Finite Group Databasebeta

← #126304

Subgroups — #126304

Order 512.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 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#32653C211yesmaximal, characteristic
2#13538C211yesmaximal, characteristic
3#13481C211yesmaximal, characteristic
4#10618C211yesmaximal, characteristic
5#7128C211yesmaximal, characteristic
6#7120C211yesmaximal, characteristic
7#7113C211yesmaximal, characteristic
8#2581—21no—
9C2×#358—21no—
10#2097—21no—
11#2085—21no—
12#2066—21no—
13#1984C2211yescharacteristic
14#1969C2211yescharacteristic
15#1942C2211yescharacteristic
16#1878—21no—
17#1863—21no—
18#1845C2211yescharacteristic
19#1803—21no—
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×Q16D411yescharacteristic
28C2×C4×Q8—21no—
29C2×C4×Q8D411yescharacteristic
30C22×C16—21no—
31C22×C16—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#106—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#103—21no—
41C2×#103—21no—
42C2×#103—21no—
43C2×#103—41no—
44C2×C4×C8C2311yescharacteristic
45#400—41no—
46#395—41no—
47#391—21no—
48#391—21no—
49#389—21no—
50#389—21no—
51#387—41no—
52#387—41no—
53#384—21no—
54#382—21no—
55#358—42no—
56#340—21no—
57#340—21no—
58#340—41no—
59#336—21no—
60#336—21no—
61C22×Q8—21no—
62C22×Q8D811yescharacteristic
63C2×Q16—21no—
64C2×Q16—41no—
65C2×Q16—41no—
66C2×Q16SD1612yes—
67C22×C8—21no—
68C22×C8—41no—
69C22×C8C2×D411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C4×Q8—41no—
72C4×Q8—41no—
73C4×Q8—81no—
74C4×Q8—81no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4D811yescharacteristic
86C2×C42—21no—
87C2×C42—41no—
88C2×C42C2×D411yescharacteristic
89C2×C16—41no—
90C2×C16—41no—
91C2×C16—41no—
92C2×C16—41no—
93C2×C16—41no—
94C2×C16—41no—
95#107—21no—
96#107—21no—
97#107—41no—
98#107—41no—
99#107—42no—
100#107—41no—
101#107—41no—
102#106—81no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106#103—41no—
107#103—41no—
108#103—81no—
109#103—81no—
110#103—82no—
111C4×C8—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—
120C2×Q8—22no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—81no—
137C22×C4#12011yescharacteristic
138C22×C4C2×D811yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140Q16—42no—
141Q16—81no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8C2×SD1611yescharacteristic
153C2×C8#13511yescharacteristic
154C2×C8#13711yescharacteristic
155C2×C8#13711yescharacteristic
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—161no—
173C42—21no—
174C42—21no—
175C42—41no—
176C42—41no—
177C42—82no—
178C16—81no—
179C16—81no—
180C23#44711yescharacteristic
181Q8—42no—
182Q8—42no—
183Q8—42no—
184Q8—81no—
185Q8—81no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—161no—
211C2×C4—161no—
212C2×C4—161no—
213C2×C4#44811yescharacteristic
214C2×C4#45211yescharacteristic
215C2×C4#45411yescharacteristic
216C2×C4#45511yescharacteristic
217C2×C4#45511yescharacteristic
218C2×C4#45811yescharacteristic
219C2×C4#50811yescharacteristic
220C2×C4#51011yescharacteristic
221C8—21no—
222C8—21no—
223C8—21no—
224C8—21no—
225C8—81no—
226C8—161no—
227C22#160711yescharacteristic
228C22#162711yescharacteristic
229C22#164111yescharacteristic
230C22#218911yescharacteristic
231C22#219011yescharacteristic
232C22#219311yescharacteristic
233C22#219511yescharacteristic
234C4—21no—
235C4—21no—
236C4—41no—
237C4—82no—
238C4—81no—
239C4—81no—
240C4—161no—
241C4—161no—
242C4—161no—
243C4—321no—
244C4#160811yescharacteristic
245C4#161511yescharacteristic
246C4#164311yescharacteristic
247C4#166211yescharacteristic
248C2#778711yescharacteristic
249C2#1228311yescharacteristic
250C2#1228611yescharacteristic
251C2#1235711yescharacteristic
252C2#1236111yescharacteristic
253C2#1238711yescharacteristic
254C2#1239111yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984 ◁ 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
#326532
#135382
#134812
#106182
#71282
#71202
#71132