Finite Group Databasebeta

← #145251

Subgroups — #145251

Order 512.

263 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×#1676C211yesmaximal, characteristic
2#12033C211yesmaximal, characteristic
3#11972C211yesmaximal, characteristic
4#11972C211yesmaximal, characteristic
5#11637C211yesmaximal, characteristic
6#8617C211yesmaximal, characteristic
7#7288C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#477—21no—
10C2×C8×Q8C2211yescharacteristic
11#2162C2211yescharacteristic
12#2162C2211yescharacteristic
13#1801—21no—
14#1676—22no—
15#1676—22no—
16#1567C2211yescharacteristic
17#1567C2211yescharacteristic
18C2×#335C2211yescharacteristic
19C2×#333—21no—
20#1372—22no—
21#1365—22no—
22#1365—22no—
23Q8⋊C16—22no—
24#1292—21no—
25C2×C4⋊Q8—21no—
26C2×C4×Q8D411yescharacteristic
27#500—21no—
28#500—21no—
29#500—21no—
30#500D412yes—
31#477—24no—
32#477—42no—
33C8×Q8—21no—
34C8×Q8—21no—
35C8×Q8—21no—
36C8×Q8—21no—
37C2×#107—21no—
38C2×#107—21no—
39C2×#107—41no—
40#423—21no—
41#423—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#103—41no—
46C2×#101—41no—
47C2×C4×C8—21no—
48C2×C4×C8C2311yescharacteristic
49C4⋊C16—21no—
50C4⋊C16D412yes—
51C22⋊C16—21no—
52C22⋊C16—21no—
53C4×C16—21no—
54C4×C16—21no—
55#343—41no—
56#335—21no—
57#335—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#335D412yes—
62#333—24no—
63C22×Q8—41no—
64C22×Q8D811yescharacteristic
65C2×M16—41no—
66C2×M16—41no—
67C22×C8—21no—
68C22×C8—41no—
69C22×C8#4111yescharacteristic
70C22×C8#4111yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×Q8—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4D811yescharacteristic
83C2×C42—21no—
84C2×C42C2×D411yescharacteristic
85C2×C16—22no—
86C2×C16—41no—
87#107—22no—
88#107—22no—
89#107—44no—
90#107—41no—
91#107—41no—
92C4⋊C8—21no—
93C4⋊C8—22no—
94C4⋊C8—21no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102#103—44no—
103#101—82no—
104C4×C8—41no—
105C4×C8—41no—
106C4×C8C2×D411yescharacteristic
107C4×C8C2×D411yescharacteristic
108C4×C8C2×D411yescharacteristic
109C4×C8#4111yescharacteristic
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—42no—
116C2×Q8—81no—
117C2×Q8—162no—
118C2×Q8SD3212yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—81no—
122C22×C4#12311yescharacteristic
123C22×C4C2×D811yescharacteristic
124C22×C4#13711yescharacteristic
125M16—81no—
126M16—81no—
127M16—81no—
128M16—81no—
129C2×C8—22no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—42no—
133C2×C8—81no—
134C2×C8—81no—
135C2×C8—81no—
136C2×C8—81no—
137C2×C8—81no—
138C2×C8#12111yescharacteristic
139C2×C8#12111yescharacteristic
140C2×C8#12311yescharacteristic
141C2×C8#12411yescharacteristic
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—81no—
149C4⋊C4—81no—
150C42—41no—
151C42—41no—
152C42#12011yescharacteristic
153C42#12111yescharacteristic
154C42#13511yescharacteristic
155C42AGL(1,ℤ/8)11yescharacteristic
156C16—42no—
157C16—81no—
158C23#48011yescharacteristic
159Q8—22no—
160Q8—22no—
161Q8—41no—
162Q8—41no—
163Q8—162no—
164Q8—162no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—82no—
174C2×C4—162no—
175C2×C4#39411yescharacteristic
176C2×C4#44711yescharacteristic
177C2×C4#45211yescharacteristic
178C2×C4#46611yescharacteristic
179C2×C4#46811yescharacteristic
180C2×C4#48211yescharacteristic
181C2×C4C2×SD3211yescharacteristic
182C2×C4#50811yescharacteristic
183C2×C4#51011yescharacteristic
184C2×C4#51011yescharacteristic
185C8—21no—
186C8—21no—
187C8—21no—
188C8—81no—
189C8—82no—
190C8—161no—
191C8—161no—
192C8#47312yes—
193C22—21no—
194C22—21no—
195C22#203311yescharacteristic
196C22#223511yescharacteristic
197C22#223711yescharacteristic
198C4—21no—
199C4—21no—
200C4—21no—
201C4—42no—
202C4—81no—
203C4—162no—
204C4#204711yescharacteristic
205C4#204911yescharacteristic
206C4#218511yescharacteristic
207C4#218911yescharacteristic
208C4#221711yescharacteristic
209C4#221911yescharacteristic
210C2—21no—
211C2—21no—
212C2#1044611yescharacteristic
213C2#1338211yescharacteristic
214C2#1338911yescharacteristic
subgroupG/H
center, socleC22#2033
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#480
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#16762
#120332
#119722
#116372
#86172
#72882