Finite Group Databasebeta

← #110947

Subgroups — #110947

Order 512.

727 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 158 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#13797C211yesmaximal, characteristic, complemented
4#8098C212yesmaximal, complemented
5#7855C211yesmaximal, characteristic, complemented
6#3444C2211yescharacteristic
7#2920C2212yes—
8#2920C2211yescharacteristic
9C2×C4×C4⋊C4C2212yes—
10C2×#376C2211yescharacteristic
11#1995C412yescomplemented
12#1963C412yescomplemented
13#1896—22no—
14#1759—22no—
15#1509—24no—
16#1504—24no—
17#1503C414yescomplemented
18#1502C414yescomplemented
19C4⋊#95—22no—
20#1434—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—22no—
36C22×C42D412yes—
37C22×C42C2311yescharacteristic
38#432—22no—
39#432—22no—
40#432—22no—
41#431—22no—
42#431—22no—
43#431C2×C414yes—
44#423—22no—
45#423C2×C412yes—
46C2×#104—24no—
47#407—22no—
48#407—21no—
49C4×M16—22no—
50C4×M16—21no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4C2×C414yes—
54C4×C4⋊C4D414yes—
55#376—22no—
56#376C2×C412yes—
57#376D412yes—
58C2×#95—22no—
59C2×#95—28no—
60C2×#95D414yes—
61#339—24no—
62#328—28no—
63#327—28no—
64C2×#41—21no—
65C2×#41—22no—
66C23×C4—24no—
67C23×C4—22no—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D412yes—
70C2×M16—22no—
71C2×M16—42no—
72C4⋊Q8C4211yescharacteristic
73C4⋊Q8C4212yes—
74#127C4211yescharacteristic
75#126—42no—
76#125—21no—
77#125—22no—
78#124—21no—
79#124—22no—
80#123—42no—
81#123—41no—
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85#121—22no—
86#121—22no—
87#121—22no—
88#120—41no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—42no—
96C4×D4—42no—
97C4×D4—42no—
98#117—21no—
99#117—22no—
100#117—42no—
101#117—41no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—48no—
105C2×C42—22no—
106C2×C42—24no—
107C2×C42—24no—
108C2×C42—44no—
109C2×C42—41no—
110C2×C42—84no—
111C2×C42C22⋊C414yes—
112C2×C42C22×C411yescharacteristic
113C2×C42C22×C412yes—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D412yes—
116C2×C42#4112yes—
117C2×C42#4112yes—
118C2×C42#4112yes—
119C4⋊C8—24no—
120C4⋊C8—24no—
121C4⋊C8—42no—
122#104—48no—
123C22⋊C8—24no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126#97—41no—
127#97—42no—
128C4×C8—24no—
129C4×C8—41no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—88no—
134#95C22⋊C418yes—
135C24#12011yescharacteristic
136#41—44no—
137#41—82no—
138#41—82no—
139C2×Q8—21no—
140C2×Q8—22no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—41no—
144C2×D4—42no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—28no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—48no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—48no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4C2×C22⋊C412yes—
161C22×C4C2×C22⋊C411yescharacteristic
162C22×C4C4×D412yes—
163C22×C4C4×D412yes—
164C22×C4C4×D412yes—
165C22×C4#12112yes—
166C22×C4#12112yes—
167C22×C4#12112yes—
168C22×C4#12311yescharacteristic
169C22×C4#12312yes—
170M16—44no—
171M16—82no—
172M16—82no—
173C2×C8—44no—
174C2×C8—42no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—48no—
179C4⋊C4—48no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C42—24no—
189C42—24no—
190C42—44no—
191C42—42no—
192C42—41no—
193C42—84no—
194C42—84no—
195C42C2×C4211yescharacteristic
196C42#11712yes—
197C42#11711yescharacteristic
198C42C4×D412yes—
199C42C4×D412yes—
200C42C4×D412yes—
201C42#12411yescharacteristic
202C42#12412yes—
203C42#12612yes—
204C42#12711yescharacteristic
205C23—22no—
206C23—22no—
207C23—22no—
208C23—41no—
209C23#38611yescharacteristic
210C23#38612yes—
211Q8—42no—
212Q8—82no—
213D4—42no—
214D4—82no—
215D4—82no—
216D4—82no—
217D4—82no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—48no—
232C2×C4—44no—
233C2×C4—48no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—84no—
237C2×C4—82no—
238C2×C4—88no—
239C2×C4—84no—
240C2×C4—88no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4C4×C22⋊C412yes—
244C2×C4C4×C22⋊C411yescharacteristic
245C2×C4#38812yes—
246C2×C4#38812yes—
247C2×C4#38812yes—
248C2×C4#39012yes—
249C2×C4#39011yescharacteristic
250C2×C4#42114yes—
251C8—84no—
252C8—82no—
253C22—22no—
254C22—24no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—81no—
260C22—82no—
261C22#143711yescharacteristic
262C22#189712yes—
263C22#190412yes—
264C22#190512yes—
265C4—22no—
266C4—22no—
267C4—22no—
268C4—22no—
269C4—22no—
270C4—88no—
271C4—84no—
272C4—81no—
273C4—82no—
274C4#175912yes—
275C4#196312yes—
276C2—22no—
277C2—22no—
278C2—22no—
279C2—161no—
280C2#1092411yescharacteristic
281C2#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. #13797C2semidirect
4. #8098C2semidirect
5. #7855C2semidirect
11. #1995C4semidirect
12. #1963C4semidirect
17. #1503C4semidirect
18. #1502C4semidirect

Maximal subgroups

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

subgroupindex
#214172
#175462
#137972
#80982
#78552