Finite Group Databasebeta

← #190105

Subgroups — #190105

Order 512.

1119 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1119 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#34057C211yesmaximal, characteristic
2#20722C212yesmaximal
3#8750C214yesmaximal, complemented
4#8747C212yesmaximal
5#8746C212yesmaximal
6#8100C214yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8#2318—24no—
9#2315C412yescomplemented
10#2313C412yescomplemented
11#2312C2212yes—
12#2308C2211yescharacteristic
13#2308C2211yescharacteristic
14#2290—24no—
15#2283C412yescomplemented
16#2280C2214yes—
17#2280C2212yes—
18C2×C4×C22⋊C4C412yescomplemented
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—24no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2×C412yes—
37C22×C22⋊C4C2×C412yes—
38C22×C22⋊C4C2311yescharacteristic
39C22×C42—22no—
40C22×C42C2×C411yescharacteristic
41C2×#99—28no—
42#388—24no—
43#388—24no—
44#387—24no—
45#387—24no—
46#386—24no—
47#386—24no—
48#385—24no—
49#385—24no—
50#383—22no—
51#382—22no—
52#381—22no—
53#381—24no—
54#381C2×C412yes—
55#381C2×C412yes—
56#380—22no—
57#380—24no—
58#380—24no—
59#380C2×C414yes—
60#380C2×C414yes—
61#379C2318yes—
62#379C2312yes—
63#379C2312yes—
64C4×C4⋊C4—22no—
65C4×C22⋊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—
73C2×#95—22no—
74#342—28no—
75C25C22×C411yescharacteristic
76C25C2411yescharacteristic
77C22×D4—24no—
78C22×D4—44no—
79C22×D4C2×D414yes—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—82no—
85C23×C4—82no—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4#4114yes—
96C23×C4#4114yes—
97C4×D4—28no—
98C4×D4—28no—
99C4×D4—28no—
100C4×D4—28no—
101#117—84no—
102#117—88no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4C22⋊C414yes—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—44no—
120C2×C22⋊C4—44no—
121C2×C22⋊C4C22⋊C414yes—
122C2×C22⋊C4C22×C414yes—
123C2×C22⋊C4C22×C414yes—
124C2×C22⋊C4C2×D414yes—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—24no—
128C2×C42—24no—
129C2×C42—44no—
130C2×C42—84no—
131C2×C42C22⋊C412yes—
132C2×C42C22⋊C412yes—
133#99—416no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—22no—
138#95—22no—
139#95—24no—
140#95—22no—
141#95—22no—
142#95—44no—
143#95—44no—
144C24—22no—
145C24—24no—
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—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—44no—
168C22×C4—48no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—48no—
175C22×C4—84no—
176C22×C4—81no—
177C22×C4—82no—
178C22×C4—84no—
179C22×C4—84no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4C2×C22⋊C412yes—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4C2×C22⋊C412yes—
190C22×C4C2×C22⋊C412yes—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C2×C22⋊C412yes—
193C22×C4#11712yes—
194C22×C4#11712yes—
195C22×C4#11714yes—
196C22×C4#12314yes—
197C22×C4#12314yes—
198C22×C4#12314yes—
199C22×C4#12314yes—
200C4⋊C4—28no—
201C4⋊C4—84no—
202C4⋊C4—84no—
203C4⋊C4—88no—
204C22⋊C4—28no—
205C22⋊C4—28no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—416no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C22⋊C4—88no—
216C22⋊C4—88no—
217C42—22no—
218C42—24no—
219C42—22no—
220C42—48no—
221C42—88no—
222C42—84no—
223C23—28no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—41no—
229C23—41no—
230C23—42no—
231C23—44no—
232C23—42no—
233C23—44no—
234C23—44no—
235C23—42no—
236C23—42no—
237C23—82no—
238C23—82no—
239C23—88no—
240C23—88no—
241C23#41014yes—
242C23C22×C22⋊C411yescharacteristic
243C23C2×#11712yes—
244C23C2×#12314yes—
245D4—416no—
246C2×C4—22no—
247C2×C4—28no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
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—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—88no—
275C2×C4—88no—
276C2×C4—88no—
277C2×C4—88no—
278C2×C4—88no—
279C2×C4#38111yescharacteristic
280C2×C4#38112yes—
281C2×C4#38114yes—
282C2×C4#38111yescharacteristic
283C22—21no—
284C22—21no—
285C22—22no—
286C22—24no—
287C22—24no—
288C22—24no—
289C22—22no—
290C22—24no—
291C22—22no—
292C22—44no—
293C22—44no—
294C22—44no—
295C22—42no—
296C22—42no—
297C22—84no—
298C22—88no—
299C22—88no—
300C22C2×#38111yescharacteristic
301C22C2×#41012yes—
302C4—21no—
303C4—22no—
304C4—24no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—84no—
309C4—84no—
310C4—168no—
311C4#182812yes—
312C2—22no—
313C2—44no—
314C2—42no—
315C2—44no—
316C2#2068811yescharacteristic
317C2C2×#182811yescharacteristic
318C2#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. #2315C4semidirect
10. #2313C4semidirect
15. #2283C4semidirect
18. C2×C4×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#340572
#207222
#87502
#87472
#87462
#81002