Finite Group Databasebeta

← #126076

Subgroups — #126076

Order 512.

307 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 66 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#31236C211yesmaximal, characteristic
2#13452C211yesmaximal, characteristic
3#13364C211yesmaximal, characteristic
4#10543C211yesmaximal, characteristic
5#7120C211yesmaximal, characteristic
6#7118C211yesmaximal, characteristic
7#7113C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#479—21no—
10#2453—21no—
11C2×#367—21no—
12C2×#358—21no—
13#2068—21no—
14#2066—21no—
15#2059—21no—
16#1987—21no—
17#1969C2211yescharacteristic
18#1942C2211yescharacteristic
19#1939C2211yescharacteristic
20#1878—21no—
21C2×#333—21no—
22#1380—21no—
23#1380—21no—
24#1374C2211yescharacteristic
25#1374C2211yescharacteristic
26#1277C2211yescharacteristic
27C22×Q16D411yescharacteristic
28C2×C4⋊Q8—21no—
29C2×#125—21no—
30C2×C4×Q8—21no—
31C22×C16—21no—
32C22×C16—21no—
33#500—42no—
34#479—42no—
35#479—42no—
36C2×#107—21no—
37C2×#107—21no—
38C2×#107D411yescharacteristic
39C2×#107D411yescharacteristic
40C2×#107D411yescharacteristic
41C2×#106—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#103—21no—
46C2×#103—21no—
47C2×#103—21no—
48C2×C4×C8C2311yescharacteristic
49#400—41no—
50#398—41no—
51#391—21no—
52#391—21no—
53#389—21no—
54#389—21no—
55#389—41no—
56#387—41no—
57C4×C4⋊C4D411yescharacteristic
58#367—42no—
59#358—42no—
60#340—21no—
61#340—21no—
62#340—41no—
63#340—41no—
64#336—21no—
65#333—42no—
66C22×Q8—21no—
67C22×Q8—41no—
68C2×Q16—21no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×Q16Q1612yes—
72C22×C8—21no—
73C22×C8—41no—
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—41no—
78C4⋊Q8—41no—
79C4⋊Q8—41no—
80#125—42no—
81#125—81no—
82C4×Q8—42no—
83C4×Q8—81no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4D811yescharacteristic
95C2×C4⋊C4D811yescharacteristic
96C2×C42—21no—
97C2×C42—41no—
98C2×C42C2×D411yescharacteristic
99C2×C16—22no—
100C2×C16—41no—
101C2×C16—41no—
102C2×C16—41no—
103C2×C16—41no—
104C2×C16—41no—
105#107—21no—
106#107—22no—
107#107—22no—
108#107—41no—
109#107—41no—
110#107—41no—
111#107—41no—
112#107SD1612yes—
113#106—42no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#103—42no—
120#103—41no—
121#103—41no—
122#103—81no—
123C4×C8—41no—
124#95—21no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
131C2×Q8—22no—
132C2×Q8—42no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—81no—
139C2×Q8—161no—
140C2×Q8—161no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—81no—
148C22×C4#12011yescharacteristic
149C22×C4C2×D811yescharacteristic
150C22×C4AGL(1,ℤ/8)11yescharacteristic
151Q16—42no—
152Q16—81no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—81no—
161C2×C8—81no—
162C2×C8—81no—
163C2×C8C2×SD1611yescharacteristic
164C2×C8C2×Q1611yescharacteristic
165C2×C8AGL(1,ℤ/8)11yescharacteristic
166C2×C8#13711yescharacteristic
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C42—21no—
187C42—21no—
188C42—41no—
189C42—41no—
190C42—82no—
191C16—42no—
192C16—81no—
193C23#44711yescharacteristic
194Q8—42no—
195Q8—42no—
196Q8—81no—
197Q8—81no—
198Q8—162no—
199Q8—162no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—82no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—161no—
225C2×C4—161no—
226C2×C4#45011yescharacteristic
227C2×C4#45111yescharacteristic
228C2×C4#45411yescharacteristic
229C2×C4#45511yescharacteristic
230C2×C4#45611yescharacteristic
231C2×C4B(4,𝔽2)11yescharacteristic
232C2×C4#50811yescharacteristic
233C2×C4#51011yescharacteristic
234C8—21no—
235C8—21no—
236C8—21no—
237C8—21no—
238C8—81no—
239C8—161no—
240C22#159811yescharacteristic
241C22#163911yescharacteristic
242C22#164911yescharacteristic
243C22#218911yescharacteristic
244C22#219111yescharacteristic
245C22#219311yescharacteristic
246C22#219311yescharacteristic
247C4—21no—
248C4—21no—
249C4—41no—
250C4—82no—
251C4—81no—
252C4—82no—
253C4—81no—
254C4—161no—
255C4—161no—
256C4—162no—
257C4#160211yescharacteristic
258C4#160411yescharacteristic
259C4#163711yescharacteristic
260C4#166411yescharacteristic
261C2#776411yescharacteristic
262C2#1216311yescharacteristic
263C2#1216511yescharacteristic
264C2#1237511yescharacteristic
265C2#1237711yescharacteristic
266C2#1240911yescharacteristic
267C2#1241311yescharacteristic
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
#312362
#134522
#133642
#105432
#71202
#71182
#71132