Finite Group Databasebeta

← #13147000

Subgroups — #13147000

Order 1728.

343 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11720738C212yesmaximal
3C3×#5386—33nomaximal
4C3×#5020—31nomaximal
5C2×#6229—41nomaximal
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7C32×C4×Q8D311yescharacteristic
8C3×#869—31no—
9C3×C4×SL(2,𝔽3)D313yes—
10C3×#859—36no—
11C3×#835—32no—
12C3×#808—32no—
13C3×C4⋊(C3⋊C8)—31no—
14C2×C4×H(3,𝔽3)—41no—
15C2×#1084—42no—
16#6229—44no—
17Q8⋊H(3,𝔽3)Q811yescharacteristic, complemented
18#5386—33no—
19C3×#475—91no—
20#5020—31no—
21C2×C32×Q8C2×D311yescharacteristic
22C2×C3×Dic6—61no—
23C2×C3×Dic6—123no—
24C2×C3×SL(2,𝔽3)C2×D313yes—
25C32×C4⋊C4—31no—
26C32×C42—31no—
27C3×C4⋊Dic3—32no—
28C3×C4⋊Dic3—61no—
29C2×C3×C3⋊C8—32no—
30C22×H(3,𝔽3)—41no—
31C4×H(3,𝔽3)—42no—
32#1084—44no—
33C3×C4⋊Q8—91no—
34C3×C4×Q8—31no—
35C3×C4×Q8C3⋊D311yescharacteristic
36#869—31no—
37C4×SL(2,𝔽3)—33no—
38#859—36no—
39C3×#106—92no—
40C3×C4⋊C8—91no—
41C3×#103—92no—
42#835—32no—
43#808—32no—
44C4⋊(C3⋊C8)—31no—
45C32×Q8—31no—
46C32×Q8Dic611yescharacteristic, complemented
47C2×C32×C4—31no—
48C2×C32×C4—31no—
49C2×C32×C4—43no—
50C2×C32×C4S411yescharacteristic
51C2×C3×Dic3—62no—
52C2×C3×Dic3—126no—
53C3×Dic6—1212no—
54C3×Dic6—122no—
55C3×SL(2,𝔽3)Dic613yescomplemented
56C3×C3⋊C8—62no—
57#475—91noSylow
58C2×H(3,𝔽3)—41no—
59C2×H(3,𝔽3)—41no—
60C2×H(3,𝔽3)—41no—
61C2×C3×Q8—31no—
62C2×C3×Q8—181no—
63C2×C3×Q8C2×C3⋊D311yescharacteristic
64C2×Dic6—61no—
65C2×Dic6—123no—
66C2×SL(2,𝔽3)—33no—
67C2×C3×C8—92no—
68C3×C4⋊C4—31no—
69C3×C4⋊C4—31no—
70C3×C4⋊C4—61no—
71C3×C4⋊C4—92no—
72C3×C4⋊C4—181no—
73C3×C42—31no—
74C3×C42—31no—
75C3×C42—61no—
76C4⋊Dic3—32no—
77C4⋊Dic3—61no—
78C2×C3⋊C8—32no—
79C22×C32—43no—
80C22×C32C2×S411yescharacteristic
81C32×C4—31no—
82C32×C4—31no—
83C32×C4—46no—
84C32×C4—61no—
85C32×C4GL(2,𝔽3)12yes—
86C3×Dic3—1212no—
87C3×Dic3—122no—
88C4⋊Q8—91no—
89C4×Q8#27211yescharacteristic
90#106—92no—
91C4⋊C8—91no—
92#103—92no—
93H(3,𝔽3)—41noSylow
94C3×Q8—31no—
95C3×Q8—31no—
96C3×Q8—31no—
97C3×Q8—61no—
98C3×Q8—362no—
99C3×Q8#63411yescharacteristic
100C2×C3×C4—31no—
101C2×C3×C4—31no—
102C2×C3×C4—31no—
103C2×C3×C4—31no—
104C2×C3×C4—31no—
105C2×C3×C4—61no—
106C2×C3×C4—61no—
107C2×C3×C4—123no—
108C2×C3×C4—182no—
109C2×C3×C4C3⋊S411yescharacteristic
110C2×Dic3—62no—
111C2×Dic3—126no—
112Dic6—122no—
113Dic6—1212no—
114SL(2,𝔽3)—33no—
115C3×C8—182no—
116C3⋊C8—62no—
117C2×C32—43no—
118C2×C32—43no—
119C2×C32—43no—
120C2×C32A4⋊Q811yescharacteristic
121C2×C32C2×GL(2,𝔽3)11yescharacteristic
122C2×C32#98411yescharacteristic
123C2×Q8—181no—
124C2×Q8C2×#27211yescharacteristic
125C2×C8—92no—
126C4⋊C4—31no—
127C4⋊C4—92no—
128C4⋊C4—181no—
129C42—31no—
130C22×C3—31no—
131C22×C3—123no—
132C22×C3C2×C3⋊S411yescharacteristic
133C3×C4—31no—
134C3×C4—31no—
135C3×C4—31no—
136C3×C4—31no—
137C3×C4—32no—
138C3×C4—61no—
139C3×C4—61no—
140C3×C4—61no—
141C3×C4—61no—
142C3×C4—121no—
143C3×C4—126no—
144C3×C4—362no—
145C3×C4C3⋊GL(2,𝔽3)12yes—
146Dic3—122no—
147Dic3—1212no—
148C32—43no—
149C32#538611yescharacteristic, complemented
150Q8—31no—
151Q8—362no—
152Q8#622911yescharacteristic, complemented
153C2×C4—31no—
154C2×C4—31no—
155C2×C4—182no—
156C2×C4#625811yescharacteristic
157C8—182no—
158C2×C3—31no—
159C2×C3—31no—
160C2×C3—31no—
161C2×C3—123no—
162C2×C3—123no—
163C2×C3—123no—
164C2×C3#6425911yescharacteristic
165C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
166C2×C3#6426511yescharacteristic
167C22C2×#625811yescharacteristic
168C4—31no—
169C4—31no—
170C4—61no—
171C4—362no—
172C4#8861112yes—
173C3—31no—
174C3—123no—
175C3#1059362011yescharacteristic
176C2#1172238011yescharacteristic
177C2C2×#8861111yescharacteristic
178C2#1172239211yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C2×C3⋊S4
commutator, prime radical, cosocleC2×Q8⋊H(3,𝔽3)C22
FittingC32×C4×Q8D3
hypercenterC2×C3×C4C3⋊S4
hypocenterQ8⋊H(3,𝔽3)Q8
LCSG ▷ C2×Q8⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
17. Q8⋊H(3,𝔽3)Q8semidirect
46. C32×Q8Dic6semidirect
55. C3×SL(2,𝔽3)Dic6semidirect
149. C32#5386semidirect
152. Q8#6229semidirect

Maximal subgroups

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

subgroupindex
C4×Q8⋊H(3,𝔽3)2
#117207382
C3×#53863
C3×#50203
C2×#62294