Finite Group Databasebeta

← #134151

Subgroups — #134151

Order 512.

804 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 82 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#35592C211yesmaximal, characteristic
2#11508C214yesmaximal, complemented
3#8551C212yesmaximal, complemented
4C2×#537C411yescharacteristic
5C22×C4⋊Q8C2211yescharacteristic
6#2591—22no—
7#2492—24no—
8#2491—24no—
9#2436—22no—
10#2311C2212yes—
11#2297C412yes—
12#2274C411yescharacteristic
13#1784—22no—
14#1463—22no—
15#1463C2214yes—
16#1319—24no—
17#1318—24no—
18C22×#41C2×C412yes—
19C23×Q8D412yes—
20#537—44no—
21C2×C4⋊Q8—24no—
22C2×C4⋊Q8—44no—
23C2×#124—24no—
24C2×C22⋊Q8—22no—
25C2×#121—22no—
26C2×#121—22no—
27C2×#117C2×C412yes—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D412yes—
30C22×C4⋊C4C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32C2×#101—44no—
33#407—24no—
34#400—24no—
35#394—24no—
36#393—28no—
37#391—24no—
38#391—22no—
39#391D414yes—
40#390—22no—
41#390D414yes—
42#388—28no—
43#386—22no—
44#386—22no—
45#386—24no—
46#380—24no—
47C4×C4⋊C4—24no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—24no—
50#324—28no—
51#324—44no—
52C2×#41—44no—
53C2×#41—44no—
54C2×#41—44no—
55C22×Q8—24no—
56C22×Q8—24no—
57C22×Q8—42no—
58C22×Q8—84no—
59C22×Q8C22⋊C412yes—
60C22×D4—22no—
61C22×D4—22no—
62C22×D4C22⋊C412yes—
63C23×C4—22no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C412yes—
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C2×M16—44no—
70C4⋊Q8—416no—
71#124—88no—
72C22⋊Q8—48no—
73#121—48no—
74#121—44no—
75#121—44no—
76#117—44no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4C2×D414yes—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—28no—
91C2×C42—22no—
92C2×C42—24no—
93C2×C42—42no—
94C2×C42C22⋊C414yes—
95#101—88no—
96C22⋊C8—44no—
97C22⋊C8—44no—
98#95—24no—
99#95—24no—
100#95—24no—
101#95—28no—
102C24—22no—
103C24C2×C22⋊C411yescharacteristic
104#41—816no—
105C2×Q8—24no—
106C2×Q8—48no—
107C2×Q8—48no—
108C2×Q8—48no—
109C2×Q8—42no—
110C2×Q8—42no—
111C2×Q8—84no—
112C2×Q8—84no—
113C2×Q8—84no—
114C2×Q8—84no—
115C2×D4—24no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—44no—
122C2×D4—84no—
123C22×C4—22no—
124C22×C4—24no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4#9914yes—
143C22×C4C2×C22⋊C412yes—
144C22×C4#12014yes—
145M16—88no—
146C2×C8—84no—
147C4⋊C4—28no—
148C4⋊C4—44no—
149C4⋊C4—48no—
150C4⋊C4—44no—
151C4⋊C4—44no—
152C22⋊C4—48no—
153C22⋊C4—48no—
154C22⋊C4—48no—
155C22⋊C4—44no—
156C22⋊C4—88no—
157C42—44no—
158C42—44no—
159C42—84no—
160C23—22no—
161C23—41no—
162C23—41no—
163C23—44no—
164C23—44no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23#37911yescharacteristic
169C23C2×#9912yes—
170Q8—84no—
171Q8—816no—
172D4—84no—
173D4—88no—
174D4—164no—
175C2×C4—28no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—88no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4#41314yes—
204C8—164no—
205C22—21no—
206C22—21no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—44no—
211C22—84no—
212C22—84no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22#178111yescharacteristic
217C22#178512yes—
218C4—24no—
219C4—44no—
220C4—42no—
221C4—44no—
222C4—42no—
223C4—84no—
224C4—82no—
225C2—22no—
226C2—42no—
227C2—162no—
228C2#855211yescharacteristic
229C2#1270711yescharacteristic
230C2#1271011yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC2×#537C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11508C2semidirect
3. #8551C2semidirect

Maximal subgroups

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

subgroupindex
#355922
#115082
#85512