Finite Group Databasebeta

← 3−1+2×#354 #13136065

Subgroups — 3−1+2×#354 #13136065

Order 1728.

280 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 91 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
13−1+2×#124C211yesmaximal, characteristic
23−1+2×#99C212yesmaximal
3C32×#354C311yesmaximal, characteristic
4C9×#354C313yesmaximal, complemented
5C2×Q8×3−1+2C411yescharacteristic, complemented
6C2×D4×3−1+2C2211yescharacteristic
7C22⋊C4×3−1+2—22no—
8C22⋊C4×3−1+2—22no—
9C42×3−1+2C411yescharacteristic, complemented
10C32×#124C2×C311yescharacteristic
11C32×#99C2×C312yes—
12C9×#124C2×C313yes—
13C9×#99C2×C316yes—
14C23×3−1+2D412yes—
15Q8×3−1+2—41no—
16D4×3−1+2—41no—
17C2×C4×3−1+2—21no—
18C2×C4×3−1+2—21no—
19C2×C4×3−1+2—42no—
20C2×C4×3−1+2C2×C411yescharacteristic
21C3×#354—31no—
22C3×#354C3211yescharacteristic
23C2×C32×Q8C3×C411yescharacteristic
24C2×C32×D4C22×C311yescharacteristic
25C32×C22⋊C4—22no—
26C32×C22⋊C4—22no—
27C32×C42C3×C411yescharacteristic
28C2×Q8×C9C3×C413yescomplemented
29C2×D4×C9C22×C313yes—
30C9×C22⋊C4—26no—
31C9×C22⋊C4—26no—
32C42×C9C3×C413yescomplemented
33C22×3−1+2—22no—
34C22×3−1+2—42no—
35C22×3−1+2C22⋊C411yescharacteristic
36C4×3−1+2—21no—
37C4×3−1+2—41no—
38C4×3−1+2—41no—
39C4×3−1+2—82no—
40C3×#124—31no—
41C3×#124C2×C3211yescharacteristic
42C3×#99—32no—
43C3×#99C2×C3212yes—
44C23×C32C3×D412yes—
45C32×Q8—41no—
46C32×D4—41no—
47C2×C32×C4—21no—
48C2×C32×C4—21no—
49C2×C32×C4—42no—
50C2×C32×C4C2×C3×C411yescharacteristic
51C23×C9C3×D416yes—
52Q8×C9—43no—
53D4×C9—43no—
54C2×C4×C9—23no—
55C2×C4×C9—23no—
56C2×C4×C9—46no—
57C2×C4×C9C2×C3×C413yes—
58#3543−1+211yescharacteristic, direct product factor, Sylow
59C2×3−1+2—21no—
60C2×3−1+2—42no—
61C2×3−1+2#9911yescharacteristic
62C2×C3×Q8—31no—
63C2×C3×Q8C32×C411yescharacteristic
64C2×C3×D4—31no—
65C2×C3×D4C22×C3211yescharacteristic
66C3×C22⋊C4—22no—
67C3×C22⋊C4—22no—
68C3×C22⋊C4—62no—
69C3×C22⋊C4—62no—
70C3×C42—31no—
71C3×C42C32×C411yescharacteristic
72C22×C32—22no—
73C22×C32—42no—
74C22×C32C3×C22⋊C411yescharacteristic
75C32×C4—21no—
76C32×C4—41no—
77C32×C4—41no—
78C32×C4—82no—
79C22×C9—26no—
80C22×C9—46no—
81C22×C9C3×C22⋊C413yes—
82C4×C9—23no—
83C4×C9—43no—
84C4×C9—43no—
85C4×C9—86no—
86#124C2×3−1+211yescharacteristic
87#99C2×3−1+212yes—
883−1+2#35411yescharacteristic, direct product factor, Sylow
89C23×C3—32no—
90C23×C3C32×D412yes—
91C3×Q8—41no—
92C3×Q8—121no—
93C3×D4—41no—
94C3×D4—121no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—31no—
98C2×C3×C4—42no—
99C2×C3×C4—61no—
100C2×C3×C4—61no—
101C2×C3×C4—122no—
102C2×C3×C4C2×C32×C411yescharacteristic
103C2×C32—21no—
104C2×C32—42no—
105C2×C32C3×#9911yescharacteristic
106C2×C9—23no—
107C2×C9—46no—
108C2×C9C3×#9913yes—
109C2×Q8C4×3−1+211yescharacteristic, complemented
110C2×D4C22×3−1+211yescharacteristic
111C22⋊C4—22no—
112C22⋊C4—22no—
113C42C4×3−1+211yescharacteristic, complemented
114C22×C3—22no—
115C22×C3—31no—
116C22×C3—42no—
117C22×C3—62no—
118C22×C3—122no—
119C22×C3C32×C22⋊C411yescharacteristic
120C3×C4—21no—
121C3×C4—41no—
122C3×C4—41no—
123C3×C4—61no—
124C3×C4—82no—
125C3×C4—121no—
126C3×C4—121no—
127C3×C4—242no—
128C32C3×#35411yescharacteristic
129C9C3×#35413yescomplemented
130C23D4×3−1+212yes—
131Q8—41no—
132D4—41no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4C2×C4×3−1+211yescharacteristic
137C2×C3—21no—
138C2×C3—31no—
139C2×C3—42no—
140C2×C3—61no—
141C2×C3—122no—
142C2×C3C32×#9911yescharacteristic
143C22—22no—
144C22—42no—
145C22C22⋊C4×3−1+211yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—82no—
150C3—31no—
151C3C32×#35411yescharacteristic
152C2—21no—
153C2—42no—
154C23−1+2×#9911yescharacteristic
subgroupG/H
center, socleC2×C3C32×#99
commutatorC2×C3×C4C2×C32×C4
Frattini, cosocleC2×C3×D4C22×C32
prime radicalC2×C32×D4C22×C3
LCSG ▷ C2×C3×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×3−1+2 ◁ C2×D4×3−1+2 ◁ G

Complements

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

NH ≅ G/Ntype
4. C9×#354C3semidirect
5. C2×Q8×3−1+2C4semidirect
9. C42×3−1+2C4semidirect
28. C2×Q8×C9C3×C4semidirect
32. C42×C9C3×C4semidirect
58. #3543−1+2direct (N × H)
88. 3−1+2#354direct (N × H)
109. C2×Q8C4×3−1+2semidirect
113. C42C4×3−1+2semidirect
129. C9C3×#354semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1242
3−1+2×#992
C32×#3543
C9×#3543