Finite Group Databasebeta

← Q8⋊#6188 #13136953

Subgroups — Q8⋊#6188 #13136953

Order 1728.

245 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 43 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
2C4⋊#6188C211yesmaximal, characteristic
3C4×#6188C211yesmaximal, characteristic
4C2×Q8×H(3,𝔽3)C411yescharacteristic
5C2×#6188—21no—
6C2×#6188—21no—
7C4⋊#1084—21no—
8C4⋊#1084C411yescharacteristic
9C4×#1084—21no—
10C4×#1084C2211yescharacteristic
11C3×Q8×Dic3—62no—
12Q8×H(3,𝔽3)—21no—
13Q8×H(3,𝔽3)C812yescomplemented
14C2×C4×H(3,𝔽3)—21no—
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16C2×#1084—21no—
17C2×#1084D411yescharacteristic
18C2×#1084D411yescharacteristic
19#6188—22no—
20#6188—41no—
21C3×Q8⋊C8—91nomaximal
22C2×C32×Q8—22no—
23C3×C4⋊Dic3—62no—
24C3×C4⋊Dic3—64no—
25C3×C4×Dic3—62no—
26C3×C4×Dic3—64no—
27C22×H(3,𝔽3)C22⋊C411yescharacteristic
28C4×H(3,𝔽3)—22no—
29C4×H(3,𝔽3)C2×C811yescharacteristic
30C4×H(3,𝔽3)M1611yescharacteristic
31#1084—21no—
32#1084—41no—
33#1084SD1611yescharacteristic
34#1084Q1611yescharacteristic
35C3×C4×Q8—91no—
36Q8×Dic3—62no—
37C3×C4⋊C8—91no—
38C3×C4×C8—91no—
39C32×Q8—24no—
40C32×Q8—24no—
41C2×C32×C4—24no—
42C2×C32×C4—22no—
43C2×C3×Dic3—62no—
44C2×C3×Dic3—62no—
45C2×C3×Dic3—64no—
46Q8⋊C8—91noSylow
47C2×H(3,𝔽3)C22⋊C811yescharacteristic
48C2×H(3,𝔽3)#10311yescharacteristic
49C2×H(3,𝔽3)#10411yescharacteristic
50C2×C3×Q8—62no—
51C2×C3×Q8#15711yescharacteristic
52C2×C3×C8—181no—
53C2×C3×C8—181no—
54C3×C4⋊C4—91no—
55C3×C4⋊C4—181no—
56C3×C42—91no—
57C3×C42—181no—
58C4⋊Dic3—62no—
59C4⋊Dic3—64no—
60C4×Dic3—62no—
61C4×Dic3—64no—
62C22×C32—22no—
63C32×C4—28no—
64C32×C4—22no—
65C32×C4—22no—
66C3×Dic3—62no—
67C3×Dic3—62no—
68C3×Dic3—122no—
69C3×Dic3—124no—
70C4×Q8—91no—
71C4⋊C8—91no—
72C4×C8—91no—
73H(3,𝔽3)Q8⋊C811yescharacteristic, complemented, Sylow
74C3×Q8—21no—
75C3×Q8—64no—
76C3×Q8—64no—
77C3×Q8#62212yes—
78C2×C3×C4—21no—
79C2×C3×C4—64no—
80C2×C3×C4—62no—
81C2×C3×C4—91no—
82C2×C3×C4—91no—
83C2×C3×C4—181no—
84C2×C3×C4C2×#15711yescharacteristic
85C2×Dic3—62no—
86C2×Dic3—62no—
87C2×Dic3—64no—
88C3×C8—182no—
89C3×C8—361no—
90C2×C32—22no—
91C2×C32—22no—
92C2×C32—22no—
93C2×Q8#108811yescharacteristic
94C2×C8—181no—
95C2×C8—181no—
96C4⋊C4—91no—
97C4⋊C4—181no—
98C42—91no—
99C42—181no—
100C22×C3—62no—
101C22×C3C22⋊#15711yescharacteristic
102C3×C4—22no—
103C3×C4—68no—
104C3×C4—62no—
105C3×C4—62no—
106C3×C4—91no—
107C3×C4—91no—
108C3×C4—181no—
109C3×C4—361no—
110C3×C4C2×#62211yescharacteristic
111C3×C4#379211yescharacteristic
112Dic3—62no—
113Dic3—62no—
114Dic3—122no—
115Dic3—124no—
116C32—22no—
117Q8—21no—
118Q8#618812yescomplemented
119C2×C4—21no—
120C2×C4—91no—
121C2×C4—91no—
122C2×C4—181no—
123C2×C4C2×#108811yescharacteristic
124C8—182no—
125C8—361no—
126C2×C3—62no—
127C2×C3—62no—
128C2×C3—62no—
129C2×C3#6378411yescharacteristic
130C2×C3#6378511yescharacteristic
131C2×C3C22⋊#62211yescharacteristic
132C22C22⋊#108811yescharacteristic
133C4—22no—
134C4—91no—
135C4—91no—
136C4—181no—
137C4—361no—
138C4C2×#618811yescharacteristic
139C4#8863111yescharacteristic
140C3—62no—
141C3Q8⋊#62211yescharacteristic
142C2#1172079011yescharacteristic
143C2#1172079111yescharacteristic
144C2C22⋊#618811yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC2×C3×C4C2×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC4×#1084C22
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)Q8⋊C8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
13. Q8×H(3,𝔽3)C8semidirect
73. H(3,𝔽3)Q8⋊C8semidirect
118. Q8#6188semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
C4⋊#61882
C4×#61882
C3×Q8⋊C89