Finite Group Databasebeta

← #237897

Subgroups — #237897

Order 512.

882 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 186 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×#2844C211yesmaximal, characteristic
2#21866C212yesmaximal
3#21323C211yesmaximal, characteristic
4#10359C218yesmaximal, complemented
5#8033C212yesmaximal
6#8030C211yesmaximal, characteristic
7C2×#533C2211yescharacteristic
8C2×#432C2212yes—
9C2×C4×M16C2211yescharacteristic
10#2844—24no—
11#2844—22no—
12#2844C2214yes—
13C2×#400—21no—
14C2×#400—22no—
15C2×#395—21no—
16C2×#395—22no—
17C2×#382C2211yescharacteristic
18C2×#382C2212yes—
19#1993C2214yes—
20#1993C2214yes—
21#1991—22no—
22#1960C2214yes—
23#1872—22no—
24#1872—22no—
25#1805—24no—
26#1736—21no—
27#1736—21no—
28#1295C2214yes—
29#1295C2218yes—
30C23×Q8—21no—
31C22×M16—21no—
32#533—22no—
33#533—24no—
34#533Q814yescomplemented
35C2×C4⋊Q8C2312yes—
36C2×#125—21no—
37C2×C22⋊Q8—22no—
38C2×C4×Q8—22no—
39C2×#117—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C42C2311yescharacteristic
47#432—24no—
48#432—24no—
49C2×C4⋊C8—22no—
50C2×#103—24no—
51C2×#103—24no—
52C2×C22⋊C8—22no—
53C4×M16—24no—
54C2×#97C2312yes—
55C2×#97C2314yes—
56C2×C4×C8—21no—
57#400—24no—
58#400—28no—
59#400—24no—
60#400—28no—
61#395—24no—
62#395—28no—
63#395—24no—
64#395—28no—
65#382—22no—
66#382—22no—
67#382—22no—
68#382D414yes—
69#382D412yes—
70#382C2314yes—
71#382C2312yes—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78#340—24no—
79#340—24no—
80#340—24no—
81#340—24no—
82C22×Q8—24no—
83C22×Q8—22no—
84C22×Q8—84no—
85C22×Q8—82no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C2×M16—24no—
93C2×M16—42no—
94C2×M16—42no—
95C22×C8—22no—
96C22×C8—24no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8C2×Q814yescomplemented
101#125—24no—
102#125—42no—
103C22⋊Q8—28no—
104C22⋊Q8—44no—
105C4×Q8—28no—
106C4×Q8—44no—
107#117—24no—
108#117—42no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—82no—
121C2×C4⋊C4—82no—
122C2×C4⋊C4—84no—
123C2×C4⋊C4—82no—
124C2×C4⋊C4C2×D412yes—
125C2×C4⋊C4C2×D412yes—
126C2×C4⋊C4C2×D414yes—
127C2×C4⋊C4C2×D414yes—
128C2×C22⋊C4—41no—
129C2×C42—41no—
130C2×C42C2×D412yes—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×Q812yes—
133C2×C42#4114yes—
134C2×C42#4112yes—
135C2×C42C2411yescharacteristic
136C4⋊C8—44no—
137#103—84no—
138#103—84no—
139C22⋊C8—44no—
140#97—24no—
141#97—24no—
142#97—24no—
143C4×C8—42no—
144#95—28no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—24no—
153C24#12011yescharacteristic
154C2×Q8—24no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—42no—
158C2×Q8—84no—
159C2×Q8—88no—
160C2×Q8—816no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4#12011yescharacteristic
181C22×C4#12012yes—
182C22×C4C22⋊Q812yes—
183C22×C4C22⋊Q814yes—
184C22×C4#12412yes—
185C22×C4#12414yes—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D412yes—
188M16—48no—
189C2×C8—24no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—44no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—48no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—84no—
209C4⋊C4—88no—
210C22⋊C4—44no—
211C42—44no—
212C42C22⋊Q814yes—
213C42C22⋊Q812yes—
214C42#12412yes—
215C42#12414yes—
216C42C22×Q811yescharacteristic
217C42C2×#4111yescharacteristic
218C42C2×#4112yes—
219C23—22no—
220C23—22no—
221C23#39312yes—
222C23#45512yes—
223C23#45612yes—
224C23C2×#12011yescharacteristic
225Q8—48no—
226Q8—84no—
227Q8—816no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—84no—
252C2×C4—88no—
253C2×C4—84no—
254C2×C4—88no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4#39312yes—
258C2×C4#39314yes—
259C2×C4C2×C22⋊Q811yescharacteristic
260C2×C4C2×C22⋊Q812yes—
261C2×C4C2×#12411yescharacteristic
262C2×C4C2×#12412yes—
263C8—44no—
264C8—84no—
265C22—22no—
266C22—22no—
267C22—24no—
268C22—24no—
269C22#202814yes—
270C22C2×#39311yescharacteristic
271C22C2×#45511yescharacteristic
272C22C2×#45611yescharacteristic
273C22#301912yes—
274C22#302012yes—
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—22no—
280C4—24no—
281C4—82no—
282C4—84no—
283C4—82no—
284C4—84no—
285C4—84no—
286C2—24no—
287C2#1899711yescharacteristic
288C2#1899911yescharacteristic
289C2#1900712yes—
290C2C2×#202811yescharacteristic
291C2#2346212yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10359C2semidirect
34. #533Q8semidirect
100. C4⋊Q8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28442
#218662
#213232
#103592
#80332
#80302