Finite Group Databasebeta

← #136107

Subgroups — #136107

Order 512.

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

normal subgroups only

This group has 310 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#23353C211yesmaximal, characteristic
2#11476C211yesmaximal, characteristic
3#8964C211yesmaximal, characteristic
4#8865C212yesmaximal, complemented
5#7286C212yesmaximal, complemented
6#2983C2211yescharacteristic
7C4×#124C2211yescharacteristic
8#1985C2212yes—
9#1924—22no—
10#1870—21no—
11#1851—22no—
12#1831—21no—
13#1573—22no—
14#1478—21no—
15#1461C2211yescharacteristic
16#1461C2212yes—
17#1324C412yes—
18#1318C412yes—
19#1302—22no—
20#1276—21no—
21C22×M16—21no—
22C2×#124Q811yescharacteristic
23C2×C4×Q8C2311yescharacteristic
24C2×C4×D4D411yescharacteristic
25#446—41no—
26#446—41no—
27#432—22no—
28#431—22no—
29#431—21no—
30#431—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8D412yes—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—21no—
36C2×#97D412yes—
37#391Q811yescharacteristic
38#390C2×C411yescharacteristic
39#388—22no—
40#382C2×C411yescharacteristic
41C4×C22⋊C4—22no—
42C43D411yescharacteristic
43#336—22no—
44#336—22no—
45#336—22no—
46#335—42no—
47#334—42no—
48D4⋊C8—41no—
49D4⋊C8—41no—
50#324—22no—
51#324—22no—
52#324C2×C414yes—
53C22×Q8C2×Q811yescharacteristic
54C22×D4C4⋊C411yescharacteristic
55C23×C4—21no—
56C2×M16—82no—
57C2×M16—81no—
58C22×C8—22no—
59C22×C8—22no—
60#124—22no—
61#124—41no—
62C4×Q8—41no—
63C4×Q8—41no—
64C4×D4—41no—
65C4×D4—41no—
66C2×C4⋊C4C4⋊C411yescharacteristic
67C2×C4⋊C4C22×C411yescharacteristic
68C2×C4⋊C4#4112yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C4⋊C411yescharacteristic
75C2×C42C4⋊C411yescharacteristic
76C2×C42C2×D411yescharacteristic
77C2×C42C2×D412yes—
78C4⋊C8—21no—
79C4⋊C8—21no—
80C4⋊C8—22no—
81C4⋊C8—22no—
82C4⋊C8—42no—
83C4⋊C8—42no—
84C4⋊C8—41no—
85C4⋊C8—81no—
86C4⋊C8—81no—
87C22⋊C8—24no—
88C22⋊C8—41no—
89C22⋊C8—41no—
90C22⋊C8—82no—
91C22⋊C8—81no—
92#97—42no—
93#97—42no—
94C4×C8—42no—
95C4×C8—41no—
96C4×C8—41no—
97#95—22no—
98#95—22no—
99C24—21no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—41no—
103C2×Q8—41no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—41no—
107C2×D4—41no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4C2×C4⋊C411yescharacteristic
116C22×C4C4×D412yes—
117C22×C4#12011yescharacteristic
118C22×C4C22⋊Q812yes—
119C22×C4#12311yescharacteristic
120M16—84no—
121C2×C8—44no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—42no—
125C2×C8—42no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—81no—
131C4⋊C4—41no—
132C4⋊C4—42no—
133C4⋊C4—41no—
134C22⋊C4—44no—
135C22⋊C4—81no—
136C22⋊C4—82no—
137C42—21no—
138C42—21no—
139C42—22no—
140C42—22no—
141C42—21no—
142C42—41no—
143C42—41no—
144C42—41no—
145C42—81no—
146C42#10812yes—
147C23—42no—
148C23—42no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23#38511yescharacteristic
153Q8—81no—
154Q8—81no—
155D4—42no—
156D4—81no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—82no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4C2×#10811yescharacteristic
180C2×C4#43011yescharacteristic
181C2×C4B(4,𝔽2)11yescharacteristic
182C2×C4#45811yescharacteristic
183C8—82no—
184C8—84no—
185C8—81no—
186C8—82no—
187C8—81no—
188C8—161no—
189C22—21no—
190C22—21no—
191C22—82no—
192C22—82no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22#185511yescharacteristic
197C22#185811yescharacteristic
198C22#186711yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—42no—
204C4—42no—
205C4—41no—
206C4—81no—
207C4—81no—
208C4#220211yescharacteristic
209C4#220311yescharacteristic
210C2—21no—
211C2—21no—
212C2—161no—
213C2#886011yescharacteristic
214C2#1296911yescharacteristic
215C2#1297011yescharacteristic
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
#233532
#114762
#89642
#88652
#72862