Finite Group Databasebeta

← #110946

Subgroups — #110946

Order 512.

887 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 887 conjugacy classes of subgroups — too many to draw. Only its 166 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#21417C211yesmaximal, characteristic
2#17546C212yesmaximal
3#13750C211yesmaximal, characteristic, complemented
4#8098C212yesmaximal, complemented
5#7855C211yesmaximal, characteristic, complemented
6#3444C2211yescharacteristic
7#2920C2212yes—
8#2920C2211yescharacteristic
9C2×C4×C4⋊C4—22no—
10C2×C4×C4⋊C4C2212yes—
11C2×#376C2211yescharacteristic
12#1995C412yescomplemented
13#1963C412yescomplemented
14#1896—22no—
15#1759—22no—
16#1509—24no—
17#1504—24no—
18#1503C414yescomplemented
19#1502C414yescomplemented
20C4×#95—24no—
21#1303—24no—
22#1274—22no—
23#533C2×C411yescharacteristic
24#533C2×C412yes—
25#532C2×C411yescharacteristic
26#532C2×C412yes—
27#529—22no—
28#529—22no—
29#529—22no—
30#525—21no—
31#525—22no—
32C4×#41—21no—
33C4×#41—22no—
34C22×C4⋊C4—22no—
35C22×C42—24no—
36C22×C42—22no—
37C22×C42D412yes—
38C22×C42C2311yescharacteristic
39#432—22no—
40#432—22no—
41#432—22no—
42#431—22no—
43#431—22no—
44#431C2×C414yes—
45#423—22no—
46#423C2×C412yes—
47C2×#104—24no—
48#407—22no—
49#407—21no—
50C4×M16—22no—
51C4×M16—21no—
52C4×C4⋊C4—216no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—22no—
55C4×C4⋊C4C2×C414yes—
56C4×C4⋊C4D414yes—
57#376—22no—
58#376C2×C412yes—
59#376D412yes—
60C2×#95—24no—
61C2×#95—22no—
62C2×#95D414yes—
63#339—24no—
64#328—28no—
65#327—28no—
66C2×#41—21no—
67C2×#41—22no—
68C23×C4—24no—
69C23×C4—22no—
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D412yes—
72C2×M16—22no—
73C2×M16—42no—
74C4⋊Q8C4211yescharacteristic, complemented
75C4⋊Q8C4212yescomplemented
76#127C4211yescharacteristic, complemented
77#126—42no—
78#125—21no—
79#125—22no—
80#124—21no—
81#124—22no—
82#123—42no—
83#123—41no—
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87#121—22no—
88#121—22no—
89#121—22no—
90#120—41no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—42no—
98C4×D4—42no—
99C4×D4—42no—
100#117—21no—
101#117—22no—
102#117—42no—
103#117—41no—
104C2×C4⋊C4—216no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C42—22no—
108C2×C42—28no—
109C2×C42—216no—
110C2×C42—24no—
111C2×C42—24no—
112C2×C42—41no—
113C2×C42—84no—
114C2×C42—88no—
115C2×C42C22⋊C414yes—
116C2×C42C22×C411yescharacteristic
117C2×C42C22×C412yes—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D412yes—
120C2×C42#4112yes—
121C2×C42#4112yes—
122C2×C42#4112yes—
123C4⋊C8—24no—
124C4⋊C8—24no—
125C4⋊C8—42no—
126#104—48no—
127C22⋊C8—24no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130#97—41no—
131#97—42no—
132C4×C8—24no—
133C4×C8—41no—
134#95—216no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95C22⋊C418yes—
139C24#12011yescharacteristic
140#41—44no—
141#41—82no—
142#41—82no—
143C2×Q8—21no—
144C2×Q8—22no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—41no—
148C2×D4—42no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—216no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—28no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—48no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4C2×C22⋊C412yes—
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4C4×D412yes—
167C22×C4C4×D412yes—
168C22×C4C4×D412yes—
169C22×C4#12112yes—
170C22×C4#12112yes—
171C22×C4#12112yes—
172C22×C4#12311yescharacteristic
173C22×C4#12312yes—
174M16—44no—
175M16—82no—
176M16—82no—
177C2×C8—44no—
178C2×C8—42no—
179C4⋊C4—216no—
180C4⋊C4—216no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C22⋊C4—41no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C42—28no—
193C42—24no—
194C42—42no—
195C42—41no—
196C42—84no—
197C42—84no—
198C42—816no—
199C42#10418yes—
200C42C2×C4211yescharacteristic
201C42#11712yes—
202C42#11711yescharacteristic
203C42C4×D412yes—
204C42C4×D412yes—
205C42C4×D412yes—
206C42#12411yescharacteristic
207C42#12412yes—
208C42#12612yes—
209C42#12711yescharacteristic
210C23—22no—
211C23—22no—
212C23—22no—
213C23—41no—
214C23#38611yescharacteristic
215C23#38612yes—
216Q8—42no—
217Q8—82no—
218D4—42no—
219D4—82no—
220D4—82no—
221D4—82no—
222D4—82no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—216no—
231C2×C4—28no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—416no—
238C2×C4—48no—
239C2×C4—416no—
240C2×C4—44no—
241C2×C4—48no—
242C2×C4—48no—
243C2×C4—48no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4—82no—
248C2×C4C4×C22⋊C412yes—
249C2×C4C4×C22⋊C411yescharacteristic
250C2×C4#38812yes—
251C2×C4#38812yes—
252C2×C4#38812yes—
253C2×C4#39012yes—
254C2×C4#39011yescharacteristic
255C2×C4C2×#10414yes—
256C8—84no—
257C8—82no—
258C22—22no—
259C22—24no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—81no—
265C22—82no—
266C22#143711yescharacteristic
267C22#189612yes—
268C22#190412yes—
269C22#190512yes—
270C4—22no—
271C4—22no—
272C4—22no—
273C4—22no—
274C4—22no—
275C4—416no—
276C4—48no—
277C4—81no—
278C4—82no—
279C4C4×#10412yes—
280C4#196212yes—
281C2—22no—
282C2—22no—
283C2—22no—
284C2—161no—
285C2#1092311yescharacteristic
286C2#1095412yes—
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13750C2semidirect
4. #8098C2semidirect
5. #7855C2semidirect
12. #1995C4semidirect
13. #1963C4semidirect
18. #1503C4semidirect
19. #1502C4semidirect
74. C4⋊Q8C42semidirect
75. C4⋊Q8C42semidirect
76. #127C42semidirect

Maximal subgroups

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

subgroupindex
#214172
#175462
#137502
#80982
#78552