Finite Group Databasebeta

← #137534

Subgroups — #137534

Order 512.

331 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 92 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#33731C211yesmaximal, characteristic
2#23162C211yesmaximal, characteristic
3#13278C211yesmaximal, characteristic
4#11940C211yesmaximal, characteristic
5#9752C211yesmaximal, characteristic
6#7174C211yesmaximal, characteristic
7#7168C211yesmaximal, characteristic
8#3160C2211yescharacteristic
9C2×#500C2211yescharacteristic
10#2369—21no—
11#2255—22no—
12#2254C412yescomplemented
13#2252C412yescomplemented
14#2159—22no—
15#2150C2211yescharacteristic
16C4×M32C2211yescharacteristic
17#1987—21no—
18#1943—21no—
19#1918—22no—
20#1776—21no—
21C4×#106C2211yescharacteristic
22#1770C2211yescharacteristic
23#1564C2211yescharacteristic
24#1503—21no—
25#1481—21no—
26#1386—21no—
27#1386—21no—
28#1364—22no—
29#533—21no—
30C2×C4⋊Q8—21no—
31C2×M32—21no—
32C2×M32—21no—
33#500—21no—
34#500—21no—
35#500C2×C411yescharacteristic
36#500C2×C411yescharacteristic
37#500C2×C412yes—
38#500D412yes—
39#499—21no—
40#499—21no—
41#498—21no—
42#431—21no—
43#427—21no—
44#427—21no—
45C2×#107—21no—
46C2×#107D412yes—
47C2×#106D411yescharacteristic
48C2×#106D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×#97—21no—
51C2×C4×C8C2311yescharacteristic
52#391—42no—
53#389—22no—
54#389—41no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—41no—
58C4×C4⋊C4D411yescharacteristic
59#367—24no—
60#366—24no—
61C4⋊C16—21no—
62C4⋊C16—21no—
63#346—21no—
64C4×C16C2×C412yes—
65#340—22no—
66#340—22no—
67#340—21no—
68#336—21no—
69#335—21no—
70#334—21no—
71#328—42no—
72Q8⋊C8—42no—
73C22×Q8—41no—
74C22×C8—21no—
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C4⋊Q8—22no—
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—22no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83#125—41no—
84C22⋊Q8—41no—
85C22⋊Q8—41no—
86C4×Q8—41no—
87C4×Q8—41no—
88#117—41no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C42—21no—
96C2×C42—41no—
97C2×C42—42no—
98C2×C42C2×D411yescharacteristic
99M32—22no—
100M32—22no—
101M32—41no—
102M32—41no—
103C2×C16—22no—
104C2×C16—41no—
105#107—22no—
106#107—22no—
107#107—22no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107—21no—
112#107C22⋊C414yes—
113#106—21no—
114#106—21no—
115#106—21no—
116#106—41no—
117#106—41no—
118#106Q1612yes—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8—41no—
122C22⋊C8—41no—
123#97—41no—
124#97—41no—
125C4×C8—41no—
126C4×C8C22×C411yescharacteristic
127C4×C8C2×D411yescharacteristic
128C4×C8#4111yescharacteristic
129C4×C8#4111yescharacteristic
130#95—42no—
131#95—42no—
132#95—41no—
133#95D812yes—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—161no—
139C2×Q8—161no—
140C22×C4—22no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4#12011yescharacteristic
145C22×C4C2×D811yescharacteristic
146C22×C4AGL(1,ℤ/8)11yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—81no—
153C2×C8C2×C22⋊C411yescharacteristic
154C2×C8C4×D411yescharacteristic
155C2×C8#12111yescharacteristic
156C2×C8#12311yescharacteristic
157C2×C8C2×Q1611yescharacteristic
158C2×C8#13511yescharacteristic
159C2×C8#13711yescharacteristic
160C2×C8#13711yescharacteristic
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—162no—
177C22⋊C4—81no—
178C42—41no—
179C42—41no—
180C42—82no—
181C42—81no—
182C42—162no—
183C42C4×D411yescharacteristic
184C42#12111yescharacteristic
185C42#13511yescharacteristic
186C42AGL(1,ℤ/8)11yescharacteristic
187C16—24no—
188C16—81no—
189C23#44711yescharacteristic
190Q8—82no—
191Q8—162no—
192Q8—162no—
193Q8—161no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4#38611yescharacteristic
211C2×C4C4×D811yescharacteristic
212C2×C4#44211yescharacteristic
213C2×C4#44911yescharacteristic
214C2×C4#45111yescharacteristic
215C2×C4#45211yescharacteristic
216C2×C4#45611yescharacteristic
217C2×C4#45611yescharacteristic
218C2×C4#45911yescharacteristic
219C2×C4#46311yescharacteristic
220C8—21no—
221C8—21no—
222C8—81no—
223C8—161no—
224C8C4×Q1611yescharacteristic
225C8#44111yescharacteristic
226C8#46211yescharacteristic
227C8#46411yescharacteristic
228C22—21no—
229C22—21no—
230C22#164711yescharacteristic
231C22#164911yescharacteristic
232C22#187911yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—82no—
237C4—82no—
238C4—84no—
239C4—161no—
240C4—161no—
241C4—161no—
242C4#164811yescharacteristic
243C4#165011yescharacteristic
244C4#187811yescharacteristic
245C4#188611yescharacteristic
246C4#218012yes—
247C2—21no—
248C2—21no—
249C2#905811yescharacteristic
250C2#1303311yescharacteristic
251C2#1303811yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#447
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2254C4semidirect
13. #2252C4semidirect

Maximal subgroups

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

subgroupindex
#337312
#231622
#132782
#119402
#97522
#71742
#71682