Finite Group Databasebeta

← #126365

Subgroups — #126365

Order 512.

263 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 62 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#32528C211yesmaximal, characteristic
2#13531C211yesmaximal, characteristic
3#13505C211yesmaximal, characteristic
4#10651C211yesmaximal, characteristic
5#7120C211yesmaximal, characteristic
6#7118C211yesmaximal, characteristic
7#7113C211yesmaximal, characteristic
8#2514—21no—
9C2×#367—21no—
10C2×#366—21no—
11C2×#358—21no—
12#2097—21no—
13#2066—21no—
14#2056—21no—
15#1969C2211yescharacteristic
16#1949—21no—
17#1948—21no—
18#1942C2211yescharacteristic
19#1939C2211yescharacteristic
20#1928—21no—
21#1845C2211yescharacteristic
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×Q16D411yescharacteristic
28C22×C16—21no—
29C22×C16—21no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107D411yescharacteristic
33C2×#107D411yescharacteristic
34C2×#107D411yescharacteristic
35C2×#106—21no—
36C2×#106—41no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#103—21no—
41C2×#103—21no—
42C2×#103—21no—
43C2×C4×C8C2311yescharacteristic
44#401—41no—
45#400—41no—
46#391—21no—
47#391—21no—
48#391—21no—
49#389—21no—
50#387—41no—
51#384—21no—
52#383—41no—
53#382—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4D411yescharacteristic
56#367—42no—
57#366—42no—
58#358—42no—
59#340—21no—
60#340—21no—
61#340—41no—
62#336—21no—
63C22×Q8—21no—
64C2×Q16—21no—
65C2×Q16—21no—
66C2×Q16—41no—
67C2×Q16—41no—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4D811yescharacteristic
80C2×C4⋊C4D811yescharacteristic
81C2×C42—21no—
82C2×C42—41no—
83C2×C42C2×D411yescharacteristic
84C2×C16—22no—
85C2×C16—22no—
86C2×C16—41no—
87C2×C16—41no—
88C2×C16—41no—
89C2×C16—41no—
90#107—21no—
91#107—21no—
92#107—41no—
93#107—41no—
94#107—41no—
95#107—41no—
96#107—41no—
97#107—41no—
98#106—82no—
99#106—81no—
100C4⋊C8—22no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105#103—41no—
106#103—41no—
107#103—81no—
108#103—81no—
109C4×C8—41no—
110#95—21no—
111#95—41no—
112#95—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—41no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—81no—
134C22×C4#12011yescharacteristic
135C22×C4C2×D811yescharacteristic
136C22×C4AGL(1,ℤ/8)11yescharacteristic
137Q16—42no—
138Q16—81no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—42no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8#13511yescharacteristic
150C2×C8#13511yescharacteristic
151C2×C8AGL(1,ℤ/8)11yescharacteristic
152C2×C8#13711yescharacteristic
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—161no—
169C42—21no—
170C42—21no—
171C42—41no—
172C42—41no—
173C42—82no—
174C16—42no—
175C16—42no—
176C23#44711yescharacteristic
177Q8—81no—
178Q8—81no—
179Q8—81no—
180Q8—81no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—161no—
205C2×C4—161no—
206C2×C4—161no—
207C2×C4#44911yescharacteristic
208C2×C4#45211yescharacteristic
209C2×C4#45311yescharacteristic
210C2×C4#45611yescharacteristic
211C2×C4#45611yescharacteristic
212C2×C4#45811yescharacteristic
213C2×C4#50811yescharacteristic
214C2×C4#50911yescharacteristic
215C8—21no—
216C8—21no—
217C8—21no—
218C8—21no—
219C8—82no—
220C8—81no—
221C22#161011yescharacteristic
222C22#162111yescharacteristic
223C22#164711yescharacteristic
224C22#218611yescharacteristic
225C22#219111yescharacteristic
226C22#219311yescharacteristic
227C22#219511yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—81no—
232C4—81no—
233C4—161no—
234C4—161no—
235C4—161no—
236C4—161no—
237C4—321no—
238C4#160911yescharacteristic
239C4#161811yescharacteristic
240C4#165211yescharacteristic
241C4#166011yescharacteristic
242C2#779311yescharacteristic
243C2#1229011yescharacteristic
244C2#1229611yescharacteristic
245C2#1233111yescharacteristic
246C2#1233911yescharacteristic
247C2#1239311yescharacteristic
248C2#1240111yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1939 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#325282
#135312
#135052
#106512
#71202
#71182
#71132