Finite Group Databasebeta

← 3−1+2×#353 #13136062

Subgroups — 3−1+2×#353 #13136062

Order 1728.

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

normal subgroups only

This group has 344 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×#127C211yesmaximal, characteristic
23−1+2×#99C212yesmaximal, complemented
3C32×#353C311yesmaximal, characteristic
4C9×#353C313yesmaximal, complemented
5C2×D4×3−1+2—22no—
6C2×D4×3−1+2C411yescharacteristic, complemented
7C2×D4×3−1+2C2211yescharacteristic
8C22⋊C4×3−1+2—22no—
9C42×3−1+2C411yescharacteristic, complemented
10C32×#127C2×C311yescharacteristic
11C32×#99C2×C312yes—
12C9×#127C2×C313yes—
13C9×#99C2×C316yescomplemented
14C23×3−1+2—21no—
15C23×3−1+2D412yes—
16D4×3−1+2—41no—
17D4×3−1+2—44no—
18D4×3−1+2—41no—
19C2×C4×3−1+2—21no—
20C2×C4×3−1+2—42no—
21C2×C4×3−1+2C2×C411yescharacteristic
22C3×#353—31no—
23C3×#353C3211yescharacteristic
24C2×C32×D4—22no—
25C2×C32×D4C3×C411yescharacteristic
26C2×C32×D4C22×C311yescharacteristic
27C32×C22⋊C4—22no—
28C32×C42C3×C411yescharacteristic
29C2×D4×C9—26no—
30C2×D4×C9C3×C413yescomplemented
31C2×D4×C9C22×C313yes—
32C9×C22⋊C4—26no—
33C42×C9C3×C413yescomplemented
34C22×3−1+2—22no—
35C22×3−1+2—42no—
36C22×3−1+2—42no—
37C22×3−1+2—41no—
38C22×3−1+2C22⋊C411yescharacteristic
39C4×3−1+2—22no—
40C4×3−1+2—21no—
41C4×3−1+2—82no—
42C3×#127—31no—
43C3×#127C2×C3211yescharacteristic
44C3×#99—32no—
45C3×#99C2×C3212yes—
46C23×C32—21no—
47C23×C32C3×D412yes—
48C32×D4—41no—
49C32×D4—41no—
50C32×D4—44no—
51C2×C32×C4—21no—
52C2×C32×C4—42no—
53C2×C32×C4C2×C3×C411yescharacteristic
54C23×C9—23no—
55C23×C9C3×D416yes—
56D4×C9—43no—
57D4×C9—43no—
58D4×C9—412no—
59C2×C4×C9—23no—
60C2×C4×C9—46no—
61C2×C4×C9C2×C3×C413yes—
62#3533−1+211yescharacteristic, direct product factor, Sylow
63C2×3−1+2—21no—
64C2×3−1+2—42no—
65C2×3−1+2—81no—
66C2×3−1+2#9911yescharacteristic
67C2×C3×D4—22no—
68C2×C3×D4—31no—
69C2×C3×D4—31no—
70C2×C3×D4—62no—
71C2×C3×D4C32×C411yescharacteristic
72C2×C3×D4C22×C3211yescharacteristic
73C3×C22⋊C4—22no—
74C3×C22⋊C4—62no—
75C3×C42—31no—
76C3×C42C32×C411yescharacteristic
77C22×C32—22no—
78C22×C32—42no—
79C22×C32—42no—
80C22×C32—41no—
81C22×C32C3×C22⋊C411yescharacteristic
82C32×C4—22no—
83C32×C4—21no—
84C32×C4—82no—
85C22×C9—26no—
86C22×C9—46no—
87C22×C9—46no—
88C22×C9—43no—
89C22×C9C3×C22⋊C413yes—
90C4×C9—26no—
91C4×C9—23no—
92C4×C9—86no—
93#127C2×3−1+211yescharacteristic
94#99C2×3−1+212yescomplemented
953−1+2#35311yescharacteristic, direct product factor, Sylow
96C23×C3—21no—
97C23×C3—32no—
98C23×C3—61no—
99C23×C3C32×D412yes—
100C3×D4—41no—
101C3×D4—41no—
102C3×D4—44no—
103C3×D4—121no—
104C3×D4—121no—
105C3×D4—124no—
106C2×C3×C4—21no—
107C2×C3×C4—31no—
108C2×C3×C4—42no—
109C2×C3×C4—61no—
110C2×C3×C4—122no—
111C2×C3×C4C2×C32×C411yescharacteristic
112C2×C32—21no—
113C2×C32—42no—
114C2×C32—81no—
115C2×C32C3×#9911yescharacteristic
116C2×C9—23no—
117C2×C9—46no—
118C2×C9—83no—
119C2×C9C3×#9913yes—
120C2×D4—22no—
121C2×D4C4×3−1+211yescharacteristic, complemented
122C2×D4C22×3−1+211yescharacteristic
123C22⋊C4—22no—
124C42C4×3−1+211yescharacteristic, complemented
125C22×C3—22no—
126C22×C3—31no—
127C22×C3—42no—
128C22×C3—42no—
129C22×C3—41no—
130C22×C3—62no—
131C22×C3—121no—
132C22×C3—122no—
133C22×C3—122no—
134C22×C3C32×C22⋊C411yescharacteristic
135C3×C4—22no—
136C3×C4—21no—
137C3×C4—61no—
138C3×C4—62no—
139C3×C4—82no—
140C3×C4—242no—
141C32C3×#35311yescharacteristic
142C9C3×#35313yescomplemented
143C23—21no—
144C23D4×3−1+212yes—
145D4—41no—
146D4—41no—
147D4—44no—
148C2×C4—21no—
149C2×C4—42no—
150C2×C4C2×C4×3−1+211yescharacteristic
151C2×C3—21no—
152C2×C3—31no—
153C2×C3—42no—
154C2×C3—61no—
155C2×C3—81no—
156C2×C3—122no—
157C2×C3—241no—
158C2×C3C32×#9911yescharacteristic
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22C22⋊C4×3−1+211yescharacteristic
164C4—22no—
165C4—21no—
166C4—82no—
167C3—31no—
168C3C32×#35311yescharacteristic
169C2—21no—
170C2—42no—
171C2—81no—
172C23−1+2×#9911yescharacteristic
subgroupG/H
center, socleC2×C3C32×#99
commutatorC2×C3×C4C2×C32×C4
Frattini, cosocleC2×C3×D4C22×C32
prime radicalC32×#127C2×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
2. 3−1+2×#99C2semidirect
4. C9×#353C3semidirect
6. C2×D4×3−1+2C4semidirect
9. C42×3−1+2C4semidirect
13. C9×#99C2×C3semidirect
30. C2×D4×C9C3×C4semidirect
33. C42×C9C3×C4semidirect
62. #3533−1+2direct (N × H)
94. #99C2×3−1+2semidirect
95. 3−1+2#353direct (N × H)
121. C2×D4C4×3−1+2semidirect
124. C42C4×3−1+2semidirect
142. C9C3×#353semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1272
3−1+2×#992
C32×#3533
C9×#3533