Finite Group Databasebeta

← #150744

Subgroups — #150744

Order 512.

241 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 50 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#29386C211yesmaximal, characteristic
2#12864C211yesmaximal, characteristic, complemented
3#11462C211yesmaximal, characteristic, complemented
4#7439C211yesmaximal, characteristic, complemented
5#7438C211yesmaximal, characteristic, complemented
6#7407C211yesmaximal, characteristic
7#7406C211yesmaximal, characteristic
8#2956C411yescharacteristic, complemented
9#2922C2211yescharacteristic
10#2162—21no—
11#2103C411yescharacteristic, complemented
12#2098C2211yescharacteristic
13#2053C411yescharacteristic, complemented
14#2017C2211yescharacteristic
15#1937—21no—
16#1906—21no—
17#1881—21no—
18#1868—21no—
19#1841C411yescharacteristic, complemented
20#1827—21no—
21#1395—21no—
22#1392C2211yescharacteristic
23#1391C2211yescharacteristic
24#1323—21no—
25#1315C2211yescharacteristic
26#1315C2211yescharacteristic
27#575—21no—
28#575C2×C411yescharacteristic
29#568—21no—
30C4×#41D411yescharacteristic
31#444—41no—
32#443—41no—
33#433—41no—
34#431—41no—
35#430C2×C411yescharacteristic
36#424—21no—
37#424C2×C411yescharacteristic
38#424D411yescharacteristic
39#421—21no—
40C2×#104—21no—
41#416—21no—
42#413—21no—
43#410—21no—
44#410—21no—
45#408—21no—
46#408D411yescharacteristic
47#408C2311yescharacteristic
48C4×M16C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C4⋊C16—41no—
51C22⋊C16—21no—
52C22⋊C16—21no—
53C4×C16—41no—
54#344—21no—
55#344—21no—
56#341—21no—
57#339—21no—
58#339—41no—
59#338C2×C411yescharacteristic
60H(3,ℤ/4)—41no—
61H(3,ℤ/4)C2×C411yescharacteristic
62C2×#41—21no—
63C2×#41C2×D411yescharacteristic
64#137—41no—
65#137—41no—
66#137—41no—
67AGL(1,ℤ/8)—41no—
68AGL(1,ℤ/8)—41no—
69AGL(1,ℤ/8)—41no—
70#135—41no—
71#135—81no—
72#131—42no—
73#131—81no—
74C2×M16—21no—
75C2×M16—21no—
76C2×M16—41no—
77C2×M16—81no—
78C2×M16C22⋊C411yescharacteristic
79C2×M16C22×C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C4×Q8—21no—
83C4×D4—21no—
84C4×D4—41no—
85#117—21no—
86#117—21no—
87#117—41no—
88#117C22⋊C411yescharacteristic
89C2×C42—21no—
90C2×C42C22⋊C411yescharacteristic
91C2×C16—41no—
92C2×C16—41no—
93#108—41no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C4⋊C8—81no—
97#104—41no—
98#104—41no—
99#104—81no—
100#103—41no—
101#102—41no—
102#101—41no—
103#100—41no—
104#99—42no—
105#99—41no—
106#99—41no—
107C22⋊C8—41no—
108C22⋊C8—41no—
109C22⋊C8—81no—
110#97—21no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114#41—41no—
115#41—41no—
116#41—41no—
117#41—41no—
118#41—41no—
119#41—41no—
120C2×Q8—21no—
121C2×Q8#13511yescharacteristic
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—41no—
125C2×D4#13511yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4C2×C22⋊C411yescharacteristic
131Q16—41no—
132Q16—81no—
133SD16—42no—
134SD16—81no—
135SD16—81no—
136D8—41no—
137D8—81no—
138M16—41no—
139M16—41no—
140M16—41no—
141M16—41no—
142M16—41no—
143M16—41no—
144M16—82no—
145M16—81no—
146M16—81no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—82no—
156C2×C8—81no—
157C4⋊C4—21no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—81no—
161C22⋊C4—21no—
162C22⋊C4—41no—
163C22⋊C4—41no—
164C22⋊C4—81no—
165C42—21no—
166C42—21no—
167C42—41no—
168C42—41no—
169C42—81no—
170C42#9911yescharacteristic
171C42#9911yescharacteristic
172C16—81no—
173C16—81no—
174C23—21no—
175C23—41no—
176C23#42111yescharacteristic
177Q8—41no—
178Q8—41no—
179Q8—41no—
180D4—41no—
181D4—41no—
182D4—41no—
183D4—81no—
184D4—81no—
185D4—81no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—161no—
201C2×C4C2×#9911yescharacteristic
202C2×C4#41611yescharacteristic
203C8—41no—
204C8—41no—
205C8—41no—
206C8—41no—
207C8—81no—
208C8—81no—
209C8—161no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—81no—
214C22—81no—
215C22—161no—
216C22#151611yescharacteristic
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—81no—
222C4—81no—
223C4—161no—
224C4—321no—
225C4#212011yescharacteristic
226C4#212811yescharacteristic
227C2—21no—
228C2—41no—
229C2—81no—
230C2—161no—
231C2#1184011yescharacteristic
subgroupG/H
centerC4#2128
commutatorC2×M16C22×C4
Frattini, cosocle#408C23
socleC2#11840
prime radical#2017C22
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C42 ◁ #2922 ◁ G
LECSG ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12864C2semidirect
3. #11462C2semidirect
4. #7439C2semidirect
5. #7438C2semidirect
8. #2956C4semidirect
11. #2103C4semidirect
13. #2053C4semidirect
19. #1841C4semidirect

Maximal subgroups

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

subgroupindex
#293862
#128642
#114622
#74392
#74382
#74072
#74062