Finite Group Databasebeta

← #10589431⋊C3 #13135709

Subgroups — #10589431⋊C3 #13135709

Order 1728.

249 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×Q16×H(3,𝔽3)C211yesmaximal, characteristic
2#63646⋊C3C211yesmaximal, characteristic
3#63639⋊C3C211yesmaximal, characteristic
4#10589431C311yesmaximal, characteristic, complemented
5C3×#4565—31nomaximal
6C2×Q8×H(3,𝔽3)—21no—
7Q16×H(3,𝔽3)—21no—
8Q16×H(3,𝔽3)C412yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10#3748⋊C3—21no—
11(C3⋊(C3⋊C16))⋊C3—21no—
12C2×C32×Q16—21no—
13C2×C32×Q16D311yescharacteristic
14C2×C32×Q16C2×C311yescharacteristic
15#63646C2×C311yescharacteristic
16#63639C2×C311yescharacteristic
17C3×#819—31no—
18C3×#812—31no—
19Q8×H(3,𝔽3)—22no—
20Q8×H(3,𝔽3)—41no—
21C2×C4×H(3,𝔽3)—41no—
22C2×C4×H(3,𝔽3)D411yescharacteristic
23C8×H(3,𝔽3)C2×C411yescharacteristic
24C8×H(3,𝔽3)D411yescharacteristic
25(C3⋊(C3⋊C8))⋊C3—41no—
26C3×#362—91no—
27#4565—31no—
28#4565—91no—
29C2×C32×Q8—22no—
30C2×C32×Q8—21no—
31C2×C32×Q8—21no—
32C32×Q16—22no—
33C32×Q16—22no—
34C32×Q16—21no—
35C32×Q16—21no—
36C32×Q16Dic312yes—
37C32×Q16C3×C412yes—
38C2×C32×C8—21no—
39C2×C32×C8C2×D311yescharacteristic
40C2×C32×C8C22×C311yescharacteristic
41#3748—21no—
42C3×#208—61no—
43C3⋊(C3⋊C16)—21no—
44C3×C3⋊C16—61no—
45C22×H(3,𝔽3)SD1611yescharacteristic
46C4×H(3,𝔽3)—42no—
47C4×H(3,𝔽3)C22⋊C411yescharacteristic
48C4×H(3,𝔽3)D811yescharacteristic
49C2×C3×Q16—31no—
50C2×C3×Q16—31no—
51C2×C3×Q16—61no—
52C2×C3×Q16C3×D311yescharacteristic
53C3×M32—91no—
54C3×#108—91no—
55#819—31no—
56#819—91no—
57#812—31no—
58#812—91no—
59C32×Q8—24no—
60C32×Q8—22no—
61C32×Q8—22no—
62C32×Q8—42no—
63C32×Q8—41no—
64C32×Q8—41no—
65C2×C32×C4—21no—
66C2×C32×C4—42no—
67C2×C32×C4—41no—
68C2×C32×C4—41no—
69C2×C32×C4#6711yescharacteristic
70C2×C32×C4C3×D411yescharacteristic
71C32×C8—21no—
72C32×C8—21no—
73C32×C8C2×Dic311yescharacteristic
74C32×C8#6711yescharacteristic
75C32×C8C2×C3×C411yescharacteristic
76C32×C8C3×D411yescharacteristic
77C3⋊(C3⋊C8)—41no—
78C3×C3⋊C8—121no—
79#362—91noSylow
80C2×H(3,𝔽3)—21no—
81C2×H(3,𝔽3)#10211yescharacteristic
82C2×C3×Q8—21no—
83C2×C3×Q8—61no—
84C2×C3×Q8—61no—
85C2×C3×Q8—62no—
86C3×Q16—21no—
87C3×Q16—32no—
88C3×Q16—32no—
89C3×Q16—61no—
90C3×Q16—61no—
91C3×Q16—62no—
92C3×Q16—62no—
93C3×Q16C3×Dic312yes—
94C3×M16—181no—
95C2×C3×C8—31no—
96C2×C3×C8—31no—
97C2×C3×C8—61no—
98C2×C3×C8C2×C3×D311yescharacteristic
99#208—61no—
100#208—181no—
101C3×C16—181no—
102C3⋊C16—61no—
103C3⋊C16—181no—
104C22×C32—21no—
105C22×C32#21411yescharacteristic
106C22×C32C3×SD1611yescharacteristic
