Finite Group Databasebeta

← #126424

Subgroups — #126424

Order 512.

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

normal subgroups only

This group has 255 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#13537C211yesmaximal, characteristic
3#13481C211yesmaximal, characteristic
4#10649C211yesmaximal, characteristic
5#7120C211yesmaximal, characteristic
6#7118C211yesmaximal, characteristic
7#7113C211yesmaximal, characteristic
8#2514—21no—
9C2×#358—21no—
10#2097—21no—
11#2066—21no—
12#2056—21no—
13#1969C2211yescharacteristic
14#1949—21no—
15#1947—21no—
16#1942C2211yescharacteristic
17#1939C2211yescharacteristic
18#1928—21no—
19#1845C2211yescharacteristic
20#1380—21no—
21#1380—21no—
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×#107—41no—
33C2×#107D411yescharacteristic
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#106—21no—
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#358—42no—
57#340—21no—
58#340—21no—
59#340—41no—
60#336—21no—
61C22×Q8—21no—
62C2×Q16—21no—
63C2×Q16—21no—
64C2×Q16—41no—
65C2×Q16—41no—
66C22×C8—21no—
67C22×C8—41no—
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4D811yescharacteristic
78C2×C4⋊C4D811yescharacteristic
79C2×C42—21no—
80C2×C42—41no—
81C2×C42C2×D411yescharacteristic
82C2×C16—41no—
83C2×C16—41no—
84C2×C16—41no—
85C2×C16—41no—
86C2×C16—41no—
87C2×C16—41no—
88#107—21no—
89#107—21no—
90#107—41no—
91#107—41no—
92#107—42no—
93#107—41no—
94#107—41no—
95#107—82no—
96#106—81no—
97C4⋊C8—22no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102#103—41no—
103#103—41no—
104#103—81no—
105#103—81no—
106C4×C8—41no—
107#95—21no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—41no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—81no—
131C22×C4#12011yescharacteristic
132C22×C4C2×D811yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134Q16—42no—
135Q16—81no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—42no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—81no—
145C2×C8—81no—
146C2×C8#13511yescharacteristic
147C2×C8#13511yescharacteristic
148C2×C8AGL(1,ℤ/8)11yescharacteristic
149C2×C8#13711yescharacteristic
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—161no—
166C42—21no—
167C42—21no—
168C42—41no—
169C42—41no—
170C42—82no—
171C16—81no—
172C16—81no—
173C23#44711yescharacteristic
174Q8—81no—
175Q8—81no—
176Q8—81no—
177Q8—81no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
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—161no—
202C2×C4—161no—
203C2×C4—161no—
204C2×C4#44911yescharacteristic
205C2×C4#45211yescharacteristic
206C2×C4#45411yescharacteristic
207C2×C4#45511yescharacteristic
208C2×C4#45611yescharacteristic
209C2×C4#45811yescharacteristic
210C2×C4#50811yescharacteristic
211C2×C4#50911yescharacteristic
212C8—21no—
213C8—21no—
214C8—21no—
215C8—21no—
216C8—82no—
217C8—81no—
218C22#161011yescharacteristic
219C22#162511yescharacteristic
220C22#164111yescharacteristic
221C22#218611yescharacteristic
222C22#219111yescharacteristic
223C22#219311yescharacteristic
224C22#219511yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—81no—
229C4—81no—
230C4—161no—
231C4—161no—
232C4—161no—
233C4—161no—
234C4—321no—
235C4#161211yescharacteristic
236C4#161511yescharacteristic
237C4#165011yescharacteristic
238C4#166411yescharacteristic
239C2#779911yescharacteristic
240C2#1229011yescharacteristic
241C2#1229611yescharacteristic
242C2#1234111yescharacteristic
243C2#1235111yescharacteristic
244C2#1238311yescharacteristic
245C2#1238911yescharacteristic
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
#135372
#134812
#106492
#71202
#71182
#71132