Finite Group Databasebeta

← #128605

Subgroups — #128605

Order 512.

351 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 118 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#21292C211yesmaximal, characteristic
2C4×#346C211yesmaximal, characteristic, complemented
3#11597C212yesmaximal
4#8267C211yesmaximal, characteristic, complemented
5#7177C212yesmaximal, complemented
6C4×#135C2211yescharacteristic
7#2922—21no—
8C2×#346—21no—
9C2×#346C2212yes—
10C2×C4×C16—21no—
11#1974C411yescharacteristic
12#1974C411yescharacteristic
13#1945C411yescharacteristic
14#1945C411yescharacteristic
15#1937—21no—
16#1937—21no—
17C4×#108C2211yescharacteristic
18C4×#104—21no—
19C42×C8C2211yescharacteristic
20#1564C2212yes—
21#1452—21no—
22#1334C414yescomplemented
23#1333C414yescomplemented
24#1276—21no—
25C2×#135C2×C411yescharacteristic
26C4×#41—21no—
27C22×C16—22no—
28C4×Q16C2×C412yes—
29C4×SD16—22no—
30C4×D8C2×C412yes—
31#433—42no—
32#431—22no—
33#431—42no—
34C2×#108C2×C411yescharacteristic
35C2×#108C2×C411yescharacteristic
36#427C2×C411yescharacteristic
37#424—21no—
38#424—21no—
39C2×#104—21no—
40C2×#104—21no—
41#416—21no—
42#408—21no—
43#408—21no—
44C4×M16—21no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8C2311yescharacteristic
51C43D411yescharacteristic
52#346—24no—
53#346—22no—
54#346C2×C414yes—
55C4×C16—24no—
56#341—21no—
57#339—21no—
58#339—21no—
59#335—22no—
60#334C2×C414yes—
61#322—42no—
62C2×#41—21no—
63#135—22no—
64#135—22no—
65C2×Q16C4211yescharacteristic
66C2×SD16—21no—
67C2×D8C4211yescharacteristic
68C2×M16—21no—
69C2×M16—21no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4×Q8—22no—
75C4×D4—22no—
76C4×D4—42no—
77#117—21no—
78#117—41no—
79C2×C42—22no—
80C2×C42—41no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C16—24no—
85C2×C16—24no—
86C2×C16—24no—
87#108—22no—
88#108—22no—
89#107C4212yes—
90#106—21no—
91C4⋊C8—24no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94#104—42no—
95#104—42no—
96#103—22no—
97#102—22no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101#97—41no—
102C4×C8—22no—
103C4×C8—22no—
104C4×C8—22no—
105C4×C8—22no—
106C4×C8—42no—
107C4×C8—41no—
108C4×C8D812yes—
109C4×C8SD1612yes—
110C4×C8C22×C412yes—
111C4×C8#4112yes—
112#41—41no—
113#41—41no—
114#41—41no—
115#41—41no—
116C2×Q8—21no—
117C2×D4—21no—
118C2×D4—41no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—41no—
122C22×C4—42no—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4#13512yes—
125Q16—22no—
126SD16—42no—
127D8—22no—
128M16—41no—
129M16—41no—
130M16—41no—
131M16—41no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—44no—
141C2×C8#10212yes—
142C2×C8#10212yes—
143C2×C8C2×C4211yescharacteristic
144C2×C8#11711yescharacteristic
145C2×C8C4×D412yes—
146C2×C8#13112yes—
147C2×C8#13112yes—
148C4⋊C4—22no—
149C4⋊C4—42no—
150C22⋊C4—41no—
151C22⋊C4—42no—
152C42—21no—
153C42—21no—
154C42—44no—
155C42—41no—
156C42—42no—
157C42#10412yes—
158C42#10412yes—
159C42C4×D412yes—
160C42C2×D811yescharacteristic
161C42C2×SD1611yescharacteristic
162C16—28no—
163C23—41no—
164C23#41611yescharacteristic
165Q8—41no—
166Q8—41no—
167D4—41no—
168D4—41no—
169D4—81no—
170D4—81no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—84no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4C4×C22⋊C411yescharacteristic
182C2×C4#40811yescharacteristic
183C2×C4#40811yescharacteristic
184C2×C4C2×#10211yescharacteristic
185C2×C4C2×#10411yescharacteristic
186C2×C4C2×#10411yescharacteristic
187C2×C4C4×D812yes—
188C2×C4C4×SD1612yes—
189C8—22no—
190C8—22no—
191C8—22no—
192C8—84no—
193C8—81no—
194C8—81no—
195C8#40512yes—
196C8#43312yes—
197C22—22no—
198C22—82no—
199C22—81no—
200C22#149411yescharacteristic
201C22#149411yescharacteristic
202C22C4×#10211yescharacteristic
203C4—24no—
204C4—22no—
205C4—22no—
206C4—42no—
207C4—82no—
208C4—81no—
209C4#149211yescharacteristic
210C4#149211yescharacteristic
211C4#175011yescharacteristic
212C4C4×#10411yescharacteristic
213C2—21no—
214C2—21no—
215C2—161no—
216C2#817711yescharacteristic
217C2#1264612yes—
subgroupG/H
centerC2×C4C2×#102
commutatorC2×C8C2×C42
Frattini, cosocleC2×C4×C8C23
socleC22C4×#102
prime radicalC2×#135C2×C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#346C2semidirect
4. #8267C2semidirect
5. #7177C2semidirect
22. #1334C4semidirect
23. #1333C4semidirect

Maximal subgroups

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

subgroupindex
#212922
C4×#3462
#115972
#82672
#71772