Finite Group Databasebeta

← #382723

Subgroups — #382723

Order 512.

2850 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2850 conjugacy classes of subgroups — too many to draw. Only its 314 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×#380C211yesmaximal, characteristic
2#14681C212yesmaximal
3#11353C218yesmaximal, complemented
4#10725C214yesmaximal
5C25×C4C2211yescharacteristic
6C23×C4⋊C4C412yescomplemented
7C23×C22⋊C4C2212yes—
8C2×#381—24no—
9C2×#380—24no—
10C2×#380—24no—
11C2×#380C414yescomplemented
12C2×#380C2214yes—
13C2×#379C2214yes—
14C22×#95C412yescomplemented
15C2×#342C22116yes—
16C22⋊#95—24no—
17C22⋊#95C2218yes—
18C26C2311yescharacteristic
19C24×C4—28no—
20C24×C4—48no—
21C24×C4C2×C411yescharacteristic
22C24×C4C2×C411yescharacteristic
23C24×C4C2×C412yes—
24C24×C4C2×C414yes—
25C24×C4D414yes—
26C24×C4D414yes—
27C24×C4C2312yes—
28C22×C4⋊C4—24no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—48no—
32C22×C4⋊C4C2×C418yes—
33C22×C22⋊C4—24no—
34C22×C22⋊C4—24no—
35C22×C22⋊C4—24no—
36C22×C22⋊C4—24no—
37C22×C22⋊C4—48no—
38C22×C22⋊C4C2×C414yes—
39C22×C22⋊C4C2318yes—
40C22×C22⋊C4C2314yes—
41#381—416no—
42#380—216no—
43#380—216no—
44#379—216no—
45C2×#95—24no—
46C2×#95—24no—
47C2×#95—216no—
48C2×#95C2×C418yes—
49#342—416no—
50C25—24no—
51C25—24no—
52C25—44no—
53C25—44no—
54C25C22×C411yescharacteristic
55C25C22×C411yescharacteristic
56C25C2×D412yes—
57C25C2×D412yes—
58C25#4118yes—
59C25C2411yescharacteristic
60C23×C4—24no—
61C23×C4—28no—
62C23×C4—28no—
63C23×C4—24no—
64C23×C4—216no—
65C23×C4—216no—
66C23×C4—28no—
67C23×C4—216no—
68C23×C4—24no—
69C23×C4—24no—
70C23×C4—28no—
71C23×C4—48no—
72C23×C4—416no—
73C23×C4—48no—
74C23×C4—44no—
75C23×C4—48no—
76C23×C4—48no—
77C23×C4—464no—
78C23×C4—48no—
79C23×C4—48no—
80C23×C4C22⋊C412yes—
81C23×C4C22⋊C412yes—
82C23×C4C22⋊C418yes—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C412yes—
85C23×C4C22×C412yes—
86C23×C4C22×C418yes—
87C23×C4C22×C412yes—
88C23×C4C2×D418yes—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—48no—
92C2×C4⋊C4—416no—
93C2×C4⋊C4—48no—
94C2×C4⋊C4—416no—
95C2×C22⋊C4—28no—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—216no—
98C2×C22⋊C4—216no—
99C2×C22⋊C4—216no—
100C2×C22⋊C4—216no—
101C2×C22⋊C4—48no—
102C2×C22⋊C4—416no—
103C2×C22⋊C4—416no—
104C2×C22⋊C4—416no—
105C2×C22⋊C4—416no—
106C2×C22⋊C4—48no—
107C2×C22⋊C4—48no—
108#95—216no—
109#95—432no—
110C24—22no—
111C24—22no—
112C24—24no—
113C24—24no—
114C24—22no—
115C24—22no—
116C24—24no—
117C24—24no—
118C24—24no—
119C24—24no—
120C24—232no—
121C24—28no—
122C24—28no—
123C24—42no—
124C24—44no—
125C24—416no—
126C24—42no—
127C24—42no—
128C24—44no—
129C24—416no—
130C24—42no—
131C24—432no—
132C24—432no—
133C24C2×C22⋊C412yes—
134C24C2×C22⋊C411yescharacteristic
135C24C2×C22⋊C411yescharacteristic
136C24#11714yes—
137C24#11714yes—
138C24#12318yes—
139C24#12318yes—
140C24C23×C411yescharacteristic
141C24C22×D412yes—
142C24C2×#4114yes—
143C24H(5,𝔽2)14yes—
144C242−1+414yes—
