Finite Group Databasebeta

← #125305

Subgroups — #125305

Order 512.

288 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 94 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#33257C211yesmaximal, characteristic
2#12590C211yesmaximal, characteristic
3#8273C211yesmaximal, characteristic
4#7121C212yesmaximal
5#7120C212yesmaximal
6#2864—21no—
7C2×C4×C16—21no—
8#1942C411yescharacteristic
9#1942C411yescharacteristic
10#1942C2211yescharacteristic
11#1942C2211yescharacteristic
12#1941—21no—
13#1941—21no—
14#1846C411yescharacteristic
15#1846C411yescharacteristic
16#1835—21no—
17#1767—21no—
18C42×C8C2211yescharacteristic
19C2×#334—21no—
20#1374—21no—
21#1374C2212yes—
22#1277C2212yes—
23C22×C16—22no—
24C2×#107C2×C411yescharacteristic
25C2×#107C2×C411yescharacteristic
26C2×#107C2×C411yescharacteristic
27C2×#107C2×C411yescharacteristic
28C2×#106—21no—
29C2×#106—21no—
30C2×C4⋊C8—22no—
31C2×#97—41no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38#389—21no—
39#389—21no—
40#389—21no—
41#389—21no—
42#387—41no—
43#387—41no—
44#387—41no—
45#387—41no—
46#382—21no—
47#382—21no—
48C43D411yescharacteristic
49C4×C16—24no—
50#340—21no—
51#340—21no—
52#340—21no—
53#340—21no—
54#336—41no—
55#334—24no—
56C22×C8—42no—
57C22×C8C22⋊C411yescharacteristic
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22×C411yescharacteristic
60C22×C8C2×D411yescharacteristic
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C42C22⋊C411yescharacteristic
70C2×C42C22⋊C411yescharacteristic
71C2×C42D811yescharacteristic
72C2×C42D811yescharacteristic
73C2×C42SD1611yescharacteristic
74C2×C42SD1611yescharacteristic
75C2×C42C2×D411yescharacteristic
76C2×C16—24no—
77C2×C16—42no—
78C2×C16—42no—
79#107—22no—
80#107—22no—
81#107—41no—
82#107—41no—
83#106—42no—
84#106—42no—
85C4⋊C8—24no—
86C4⋊C8—42no—
87#97—82no—
88C4×C8—22no—
89C4×C8—22no—
90C4×C8—22no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—41no—
94#95—41no—
95#95—41no—
96#95—41no—
97#95—41no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4#10211yescharacteristic
107C22×C4#10211yescharacteristic
108C22×C4#10211yescharacteristic
109C22×C4#10211yescharacteristic
110C22×C4C2×C22⋊C411yescharacteristic
111C22×C4C2×D811yescharacteristic
112C22×C4C2×SD1611yescharacteristic
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—44no—
121C2×C8—82no—
122C2×C8—82no—
123C2×C8#10112yes—
124C2×C8#10412yes—
125C2×C8#10412yes—
126C2×C8#13112yes—
127C2×C8#13712yes—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—82no—
137C4⋊C4—82no—
138C4⋊C4—82no—
139C4⋊C4—82no—
140C42—22no—
141C42—22no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—22no—
149C42—41no—
150C16—44no—
151C23C2×#10211yescharacteristic
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4#35912yes—
180C2×C4#36012yes—
181C2×C4#40811yescharacteristic
182C2×C4#41011yescharacteristic
183C2×C4C2×#10111yescharacteristic
184C2×C4#41911yescharacteristic
185C2×C4C2×#10411yescharacteristic
186C2×C4C2×#10411yescharacteristic
187C2×C4#50812yes—
188C8—22no—
189C8—22no—
190C8—41no—
191C8—84no—
192C8#33014yes—
193C22#149211yescharacteristic
194C22#150211yescharacteristic
195C22#154611yescharacteristic
196C22#213811yescharacteristic
197C22C2×#35911yescharacteristic
198C22C2×#36011yescharacteristic
199C22#214811yescharacteristic
200C4—22no—
201C4—22no—
202C4—22no—
203C4—41no—
204C4—41no—
205C4—41no—
206C4—161no—
207C4—161no—
208C4—161no—
209C4—161no—
210C4#147011yescharacteristic
211C4#148511yescharacteristic
212C4#150611yescharacteristic
213C4C2×#33011yescharacteristic
214C2#768011yescharacteristic
215C2#1167311yescharacteristic
216C2#1167711yescharacteristic
217C2#1174111yescharacteristic
218C2#1174911yescharacteristic
219C2#1196811yescharacteristic
220C2#1197311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ 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
#332572
#125902
#82732
#71212
#71202