Finite Group Databasebeta

← #142019

Subgroups — #142019

Order 512.

438 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 118 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#33208C211yesmaximal, characteristic
2C4×C22⋊C16C211yesmaximal, characteristic
3#10842C211yesmaximal, characteristic
4#7258C214yesmaximal
5C2×#499C411yescharacteristic, complemented
6C22×C4×C8C2211yescharacteristic
7#2851—21no—
8C2×#382—21no—
9C2×C22⋊C16C2212yes—
10C2×C4×C16—21no—
11#1991—21no—
12#1942C412yescomplemented
13#1941—21no—
14#1846C411yescharacteristic, complemented
15#1808C2212yes—
16#1807—21no—
17C2×#340—21no—
18C2×#340C2212yes—
19#1374—21no—
20C23×C8D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#125—21no—
23C22×C4⋊C4—22no—
24C22×C42D411yescharacteristic
25C22×C16—22no—
26#499—42no—
27C2×#107C2×C412yes—
28C2×#107C2×C412yes—
29C2×#106—21no—
30C2×#106—21no—
31C2×C4×C8—42no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36#389—22no—
37#387—42no—
38#385—42no—
39#382—24no—
40#382—21no—
41#382—42no—
42#380—22no—
43C2×#95—42no—
44C22⋊C16—22no—
45C22⋊C16Q814yes—
46C4×C16—42no—
47#340—22no—
48#340—22no—
49#340—24no—
50#340—42no—
51C23×C4—42no—
52C23×C4D812yes—
53C23×C4C2×D411yescharacteristic
54C22×C8—41no—
55C22×C8—41no—
56C22×C8—41no—
57C22×C8—41no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8SD1612yes—
61C22×C8Q1612yes—
62C22×C8C22×C411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C22×C8C2×Q812yes—
65C22×C8#4112yes—
66#125—42no—
67#125—42no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42SD1612yes—
84C2×C16—24no—
85C2×C16—42no—
86C2×C16—42no—
87#107—42no—
88#107—42no—
89#106—42no—
90#106—42no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—41no—
94C4×C8—41no—
95#95—48no—
96#95—42no—
97#95—42no—
98#95—42no—
99C24C2×D811yescharacteristic
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—42no—
107C22×C4—44no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—82no—
111C22×C4—84no—
112C22×C4—82no—
113C22×C4—84no—
114C22×C4#10212yes—
115C22×C4#10212yes—
116C22×C4C2×C22⋊C411yescharacteristic
117C22×C4#12411yescharacteristic
118C22×C4C4⋊Q811yescharacteristic
119C22×C4C2×SD1611yescharacteristic
120C22×C4C2×SD1611yescharacteristic
121C22×C4C2×Q1611yescharacteristic
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8#10312yes—
135C2×C8#10312yes—
136C2×C8C4×Q812yes—
137C2×C8C22⋊Q812yes—
138C4⋊C4—44no—
139C4⋊C4—44no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—84no—
145C4⋊C4—82no—
146C4⋊C4—82no—
147C4⋊C4—84no—
148C22⋊C4—84no—
149C42—24no—
150C42—21no—
151C42—21no—
152C42#9912yes—
153C42#10412yes—
154C16—44no—
155C23—22no—
156C23—21no—
157C23—21no—
158C23C2×#10211yescharacteristic
159C23#48611yescharacteristic
160C23#50011yescharacteristic
161C23#50814yes—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—48no—
178C2×C4—84no—
179C2×C4—88no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—164no—
187C2×C4—162no—
188C2×C4—162no—
189C2×C4#32714yes—
190C2×C4#39111yescharacteristic
191C2×C4C2×#9911yescharacteristic
192C2×C4#41011yescharacteristic
193C2×C4C2×#10311yescharacteristic
194C2×C4C2×#10411yescharacteristic
195C2×C4#42111yescharacteristic
196C2×C4#48711yescharacteristic
197C2×C4#49811yescharacteristic
198C8—22no—
199C8—22no—
200C8—22no—
201C8—22no—
202C22—24no—
203C22—21no—
204C22—21no—
205C22—24no—
206C22—21no—
207C22—21no—
208C22C2×#32711yescharacteristic
209C22#151711yescharacteristic
210C22#199011yescharacteristic
211C22#213812yes—
212C22#214512yes—
213C22#224212yes—
214C22#225312yes—
215C4—21no—
216C4—21no—
217C4—24no—
218C4—42no—
219C4—164no—
220C4—162no—
221C4—162no—
222C4#150311yescharacteristic
223C4#152011yescharacteristic
224C4#198611yescharacteristic
225C4#199611yescharacteristic
226C2—22no—
227C2—22no—
228C2#988511yescharacteristic
229C2#1170312yes—
230C2#1185512yes—
231C2#1328111yescharacteristic
232C2#1328811yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#499C4semidirect
12. #1942C4semidirect
14. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332082
C4×C22⋊C162
#108422
#72582