Finite Group Databasebeta

← #150709

Subgroups — #150709

Order 512.

241 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 52 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#30086C211yesmaximal, characteristic
2#12804C211yesmaximal, characteristic
3#11536C211yesmaximal, characteristic
4#7436C212yesmaximal
5#7413C212yesmaximal
6#2647C411yescharacteristic
7#2327C2211yescharacteristic
8C2×#356—21no—
9#2096C411yescharacteristic
10#2095C2211yescharacteristic
11#2085—21no—
12#2074C411yescharacteristic
13#2074C2211yescharacteristic
14#2043—21no—
15#1865—21no—
16#1863—21no—
17#1840C411yescharacteristic
18#1801—21no—
19#1467—21no—
20#1394C2212yes—
21#1324—21no—
22#1319C2212yes—
23C2×C4⋊Q8D411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26#432—41no—
27C2×#107—21no—
28C2×#106—21no—
29#423—41no—
30C2×#103—21no—
31C2×#103—21no—
32C2×#103—21no—
33C2×#103—21no—
34C2×#101—22no—
35#400—41no—
36#398—41no—
37#391C2×C411yescharacteristic
38#391C2311yescharacteristic
39#389—21no—
40#389—21no—
41#387—41no—
42#387—41no—
43#384C2×C411yescharacteristic
44#382C2×C411yescharacteristic
45#382D411yescharacteristic
46#376C2×C411yescharacteristic
47#356—42no—
48#343—41no—
49#340—21no—
50#340—21no—
51#336—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#324—41no—
57C22×Q8C22⋊C411yescharacteristic
58C22×Q8C2×D411yescharacteristic
59C2×M16—42no—
60C22×C8—21no—
61C22×C8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8—41no—
65C4×Q8—81no—
66C4×Q8—81no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4C22⋊C411yescharacteristic
76C2×C4⋊C4C2×D411yescharacteristic
77C2×C42—21no—
78C2×C42—21no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22×C411yescharacteristic
82#107—81no—
83#106—81no—
84C4⋊C8—82no—
85C4⋊C8—81no—
86#103—42no—
87#103—41no—
88#103—41no—
89#103—81no—
90#103—81no—
91#101—42no—
92#101—42no—
93C22⋊C8—42no—
94#97—81no—
95#95—21no—
96#95—21no—
97#95—41no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102#95—41no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—41no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8—41no—
110C2×Q8AGL(1,ℤ/8)11yescharacteristic
111C2×Q8#13711yescharacteristic
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4#9911yescharacteristic
121C22×C4#9911yescharacteristic
122C22×C4C2×C22⋊C411yescharacteristic
123M16—82no—
124M16—82no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—82no—
132C2×C8—82no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—81no—
142C4⋊C4—81no—
143C4⋊C4—161no—
144C4⋊C4—161no—
145C4⋊C4—161no—
146C42—21no—
147C42—21no—
148C42—41no—
149C42—41no—
150C42—41no—
151C42—42no—
152C42—41no—
153C23C2×#9911yescharacteristic
154Q8—41no—
155Q8—41no—
156Q8—42no—
157Q8—81no—
158Q8—81no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4#41711yescharacteristic
189C2×C4#42111yescharacteristic
190C8—81no—
191C8—81no—
192C8—162no—
193C8—162no—
194C22—21no—
195C22—21no—
196C22#151311yescharacteristic
197C22#212311yescharacteristic
198C22#212711yescharacteristic
199C4—21no—
200C4—21no—
201C4—41no—
202C4—41no—
203C4—41no—
204C4—41no—
205C4—81no—
206C4—81no—
207C4—81no—
208C4—81no—
209C4—81no—
210C4—161no—
211C4—161no—
212C2—21no—
213C2—21no—
214C2#1182711yescharacteristic
215C2#1360611yescharacteristic
216C2#1361211yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC23C2×#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2327 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#300862
#128042
#115362
#74362
#74132