Finite Group Databasebeta

← #237952

Subgroups — #237952

Order 512.

903 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 903 conjugacy classes of subgroups — too many to draw. Only its 182 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
1C2×#2864C211yesmaximal, characteristic, complemented
2#21415C211yesmaximal, characteristic
3#17588C212yesmaximal
4#10369C214yesmaximal, complemented
5#10368C214yesmaximal, complemented
6#7959C212yesmaximal, complemented
7#7867C211yesmaximal, characteristic, complemented
8#3444C2211yescharacteristic
9#2920C2212yes—
10#2920C2211yescharacteristic
11#2864—24no—
12#2864—22no—
13#2864C2214yes—
14C2×#389—22no—
15C2×#387—24no—
16C2×#382C2212yes—
17C2×C43C2211yescharacteristic
18#1994C2212yes—
19#1962C2212yes—
20#1896—22no—
21C4×#104—22no—
22#1518C2214yes—
23#1517C2214yes—
24#1512—24no—
25#1504—24no—
26#1286C2218yes—
27#1276C2214yes—
28#533Q812yes—
29#533C2311yescharacteristic
30#532Q812yes—
31#532C2311yescharacteristic
32#529—22no—
33#529—22no—
34#529—22no—
35#525—21no—
36#525—22no—
37C4×#41—21no—
38C4×#41—22no—
39C22×C4⋊C4—24no—
40C22×C4⋊C4—22no—
41C22×C42—22no—
42C22×C42D412yes—
43C22×C42C2311yescharacteristic
44#432—22no—
45#432—22no—
46#432—22no—
47#431—22no—
48#431—22no—
49#431C2314yes—
50#423—22no—
51#423C2312yes—
52C2×#104—24no—
53#407—22no—
54#407—21no—
55C4×M16—22no—
56C4×M16—21no—
57#389—28no—
58#389—28no—
59#387—216no—
60#387—216no—
61#382—22no—
62#382—22no—
63#382D414yes—
64#382C2314yes—
65C2×#95—24no—
66C2×#95—22no—
67C2×#95D414yes—
68C43—22no—
69C43D412yes—
70C43C2312yes—
71#339—24no—
72#328—28no—
73#327—28no—
74C2×#41—22no—
75C2×#41—21no—
76C23×C4—22no—
77C23×C4—24no—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D412yes—
80C2×M16—22no—
81C2×M16—42no—
82C4⋊Q8C2×Q812yes—
83C4⋊Q8C2×Q811yescharacteristic
84#127C2×Q811yescharacteristic
85#126—42no—
86#125—22no—
87#125—21no—
88#124—22no—
89#124—21no—
90#123—41no—
91#123—42no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95#121—22no—
96#121—22no—
97#121—22no—
98#120—41no—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—42no—
106C4×D4—42no—
107C4×D4—42no—
108#117—22no—
109#117—21no—
110#117—41no—
111#117—42no—
112C2×C4⋊C4—216no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—416no—
116C2×C4⋊C4—48no—
117C2×C42—22no—
118C2×C42—24no—
119C2×C42—24no—
120C2×C42—44no—
121C2×C42—48no—
122C2×C42—41no—
123C2×C42C2×D414yes—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D412yes—
126C2×C42C2×Q812yes—
127C2×C42#4112yes—
128C2×C42#4112yes—
129C2×C42#4112yes—
130C2×C42C2411yescharacteristic
131C4⋊C8—24no—
132C4⋊C8—24no—
133C4⋊C8—42no—
134#104—48no—
135C22⋊C8—24no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#97—42no—
139#97—41no—
140C4×C8—24no—
141C4×C8—41no—
142#95—216no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95C2×D418yes—
147C24#12011yescharacteristic
148#41—44no—
149#41—84no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×D4—21no—
153C2×D4—22no—
154C2×D4—41no—
155C2×D4—42no—
156C22×C4—24no—
157C22×C4—28no—
158C22×C4—24no—
159C22×C4—28no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—416no—
164C22×C4—48no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4#12011yescharacteristic
168C22×C4#12012yes—
169C22×C4C22⋊Q812yes—
170C22×C4C22⋊Q812yes—
171C22×C4C22⋊Q812yes—
172C22×C4#12412yes—
173C22×C4#12412yes—
174C22×C4#12412yes—
175C22×C4C22×D412yes—
176C22×C4C22×D411yescharacteristic
177M16—44no—
178M16—84no—
179C2×C8—44no—
180C2×C8—42no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—48no—
185C4⋊C4—432no—
186C4⋊C4—42no—
187C4⋊C4—48no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C22⋊C4—41no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C42—24no—
196C42—24no—
197C42—44no—
198C42—48no—
199C42—48no—
200C42—41no—
201C42—42no—
202C42C22⋊Q812yes—
203C42C22⋊Q812yes—
204C42C22⋊Q812yes—
205C42#12411yescharacteristic
206C42#12412yes—
207C42#12411yescharacteristic
208C42#12412yes—
209C42C22×Q811yescharacteristic
210C42C2×#4111yescharacteristic
211C42C2×#4112yes—
212C23—22no—
213C23—24no—
214C23—41no—
215C23#39312yes—
216C23C2×#12011yescharacteristic
217Q8—42no—
218Q8—82no—
219D4—42no—
220D4—82no—
221D4—82no—
222D4—84no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—28no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—416no—
235C2×C4—44no—
236C2×C4—416no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—48no—
240C2×C4—84no—
241C2×C4—816no—
242C2×C4—82no—
243C2×C4#39312yes—
244C2×C4#39312yes—
245C2×C4#39312yes—
246C2×C4C2×C22⋊Q812yes—
247C2×C4C2×C22⋊Q811yescharacteristic
248C2×C4C2×#12412yes—
249C2×C4C2×#12411yescharacteristic
250C2×C4#57514yes—
251C8—84no—
252C8—82no—
253C22—28no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—81no—
258C22—82no—
259C22#203114yes—
260C22C2×#39311yescharacteristic
261C22#300512yes—
262C4—22no—
263C4—22no—
264C4—22no—
265C4—22no—
266C4—22no—
267C4—48no—
268C4—88no—
269C4—81no—
270C4—82no—
271C4#308012yes—
272C4#314212yes—
273C2—22no—
274C2—24no—
275C2—161no—
276C2#2345311yescharacteristic
277C2C2×#203112yes—
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2864C2semidirect
4. #10369C2semidirect
5. #10368C2semidirect
6. #7959C2semidirect
7. #7867C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28642
#214152
#175882
#103692
#103682
#79592
#78672