107C32×C4—21no—
108C32×C4—21no—
109C32×C4—44no—
110C32×C4—42no—
111C32×C4—42no—
112C32×C4#21311yescharacteristic
113C32×C4#21711yescharacteristic
114C32×C4C3×C22⋊C411yescharacteristic
115C32×C4C3×D811yescharacteristic
116C2×Q16(C3⋊D3)⋊C311yescharacteristic
117M32—91no—
118#108—91no—
119H(3,𝔽3)#36211yescharacteristic, complemented, Sylow
120C3×Q8—22no—
121C3×Q8—41no—
122C3×Q8—62no—
123C3×Q8—62no—
124C3×Q8—64no—
125C3×Q8—121no—
126C3×Q8—121no—
127C3×Q8—122no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—41no—
131C2×C3×C4—61no—
132C2×C3×C4—121no—
133C2×C3×C4—121no—
134C2×C3×C4—122no—
135C2×C3×C4C3×#6711yescharacteristic
136C3×C8—31no—
137C3×C8—31no—
138C3×C8—31no—
139C3×C8—31no—
140C3×C8—61no—
141C3×C8—61no—
142C3×C8—361no—
143C3×C8C2×C3×Dic311yescharacteristic
144C3×C8C3×#6711yescharacteristic
145C3⋊C8—121no—
146C3⋊C8—361no—
147C2×C32—22no—
148C2×C32—21no—
149C2×C32—21no—
150C2×C32—21no—
151C2×C32#83211yescharacteristic
152C2×C32C3×#10211yescharacteristic
153C2×Q8—21no—
154Q16—21no—
155Q16(C3⋊Dic3)⋊C312yes—
156M16—181no—
157C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
158C16—181no—
159C22×C3—31no—
160C22×C3—31no—
161C22×C3—61no—
162C22×C3C3×#21411yescharacteristic
163C3×C4—31no—
164C3×C4—31no—
165C3×C4—31no—
166C3×C4—31no—
167C3×C4—42no—
168C3×C4—61no—
169C3×C4—61no—
170C3×C4—122no—
171C3×C4—122no—
172C3×C4—124no—
173C3×C4C3×#21311yescharacteristic
174C3×C4C3×#21711yescharacteristic
175C32—21no—
176C32#456511yescharacteristic, complemented
177C32C3×#36211yescharacteristic, complemented
178Q8—22no—
179Q8—41no—
180C2×C4—41no—
181C2×C4#638⋊C311yescharacteristic
182C8—361no—
183C8C2×(C3⋊Dic3)⋊C311yescharacteristic
184C8#638⋊C311yescharacteristic
185C2×C3—21no—
186C2×C3—31no—
187C2×C3—31no—
188C2×C3—61no—
189C2×C3—61no—
190C2×C3—62no—
191C2×C3—61no—
192C2×C3C3×#83211yescharacteristic
193C22#3754⋊C311yescharacteristic
194C4—42no—
195C4#3753⋊C311yescharacteristic
196C4#3757⋊C311yescharacteristic
197C3—31no—
198C3—31no—
199C3—61no—
200C3C3×#456511yescharacteristic
201C2—21no—
202C2#63659⋊C311yescharacteristic
subgroupG/H
centerC2#63659⋊C3
commutatorC32×C8C2×C3×C4
FrattiniC2×C3×C8C2×C3×D3
FittingC2×Q16×H(3,𝔽3)C2
socleC2×C3C3×#832
prime radicalC22×H(3,𝔽3)SD16
cosocleC2×C32×C8C22×C3
hypercenterC2×Q16(C3⋊D3)⋊C3
hypocenterC32C3×#362
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
4. #10589431C3semidirect
119. H(3,𝔽3)#362semidirect
176. C32#4565semidirect
177. C32C3×#362semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×H(3,𝔽3)2
#63646⋊C32
#63639⋊C32
#105894313
C3×#45653