Finite Group Databasebeta

← #152042

Subgroups — #152042

Order 512.

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

normal subgroups only

This group has 952 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
1C22×#494C211yesmaximal, characteristic
2C2×#1411C214yesmaximal, complemented
3#12700C212yesmaximal
4C23×Q16—22no—
5C22×C4⋊Q8C411yescharacteristic
6C22×C4⋊Q8C2211yescharacteristic
7C2×#494—28no—
8C2×#494—22no—
9C2×#494C412yes—
10C2×#494C2212yes—
11C22×C4×C8C411yescharacteristic
12C2×#356—22no—
13C2×#356C2214yes—
14#1784—22no—
15#1411—28no—
16C23×Q8—21no—
17C23×Q8D412yes—
18C22×Q16—216no—
19C22×Q16—24no—
20C22×Q16—24no—
21C22×Q16—48no—
22C22×Q16—48no—
23C23×C8—21no—
24C2×C4⋊Q8—24no—
25C2×C4⋊Q8—24no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—42no—
28C2×C4⋊Q8—41no—
29C2×C4⋊Q8—41no—
30C2×C4⋊Q8—42no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×C4⋊Q8C2×C412yes—
33C2×C4⋊Q8D412yes—
34C2×C4⋊Q8C2311yescharacteristic
35C22×C4⋊C4—41no—
36C22×C4⋊C4—41no—
37C22×C42C2×C411yescharacteristic
38#494—216no—
39#494—24no—
40#494—24no—
41#494—24no—
42#494D418yes—
43C2×#101—24no—
44C2×#101—42no—
45#407—42no—
46C2×C4×C8—21no—
47C2×C4×C8—24no—
48C2×C4×C8C2×C412yes—
49#356—28no—
50#356—44no—
51C22×Q8—22no—
52C22×Q8—24no—
53C22×Q8—24no—
54C22×Q8—22no—
55C22×Q8—42no—
56C22×Q8—42no—
57C22×Q8—44no—
58C22×Q8—44no—
59C22×Q8—44no—
60C22×Q8—44no—
61C22×Q8—41no—
62C22×Q8C22⋊C412yes—
63C22×Q8C2×D412yes—
64C23×C4—21no—
65C23×C4—22no—
66C23×C4—41no—
67C23×C4C22⋊C411yescharacteristic
68C2×Q16—216no—
69C2×Q16—216no—
70C2×Q16—48no—
71C2×Q16—48no—
72C2×Q16—48no—
73C2×Q16—48no—
74C2×Q16—464no—
75C2×M16—44no—
76C22×C8—24no—
77C22×C8—24no—
78C22×C8—24no—
79C22×C8—22no—
80C4⋊Q8—24no—
81C4⋊Q8—24no—
82C4⋊Q8—24no—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—42no—
88C4⋊Q8—48no—
89C4⋊Q8—48no—
90C4⋊Q8—42no—
91C4⋊Q8C22⋊C414yes—
92C4⋊Q8C2×D414yes—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—48no—
98C2×C4⋊C4—48no—
99C2×C4⋊C4—42no—
100C2×C42—21no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C22×C411yescharacteristic
105#101—44no—
106#101—44no—
107#101—84no—
108C22⋊C8—84no—
109C4×C8—24no—
110C4×C8—22no—
111C4×C8C22⋊C414yes—
112C24#9911yescharacteristic
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—24no—
116C2×Q8—22no—
117C2×Q8—44no—
118C2×Q8—44no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—416no—
124C2×Q8—416no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—42no—
129C2×Q8—82no—
130C2×Q8—88no—
131C2×Q8—88no—
132C2×Q8—82no—
133C2×Q8—82no—
134C2×Q8#12014yes—
135C22×C4—24no—
136C22×C4—28no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—82no—
150C22×C4—82no—
151C22×C4—82no—
152C22×C4#9911yescharacteristic
153C22×C4C2×C22⋊C411yescharacteristic
154Q16—432no—
155Q16—432no—
156M16—84no—
157M16—84no—
158C2×C8—216no—
159C2×C8—24no—
160C2×C8—24no—
161C2×C8—24no—
162C2×C8—84no—
163C4⋊C4—44no—
164C4⋊C4—48no—
165C4⋊C4—44no—
166C4⋊C4—48no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C42—24no—
170C42—21no—
171C42—21no—
172C42#9912yes—
173C42C2×C22⋊C412yes—
174C23—21no—
175C23—24no—
176C23—21no—
177C23#35312yes—
178C23C2×#9911yescharacteristic
179Q8—44no—
180Q8—44no—
181Q8—48no—
182Q8—82no—
183Q8—88no—
184Q8—82no—
185Q8—88no—
186Q8—84no—
187C2×C4—24no—
188C2×C4—216no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—88no—
206C2×C4—88no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4#35312yes—
210C2×C4#37911yescharacteristic
211C2×C4C2×#9911yescharacteristic
212C8—28no—
213C8—164no—
214C22—24no—
215C22—24no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22#141114yescomplemented
220C22#178111yescharacteristic
221C22C2×#35311yescharacteristic
222C22C2×#35311yescharacteristic
223C4—24no—
224C4—24no—
225C4—21no—
226C4—22no—
227C4—21no—
228C4—44no—
229C4—82no—
230C4—84no—
231C2—21no—
232C2—21no—
233C2—24no—
234C2#1269311yescharacteristic
235C2C2×#141112yes—
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4⋊Q8C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1411C2semidirect
219. C22#1411semidirect

Maximal subgroups

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

subgroupindex
C22×#4942
C2×#14112
#127002