Finite Group Databasebeta

← #134108

Subgroups — #134108

Order 512.

798 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2568C211yesmaximal, characteristic
2#11511C212yesmaximal
3#11506C212yesmaximal
4#8551C212yesmaximal
5C22×C4⋊Q8C2211yescharacteristic
6C22×C4×Q8C411yescharacteristic
7#2568—24no—
8#2568—24no—
9C2×#400—22no—
10C2×#391—22no—
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#382C411yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C4×C4⋊C4C411yescharacteristic
16#1784—22no—
17#1463—22no—
18#1463C2214yes—
19#1324—24no—
20#1319—24no—
21C23×Q8C2×C411yescharacteristic
22C23×Q8D412yes—
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—42no—
26C2×C4⋊Q8—42no—
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—42no—
30C2×C4×Q8—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4D412yes—
36C22×C4⋊C4C2311yescharacteristic
37C22×C42—21no—
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C2×#101—44no—
42#407—24no—
43#400—216no—
44#391—216no—
45#391—22no—
46#391—22no—
47#391—22no—
48#391D414yes—
49#382—22no—
50#382—22no—
51#382—22no—
52#382D414yes—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—22no—
55C2×#95—21no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60#324—24no—
61#324—24no—
62#324—44no—
63C22×Q8—24no—
64C22×Q8—24no—
65C22×Q8—21no—
66C22×Q8—24no—
67C22×Q8—42no—
68C22×Q8—42no—
69C22×Q8—48no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8C22⋊C412yes—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C412yes—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D412yes—
82C2×M16—44no—
83C4⋊Q8—48no—
84C4⋊Q8—48no—
85C4×Q8—48no—
86C4×Q8—48no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4C22⋊C412yes—
105C2×C4⋊C4C2×D412yes—
106C2×C4⋊C4C2×D412yes—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—24no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42C22⋊C412yes—
119C2×C42C22⋊C412yes—
120#101—88no—
121C22⋊C8—44no—
122C22⋊C8—44no—
123#95—24no—
124#95—28no—
125#95—28no—
126#95—28no—
127#95—28no—
128C24C2×C22⋊C411yescharacteristic
129C2×Q8—24no—
130C2×Q8—22no—
131C2×Q8—48no—
132C2×Q8—42no—
133C2×Q8—48no—
134C2×Q8—48no—
135C2×Q8—416no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—48no—
139C2×Q8—42no—
140C2×Q8—84no—
141C2×Q8—88no—
142C2×Q8#10114yescomplemented
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4#9912yes—
178C22×C4#9912yes—
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4C2×C22⋊C411yescharacteristic
181C22×C4#12014yes—
182M16—88no—
183C2×C8—84no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—48no—
187C4⋊C4—42no—
188C4⋊C4—48no—
189C4⋊C4—42no—
190C4⋊C4—48no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C42—24no—
194C42—24no—
195C42—24no—
196C42—42no—
197C42—42no—
198C42—48no—
199C42—42no—
200C23—22no—
201C23—22no—
202C23—21no—
203C23—21no—
204C23#37911yescharacteristic
205C23C2×#9911yescharacteristic
206C23C2×#9911yescharacteristic
207Q8—44no—
208Q8—44no—
209Q8—48no—
210Q8—816no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—84no—
246C2×C4—84no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4#35414yes—
251C2×C4C2×#10112yes—
252C2×C4#41312yes—
253C8—164no—
254C22—24no—
255C22—24no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22#178111yescharacteristic
263C22#178411yescharacteristic
264C22#178511yescharacteristic
265C22C2×#35412yes—
266C22#213312yes—
267C4—22no—
268C4—22no—
269C4—24no—
270C4—44no—
271C4—44no—
272C4—42no—
273C4—82no—
274C4—82no—
275C4—82no—
276C4—82no—
277C2—21no—
278C2—22no—
279C2—22no—
280C2—21no—
281C2#855011yescharacteristic
282C2#1269411yescharacteristic
283C2#1270211yescharacteristic
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×C4⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
142. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
#115112
#115062
#85512