Finite Group Databasebeta

← #161155

Subgroups — #161155

Order 512.

1102 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 250 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×C4×C4⋊Q8C211yesmaximal, characteristic
2#20778C212yesmaximal
3#7624C218yesmaximal, complemented
4#7595C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C4×Q8C412yes—
7C4×C4⋊Q8—22no—
8C4×C4⋊Q8—22no—
9C4×C4⋊Q8C414yes—
10C4×C4⋊Q8C2214yes—
11C2×#391C2212yes—
12C2×#389—22no—
13C2×#384C411yescharacteristic
14C2×C4×C4⋊C4—22no—
15C2×C43C2211yescharacteristic
16#1839C2214yes—
17#1831—22no—
18#1817C2214yes—
19#1797—22no—
20#1767C2214yes—
21#1749—22no—
22#1745—22no—
23#1272C22116yes—
24C23×Q8C2×C412yes—
25C2×C4⋊Q8—28no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8C2×C414yes—
29C2×C4×Q8—24no—
30C2×C4×Q8—48no—
31C2×C4×Q8C2×C418yes—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4C2×C412yes—
36C22×C42—22no—
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39C22×C42D412yes—
40C22×C42D412yes—
41C22×C42C2311yescharacteristic
42C2×C4⋊C8—24no—
43C2×C22⋊C8—24no—
44C2×#97—24no—
45#391—24no—
46#391C2×C414yes—
47#391C2314yes—
48#389—24no—
49#389—24no—
50#389—24no—
51#389—24no—
52#384—21no—
53#384—21no—
54#384C2×C414yes—
55C4×C4⋊C4—24no—
56C4×C4⋊C4—24no—
57C4×C4⋊C4—24no—
58C4×C4⋊C4—24no—
59C2×#95—24no—
60C43C2×C412yes—
61C43D412yes—
62C43D412yes—
63C43C2312yes—
64#336—24no—
65#336—24no—
66#336—24no—
67#336C2318yes—
68C22×Q8—22no—
69C22×Q8—24no—
70C22×Q8—44no—
71C22×Q8—44no—
72C22×Q8M1618yes—
73C22×Q8C22×C414yes—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4C22⋊C412yes—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D412yes—
81C22×C8—28no—
82C4⋊Q8—28no—
83C4⋊Q8—28no—
84C4⋊Q8—216no—
85C4×Q8—416no—
86C4×Q8—416no—
87C2×C4⋊C4—216no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—48no—
96C2×C4⋊C4—48no—
97C2×C4⋊C4C22×C414yes—
98C2×C42—24no—
99C2×C42—24no—
100C2×C42—28no—
101C2×C42—28no—
102C2×C42—44no—
103C2×C42—42no—
104C2×C42—48no—
105C2×C42C22⋊C412yes—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C22⋊C411yescharacteristic
108C2×C42C22⋊C412yes—
109C2×C42C22⋊C412yes—
110C2×C42C22⋊C412yes—
111C2×C42C22⋊C412yes—
112C2×C42C22×C412yes—
113C2×C42C22×C412yes—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D412yes—
116C2×C42C2×D412yes—
117C2×C42C2×D412yes—
118C2×C42C2×D412yes—
119C2×C42C2×D412yes—
120C2×C42C2411yescharacteristic
121C4⋊C8—48no—
122C22⋊C8—48no—
123#97—48no—
124#95—28no—
125#95—28no—
126C24C2×C22⋊C411yescharacteristic
127C2×Q8—24no—
128C2×Q8—28no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—416no—
132C2×Q8—44no—
133C2×Q8—416no—
134C2×Q8—44no—
135C2×Q8C2×M1618yes—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—28no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—48no—
153C22×C4—44no—
154C22×C4#10014yes—
155C22×C4C2×C22⋊C412yes—
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4C2×C22⋊C412yes—
158C22×C4C2×C22⋊C412yes—
159C22×C4C2×C22⋊C412yes—
160C22×C4C2×C22⋊C412yes—
161C22×C4C2×M1614yes—
162C22×C4C23×C411yescharacteristic
163C22×C4C22×D412yes—
164C2×C8—48no—
165C2×C8—48no—
166C2×C8—48no—
167C4⋊C4—216no—
168C4⋊C4—216no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C4⋊C4—44no—
172C4⋊C4—416no—
173C4⋊C4—48no—
174C42—24no—
175C42—24no—
176C42—22no—
177C42—22no—
178C42—24no—
179C42—24no—
180C42—24no—
181C42—24no—
182C42—24no—
183C42—24no—
184C42—42no—
185C42—41no—
186C42—42no—
187C42—42no—
188C42—41no—
189C42—416no—
190C42#13718yes—
191C23—22no—
192C23—22no—
193C23#40712yes—
194C23#41012yes—
195C23C2×#10012yes—
196C23C22×C22⋊C411yescharacteristic
197Q8—48no—
198Q8—48no—
199Q8—416no—
200C2×C4—216no—
201C2×C4—28no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—28no—
217C2×C4—28no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—44no—
222C2×C4—48no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—416no—
226C2×C4—44no—
227C2×C4—48no—
228C2×C4—88no—
229C2×C4#40712yes—
230C2×C4C2×#10014yes—
231C2×C4#41914yes—
232C2×C4#41914yes—
233C2×C4#42114yes—
234C2×C4C22×M1612yes—
235C2×C4C2×#13714yes—
236C8—88no—
237C22—22no—
238C22—24no—
239C22—22no—
240C22—24no—
241C22#146614yes—
242C22C2×#40711yescharacteristic
243C22C2×#41011yescharacteristic
244C22C22×#10011yescharacteristic
245C22C2×#41912yes—
246C22C2×#42112yes—
247C4—24no—
248C4—24no—
249C4—24no—
250C4—24no—
251C4—24no—
252C4—22no—
253C4—22no—
254C4—42no—
255C4—48no—
256C4—84no—
257C2—24no—
258C2C2×#146611yescharacteristic
259C2#1744912yes—
260C2#1756212yes—
261C2#1767511yescharacteristic
262C2#1774611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7624C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
#207782
#76242
#75952