145C22×C4—232no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—28no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—216no—
154C22×C4—216no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—28no—
158C22×C4—216no—
159C22×C4—216no—
160C22×C4—28no—
161C22×C4—216no—
162C22×C4—48no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—48no—
166C22×C4—464no—
167C22×C4—432no—
168C22×C4—464no—
169C22×C4—432no—
170C22×C4—416no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4—48no—
175C22×C4—48no—
176C22×C4—48no—
177C22×C4—48no—
178C22×C4—48no—
179C22×C4—416no—
180C22×C4—416no—
181C22×C4—48no—
182C22×C4—48no—
183C22×C4—48no—
184C22×C4—832no—
185C22×C4#9918yes—
186C22×C4C2×C22⋊C414yes—
187C22×C4C2×C22⋊C414yes—
188C4⋊C4—416no—
189C4⋊C4—416no—
190C22⋊C4—416no—
191C22⋊C4—416no—
192C22⋊C4—432no—
193C22⋊C4—816no—
194C22⋊C4—816no—
195C23—216no—
196C23—216no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—28no—
202C23—28no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—24no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—22no—
211C23—22no—
212C23—28no—
213C23—216no—
214C23—216no—
215C23—216no—
216C23—216no—
217C23—24no—
218C23—24no—
219C23—44no—
220C23—48no—
221C23—44no—
222C23—44no—
223C23—48no—
224C23—44no—
225C23—48no—
226C23—48no—
227C23—416no—
228C23—416no—
229C23—416no—
230C23—416no—
231C23—4128no—
232C23—48no—
233C23—48no—
234C23#38114yes—
235C23#38112yes—
236C23#38112yes—
237C23C2×#9914yes—
238C23#41014yes—
239C23C22×C22⋊C411yescharacteristic
240C23C2×#11712yes—
241C23#51812yes—
242C23#51912yes—
243C23C2×#12314yes—
244C23#53514yes—
245C23#53614yes—
246C23#539116yes—
247C2×C4—216no—
248C2×C4—216no—
249C2×C4—216no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—28no—
253C2×C4—24no—
254C2×C4—432no—
255C2×C4—432no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—816no—
263C2×C4—816no—
264C2×C4—816no—
265C2×C4—88no—
266C2×C4—88no—
267C2×C4—816no—
268C2×C4—816no—
269C2×C4—88no—
270C2×C4C2×#9918yes—
271C22—232no—
272C22—22no—
273C22—22no—
274C22—24no—
275C22—24no—
276C22—24no—
277C22—28no—
278C22—28no—
279C22—24no—
280C22—24no—
281C22—24no—
282C22—22no—
283C22—22no—
284C22—24no—
285C22—24no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—24no—
290C22—24no—
291C22—416no—
292C22—432no—
293C22—432no—
294C22—416no—
295C22—44no—
296C22—44no—
297C22#181914yes—
298C22#182014yes—
299C22C2×#38111yescharacteristic
300C22#230911yescharacteristic
301C22#231011yescharacteristic
302C22#231314yes—
303C22#262318yes—
304C22C22×#9912yes—
305C22C2×#41012yes—
306C4—28no—
307C4—44no—
308C4—88no—
309C4—168no—
310C2—28no—
311C2—24no—
312C2—24no—
313C2—24no—
314C2—44no—
315C2#1434211yescharacteristic
316C2C2×#181911yescharacteristic
317C2C2×#182011yescharacteristic
318C2#2179312yes—
319C2#2182512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11353C2semidirect
6. C23×C4⋊C4C4semidirect
11. C2×#380C4semidirect
14. C22×#95C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
#146812
#113532
#107252