Finite Group Databasebeta

← 3−1+2×#379 #13144897

Subgroups — 3−1+2×#379 #13144897

Order 1728.

2024 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 2024 conjugacy classes of subgroups — too many to draw. Only its 455 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
1C25×3−1+2C211yesmaximal, characteristic
2C2×C22⋊C4×3−1+2C216yesmaximal, complemented
3C32×#379C311yesmaximal, characteristic
4C9×#379C313yesmaximal, complemented
5C24×3−1+2—212no—
6C24×3−1+2C414yescomplemented
7C24×3−1+2C2213yes—
8C22×C4×3−1+2C2214yescomplemented
9C22⋊C4×3−1+2—212no—
10C25×C32C2×C311yescharacteristic
11C25×C9C2×C313yes—
12C2×C32×C22⋊C4C2×C316yes—
13C2×C9×C22⋊C4C2×C3118yescomplemented
14C23×3−1+2—232no—
15C23×3−1+2—224no—
16C23×3−1+2—212no—
17C23×3−1+2C2×C416yes—
18C23×3−1+2D4112yes—
19C23×3−1+2C2311yescharacteristic
20C2×C4×3−1+2—212no—
21C3×#379—31no—
22C3×#379C3211yescharacteristic
23C24×C32—212no—
24C24×C32C3×C414yes—
25C24×C32C22×C313yes—
26C22×C32×C4C22×C314yes—
27C24×C9—236no—
28C24×C9C3×C4112yescomplemented
29C24×C9C22×C319yes—
30C32×C22⋊C4—212no—
31C22×C4×C9C22×C3112yescomplemented
32C9×C22⋊C4—236no—
33C22×3−1+2—232no—
34C22×3−1+2—212no—
35C22×3−1+2—224no—
36C22×3−1+2C22⋊C4112yescomplemented
37C22×3−1+2C22×C411yescharacteristic
38C22×3−1+2C2×D416yes—
39C4×3−1+2—44no—
40C25×C3—31no—
41C25×C3C2×C3211yescharacteristic
42C2×C3×C22⋊C4—36no—
43C2×C3×C22⋊C4C2×C3216yes—
44C23×C32—232no—
45C23×C32—224no—
46C23×C32—212no—
47C23×C32C2×C3×C416yes—
48C23×C32C3×D4112yes—
49C23×C32C23×C311yescharacteristic
50C2×C32×C4—212no—
51C23×C9—296no—
52C23×C9—272no—
53C23×C9—236no—
54C23×C9C2×C3×C4118yes—
55C23×C9C3×D4136yes—
56C23×C9C23×C313yes—
57C2×C4×C9—236no—
58#3793−1+211yescharacteristic, direct product factor, Sylow
59C2×3−1+2—212no—
60C2×3−1+2C2×C22⋊C413yes—
61C2×3−1+2#12014yes—
62C24×C3—212no—
63C24×C3—34no—
64C24×C3—33no—
65C24×C3—612no—
66C24×C3C32×C414yes—
67C24×C3C22×C3213yes—
68C22×C3×C4—34no—
69C22×C3×C4C22×C3214yes—
70C3×C22⋊C4—212no—
71C3×C22⋊C4—612no—
72C22×C32—232no—
73C22×C32—212no—
74C22×C32—224no—
75C22×C32C3×C22⋊C4112yes—
76C22×C32C22×C3×C411yescharacteristic
77C22×C32C2×C3×D416yes—
78C32×C4—44no—
79C22×C9—296no—
80C22×C9—236no—
81C22×C9—272no—
82C22×C9C3×C22⋊C4136yescomplemented
83C22×C9C22×C3×C413yes—
84C22×C9C2×C3×D4118yes—
85C4×C9—412no—
86C25C2×3−1+211yescharacteristic
87C2×C22⋊C4C2×3−1+216yescomplemented
883−1+2#37911yescharacteristic, direct product factor, Sylow
89C23×C3—232no—
90C23×C3—224no—
91C23×C3—212no—
92C23×C3—36no—
93C23×C3—312no—
94C23×C3—31no—
95C23×C3—612no—
96C23×C3—624no—
97C23×C3—632no—
98C23×C3C2×C32×C416yes—
99C23×C3C32×D4112yes—
100C23×C3C23×C3211yescharacteristic
101C2×C3×C4—212no—
102C2×C3×C4—612no—
103C2×C32—212no—
104C2×C32C2×C3×C22⋊C413yes—
105C2×C32C3×#12014yes—
106C2×C9—236no—
107C2×C9C2×C3×C22⋊C419yes—
108C2×C9C3×#120112yes—
109C24—212no—
110C24C4×3−1+214yescomplemented
111C24C22×3−1+213yes—
112C22×C4C22×3−1+214yescomplemented
113C22⋊C4—212no—
114C22×C3—232no—
115C22×C3—212no—
116C22×C3—224no—
117C22×C3—31no—
118C22×C3—36no—
119C22×C3—312no—
120C22×C3—612no—
121C22×C3—624no—
122C22×C3—632no—
123C22×C3C32×C22⋊C4112yes—
124C22×C3C22×C32×C411yescharacteristic
125C22×C3C2×C32×D416yes—
126C3×C4—44no—
127C3×C4—124no—
128C32C3×#37911yescharacteristic
129C9C3×#37913yescomplemented
130C23—232no—
131C23—212no—
132C23—224no—
133C23C2×C4×3−1+216yes—
134C23D4×3−1+2112yes—
135C23C23×3−1+211yescharacteristic
136C2×C4—212no—
137C2×C3—212no—
138C2×C3—33no—
139C2×C3—34no—
140C2×C3—612no—
141C2×C3C2×C32×C22⋊C413yes—
142C2×C3C32×#12014yes—
143C22—232no—
144C22—212no—
145C22—224no—
146C22C22⋊C4×3−1+2112yescomplemented
147C22C22×C4×3−1+211yescharacteristic
148C22C2×D4×3−1+216yes—
149C4—44no—
150C3—31no—
151C3C32×#37911yescharacteristic
152C2—212no—
153C2C2×C22⋊C4×3−1+213yes—
154C23−1+2×#12014yes—
subgroupG/H
center, Frattini, socle, cosocleC23×C3C23×C32
commutatorC22×C3C22×C32×C4
prime radicalC25×C32C2×C3
LCSG ▷ C22×C3 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×3−1+2C2semidirect
4. C9×#379C3semidirect
6. C24×3−1+2C4semidirect
8. C22×C4×3−1+2C22semidirect
13. C2×C9×C22⋊C4C2×C3semidirect
28. C24×C9C3×C4semidirect
31. C22×C4×C9C22×C3semidirect
36. C22×3−1+2C22⋊C4semidirect
58. #3793−1+2direct (N × H)
82. C22×C9C3×C22⋊C4semidirect
87. C2×C22⋊C4C2×3−1+2semidirect
88. 3−1+2#379direct (N × H)
110. C24C4×3−1+2semidirect
112. C22×C4C22×3−1+2semidirect
129. C9C3×#379semidirect
146. C22C22⋊C4×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C25×3−1+22
C2×C22⋊C4×3−1+22
C32×#3793
C9×#3793