Finite Group Databasebeta

← #136103

Subgroups — #136103

Order 512.

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

normal subgroups only

This group has 329 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#11477C211yesmaximal, characteristic
3#8961C211yesmaximal, characteristic
4#8860C212yesmaximal
5#7285C212yesmaximal
6#2975C2211yescharacteristic
7C4×#124C2211yescharacteristic
8#1985—21no—
9#1924—22no—
10#1870C2212yes—
11#1855—22no—
12#1851—22no—
13#1831—21no—
14#1479—21no—
15#1460C2211yescharacteristic
16#1459C2212yes—
17#1324C412yes—
18#1319C412yes—
19#1298—22no—
20#1276—21no—
21C22×M16D412yes—
22C2×#124Q811yescharacteristic
23C2×C4×Q8D411yescharacteristic
24C2×C4×D4C2311yescharacteristic
25#436—41no—
26#436—41no—
27#431—21no—
28#431—21no—
29C2×#108—42no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33#407—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8D412yes—
37C2×#97—21no—
38#391C2×C411yescharacteristic
39#390Q811yescharacteristic
40#388—22no—
41#382C2×C411yescharacteristic
42C4×C22⋊C4—22no—
43C43D411yescharacteristic
44#341—42no—
45#336—22no—
46#336—22no—
47#336—22no—
48Q8⋊C8—41no—
49Q8⋊C8—41no—
50#324C2×C414yes—
51#323—22no—
52#323—22no—
53C22×Q8C4⋊C411yescharacteristic
54C22×D4C2×Q811yescharacteristic
55C23×C4C2×D412yes—
56C2×M16—22no—
57C2×M16—22no—
58C2×M16—42no—
59C2×M16—82no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—42no—
63#124—22no—
64#124—41no—
65C4×Q8—41no—
66C4×Q8—41no—
67C4×D4—41no—
68C4×D4—41no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C4⋊C411yescharacteristic
71C2×C4⋊C4C22×C411yescharacteristic
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4#4112yes—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C4⋊C411yescharacteristic
79C2×C42C4⋊C411yescharacteristic
80C2×C42C2×D411yescharacteristic
81#108—84no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84C4⋊C8—41no—
85C4⋊C8—81no—
86C4⋊C8—81no—
87C4⋊C8—81no—
88C22⋊C8—24no—
89C22⋊C8—42no—
90C22⋊C8—42no—
91C22⋊C8—41no—
92C22⋊C8—42no—
93C22⋊C8—41no—
94C22⋊C8—82no—
95#97—81no—
96C4×C8—41no—
97C4×C8—41no—
98#95—22no—
99#95—22no—
100C24C22⋊Q812yes—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—41no—
108C2×D4—41no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C2×C4⋊C411yescharacteristic
118C22×C4C4×D412yes—
119C22×C4#12011yescharacteristic
120C22×C4#12311yescharacteristic
121M16—44no—
122M16—82no—
123M16—82no—
124M16—82no—
125C2×C8—44no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—82no—
135C2×C8—82no—
136C2×C8—81no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—81no—
140C22⋊C4—42no—
141C22⋊C4—44no—
142C22⋊C4—82no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—41no—
147C42—41no—
148C42—41no—
149C42—41no—
150C42—41no—
151C42—81no—
152C42#10812yescomplemented
153C23—22no—
154C23—22no—
155C23—22no—
156C23—42no—
157C23—42no—
158C23#38511yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—81no—
162D4—81no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—82no—
185C2×C4C2×#10811yescharacteristic
186C2×C4#43011yescharacteristic
187C2×C4B(4,𝔽2)11yescharacteristic
188C2×C4#45811yescharacteristic
189C8—82no—
190C8—84no—
191C8—82no—
192C8—81no—
193C8—81no—
194C8—161no—
195C22—21no—
196C22—21no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—82no—
201C22—82no—
202C22#185511yescharacteristic
203C22#185811yescharacteristic
204C22#186711yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—42no—
210C4—41no—
211C4—81no—
212C4—81no—
213C4—81no—
214C4#219911yescharacteristic
215C4#220111yescharacteristic
216C2—21no—
217C2—21no—
218C2—82no—
219C2#886011yescharacteristic
220C2#1296711yescharacteristic
221C2#1297411yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC2×C4×D4C23
socleC22#1867
prime radicalC22×D4C2×Q8
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
152. C42#108semidirect

Maximal subgroups

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

subgroupindex
#233532
#114772
#89612
#88602
#72852