Finite Group Databasebeta

← #136094

Subgroups — #136094

Order 512.

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

normal subgroups only

This group has 323 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#23273C211yesmaximal, characteristic
2#11476C211yesmaximal, characteristic
3#8964C211yesmaximal, characteristic
4#8865C212yesmaximal, complemented
5#7286C212yesmaximal, complemented
6#2983C2211yescharacteristic
7C4×#124C2211yescharacteristic
8#1985C2212yes—
9#1924—22no—
10#1869—21no—
11#1752—22no—
12#1745—21no—
13#1573—22no—
14#1478—21no—
15#1461C2211yescharacteristic
16#1461C2212yes—
17#1324C412yes—
18#1318C412yes—
19#1302—22no—
20#1276—21no—
21C23×C8—21no—
22C2×#124Q811yescharacteristic
23C2×C4×Q8C2311yescharacteristic
24C2×C4×D4D411yescharacteristic
25#446—42no—
26#432—22no—
27#431—22no—
28#431—21no—
29#431—21no—
30C2×C4⋊C8D412yes—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—21no—
33C2×#97—22no—
34C2×#97D412yes—
35#391Q811yescharacteristic
36#390C2×C411yescharacteristic
37#388—22no—
38#382C2×C411yescharacteristic
39C4×C22⋊C4—22no—
40C43D411yescharacteristic
41#336—21no—
42#336—21no—
43#336—22no—
44#336—22no—
45#335—42no—
46#334—42no—
47D4⋊C8—42no—
48#324—22no—
49#324—22no—
50#324C2×C414yes—
51C22×Q8C2×Q811yescharacteristic
52C22×D4C4⋊C411yescharacteristic
53C23×C4—21no—
54C22×C8—22no—
55C22×C8—22no—
56C22×C8—44no—
57C22×C8—81no—
58#124—22no—
59#124—41no—
60C4×Q8—41no—
61C4×Q8—41no—
62C4×D4—41no—
63C4×D4—41no—
64C2×C4⋊C4C4⋊C411yescharacteristic
65C2×C4⋊C4C22×C411yescharacteristic
66C2×C4⋊C4#4112yes—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42C4⋊C411yescharacteristic
73C2×C42C4⋊C411yescharacteristic
74C2×C42C2×D411yescharacteristic
75C2×C42C2×D412yes—
76C4⋊C8—22no—
77C4⋊C8—22no—
78C4⋊C8—22no—
79C4⋊C8—42no—
80C4⋊C8—42no—
81C4⋊C8—41no—
82C22⋊C8—24no—
83C22⋊C8—44no—
84C22⋊C8—41no—
85C22⋊C8—41no—
86C22⋊C8—81no—
87#97—42no—
88#97—42no—
89#97—44no—
90C4×C8—42no—
91C4×C8—41no—
92C4×C8—41no—
93#95—22no—
94#95—22no—
95C24—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—41no—
99C2×Q8—41no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4#10812yes—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4C2×C4⋊C411yescharacteristic
112C22×C4C4×D412yes—
113C22×C4#12011yescharacteristic
114C22×C4C22⋊Q812yes—
115C22×C4#12311yescharacteristic
116C2×C8—24no—
117C2×C8—42no—
118C2×C8—42no—
119C2×C8—44no—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—84no—
125C2×C8—81no—
126C4⋊C4—41no—
127C4⋊C4—42no—
128C4⋊C4—41no—
129C22⋊C4—44no—
130C22⋊C4—81no—
131C22⋊C4—82no—
132C42—21no—
133C42—22no—
134C42—22no—
135C42—21no—
136C42—21no—
137C42—21no—
138C42—41no—
139C42—42no—
140C42—81no—
141C23—22no—
142C23—42no—
143C23—42no—
144C23—41no—
145C23—41no—
146C23#38511yescharacteristic
147Q8—81no—
148Q8—81no—
149D4—42no—
150D4—81no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—82no—
172C2×C4C2×#10811yescharacteristic
173C2×C4#43011yescharacteristic
174C2×C4B(4,𝔽2)11yescharacteristic
175C2×C4#45811yescharacteristic
176C8—44no—
177C8—84no—
178C8—82no—
179C8—82no—
180C8—161no—
181C22—21no—
182C22—21no—
183C22—44no—
184C22—42no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22#185511yescharacteristic
189C22#185811yescharacteristic
190C22#186711yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—42no—
195C4—42no—
196C4—41no—
197C4—81no—
198C4—81no—
199C4—81no—
200C4#220412yes—
201C2—21no—
202C2—21no—
203C2—82no—
204C2#885911yescharacteristic
205C2#1296911yescharacteristic
206C2#1297211yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC2×C4×Q8C23
socleC22#1867
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #8865C2semidirect
5. #7286C2semidirect

Maximal subgroups

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

subgroupindex
#232732
#114762
#89642
#88652
#72862