Finite Group Databasebeta

← #126407

Subgroups — #126407

Order 512.

301 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 68 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#31239C211yesmaximal, characteristic
2#13531C211yesmaximal, characteristic
3#13506C211yesmaximal, characteristic
4#10649C211yesmaximal, characteristic
5#7121C211yesmaximal, characteristic
6#7118C211yesmaximal, characteristic
7#7113C211yesmaximal, characteristic
8C2×#477—21no—
9#2451—21no—
10C2×#367—21no—
11C2×#367—21no—
12C2×#358—21no—
13#2068—21no—
14#2068—21no—
15#2057—21no—
16#1969C2211yescharacteristic
17#1947—21no—
18#1942C2211yescharacteristic
19#1939C2211yescharacteristic
20#1878—21no—
21#1845C2211yescharacteristic
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×Q16D411yescharacteristic
28C2×C4⋊Q8—21no—
29C2×C4×Q8—21no—
30C22×C16—21no—
31C22×C16—21no—
32#477—42no—
33#477—42no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107—21no—
37C2×#107—41no—
38C2×#107D411yescharacteristic
39C2×#107D411yescharacteristic
40C2×#107D411yescharacteristic
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×C4×C8C2311yescharacteristic
48#398—41no—
49#395—41no—
50#391—21no—
51#391—21no—
52#391—41no—
53#389—21no—
54#389—21no—
55#387—41no—
56#384—21no—
57C4×C4⋊C4D411yescharacteristic
58#367—42no—
59#367—42no—
60#358—42no—
61#340—21no—
62#340—21no—
63#340—41no—
64#336—21no—
65C22×Q8—21no—
66C22×Q8—41no—
67C2×Q16—21no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×Q16Q1612yes—
71C22×C8—21no—
72C22×C8—41no—
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C4⋊Q8—42no—
76C4⋊Q8—81no—
77C4×Q8—42no—
78C4×Q8—81no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4D811yescharacteristic
90C2×C4⋊C4D811yescharacteristic
91C2×C42—21no—
92C2×C42—41no—
93C2×C42C2×D411yescharacteristic
94C2×C16—22no—
95C2×C16—22no—
96C2×C16—41no—
97C2×C16—41no—
98C2×C16—41no—
99C2×C16—41no—
100#107—21no—
101#107—42no—
102#107—41no—
103#107—41no—
104#107—41no—
105#107—41no—
106#107—41no—
107#107—41no—
108#107—82no—
109#107D812yes—
110C4⋊C8—22no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115#103—42no—
116#103—41no—
117#103—41no—
118#103—81no—
119C4×C8—41no—
120#95—21no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127C2×Q8—22no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—82no—
134C2×Q8—82no—
135C2×Q8—161no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—81no—
143C22×C4#12011yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4AGL(1,ℤ/8)11yescharacteristic
146Q16—42no—
147Q16—81no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—81no—
157C2×C8—81no—
158C2×C8C2×D811yescharacteristic
159C2×C8C2×Q1611yescharacteristic
160C2×C8AGL(1,ℤ/8)11yescharacteristic
161C2×C8#13711yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—161no—
177C4⋊C4—161no—
178C4⋊C4—161no—
179C4⋊C4SD3212yes—
180C42—21no—
181C42—21no—
182C42—41no—
183C42—41no—
184C42—82no—
185C16—42no—
186C16—42no—
187C23#44711yescharacteristic
188Q8—42no—
189Q8—42no—
190Q8—81no—
191Q8—81no—
192Q8—164no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—161no—
217C2×C4—161no—
218C2×C4—161no—
219C2×C4#44711yescharacteristic
220C2×C4#45111yescharacteristic
221C2×C4#45311yescharacteristic
222C2×C4#45611yescharacteristic
223C2×C4#45611yescharacteristic
224C2×C4#45811yescharacteristic
225C2×C4C2×SD3211yescharacteristic
226C2×C4#51011yescharacteristic
227C8—21no—
228C8—21no—
229C8—21no—
230C8—21no—
231C8—82no—
232C8—81no—
233C22#161011yescharacteristic
234C22#161911yescharacteristic
235C22#164911yescharacteristic
236C22#218511yescharacteristic
237C22#219111yescharacteristic
238C22#219311yescharacteristic
239C22#219411yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—42no—
244C4—81no—
245C4—82no—
246C4—82no—
247C4—161no—
248C4—161no—
249C4—321no—
250C4#161011yescharacteristic
251C4#161411yescharacteristic
252C4#164911yescharacteristic
253C4#166011yescharacteristic
254C2#779711yescharacteristic
255C2#1228911yescharacteristic
256C2#1229511yescharacteristic
257C2#1231811yescharacteristic
258C2#1232611yescharacteristic
259C2#1240611yescharacteristic
260C2#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
#312392
#135312
#135062
#106492
#71212
#71182
#71132