
54#324—41no—
55#323—41no—
56C22×Q8—21no—
57C23×C4—21no—
58C23×C4C22⋊C411yescharacteristic
59C2×M16—81no—
60C22×C8—21no—
61C22×C8—21no—
62C4⋊Q8—22no—
63C4⋊Q8—41no—
64C22⋊Q8—42no—
65C22⋊Q8—42no—
66#117—22no—
67#117—41no—
68#117—81no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—81no—
77C2×C4⋊C4C22⋊C411yescharacteristic
78C2×C4⋊C4C22⋊C411yescharacteristic
79C2×C4⋊C4C2×D411yescharacteristic
80C2×C4⋊C4C2×D411yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—41no—
84C2×C42—21no—
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22×C411yescharacteristic
87C4⋊C8—81no—
88#103—81no—
89#103—81no—
90#103—81no—
91#103—81no—
92C22⋊C8—42no—
93C22⋊C8—42no—
94C22⋊C8—42no—
95C22⋊C8—81no—
96#97—22no—
97C4×C8—41no—
98C4×C8—81no—
99#95—41no—
100#95—41no—
101#95—41no—
102#95—42no—
103C24#9911yescharacteristic
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8—41no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—161no—
125C22×C4#9911yescharacteristic
126C22×C4C2×C22⋊C411yescharacteristic
127M16—162no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—82no—
135C2×C8—82no—
136C4⋊C4—42no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—81no—
144C4⋊C4—82no—
145C4⋊C4—82no—
146C4⋊C4—161no—
147C4⋊C4—161no—
148C22⋊C4—42no—
149C22⋊C4—81no—
150C22⋊C4—82no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—41no—
155C42—41no—
156C42#13112yes—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—41no—
161C23—41no—
162C23C2×#9911yescharacteristic
163Q8—81no—
164Q8—81no—
165Q8—81no—
166Q8—81no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—82no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—161no—
195C2×C4#35512yes—
196C2×C4#40811yescharacteristic
197C2×C4#41311yescharacteristic
198C8—81no—
199C8—81no—
200C8—162no—
201C8—162no—
202C22—21no—
203C22—21no—
204C22—42no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—81no—
209C22#146811yescharacteristic
210C22#212711yescharacteristic
211C22C2×#35511yescharacteristic
212C4—21no—
213C4—21no—
214C4—22no—
215C4—42no—
216C4—81no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4—81no—
221C4—161no—
222C4—161no—
223C2—21no—
224C2—21no—
225C2—81no—
226C2#1155311yescharacteristic
227C2#1361011yescharacteristic
228C2#1361211yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2317 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7367C2semidirect
5. #7355C2semidirect

Maximal subgroups

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

subgroupindex
#279852
#115142
#114502
#73672
#73552