Finite Group Databasebeta

← #11720328

Subgroups — #11720328

Order 864.

194 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 38 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#88740C211yesmaximal, characteristic
2C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3#3748⋊C3C211yesmaximal, characteristic, complemented
4#63569—31nomaximal
5#63568—31nomaximal
6C2×C4×H(3,𝔽3)C2211yescharacteristic
7#6235—21no—
8C4⋊#272C411yescharacteristic, complemented
9C4×#272—21no—
10#6229C411yescharacteristic, complemented
11C2×(C3⋊Dic3)⋊C3—21no—
12(C3⋊(C3⋊C8))⋊C3—21no—
13C3×#235—31no—
14C3×#235—31no—
15C3×#235—61no—
16C4×C3⋊Dic3D311yescharacteristic, complemented
17#3748D311yescharacteristic, complemented
18C3×C4×Dic3—31no—
19C3×#208—31no—
20C22×H(3,𝔽3)D411yescharacteristic
21C2×#272—21no—
22C4×H(3,𝔽3)C2×C411yescharacteristic
23C4×H(3,𝔽3)D411yescharacteristic
24#1084—21no—
25(C3⋊Dic3)⋊C3—22no—
26#837—91no—
27#825—91no—
28#805—91no—
29C2×C32×C4—21no—
30C2×C32×C4C2×D311yescharacteristic
31C2×C32×C4C2×D311yescharacteristic
32C2×C3⋊Dic3—21no—
33C3×#67—62no—
34C3×#67—61no—
35C3×#67—61no—
36C2×C3×Dic3—61no—
37C3×D12—31no—
38C3×D12—31no—
39C3×D12—61no—
40C3×C4×D3—62no—
41C3×C4×D3—61no—
42C3×C4×D3—61no—
43C3×Dic6—31no—
44C3×Dic6—31no—
45C3×Dic6—61no—
46C3⋊(C3⋊C8)—21no—
47C3×C3⋊C8—61no—
48C2×H(3,𝔽3)—21no—
49C2×H(3,𝔽3)C22⋊C411yescharacteristic
50#272—41no—
51C3×#41—91no—
52#235—31no—
53#235—31no—
54#235—61no—
55C3×M16—91no—
56C3×C42—91no—
57C4×Dic3—31no—
58C4×Dic3—181no—
59#208—31no—
60#208—91no—
61C22×C32—21no—
62C22×C32D1211yescharacteristic
63C22×C32#6711yescharacteristic
64C2×C3×D3—62no—
65C2×C3×D3—61no—
66C2×C3×D3—61no—
67C32×C4—21no—
68C32×C4—21no—
69C32×C4C4×D311yescharacteristic
70C32×C4C4×D311yescharacteristic
71C32×C4D1211yescharacteristic
72C32×C4#6711yescharacteristic
73C3⋊Dic3—22no—
74C3×Dic3—62no—
75C3×Dic3—62no—
76C3×Dic3—61no—
77C3×Dic3—61no—
78#104—271noSylow
79H(3,𝔽3)#10411yescharacteristic, complemented, Sylow
80C3×Q8—91no—
81C3×D4—91no—
82C3×D4—181no—
83C2×C3×C4—31no—
84C2×C3×C4—61no—
85C2×C3×C4—121no—
86C2×C3×C4—181no—
87C2×C3×C4—181no—
88C2×C3×C4D3211yescharacteristic
89#67—61no—
90#67—61no—
91#67—62no—
92C2×Dic3—61no—
93C2×Dic3—181no—
94D12—31no—
95D12—31no—
96D12—61no—
97C4×D3—61no—
98C4×D3—61no—
99C4×D3—62no—
100Dic6—31no—
101Dic6—31no—
102Dic6—61no—
103C3×C8—181no—
104C3⋊C8—61no—
105C3⋊C8—181no—
106C2×C32—21no—
107C2×C32—21no—
108C2×C32—21no—
109C2×C32—41no—
110C2×C32(C2×D3)⋊C411yescharacteristic
111C2×C32(C2×D3)⋊C411yescharacteristic
112C3×D3—122no—
113C3×D3—121no—
114C3×D3—121no—
115#41—91no—
116M16—91no—
117C42—91no—
118C22×C3—31no—
119C22×C3—61no—
120C22×C3—121no—
121C22×C3—181no—
122C22×C3#62611yescharacteristic
123C2×D3—61no—
124C2×D3—61no—
125C2×D3—62no—
126C3×C4—31no—
127C3×C4—31no—
128C3×C4—61no—
129C3×C4—61no—
130C3×C4—121no—
131C3×C4—121no—
132C3×C4—182no—
133C3×C4—181no—
134C3×C4#62411yescharacteristic
135C3×C4#62611yescharacteristic
136Dic3—62no—
137Dic3—61no—
138Dic3—61no—
139Dic3—62no—
140Dic3—182no—
141C32—21no—
142C32#80511yescharacteristic, complemented
143C32#82511yescharacteristic, complemented
144Q8—91no—
145D4—91no—
146D4—181no—
147C2×C4—181no—
148C2×C4—181no—
149C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
150C8—181no—
151C2×C3—21no—
152C2×C3—31no—
153C2×C3—61no—
154C2×C3—61no—
155C2×C3—61no—
156C2×C3—121no—
157C2×C3—121no—
158C2×C3—361no—
159C2×C3#372211yescharacteristic
160D3—121no—
161D3—121no—
162D3—122no—
163C22—181no—
164C22#620011yescharacteristic
165C4—182no—
166C4—181no—
167C4#619911yescharacteristic
168C4#620011yescharacteristic
169C3—31no—
170C3—61no—
171C3—121no—
172C3#6357011yescharacteristic
173C2—21no—
174C2—361no—
175C2#8845111yescharacteristic
subgroupG/H
centerC4#6200
commutatorC4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4D32
Fitting, cosocleC2×C4×H(3,𝔽3)C22
socleC2×C3#3722
prime radical#88740C2
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#104
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×(C3⋊Dic3)⋊C3C2semidirect
3. #3748⋊C3C2semidirect
8. C4⋊#272C4semidirect
10. #6229C4semidirect
16. C4×C3⋊Dic3D3semidirect
17. #3748D3semidirect
79. H(3,𝔽3)#104semidirect
142. C32#805semidirect
143. C32#825semidirect

Maximal subgroups

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

subgroupindex
#887402
C4×(C3⋊Dic3)⋊C32
#3748⋊C32
#635693
#635683