Finite Group Databasebeta

← #129248

Subgroups — #129248

Order 512.

1244 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 158 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×#382C211yesmaximal, characteristic
2C2×#1324C214yesmaximal, complemented
3#8299C212yesmaximal
4C23×C4⋊C4C411yescharacteristic
5C23×C4⋊C4C2211yescharacteristic
6C23×C42C411yescharacteristic
7C2×#382—28no—
8C2×#382—22no—
9C2×#382C412yes—
10C2×#382C2212yes—
11C22×#95—22no—
12#1779—22no—
13C2×#324—22no—
14C2×#324C2214yes—
15#1324—28no—
16C24×C4—21no—
17C24×C4—21no—
18C24×C4C2×C411yescharacteristic
19C24×C4D412yes—
20C22×C4⋊C4—21no—
21C22×C4⋊C4—24no—
22C22×C4⋊C4—24no—
23C22×C4⋊C4—42no—
24C22×C4⋊C4—42no—
25C22×C4⋊C4—42no—
26C22×C4⋊C4—42no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D412yes—
30C22×C4⋊C4C2311yescharacteristic
31C22×C42—21no—
32C22×C42—22no—
33C22×C42—24no—
34C22×C42—24no—
35C22×C42C814yescomplemented
36C22×C42C2×C412yes—
37C2×C22⋊C8—24no—
38C2×C22⋊C8—44no—
39#382—216no—
40#382—24no—
41#382—24no—
42#382—24no—
43#382D418yes—
44C2×#95—216no—
45C2×#95—24no—
46C2×#95—24no—
47#324—28no—
48#324—44no—
49C25C22⋊C411yescharacteristic
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—24no—
57C23×C4—24no—
58C23×C4—24no—
59C23×C4—22no—
60C23×C4—24no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—42no—
65C23×C4—44no—
66C23×C4—44no—
67C23×C4—48no—
68C23×C4—42no—
69C23×C4—48no—
70C23×C4—42no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C412yes—
73C23×C4C2×C812yes—
74C23×C4M1612yes—
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D412yes—
77C22×C8—44no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—416no—
91C2×C4⋊C4—416no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4C22⋊C414yes—
95C2×C4⋊C4C2×D414yes—
96C2×C42—24no—
97C2×C42—24no—
98C2×C42—216no—
99C2×C42—24no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—216no—
104C2×C42C22⋊C414yes—
105C2×C42C2×C814yes—
106C2×C42M1614yes—
107C22⋊C8—44no—
108C22⋊C8—44no—
109C22⋊C8—88no—
110#95—216no—
111#95—216no—
112C24—24no—
113C24—24no—
114C24—21no—
115C24—21no—
116C24—21no—
117C24—21no—
118C24C22⋊C812yes—
119C24#9912yes—
120C24#10112yes—
121C24C2×C22⋊C411yescharacteristic
122C22×C4—28no—
123C22×C4—28no—
124C22×C4—28no—
125C22×C4—28no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—216no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—48no—
151C22×C4—48no—
152C22×C4—48no—
153C22×C4—416no—
154C22×C4—416no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—48no—
160C22×C4—416no—
161C22×C4—416no—
162C22×C4—48no—
163C22×C4—48no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—84no—
167C22×C4—88no—
168C22×C4—88no—
169C22×C4—84no—
170C22×C4—84no—
171C22×C4C22⋊C812yes—
172C22×C4#9912yes—
173C22×C4#10112yes—
174C22×C4C2×C22⋊C412yes—
175C22×C4#12014yes—
176C22×C4C22×C811yescharacteristic
177C22×C4C2×M1612yes—
178C22×C4C2×M1611yescharacteristic
179C2×C8—84no—
180C2×C8—88no—
181C4⋊C4—44no—
182C4⋊C4—48no—
183C4⋊C4—44no—
184C4⋊C4—48no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C42—216no—
188C42—24no—
189C42—24no—
190C42—24no—
191C42C22⋊C818yescomplemented
192C23—24no—
193C23—24no—
194C23—28no—
195C23—24no—
196C23—24no—
197C23—28no—
198C23—28no—
199C23—28no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23#32414yes—
211C23#35414yes—
212C23#35514yes—
213C23#37911yescharacteristic
214C23C2×C22⋊C811yescharacteristic
215C23#40711yescharacteristic
216C23C2×#9911yescharacteristic
217C23C2×#9911yescharacteristic
218C23C2×#10111yescharacteristic
219C23C2×#10111yescharacteristic
220C2×C4—216no—
221C2×C4—216no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—28no—
237C2×C4—24no—
238C2×C4—432no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—48no—
242C2×C4—48no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—48no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—82no—
251C2×C4—816no—
252C2×C4—816no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—82no—
256C2×C4#32414yes—
257C2×C4C2×C22⋊C812yes—
258C2×C4#40712yes—
259C8—164no—
260C22—28no—
261C22—28no—
262C22—28no—
263C22—28no—
264C22—24no—
265C22—21no—
266C22—24no—
267C22—21no—
268C22—21no—
269C22—24no—
270C22—21no—
271C22—22no—
272C22—24no—
273C22—22no—
274C22—22no—
275C22—24no—
276C22—24no—
277C22—22no—
278C22#132418yescomplemented
279C22C2×#32411yescharacteristic
280C22C2×#32411yescharacteristic
281C22#146311yescharacteristic
282C22#146311yescharacteristic
283C22#177911yescharacteristic
284C22#178111yescharacteristic
285C22#178411yescharacteristic
286C22C2×#35412yes—
287C22C2×#35512yes—
288C4—24no—
289C4—22no—
290C4—22no—
291C4—48no—
292C4—44no—
293C4—82no—
294C4—84no—
295C2—21no—
296C2—28no—
297C2—21no—
298C2—21no—
299C2—21no—
300C2#829911yescharacteristic
301C2C2×#132412yes—
302C2#1151112yes—
303C2#1269411yescharacteristic
304C2#1269911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1324C2semidirect
35. C22×C42C8semidirect
191. C42C22⋊C8semidirect
278. C22#1324semidirect

Maximal subgroups

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

subgroupindex
C22×#3822
C2×#13242
#82992