Finite Group Databasebeta

← #152237

Subgroups — #152237

Order 512.

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

normal subgroups only

This group has 376 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#21705C211yesmaximal, characteristic
2#13583C212yesmaximal
3#12807C212yesmaximal, complemented
4#7500C212yesmaximal, complemented
5#3147C2211yescharacteristic
6C2×#443—22no—
7#2931C411yescharacteristic, complemented
8#2931C2211yescharacteristic
9#2126C2214yes—
10#1974—22no—
11#1962C411yescharacteristic, complemented
12#1962C2211yescharacteristic
13#1848C411yescharacteristic, complemented
14#1812—22no—
15C42×C8C411yescharacteristic, complemented
16#1409—22no—
17#1408—22no—
18#1391—22no—
19C2×#135—22no—
20C2×#131—21no—
21C2×#131D412yes—
22#532C2×C411yescharacteristic
23#532C2311yescharacteristic
24#495—22no—
25#494Q811yescharacteristic
26#493D411yescharacteristic
27#492D411yescharacteristic
28#492Q811yescharacteristic
29#443—44no—
30#443—44no—
31#436—41no—
32#436—41no—
33C8×D4—41no—
34C8×D4—41no—
35C2×#108—22no—
36#427—42no—
37#423C2×C411yescharacteristic
38#423C2×C411yescharacteristic
39C2×#104—22no—
40C2×#104—22no—
41#410—24no—
42#408—21no—
43#408D412yes—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C43C2×C411yescharacteristic
49#354—24no—
50#353—24no—
51H(3,ℤ/4)—42no—
52C2×#41—21no—
53C2×#41C2×D412yes—
54#135—44no—
55#135—44no—
56C2×Q16—22no—
57C2×SD16—22no—
58C2×SD16—22no—
59C2×D8—22no—
60#131—42no—
61#131—42no—
62#131—42no—
63#131—42no—
64C2×M16—22no—
65C2×M16—21no—
66C2×M16—41no—
67C2×M16C22⋊C412yes—
68C22×C8—21no—
69C22×C8—22no—
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C4⋊Q8C4⋊C411yescharacteristic
74C4⋊Q8C2×Q811yescharacteristic
75#127C4⋊C411yescharacteristic
76#127C2×D411yescharacteristic
77#124—21no—
78#124#4112yes—
79#121—41no—
80#121—41no—
81C4×D4—41no—
82C4×D4—41no—
83#117—44no—
84C2×C42—21no—
85C2×C42—41no—
86C2×C42C22×C411yescharacteristic
87#108—44no—
88#107—82no—
89#106—82no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92#104—44no—
93#104—44no—
94#99—44no—
95#99—44no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98C4×C8—22no—
99C4×C8—22no—
100C4×C8—21no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—42no—
104C4×C8C4⋊C411yescharacteristic
105C4×C8C4⋊C411yescharacteristic
106#41—42no—
107#41—42no—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8C22⋊Q812yes—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4#12112yes—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4C2×C22⋊C411yescharacteristic
123Q16—44no—
124SD16—44no—
125SD16—44no—
126D8—44no—
127M16—42no—
128M16—42no—
129M16—82no—
130M16—82no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—22no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8#9911yescharacteristic
146C2×C8#9911yescharacteristic
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—84no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—84no—
153C42—21no—
154C42—22no—
155C42—41no—
156C42—44no—
157C42—84no—
158C42C2×C4⋊C411yescharacteristic
159C42#11711yescharacteristic
160C23—22no—
161C23—41no—
162C23#41011yescharacteristic
163Q8—42no—
164Q8—42no—
165D4—42no—
166D4—42no—
167D4—82no—
168D4—82no—
169D4—82no—
170D4—82no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—164no—
185C2×C4#38011yescharacteristic
186C2×C4C2×#9911yescharacteristic
187C8—22no—
188C8—22no—
189C8—21no—
190C8—42no—
191C8—41no—
192C8—81no—
193C8#35311yescharacteristic
194C8#35311yescharacteristic
195C22—21no—
196C22—41no—
197C22—42no—
198C22—82no—
199C22—82no—
200C22—81no—
201C22#179911yescharacteristic
202C4—22no—
203C4—21no—
204C4—41no—
205C4—44no—
206C4—42no—
207C4—81no—
208C4—324no—
209C4C2×#35311yescharacteristic
210C4#212611yescharacteristic
211C2—21no—
212C2—41no—
213C2—82no—
214C2—161no—
215C2#1279711yescharacteristic
subgroupG/H
centerC4#2126
commutatorC2×C42C22×C4
Frattini, cosocle#532C23
socleC2#12797
prime radical#3147C22
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G
LECSG ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12807C2semidirect
4. #7500C2semidirect
7. #2931C4semidirect
11. #1962C4semidirect
13. #1848C4semidirect
15. C42×C8C4semidirect

Maximal subgroups

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

subgroupindex
#217052
#135832
#128072
#75002