Finite Group Databasebeta

← #13147421

Subgroups — #13147421

Order 1728.

281 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×#1084C211yesmaximal, characteristic
2#11720785C212yesmaximal
3#11720783C212yesmaximal
4C4⋊#6188C211yesmaximal, characteristic
5C4×#6188C211yesmaximal, characteristic
6C2×Q8×H(3,𝔽3)C411yescharacteristic
7C2×#6188C2211yescharacteristic
8C2×#6188C2212yes—
9C2×#6188C2211yescharacteristic
10C4⋊#1084C412yes—
11C4⋊#1084C411yescharacteristic
12C4×#1084C2211yescharacteristic
13C4×#1084C2212yes—
14C3×Q8×Dic3—62no—
15Q8×H(3,𝔽3)—22no—
16C2×C4×H(3,𝔽3)C2×C412yes—
17C2×C4×H(3,𝔽3)C2×C411yescharacteristic
18C2×#1084C2×C411yescharacteristic
19C2×#1084C2×C412yes—
20C2×#1084C2311yescharacteristic
21#6188—21no—
22#6188—22no—
23#6188Q812yescomplemented
24C3×#446—91nomaximal
25C2×C32×Q8—22no—
26C3×C4⋊Dic3—64no—
27C3×C4⋊Dic3—62no—
28C3×C4×Dic3—62no—
29C3×C4×Dic3—64no—
30C22×H(3,𝔽3)C22×C411yescharacteristic
31C4×H(3,𝔽3)—22no—
32C4×H(3,𝔽3)M1612yes—
33#1084—21no—
34#1084—22no—
35#1084C2×Q811yescharacteristic
36#1084#4111yescharacteristic
37C3×C4×Q8—91no—
38Q8×Dic3—62no—
39C3×C4⋊C8—91no—
40C3×C4⋊C8—92no—
41C3×#97—92no—
42C3×C4×C8—91no—
43C32×Q8—28no—
44C2×C32×C4—24no—
45C2×C32×C4—22no—
46C2×C3×Dic3—62no—
47C2×C3×Dic3—62no—
48C2×C3×Dic3—64no—
49#446—91noSylow
50C2×H(3,𝔽3)C4×Q811yescharacteristic
51C2×H(3,𝔽3)C2×M1611yescharacteristic
52C2×H(3,𝔽3)#13111yescharacteristic
53C2×C3×Q8—62no—
54C2×C3×Q8#15711yescharacteristic
55C2×C3×C8—91no—
56C2×C3×C8—91no—
57C2×C3×C8—92no—
58C3×C4⋊C4—92no—
59C3×C4⋊C4—91no—
60C3×C42—91no—
61C3×C42—92no—
62C4⋊Dic3—64no—
63C4⋊Dic3—62no—
64C4×Dic3—62no—
65C4×Dic3—64no—
66C22×C32—22no—
67C32×C4—28no—
68C32×C4—24no—
69C3×Dic3—62no—
70C3×Dic3—62no—
71C3×Dic3—122no—
72C3×Dic3—124no—
73C4×Q8—91no—
74C4⋊C8—91no—
75C4⋊C8—92no—
76#97—92no—
77C4×C8—91no—
78H(3,𝔽3)#44611yescharacteristic, complemented, Sylow
79C3×Q8—22no—
80C3×Q8—68no—
81C2×C3×C4—64no—
82C2×C3×C4—62no—
83C2×C3×C4—91no—
84C2×C3×C4—91no—
85C2×C3×C4—92no—
86C2×C3×C4C2×#15712yes—
87C2×C3×C4C2×#15711yescharacteristic
88C2×Dic3—62no—
89C2×Dic3—62no—
90C2×Dic3—64no—
91C3×C8—92no—
92C3×C8—181no—
93C3×C8—182no—
94C2×C32—22no—
95C2×C32—22no—
96C2×C32—22no—
97C2×Q8#108811yescharacteristic
98C2×C8—91no—
99C2×C8—91no—
100C2×C8—92no—
101C4⋊C4—92no—
102C4⋊C4—91no—
103C42—91no—
104C42—92no—
105C22×C3—62no—
106C22×C3C22×#15711yescharacteristic
107C3×C4—22no—
108C3×C4—68no—
109C3×C4—64no—
110C3×C4—91no—
111C3×C4—91no—
112C3×C4—181no—
113C3×C4—182no—
114C3×C4#379212yes—
115Dic3—62no—
116Dic3—62no—
117Dic3—122no—
118Dic3—124no—
119C32—22no—
120Q8—22no—
121C2×C4—91no—
122C2×C4—91no—
123C2×C4—92no—
124C2×C4C2×#108812yes—
125C2×C4C2×#108811yescharacteristic
126C8—92no—
127C8—181no—
128C8—182no—
129C2×C3—62no—
130C2×C3—62no—
131C2×C3—62no—
132C2×C3#6428911yescharacteristic
133C2×C3Q8×#15711yescharacteristic
134C2×C3C2×#379211yescharacteristic
135C22C22×#108811yescharacteristic
136C4—22no—
137C4—91no—
138C4—91no—
139C4—181no—
140C4—182no—
141C4#8863112yes—
142C3—62no—
143C3#1059383511yescharacteristic
144C2#1172245111yescharacteristic
145C2Q8×#108811yescharacteristic
146C2C2×#8863111yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutator, prime radicalC22×H(3,𝔽3)C22×C4
FittingC2×Q8×H(3,𝔽3)C4
cosocleC2×#1084C23
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#446
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
23. #6188Q8semidirect
78. H(3,𝔽3)#446semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
#117207852
#117207832
C4⋊#61882
C4×#61882
C3×#4469