Finite Group Databasebeta

← #136101

Subgroups — #136101

Order 512.

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

normal subgroups only

This group has 344 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#23272C211yesmaximal, characteristic
2#11477C211yesmaximal, characteristic
3#8961C211yesmaximal, characteristic
4#8860C212yesmaximal
5#7285C212yesmaximal
6#2975C2211yescharacteristic
7C4×#124C2211yescharacteristic
8#1985—21no—
9#1923—22no—
10#1870C2212yes—
11#1855—22no—
12#1753—22no—
13#1744—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—42no—
26#431—21no—
27#431—21no—
28C2×#108—42no—
29C2×C4⋊C8—21no—
30#407—22no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8D412yes—
34C2×#97—21no—
35C2×C4×C8—22no—
36#391C2×C411yescharacteristic
37#390Q811yescharacteristic
38#388—22no—
39#382C2×C411yescharacteristic
40C4×C22⋊C4—22no—
41C43D411yescharacteristic
42#341—42no—
43#336—21no—
44#336—22no—
45#336—21no—
46#336—22no—
47Q8⋊C8—42no—
48#324C2×C414yes—
49#323—22no—
50#323—22no—
51C22×Q8C4⋊C411yescharacteristic
52C22×D4C2×Q811yescharacteristic
53C23×C4C2×D412yes—
54C2×M16—22no—
55C2×M16—22no—
56C2×M16—44no—
57C2×M16—42no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—42no—
61#124—22no—
62#124—41no—
63C4×Q8—41no—
64C4×Q8—41no—
65C4×D4—41no—
66C4×D4—41no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4C4⋊C411yescharacteristic
69C2×C4⋊C4C22×C411yescharacteristic
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4#4112yes—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C4⋊C411yescharacteristic
77C2×C42C4⋊C411yescharacteristic
78C2×C42C2×D411yescharacteristic
79#108—84no—
80C4⋊C8—22no—
81C4⋊C8—41no—
82C4⋊C8—81no—
83C22⋊C8—24no—
84C22⋊C8—44no—
85C22⋊C8—42no—
86C22⋊C8—42no—
87C22⋊C8—41no—
88C22⋊C8—42no—
89C22⋊C8—41no—
90#97—81no—
91C4×C8—41no—
92C4×C8—41no—
93C4×C8—44no—
94#95—22no—
95#95—22no—
96C24C22⋊Q812yes—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—41no—
104C2×D4—41no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4C2×C4⋊C411yescharacteristic
114C22×C4C4×D412yes—
115C22×C4#12011yescharacteristic
116C22×C4#12311yescharacteristic
117M16—44no—
118M16—82no—
119M16—82no—
120M16—82no—
121C2×C8—24no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—42no—
125C2×C8—44no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—82no—
130C2×C8—82no—
131C2×C8—81no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—81no—
135C22⋊C4—42no—
136C22⋊C4—44no—
137C22⋊C4—44no—
138C42—21no—
139C42—21no—
140C42—21no—
141C42—41no—
142C42—42no—
143C42—42no—
144C42—41no—
145C42—41no—
146C42#10812yescomplemented
147C23—22no—
148C23—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23#38511yescharacteristic
153Q8—42no—
154Q8—81no—
155D4—81no—
156D4—81no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—82no—
178C2×C4C2×#10811yescharacteristic
179C2×C4#43011yescharacteristic
180C2×C4B(4,𝔽2)11yescharacteristic
181C2×C4#45811yescharacteristic
182C8—44no—
183C8—82no—
184C8—84no—
185C8—82no—
186C8—161no—
187C22—21no—
188C22—21no—
189C22—42no—
190C22—42no—
191C22—42no—
192C22—82no—
193C22—82no—
194C22#185511yescharacteristic
195C22#185811yescharacteristic
196C22#186711yescharacteristic
197C4—21no—
198C4—21no—
199C4—21no—
200C4—42no—
201C4—42no—
202C4—42no—
203C4—41no—
204C4—81no—
205C4—81no—
206C4#219712yes—
207C2—21no—
208C2—21no—
209C2—82no—
210C2#886011yescharacteristic
211C2#1296511yescharacteristic
212C2#1297511yescharacteristic
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
146. C42#108semidirect

Maximal subgroups

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

subgroupindex
#232722
#114772
#89612
#88602
#72852