Finite Group Databasebeta

← #146060

Subgroups — #146060

Order 512.

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

normal subgroups only

This group has 218 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#13197C211yesmaximal, characteristic
2#12886C211yesmaximal, characteristic, complemented
3#12823C211yesmaximal, characteristic
4#12822C211yesmaximal, characteristic
5#7272C211yesmaximal, characteristic, complemented
6#7268C211yesmaximal, characteristic, complemented
7#7162C211yesmaximal, characteristic, complemented
8C2×#443C2211yescharacteristic
9C2×#364C2211yescharacteristic
10C2×#364C2211yescharacteristic
11C2×#364C2211yescharacteristic
12C2×#360—21no—
13#2143—21no—
14#2117—21no—
15#2115—21no—
16C4×M32C2211yescharacteristic
17#1848C2211yescharacteristic
18#1844C2211yescharacteristic
19#1388—21no—
20#1383—21no—
21#1383—21no—
22#1383—21no—
23C2×#135D411yescharacteristic
24C2×#131—21no—
25C2×M32—21no—
26C2×M32—21no—
27C2×M32—21no—
28C2×M32D411yescharacteristic
29C2×M32D411yescharacteristic
30#443—21no—
31#443—21no—
32#443—21no—
33#443—21no—
34C2×#108—21no—
35C2×#108—21no—
36C2×#108D411yescharacteristic
37C2×#106—21no—
38C2×#106—21no—
39#423—21no—
40C2×#104—21no—
41C2×C4×C8C2311yescharacteristic
42#384—21no—
43#364—21no—
44#364—21no—
45#364—21no—
46#364—21no—
47#364—21no—
48#364—21no—
49#360—42no—
50#359—42no—
51#350—42no—
52#349—42no—
53#346—21no—
54C4×C16—21no—
55C2×#41—21no—
56#135—21no—
57#135—21no—
58#135—21no—
59#135—21no—
60C2×Q16Q1611yescharacteristic
61C2×SD16—21no—
62C2×D8Q1611yescharacteristic
63#131—41no—
64#131—41no—
65#131—41no—
66#131—41no—
67C2×M16—21no—
68C2×M16—41no—
69C2×M16—41no—
70C22×C8—41no—
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C2×C4⋊C4—41no—
74C2×C4⋊C4—81no—
75C2×C42#4111yescharacteristic
76M32—21no—
77M32—21no—
78M32—21no—
79M32—21no—
80M32—41no—
81M32—41no—
82M32—41no—
83M32—41no—
84M32—41no—
85M32—41no—
86C2×C16—21no—
87C2×C16—21no—
88C2×C16—41no—
89C2×C16—41no—
90C2×C16—41no—
91#108—21no—
92#108—21no—
93#108—41no—
94#108—41no—
95#108—41no—
96#108—41no—
97#106—41no—
98#106—41no—
99#106—41no—
100#106—41no—
101C4⋊C8—81no—
102#104—41no—
103#104—41no—
104C4×C8#4111yescharacteristic
105C4×C8#4111yescharacteristic
106C4×C8#4111yescharacteristic
107C4×C8#4111yescharacteristic
108#41—41no—
109#41—41no—
110#41—41no—
111#41—41no—
112C2×Q8—21no—
113C2×D4—21no—
114C2×D4—41no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4—81no—
118C22×C4#12111yescharacteristic
119Q16—21no—
120Q16—21no—
121SD16—41no—
122SD16—41no—
123D8—21no—
124D8—21no—
125M16—41no—
126M16—41no—
127M16—81no—
128M16—81no—
129M16—81no—
130M16—81no—
131C2×C8—41no—
132C2×C8—81no—
133C2×C8—81no—
134C2×C8—81no—
135C2×C8#12311yescharacteristic
136C2×C8#12311yescharacteristic
137C2×C8#12311yescharacteristic
138C2×C8#12411yescharacteristic
139C2×C8C2×Q1611yescharacteristic
140C2×C8#13511yescharacteristic
141C2×C8#13511yescharacteristic
142C2×C8#13511yescharacteristic
143C2×C8AGL(1,ℤ/8)11yescharacteristic
144C2×C8AGL(1,ℤ/8)11yescharacteristic
145C2×C8AGL(1,ℤ/8)11yescharacteristic
146C2×C8AGL(1,ℤ/8)11yescharacteristic
147C4⋊C4—81no—
148C4⋊C4—81no—
149C4⋊C4—162no—
150C42—21no—
151C42#12611yescharacteristic
152C42#12611yescharacteristic
153C16—41no—
154C16—41no—
155C16—81no—
156C16—81no—
157C16—81no—
158C23—41no—
159C23#47111yescharacteristic
160Q8—41no—
161Q8—41no—
162D4—41no—
163D4—41no—
164D4—81no—
165D4—81no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—162no—
173C2×C4—161no—
174C2×C4#39711yescharacteristic
175C2×C4#46311yescharacteristic
176C2×C4#46311yescharacteristic
177C2×C4#46711yescharacteristic
178C2×C4#46911yescharacteristic
179C2×C4#47111yescharacteristic
180C8—81no—
181C8—161no—
182C8#48211yescharacteristic
183C8#48211yescharacteristic
184C8#48211yescharacteristic
185C8#48211yescharacteristic
186C8#48411yescharacteristic
187C8#48411yescharacteristic
188C8#48811yescharacteristic
189C8#49011yescharacteristic
190C22—41no—
191C22—81no—
192C22—161no—
193C22#168311yescharacteristic
194C22#170111yescharacteristic
195C22#207911yescharacteristic
196C4—22no—
197C4—41no—
198C4—81no—
199C4—321no—
200C4#168311yescharacteristic
201C4#170111yescharacteristic
202C4#207311yescharacteristic
203C4#207611yescharacteristic
204C2—21no—
205C2—21no—
206C2—21no—
207C2—161no—
208C2#1062211yescharacteristic
subgroupG/H
center, socleC2#10622
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC2×#135D4
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×C4×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. #12886C2semidirect
5. #7272C2semidirect
6. #7268C2semidirect
7. #7162C2semidirect

Maximal subgroups

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

subgroupindex
#131972
#128862
#128232
#128222
#72722
#72682
#71622