Finite Group Databasebeta

← #128013

Subgroups — #128013

Order 512.

1354 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 150 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×#1784C212yesmaximal
2#14336C211yesmaximal, characteristic
3#7308C214yesmaximal, complemented
4C24×Q8C2211yescharacteristic
5C22×#101—22no—
6C22×#101C414yescomplemented
7C2×#407—22no—
8C2×#381—22no—
9C2×#381C412yes—
10#1784—22no—
11#1784—22no—
12#1784C414yescomplemented
13#1784C2214yes—
14#1446—22no—
15#1446C412yes—
16#1446C2212yes—
17C23×Q8—44no—
18C23×Q8—48no—
19C23×Q8C2×C412yes—
20C23×Q8C2×C412yes—
21C23×Q8C2×C411yescharacteristic
22C23×Q8D411yescharacteristic
23C23×Q8D411yescharacteristic
24C23×Q8D412yes—
25C23×Q8Q811yescharacteristic
26C23×Q8Q811yescharacteristic
27C23×Q8C2311yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4D412yes—
30C22×M16—24no—
31C22×C22⋊C4—22no—
32C2×#101—24no—
33C2×#101—24no—
34C2×#101—24no—
35C2×#101—48no—
36C2×#101—48no—
37C2×#101C2×C418yes—
38#407—24no—
39#407—24no—
40#407—44no—
41C2×C22⋊C8—44no—
42#381—48no—
43#381—44no—
44C2×#95—22no—
45C2×#95—24no—
46C2×#95—24no—
47C2×#95C2×C414yes—
48C25C22⋊C411yescharacteristic
49C22×Q8—28no—
50C22×Q8—28no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8—44no—
54C22×Q8—44no—
55C22×Q8—42no—
56C22×Q8—42no—
57C22×Q8—42no—
58C22×Q8—44no—
59C22×Q8—42no—
60C22×Q8—42no—
61C22×Q8—44no—
62C22×Q8—42no—
63C22×Q8—44no—
64C22×Q8—44no—
65C22×Q8—44no—
66C22×Q8—44no—
67C22×Q8—44no—
68C22×Q8—44no—
69C22×Q8—44no—
70C22×Q8—44no—
71C22×Q8—464no—
72C22×Q8C4212yes—
73C22×Q8C22⋊C412yes—
74C22×Q8C22⋊C412yes—
75C22×Q8C4⋊C411yescharacteristic
76C22×Q8C4⋊C412yes—
77C22×Q8C4⋊C412yes—
78C22×Q8C4⋊C411yescharacteristic
79C22×Q8C22×C412yes—
80C22×Q8C2×D411yescharacteristic
81C22×Q8C2×Q811yescharacteristic
82C23×C4—24no—
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—28no—
86C23×C4—24no—
87C23×C4—44no—
88C23×C4—44no—
89C23×C4C4211yescharacteristic
90C23×C4C4211yescharacteristic
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C4⋊C411yescharacteristic
96C23×C4C4⋊C412yes—
97C23×C4C4⋊C412yes—
98C23×C4C4⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D412yes—
101C2×M16—28no—
102C2×M16—44no—
103C2×M16—48no—
104C2×M16—48no—
105C22×C8—44no—
106C2×C22⋊C4—48no—
107C2×C22⋊C4—44no—
108#101—48no—
109#101—48no—
110#101—48no—
111#101—48no—
112C22⋊C8—48no—
113C22⋊C8—48no—
114#95—44no—
115#95—44no—
116#95—48no—
117#95—48no—
118C24—24no—
119C24—44no—
120C24#9511yescharacteristic
121C24#9511yescharacteristic
122C24#9914yes—
123C24C2×C22⋊C411yescharacteristic
124C2×Q8—24no—
125C2×Q8—24no—
126C2×Q8—24no—
127C2×Q8—24no—
128C2×Q8—24no—
129C2×Q8—24no—
130C2×Q8—24no—
131C2×Q8—24no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—41no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—41no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—41no—
148C2×Q8—42no—
149C2×Q8—41no—
150C2×Q8—416no—
151C2×Q8—416no—
152C2×Q8—416no—
153C2×Q8—432no—
154C2×Q8—416no—
155C2×Q8—432no—
156C2×Q8—416no—
157C2×Q8—416no—
158C2×Q8—416no—
159C2×Q8—416no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—416no—
192C22×C4—416no—
193C22×C4—416no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—44no—
198C22×C4#9511yescharacteristic
199C22×C4#9511yescharacteristic
200C22×C4#9512yes—
201C22×C4#9512yes—
202C22×C4#9918yes—
203C22×C4C2×C4211yescharacteristic
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C4⋊C411yescharacteristic
206C22×C4C2×C4⋊C412yes—
207M16—48no—
208M16—48no—
209M16—88no—
210C2×C8—48no—
211C2×C8—88no—
212C22⋊C4—88no—
213C23—24no—
214C23—22no—
215C23—28no—
216C23—28no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—44no—
221C23—416no—
222C23#34212yes—
223C23#34212yes—
224C23#34214yes—
225C23C2×#9511yescharacteristic
226C23C2×#9912yes—
227Q8—44no—
228Q8—48no—
229Q8—44no—
230Q8—48no—
231Q8—48no—
232Q8—48no—
233Q8—48no—
234Q8—44no—
235Q8—48no—
236Q8—44no—
237C2×C4—28no—
238C2×C4—28no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—84no—
270C2×C4—816no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4#34214yes—
274C2×C4C2×#9914yes—
275C8—88no—
276C22—28no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—22no—
281C22—24no—
282C22—22no—
283C22—44no—
284C22—416no—
285C22—44no—
286C22#130414yes—
287C22C2×#34212yes—
288C22C2×#34211yescharacteristic
289C22#213314yes—
290C4—24no—
291C4—24no—
292C4—42no—
293C4—42no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—164no—
299C2—24no—
300C2—22no—
301C2—24no—
302C2—42no—
303C2C2×#130411yescharacteristic
304C2#1258612yes—
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC23×Q8C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C24×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7308C2semidirect
6. C22×#101C4semidirect
12. #1784C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17842
#143362
#73082