Finite Group Databasebeta

← #129249

Subgroups — #129249

Order 512.

1548 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1548 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×#384C211yesmaximal, characteristic
2C2×#1325C214yesmaximal, complemented
3#8299C212yesmaximal
4C23×C4⋊C4—22no—
5C23×C4⋊C4C411yescharacteristic
6C23×C4⋊C4C2211yescharacteristic
7C23×C42C411yescharacteristic
8C2×#384—28no—
9C2×#384—22no—
10C2×#384C412yes—
11C2×#384C2212yes—
12#1779—22no—
13C2×#324—22no—
14C2×#324C2214yes—
15#1325—28no—
16C24×C4—21no—
17C24×C4—21no—
18C24×C4C2×C411yescharacteristic
19C24×C4D412yes—
20C22×C4⋊C4—216no—
21C22×C4⋊C4—24no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—24no—
24C22×C4⋊C4—24no—
25C22×C4⋊C4—24no—
26C22×C4⋊C4—42no—
27C22×C4⋊C4—42no—
28C22×C4⋊C4—42no—
29C22×C4⋊C4—416no—
30C22×C4⋊C4—42no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4D412yes—
34C22×C4⋊C4C2311yescharacteristic
35C22×C42—21no—
36C22×C42—22no—
37C22×C42—24no—
38C22×C42—24no—
39C22×C42C814yescomplemented
40C22×C42C2×C412yes—
41C2×C22⋊C8—24no—
42C2×C22⋊C8—44no—
43#384—216no—
44#384—24no—
45#384—24no—
46#384—24no—
47#384D418yes—
48#324—28no—
49#324—44no—
50C25C22⋊C411yescharacteristic
51C23×C4—24no—
52C23×C4—28no—
53C23×C4—28no—
54C23×C4—24no—
55C23×C4—24no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—24no—
64C23×C4—22no—
65C23×C4—24no—
66C23×C4—24no—
67C23×C4—42no—
68C23×C4—48no—
69C23×C4—48no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C412yes—
74C23×C4C2×C812yes—
75C23×C4M1612yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D412yes—
78C22×C8—44no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—216no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—216no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—416no—
94C2×C4⋊C4—416no—
95C2×C4⋊C4—416no—
96C2×C4⋊C4—416no—
97C2×C4⋊C4—464no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4C22⋊C414yes—
101C2×C4⋊C4C2×D414yes—
102C2×C42—24no—
103C2×C42—24no—
104C2×C42—216no—
105C2×C42—24no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—216no—
110C2×C42C22⋊C414yes—
111C2×C42C2×C814yes—
112C2×C42M1614yes—
113C22⋊C8—44no—
114C22⋊C8—44no—
115C22⋊C8—88no—
116C24—24no—
117C24—24no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24C22⋊C812yes—
123C24#9912yes—
124C24#10112yes—
125C24C2×C22⋊C411yescharacteristic
126C22×C4—216no—
127C22×C4—216no—
128C22×C4—28no—
129C22×C4—28no—
130C22×C4—216no—
131C22×C4—232no—
132C22×C4—232no—
133C22×C4—28no—
134C22×C4—28no—
135C22×C4—28no—
136C22×C4—28no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—24no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—48no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—416no—
164C22×C4—416no—
165C22×C4—48no—
166C22×C4—48no—
167C22×C4—48no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—84no—
171C22×C4—88no—
172C22×C4—88no—
173C22×C4—84no—
174C22×C4—84no—
175C22×C4C22⋊C812yes—
176C22×C4#9912yes—
177C22×C4#10112yes—
178C22×C4C2×C22⋊C412yes—
179C22×C4#12014yes—
180C22×C4C22×C811yescharacteristic
181C22×C4C2×M1612yes—
182C22×C4C2×M1611yescharacteristic
183C2×C8—84no—
184C2×C8—88no—
185C4⋊C4—44no—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—432no—
189C4⋊C4—432no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C42—216no—
194C42—24no—
195C42—24no—
196C42—24no—
197C42C22⋊C818yescomplemented
198C23—24no—
199C23—24no—
200C23—28no—
201C23—24no—
202C23—24no—
203C23—28no—
204C23—28no—
205C23—28no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23#32414yes—
217C23#35314yes—
218C23#35614yes—
219C23#37911yescharacteristic
220C23C2×C22⋊C811yescharacteristic
221C23#40711yescharacteristic
222C23C2×#9911yescharacteristic
223C23C2×#9911yescharacteristic
224C23C2×#10111yescharacteristic
225C23C2×#10111yescharacteristic
226C2×C4—28no—
227C2×C4—264no—
228C2×C4—28no—
229C2×C4—216no—
230C2×C4—216no—
231C2×C4—216no—
232C2×C4—216no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—48no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—82no—
257C2×C4—816no—
258C2×C4—816no—
259C2×C4—84no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4#32414yes—
263C2×C4C2×C22⋊C812yes—
264C2×C4#40712yes—
265C8—164no—
266C22—28no—
267C22—28no—
268C22—28no—
269C22—28no—
270C22—24no—
271C22—21no—
272C22—24no—
273C22—21no—
274C22—21no—
275C22—24no—
276C22—21no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—22no—
281C22—24no—
282C22—24no—
283C22—22no—
284C22#132518yescomplemented
285C22C2×#32411yescharacteristic
286C22C2×#32411yescharacteristic
287C22#146311yescharacteristic
288C22#146311yescharacteristic
289C22#177911yescharacteristic
290C22#178111yescharacteristic
291C22#178411yescharacteristic
292C22C2×#35312yes—
293C22C2×#35612yes—
294C4—216no—
295C4—24no—
296C4—22no—
297C4—22no—
298C4—44no—
299C4—82no—
300C4—84no—
301C2—21no—
302C2—28no—
303C2—21no—
304C2—21no—
305C2—21no—
306C2#829911yescharacteristic
307C2C2×#132512yes—
308C2#1151212yes—
309C2#1269311yescharacteristic
310C2#1270011yescharacteristic
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×#1325C2semidirect
39. C22×C42C8semidirect
197. C42C22⋊C8semidirect
284. C22#1325semidirect

Maximal subgroups

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

subgroupindex
C22×#3842
C2×#13252
#82992