Finite Group Databasebeta

← #190106

Subgroups — #190106

Order 512.

1097 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1097 conjugacy classes of subgroups — too many to draw. Only its 232 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
1#34058C211yesmaximal, characteristic
2#20722C212yesmaximal
3#8750C214yesmaximal, complemented
4#8749C212yesmaximal
5#8748C212yesmaximal
6#8100C214yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8#2319—24no—
9#2316C412yescomplemented
10#2312C412yescomplemented
11#2312C2212yes—
12#2309C2211yescharacteristic
13#2309C2211yescharacteristic
14#2293—24no—
15#2280C412yescomplemented
16#2280C2214yes—
17#2280C2212yes—
18C4×#117C412yescomplemented
19#1819C2214yes—
20#1799—24no—
21#1781C2214yes—
22C4×#99—24no—
23#1304C2218yes—
24#1304C2218yes—
25C23×D4C2311yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2311yescharacteristic
28C2×C4×D4—24no—
29C2×C4×D4—24no—
30C2×C4×D4D414yes—
31C2×C4×D4D414yes—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—24no—
35C2×#117C2×C412yes—
36C2×#117C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C22⋊C4C2×C411yescharacteristic
39C22×C22⋊C4C2311yescharacteristic
40C22×C42C2×C411yescharacteristic
41C2×#99—28no—
42#390—24no—
43#390—24no—
44#389—24no—
45#389—24no—
46#388—24no—
47#388—24no—
48#388—24no—
49#388—24no—
50#381—22no—
51#381—22no—
52#381—24no—
53#380—22no—
54#380—22no—
55#380—24no—
56#380—24no—
57#380C2×C414yes—
58#380C2×C414yes—
59#379C2×C412yes—
60#379C2×C412yes—
61#379C2318yes—
62#379C2312yes—
63#379C2312yes—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—24no—
69C4×C22⋊C4—24no—
70C4×C22⋊C4C2×C414yes—
71C4×C22⋊C4C2×C414yes—
72#376—22no—
73C43—22no—
74#342—28no—
75C25C22×C411yescharacteristic
76C25C2411yescharacteristic
77C22×D4—24no—
78C22×D4—44no—
79C22×D4C2×D414yes—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—82no—
83C23×C4—82no—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C412yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4#4114yes—
94C23×C4#4114yes—
95C4×D4—28no—
96C4×D4—28no—
97C4×D4—28no—
98C4×D4—28no—
99#117—44no—
100#117—44no—
101#117—84no—
102#117—84no—
103#117—88no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4C22⋊C414yes—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—28no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—28no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4C22⋊C414yes—
125C2×C22⋊C4C22×C414yes—
126C2×C22⋊C4C22×C414yes—
127C2×C22⋊C4C2×D414yes—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—24no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—24no—
134C2×C42—44no—
135C2×C42C22⋊C412yes—
136C2×C42C22⋊C412yes—
137#99—416no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—44no—
143#95—44no—
144C24—22no—
145C24—42no—
146C24—42no—
147C24—81no—
148C24—81no—
149C24—81no—
150C24—81no—
151C24C2×C22⋊C412yes—
152C24C23×C411yescharacteristic
153C24C22×D412yes—
154C24C2×#4114yes—
155C2×D4—28no—
156C2×D4—28no—
157C2×D4—44no—
158C2×D4—416no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—44no—
169C22×C4—48no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—48no—
173C22×C4—84no—
174C22×C4—81no—
175C22×C4—82no—
176C22×C4—84no—
177C22×C4—84no—
178C22×C4—82no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C2×C22⋊C412yes—
183C22×C4C2×C22⋊C412yes—
184C22×C4C2×C22⋊C412yes—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C2×C22⋊C412yes—
187C22×C4#11712yes—
188C22×C4#11712yes—
189C22×C4#11714yes—
190C22×C4#12314yes—
191C22×C4#12314yes—
192C22×C4#12314yes—
193C22×C4#12314yes—
194C4⋊C4—28no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C4⋊C4—88no—
200C22⋊C4—28no—
201C22⋊C4—28no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—416no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C22⋊C4—88no—
214C22⋊C4—88no—
215C42—22no—
216C42—24no—
217C42—22no—
218C42—44no—
219C42—48no—
220C42—44no—
221C42—88no—
222C42—84no—
223C23—22no—
224C23—24no—
225C23—22no—
226C23—24no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—44no—
231C23—42no—
232C23—44no—
233C23—44no—
234C23—44no—
235C23—42no—
236C23—42no—
237C23—82no—
238C23—82no—
239C23—88no—
240C23—88no—
241C23C22×C22⋊C411yescharacteristic
242C23C2×#11712yes—
243C23C2×#12314yes—
244D4—416no—
245C2×C4—28no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—28no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—88no—
274C2×C4—88no—
275C2×C4—88no—
276C2×C4#38111yescharacteristic
277C2×C4#38112yes—
278C2×C4#38114yes—
279C2×C4#38111yescharacteristic
280C2×C4#41014yes—
281C22—21no—
282C22—21no—
283C22—22no—
284C22—24no—
285C22—24no—
286C22—22no—
287C22—24no—
288C22—22no—
289C22—44no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—84no—
294C22—84no—
295C22—88no—
296C22—88no—
297C22C2×#38111yescharacteristic
298C22C2×#41012yes—
299C4—24no—
300C4—21no—
301C4—22no—
302C4—24no—
303C4—42no—
304C4—42no—
305C4—84no—
306C4—84no—
307C4—168no—
308C4#182812yes—
309C2—22no—
310C2—44no—
311C2—42no—
312C2—82no—
313C2#2068811yescharacteristic
314C2C2×#182811yescharacteristic
315C2#2071011yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC24C23×C4
Frattini, cosocleC25C24
socleC22C2×#381
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8750C2semidirect
6. #8100C2semidirect
9. #2316C4semidirect
10. #2312C4semidirect
15. #2280C4semidirect
18. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
#340582
#207222
#87502
#87492
#87482
#81002