Finite Group Databasebeta

← #214043

Subgroups — #214043

Order 512.

810 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 186 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#37390C211yesmaximal, characteristic
2#17410C212yesmaximal, complemented
3#9452C214yesmaximal, complemented
4#9448C214yesmaximal, complemented
5#8745C211yesmaximal, characteristic, complemented
6#8744C211yesmaximal, characteristic, complemented
7#8524C212yesmaximal
8C2×#541C2211yescharacteristic
9C2×#536C2211yescharacteristic
10#2983C2214yes—
11#2868C411yescharacteristic
12#2861C411yescharacteristic
13#2720—24no—
14#2718—24no—
15#2658C411yescharacteristic
16#2642C411yescharacteristic
17#2546C2212yes—
18#2545C2212yes—
19#2434C412yes—
20#2432C412yes—
21#2298C2211yescharacteristic
22#1558C2214yes—
23#1533C2214yes—
24#1473—24no—
25#1470—24no—
26#1467C2214yes—
27#1464C2212yes—
28#1464C2212yes—
29#1285C2214yes—
30#1285C2214yes—
31C22×#41C2×C411yescharacteristic
32C23×Q8C2×C411yescharacteristic
33#541—44no—
34#536—44no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4⋊Q8C2×C412yes—
37C2×C4⋊Q8C2×C412yes—
38C2×C4⋊Q8C2×C412yes—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2311yescharacteristic
41C2×#126D414yes—
42C2×#125C2×C411yescharacteristic
43C2×#125D414yes—
44C2×#124C2×C411yescharacteristic
45C2×#124C2×C412yes—
46C2×#124C2311yescharacteristic
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8C2312yes—
51C2×#117C2311yescharacteristic
52#446—44no—
53#446—44no—
54#432—24no—
55#432C2314yes—
56#432C2314yes—
57#431—24no—
58#400—24no—
59#399—24no—
60#398—24no—
61#396—24no—
62#395—24no—
63#394—24no—
64#391C2×C412yes—
65#391C2×C412yes—
66#391C2×C412yes—
67#390C2×C412yes—
68#389—24no—
69#388—24no—
70#388—22no—
71#388—24no—
72#387—22no—
73#385—24no—
74#385—24no—
75#384—24no—
76C4×C4⋊C4C2312yes—
77C4×C22⋊C4C2×C412yes—
78#376C2×C411yescharacteristic
79C43C2×C411yescharacteristic
80#333—44no—
81#329—44no—
82#324—22no—
83#324—22no—
84#324—28no—
85#323—22no—
86#323—22no—
87C2×#41—42no—
88C2×#41—82no—
89C22×Q8—21no—
90C22×Q8—42no—
91C22×Q8—82no—
92C22×Q8C22×C411yescharacteristic
93C22×Q8C22×C411yescharacteristic
94C22×Q8C22×C411yescharacteristic
95C22×Q8C22×C412yes—
96C22×D4—21no—
97C22×D4C22×C411yescharacteristic
98C23×C4—21no—
99C23×C4—21no—
100C23×C4C22×C411yescharacteristic
101C4⋊Q8—24no—
102C4⋊Q8—42no—
103C4⋊Q8—42no—
104C4⋊Q8—82no—
105C4⋊Q8—82no—
106C4⋊Q8—82no—
107#126—44no—
108#126—44no—
109#125—42no—
110#125—44no—
111#125—44no—
112#124—24no—
113#124—42no—
114#124—82no—
115#123—48no—
116C22⋊Q8—42no—
117C22⋊Q8—42no—
118C22⋊Q8—48no—
119C4×Q8—24no—
120C4×Q8—24no—
121#117—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4C22⋊C414yes—
129C2×C4⋊C4C22⋊C414yes—
130C2×C4⋊C4C22⋊C414yes—
131C2×C4⋊C4C22×C412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D414yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4C22⋊C414yes—
140C2×C22⋊C4C2×D412yes—
141C2×C42—22no—
142C2×C42C22×C411yescharacteristic
143C2×C42C22×C412yes—
144C2×C42C22×C412yes—
145C2×C42C2×D414yes—
146C2×C42C2411yescharacteristic
147C4⋊C8—24no—
148C4⋊C8—24no—
149C4⋊C8—44no—
150C4⋊C8—44no—
151C22⋊C8—24no—
152C22⋊C8—24no—
153#97—24no—
154#97—24no—
155#97—44no—
156C4×C8—44no—
157#95—22no—
158#95—22no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—22no—
163C24—21no—
164C24C2×C22⋊C411yescharacteristic
165#41—88no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—82no—
178C2×Q8—88no—
179C2×D4—22no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—82no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C2×C22⋊C414yes—
196C22×C4C2×C22⋊C414yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4C23×C411yescharacteristic
199C22×C4C22×D412yes—
200C2×C8—44no—
201C2×C8—44no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—84no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C22⋊C4—24no—
217C22⋊C4—44no—
218C22⋊C4—48no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—84no—
223C42—22no—
224C42—24no—
225C42—24no—
226C42—42no—
227C42—42no—
228C42—44no—
229C42H(5,𝔽2)14yes—
230C23—21no—
231C23—22no—
232C23—42no—
233C23—44no—
234C23—41no—
235C23—42no—
236C23C22×C22⋊C411yescharacteristic
237Q8—82no—
238Q8—82no—
239Q8—82no—
240Q8—82no—
241Q8—82no—
242Q8—84no—
243D4—82no—
244D4—84no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4#51812yes—
272C2×C4#53414yes—
273C8—84no—
274C8—84no—
275C22—22no—
276C22—42no—
277C22—41no—
278C22—82no—
279C22—84no—
280C22—82no—
281C22—81no—
282C22#230811yescharacteristic
283C22#288812yes—
284C4—24no—
285C4—22no—
286C4—44no—
287C4—42no—
288C4—44no—
289C4—41no—
290C4—44no—
291C4—82no—
292C4—81no—
293C4—81no—
294C4—81no—
295C2—22no—
296C2—81no—
297C2—161no—
298C2#2175511yescharacteristic
299C2#2178312yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17410C2semidirect
3. #9452C2semidirect
4. #9448C2semidirect
5. #8745C2semidirect
6. #8744C2semidirect

Maximal subgroups

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

subgroupindex
#373902
#174102
#94522
#94482
#87452
#87442
#85242