Finite Group Databasebeta

← D4⋊#6188 #13136947

Subgroups — D4⋊#6188 #13136947

Order 1728.

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

normal subgroups only

This group has 311 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
1D4×#1084C211yesmaximal, characteristic
2C4⋊#6188C211yesmaximal, characteristic, complemented
3C4×#6188C211yesmaximal, characteristic, complemented
4C2×D4×H(3,𝔽3)C411yescharacteristic
5C22×#1084—21no—
6C2×#6188—21no—
7C2×#6188—21no—
8#88552—21no—
9C4⋊#1084C411yescharacteristic
10C4×#1084C2211yescharacteristic
11C3×D4×Dic3—62no—
12C23×H(3,𝔽3)—21no—
13D4×H(3,𝔽3)—21no—
14D4×H(3,𝔽3)C812yescomplemented
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16C2×#1084—21no—
17C2×#1084—41no—
18C2×#1084—41no—
19C2×#1084D411yescharacteristic
20C2×#1084D411yescharacteristic
21#6188—22no—
22#6188—41no—
23C3×D4⋊C8—91nomaximal
24C2×C32×D4—22no—
25C22×C3×Dic3—64no—
26C3×#217—64no—
27C3×C4⋊Dic3—62no—
28C3×C4×Dic3—62no—
29C22×H(3,𝔽3)—22no—
30C22×H(3,𝔽3)—41no—
31C22×H(3,𝔽3)—41no—
32C22×H(3,𝔽3)C22⋊C411yescharacteristic
33C4×H(3,𝔽3)C2×C811yescharacteristic
34C4×H(3,𝔽3)M1611yescharacteristic
35#1084—21no—
36#1084—41no—
37#1084D811yescharacteristic
38#1084SD1611yescharacteristic
39C3×C4×D4—91no—
40D4×Dic3—62no—
41C3×C4⋊C8—91no—
42C3×C4×C8—91no—
43C23×C32—24no—
44C32×D4—24no—
45C32×D4—24no—
46C2×C32×C4—22no—
47C2×C3×Dic3—62no—
48C2×C3×Dic3—62no—
49C2×C3×Dic3—64no—
50C2×C3×Dic3—124no—
51C2×C3×Dic3—124no—
52D4⋊C8—91noSylow
53C2×H(3,𝔽3)—42no—
54C2×H(3,𝔽3)C22⋊C811yescharacteristic
55C2×H(3,𝔽3)#10211yescharacteristic
56C2×H(3,𝔽3)#10411yescharacteristic
57C2×C3×D4—62no—
58C2×C3×D4#15711yescharacteristic
59C22×C3×C4—181no—
60C22×Dic3—64no—
61C2×C3×C8—181no—
62C2×C3×C8—181no—
63C3×C4⋊C4—91no—
64C3×C22⋊C4—181no—
65C3×C42—91no—
66#217—64no—
67C4⋊Dic3—62no—
68C4×Dic3—62no—
69C22×C32—28no—
70C22×C32—22no—
71C22×C32—44no—
72C22×C32—44no—
73C32×C4—22no—
74C32×C4—22no—
75C3×Dic3—62no—
76C3×Dic3—62no—
77C3×Dic3—122no—
78C3×Dic3—124no—
79C4×D4—91no—
80C4⋊C8—91no—
81C4×C8—91no—
82H(3,𝔽3)D4⋊C811yescharacteristic, complemented, Sylow
83C23×C3—21no—
84C23×C3—64no—
85C3×D4—21no—
86C3×D4—64no—
87C3×D4—64no—
88C3×D4#62212yes—
89C2×C3×C4—62no—
90C2×C3×C4—91no—
91C2×C3×C4—91no—
92C2×C3×C4—181no—
93C2×C3×C4—361no—
94C2×C3×C4—361no—
95C2×C3×C4C2×#15711yescharacteristic
96C2×Dic3—62no—
97C2×Dic3—62no—
98C2×Dic3—64no—
99C2×Dic3—124no—
100C2×Dic3—124no—
101C3×C8—182no—
102C3×C8—361no—
103C2×C32—22no—
104C2×C32—22no—
105C2×C32—22no—
106C2×C32—48no—
107C2×D4#108811yescharacteristic
108C22×C4—181no—
109C2×C8—181no—
110C2×C8—181no—
111C4⋊C4—91no—
112C22⋊C4—181no—
113C42—91no—
114C22×C3—22no—
115C22×C3—41no—
116C22×C3—41no—
117C22×C3—68no—
118C22×C3—62no—
119C22×C3—124no—
120C22×C3—124no—
121C22×C3C22⋊#15711yescharacteristic
122C3×C4—62no—
123C3×C4—62no—
124C3×C4—91no—
125C3×C4—91no—
126C3×C4—181no—
127C3×C4—361no—
128C3×C4C2×#62211yescharacteristic
129C3×C4#379211yescharacteristic
130Dic3—62no—
131Dic3—62no—
132Dic3—122no—
133Dic3—124no—
134C32—22no—
135C23—21no—
136D4—21no—
137D4#618812yescomplemented
138C2×C4—91no—
139C2×C4—91no—
140C2×C4—181no—
141C2×C4—361no—
142C2×C4—361no—
143C2×C4C2×#108811yescharacteristic
144C8—182no—
145C8—361no—
146C2×C3—42no—
147C2×C3—62no—
148C2×C3—62no—
149C2×C3—62no—
150C2×C3—128no—
151C2×C3#6378211yescharacteristic
152C2×C3#6378311yescharacteristic
153C2×C3C22⋊#62211yescharacteristic
154C22—22no—
155C22—41no—
156C22—41no—
157C22C22⋊#108811yescharacteristic
158C4—91no—
159C4—91no—
160C4—181no—
161C4—361no—
162C4C2×#618811yescharacteristic
163C4#8863111yescharacteristic
164C3—62no—
165C3D4⋊#62211yescharacteristic
166C2—42no—
167C2#1172078811yescharacteristic
168C2#1172078911yescharacteristic
169C2C22⋊#618811yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC2×C3×C4C2×#157
Fitting, prime radicalC2×D4×H(3,𝔽3)C4
cosocleC4×#1084C22
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)D4⋊C8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C4⋊#6188C2semidirect
3. C4×#6188C2semidirect
14. D4×H(3,𝔽3)C8semidirect
82. H(3,𝔽3)D4⋊C8semidirect
137. D4#6188semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
C4⋊#61882
C4×#61882
C3×D4⋊C89