Finite Group Databasebeta

← #139579

Subgroups — #139579

Order 512.

313 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 78 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#33493C211yesmaximal, characteristic
2#27693C211yesmaximal, characteristic
3#12877C212yesmaximal, complemented
4#12807C211yesmaximal, characteristic
5#7228C212yesmaximal, complemented
6#2931C2211yescharacteristic
7#2916C2211yescharacteristic
8#2169—22no—
9D4×C16—22no—
10#2162—21no—
11#2126—21no—
12#2113—21no—
13#2113C2212yes—
14#2102C412yescomplemented
15#2098C412yescomplemented
16#1945—21no—
17#1884—21no—
18#1868—21no—
19#1814—21no—
20#1812C2212yes—
21#1812C2211yescharacteristic
22#1391—21no—
23#1389—22no—
24#1382—22no—
25#568—21no—
26C2×#131D411yescharacteristic
27C2×#131C2311yescharacteristic
28#532D411yescharacteristic
29C2×M32—21no—
30C2×M32D412yes—
31C22×C16—21no—
32C22×C16D412yes—
33#436—22no—
34C8×D4—22no—
35#430—22no—
36#430C2×C412yes—
37C2×#108—22no—
38C2×#108—21no—
39#427—21no—
40#427—22no—
41#424—22no—
42#423C2×C411yescharacteristic
43#421—22no—
44#410—21no—
45#410—22no—
46#408C2×C411yescharacteristic
47#408D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C4⋊C16—21no—
50C4⋊C16—21no—
51#354—41no—
52#353—41no—
53C22⋊C16—22no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56C4×C16—22no—
57#344—22no—
58#339—22no—
59H(3,ℤ/4)—22no—
60H(3,ℤ/4)C2×C412yes—
61C2×#41C22⋊C411yescharacteristic
62C2×#41C2×D411yescharacteristic
63#131—41no—
64#131—41no—
65#131—41no—
66#131—41no—
67#131—82no—
68C2×M16—21no—
69C2×M16—22no—
70C2×M16—42no—
71C2×M16C22⋊C411yescharacteristic
72C2×M16C22×C411yescharacteristic
73C2×M16#4112yes—
74C22×C8—21no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C2×D412yes—
77C22×C8C2×D411yescharacteristic
78C4⋊Q8Q1611yescharacteristic
79#127SD1611yescharacteristic
80#124SD1611yescharacteristic
81#124Q1611yescharacteristic
82#121—22no—
83C4×D4—22no—
84#117—22no—
85#117—42no—
86C2×C42C22⋊C411yescharacteristic
87M32—42no—
88M32—81no—
89C2×C16—22no—
90C2×C16—22no—
91C2×C16—42no—
92C2×C16—81no—
93#108—24no—
94#108—41no—
95#108—41no—
96#107—41no—
97#107—42no—
98#106—41no—
99#106—42no—
100C4⋊C8—22no—
101C4⋊C8—82no—
102#104—82no—
103#99—42no—
104#99—42no—
105#99—42no—
106C22⋊C8—22no—
107C22⋊C8—22no—
108C4×C8—21no—
109C4×C8—21no—
110#41—41no—
111#41—41no—
112#41—41no—
113#41—41no—
114C2×Q8#10311yescharacteristic
115C2×Q8C2×Q1611yescharacteristic
116C2×D4—22no—
117C2×D4—21no—
118C2×D4#10311yescharacteristic
119C2×D4C2×SD1611yescharacteristic
120C2×D4#13512yes—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4#12112yes—
125M16—41no—
126M16—42no—
127M16—41no—
128M16—42no—
129M16—42no—
130M16—81no—
131M16—82no—
132C2×C8—22no—
133C2×C8—21no—
134C2×C8—22no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—42no—
142C2×C8—82no—
143C2×C8C4×D412yes—
144C2×C8#12011yescharacteristic
145C2×C8#12311yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—82no—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C22⋊C4—42no—
152C22⋊C4—82no—
153C42—21no—
154C42—42no—
155C42—82no—
156C42#10311yescharacteristic
157C42#10311yescharacteristic
158C16—42no—
159C16—42no—
160C23—21no—
161C23#41611yescharacteristic
162C23#46512yes—
163Q8—41no—
164Q8—41no—
165D4—41no—
166D4—42no—
167D4—41no—
168D4—44no—
169D4—81no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—82no—
181C2×C4—162no—
182C2×C4#38611yescharacteristic
183C2×C4C2×#10311yescharacteristic
184C2×C4#46712yes—
185C8—22no—
186C8—21no—
187C8—41no—
188C8—41no—
189C8—82no—
190C8#45811yescharacteristic
191C8#45811yescharacteristic
192C22—21no—
193C22—22no—
194C22—41no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22#189211yescharacteristic
199C4—22no—
200C4—21no—
201C4—22no—
202C4—41no—
203C4—41no—
204C4—162no—
205C4—322no—
206C4#189011yescharacteristic
207C4#191011yescharacteristic
208C2—21no—
209C2—41no—
210C2—42no—
211C2—81no—
212C2#946911yescharacteristic
subgroupG/H
centerC4#1910
commutatorC2×M16C22×C4
Frattini, cosocleC2×#131C23
socleC2#9469
prime radical#532D4
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G
LECSG ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12877C2semidirect
5. #7228C2semidirect
14. #2102C4semidirect
15. #2098C4semidirect

Maximal subgroups

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

subgroupindex
#334932
#276932
#128772
#128072
#72282