Finite Group Databasebeta

← #133217

Subgroups — #133217

Order 512.

696 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×#2642C211yesmaximal, characteristic
2#11814C214yesmaximal
3#8554C211yesmaximal, characteristic
4#8554C211yesmaximal, characteristic
5C22×C4⋊Q8C411yescharacteristic
6C22×#125C2211yescharacteristic
7#2642—28no—
8C2×#400—21no—
9C2×#400—21no—
10C2×#400C412yes—
11C2×#400C2212yes—
12C2×#391—21no—
13C2×#391—21no—
14C2×#376C411yescharacteristic
15#1821—21no—
16#1821C2212yes—
17#1463—21no—
18#1463—21no—
19#1463—21no—
20#1463C2212yes—
21#1353—28no—
22C23×Q8—21no—
23C2×C4⋊Q8—24no—
24C2×C4⋊Q8—42no—
25C2×C4⋊Q8—42no—
26C2×#125—24no—
27C2×#125—42no—
28C2×#125—42no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C412yes—
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4C2311yescharacteristic
37C22×C42—21no—
38C22×C42C2×C411yescharacteristic
39#407—22no—
40#407—22no—
41#400—28no—
42#400—28no—
43#400—24no—
44#400—24no—
45#400—24no—
46#400D418yes—
47#391—28no—
48#391—28no—
49#376—24no—
50C2×#95—21no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95C2×C412yes—
56#343—24no—
57#343—42no—
58#324—24no—
59#324—42no—
60#324—42no—
61#324—42no—
62C22×Q8—24no—
63C22×Q8—42no—
64C22×Q8—48no—
65C22×Q8—42no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×M16—42no—
76C2×M16—42no—
77C4⋊Q8—416no—
78#125—416no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4C22⋊C414yes—
103C2×C4⋊C4C2×D414yes—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42—22no—
107C2×C42—24no—
108C2×C42—44no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#95—24no—
114#95—28no—
115#95—28no—
116#95—22no—
117#95—28no—
118#95—28no—
119#95C22⋊C414yes—
120C24C2×C22⋊C411yescharacteristic
121C2×Q8—48no—
122C2×Q8—416no—
123C2×Q8—88no—
124C2×Q8—84no—
125C2×Q8—84no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—24no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—24no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—48no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4#9912yes—
158C22×C4#9912yes—
159C22×C4C2×C22⋊C412yes—
160C22×C4#12012yes—
161C22×C4#12012yes—
162M16—84no—
163M16—84no—
164C2×C8—82no—
165C2×C8—82no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—48no—
174C4⋊C4—44no—
175C42—28no—
176C42—48no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23#37911yescharacteristic
181C23C2×#9911yescharacteristic
182C23C2×#9911yescharacteristic
183Q8—816no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—48no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—44no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—88no—
215C2×C4—84no—
216C2×C4—84no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4#35514yes—
221C8—162no—
222C8—162no—
223C22—28no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22#178111yescharacteristic
231C22#179612yes—
232C22#212312yes—
233C22C2×#35512yes—
234C4—24no—
235C4—42no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—84no—
240C4—81no—
241C4—81no—
242C4—84no—
243C2—22no—
244C2—22no—
245C2—22no—
246C2#852011yescharacteristic
247C2#1269211yescharacteristic
248C2#1269911yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#26422
#118142
#85542