Finite Group Databasebeta

← #142017

Subgroups — #142017

Order 512.

381 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 102 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#33211C211yesmaximal, characteristic
2C4×#349C211yesmaximal, characteristic
3C4⋊#338C211yesmaximal, characteristic
4#7255C214yesmaximal
5C2×#499C411yescharacteristic
6C2×#423—21no—
7C2×C4×M16C2211yescharacteristic
8#2851—21no—
9C2×#349C2212yes—
10C4×M32—21no—
11#1991—21no—
12#1944C412yes—
13#1943—21no—
14#1846C411yescharacteristic
15#1809—21no—
16#1809C2212yes—
17C2×#338—21no—
18C2×#338C2212yes—
19#1377—21no—
20C22×M16—22no—
21C22×M16D411yescharacteristic
22C22×M16C2311yescharacteristic
23C2×#125—21no—
24C22×C42D411yescharacteristic
25C2×M32—22no—
26#499—41no—
27#499—41no—
28C2×#107—21no—
29C2×#107C2×C412yes—
30C2×#106—21no—
31C2×#106C2×C412yes—
32#423—41no—
33#423—41no—
34#423—42no—
35C2×C4⋊C8—24no—
36C4×M16—41no—
37C4×M16—41no—
38C2×#97C2×C411yescharacteristic
39C2×#97D411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#389—22no—
43#387—42no—
44#385—42no—
45#382—21no—
46#380—22no—
47#349—22no—
48#349Q814yes—
49#346—41no—
50C4×C16—41no—
51#340—22no—
52#340—22no—
53#338—24no—
54#338—42no—
55C23×C4SD1612yes—
56C23×C4C2×D411yescharacteristic
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—42no—
62C2×M16—42no—
63C2×M16—42no—
64C2×M16—42no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16SD1612yes—
68C2×M16Q1612yes—
69C2×M16C2×Q812yes—
70C2×M16#4112yes—
71C22×C8—24no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76#125—41no—
77#125—41no—
78#125—42no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—22no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22⋊C411yescharacteristic
89C2×C42D812yes—
90M32—24no—
91M32—42no—
92C2×C16—42no—
93#107—42no—
94#107—42no—
95#106—42no—
96#106—42no—
97C4⋊C8—44no—
98C4⋊C8—44no—
99#97—41no—
100#97—41no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105#95—42no—
106#95—42no—
107#95—42no—
108C24C2×SD1611yescharacteristic
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—82no—
118C22×C4#10212yes—
119C22×C4#10212yes—
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4#12411yescharacteristic
122C22×C4C4⋊Q811yescharacteristic
123C22×C4C2×D811yescharacteristic
124C22×C4C2×SD1611yescharacteristic
125C22×C4C2×Q1611yescharacteristic
126M16—42no—
127M16—42no—
128M16—42no—
129M16—42no—
130M16—82no—
131M16—82no—
132M16—82no—
133M16—82no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—44no—
143C2×C8—44no—
144C2×C8—44no—
145C2×C8#10312yes—
146C2×C8#10312yes—
147C2×C8C4×Q812yes—
148C2×C8C22⋊Q812yes—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—84no—
154C4⋊C4—82no—
155C4⋊C4—82no—
156C22⋊C4—84no—
157C42—22no—
158C42—22no—
159C42—21no—
160C42—21no—
161C42#9911yescharacteristic
162C42#9911yescharacteristic
163C42#10412yes—
164C16—44no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23C2×#10211yescharacteristic
169C23#48711yescharacteristic
170C23#49811yescharacteristic
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4#32712yes—
193C2×C4#32712yes—
194C2×C4#39111yescharacteristic
195C2×C4C2×#9911yescharacteristic
196C2×C4#41011yescharacteristic
197C2×C4C2×#10311yescharacteristic
198C2×C4C2×#10411yescharacteristic
199C2×C4#42111yescharacteristic
200C2×C4#48611yescharacteristic
201C2×C4#50011yescharacteristic
202C8—22no—
203C8—22no—
204C8—22no—
205C8—22no—
206C8—84no—
207C22—22no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—42no—
213C22—81no—
214C22C2×#32711yescharacteristic
215C22#151711yescharacteristic
216C22#199011yescharacteristic
217C4—21no—
218C4—21no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—162no—
223C4—162no—
224C4#150311yescharacteristic
225C4#152011yescharacteristic
226C4#198611yescharacteristic
227C4#199611yescharacteristic
228C2—21no—
229C2—21no—
230C2—41no—
231C2—41no—
232C2#988511yescharacteristic
233C2#1168812yes—
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332112
C4×#3492
C4⋊#3382
#72552