Finite Group Databasebeta

← #190170

Subgroups — #190170

Order 512.

1009 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1009 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#17364C212yesmaximal
3#8759C214yesmaximal, complemented
4#8756C214yesmaximal, complemented
5#8106C214yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7#2920C2214yes—
8#2318—24no—
9#2315C2212yes—
10#2313C412yes—
11#2312C2212yes—
12#2308C411yescharacteristic
13#2308C411yescharacteristic
14#2290—24no—
15#2283C2212yes—
16#2280C412yes—
17C2×C4×C22⋊C4C412yes—
18#1466—24no—
19#1462C2214yes—
20#1462C2214yes—
21#1460—24no—
22#1305C22116yes—
23C23×D4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C2×C4×D4—24no—
27C2×C4×D4—24no—
28C2×C4×D4D414yes—
29C2×C4×D4D414yes—
30C2×#117—22no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4C2×C412yes—
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4C2311yescharacteristic
35C22×C22⋊C4C2311yescharacteristic
36C22×C42—22no—
37C22×C42C2311yescharacteristic
38#432—24no—
39#432—24no—
40#431—24no—
41#431—24no—
42#423—24no—
43#407C2318yes—
44C4×M16—24no—
45#388—24no—
46#388—24no—
47#387—24no—
48#387—24no—
49#386—24no—
50#386—24no—
51#385—24no—
52#385—24no—
53#383—22no—
54#382—22no—
55#381—22no—
56#381C2×C412yes—
57#381C2×C412yes—
58#380—22no—
59#380C2×C414yes—
60#380C2314yes—
61#379C2×C412yes—
62#379C2×C412yes—
63C4×C4⋊C4—22no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4C2×C414yes—
68C4×C22⋊C4C2×C414yes—
69#376—22no—
70C2×#95—22no—
71#323—28no—
72#323—28no—
73C25C22×C411yescharacteristic
74C25C22×C411yescharacteristic
75C22×D4—24no—
76C22×D4—44no—
77C22×D4C22⋊C414yes—
78C23×C4—22no—
79C23×C4—22no—
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⋊C412yes—
89C23×C4C22⋊C412yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2411yescharacteristic
95C2×M16—28no—
96C4×D4—28no—
97C4×D4—28no—
98C4×D4—28no—
99C4×D4—28no—
100#117—84no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4C22⋊C414yes—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4C22×C414yes—
116C2×C22⋊C4C22×C414yes—
117C2×C22⋊C4C2×D414yes—
118C2×C22⋊C4C2×D414yes—
119C2×C42—24no—
120C2×C42—24no—
121C2×C42—84no—
122C2×C42C2×D412yes—
123C2×C42C2×D412yes—
124C2×C42#4114yes—
125C2×C42#4114yes—
126C4⋊C8—44no—
127C4⋊C8—44no—
128C22⋊C8—28no—
129C22⋊C8—28no—
130#97—44no—
131C4×C8—44no—
132#95—24no—
133#95—24no—
134#95—22no—
135#95—22no—
136#95—24no—
137#95—22no—
138#95—22no—
139#95—24no—
140C24—22no—
141C24—24no—
142C24—22no—
143C24—42no—
144C24—81no—
145C24—81no—
146C24—81no—
147C24—81no—
148C24C2×C22⋊C412yes—
149C24C2×C22⋊C412yes—
150C24C23×C411yescharacteristic
151C2×D4—28no—
152C2×D4—28no—
153C2×D4—44no—
154C2×D4—416no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—84no—
172C22×C4—84no—
173C22×C4—81no—
174C22×C4—84no—
175C22×C4—84no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4C2×C22⋊C412yes—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C412yes—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4#11712yes—
190C22×C4#11714yes—
191C22×C4#11712yes—
192C22×C4C22×D412yes—
193C22×C4C2×#4114yes—
194M16—88no—
195C2×C8—48no—
196C4⋊C4—28no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C22⋊C4—28no—
200C22⋊C4—28no—
201C22⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—44no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C42—48no—
210C42—84no—
211C42#12314yes—
212C42#12314yes—
213C42#12314yes—
214C42#12314yes—
215C23—28no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—42no—
225C23—44no—
226C23—42no—
227C23—42no—
228C23—82no—
229C23—82no—
230C23—82no—
231C23—82no—
232C23—88no—
233C23—88no—
234C23#41314yes—
235C23C22×C22⋊C411yescharacteristic
236C23C2×#11712yes—
237D4—416no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—84no—
261C2×C4—84no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—88no—
265C2×C4—88no—
266C2×C4#38114yes—
267C2×C4#38111yescharacteristic
268C2×C4#38112yes—
269C2×C4#38111yescharacteristic
270C2×C4C2×#12314yes—
271C8—88no—
272C22—21no—
273C22—21no—
274C22—22no—
275C22—24no—
276C22—22no—
277C22—22no—
278C22—44no—
279C22—44no—
280C22—44no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—84no—
287C22—88no—
288C22—88no—
289C22C2×#38111yescharacteristic
290C22C2×#41312yes—
291C4—24no—
292C4—24no—
293C4—21no—
294C4—22no—
295C4—42no—
296C4—42no—
297C4—84no—
298C4—84no—
299C4#182912yes—
300C2—22no—
301C2—42no—
302C2—44no—
303C2—82no—
304C2#2069411yescharacteristic
305C2#2071011yescharacteristic
306C2C2×#182911yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8759C2semidirect
4. #8756C2semidirect
5. #8106C2semidirect

Maximal subgroups

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

subgroupindex
#340572
#173642
#87592
#87562
#81062