Finite Group Databasebeta

← 3−1+2×#352 #13136059

Subgroups — 3−1+2×#352 #13136059

Order 1728.

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

normal subgroups only

This group has 272 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×#123C211yesmaximal, characteristic
23−1+2×#100C211yesmaximal, characteristic
33−1+2×#99C211yesmaximal, characteristic
4C32×#352C311yesmaximal, characteristic
5C9×#352C313yesmaximal, complemented
6C2×D4×3−1+2C2211yescharacteristic
7C22×C4×3−1+2C411yescharacteristic, complemented
8M16×3−1+2—21no—
9C4⋊C4×3−1+2—21no—
10C22⋊C4×3−1+2—21no—
11C22⋊C4×3−1+2—21no—
12C22⋊C4×3−1+2C411yescharacteristic, complemented
13C32×#123C2×C311yescharacteristic
14C32×#100C2×C311yescharacteristic
15C32×#99C2×C311yescharacteristic
16C9×#123C2×C313yes—
17C9×#100C2×C313yes—
18C9×#99C2×C313yes—
19C23×3−1+2C2×C411yescharacteristic
20C23×3−1+2D411yescharacteristic
21D4×3−1+2—41no—
22C2×C4×3−1+2—21no—
23C2×C4×3−1+2—21no—
24C2×C4×3−1+2—41no—
25C2×C4×3−1+2—41no—
26C2×C4×3−1+2D411yescharacteristic
27C8×3−1+2—41no—
28C3×#352—31no—
29C3×#352C3211yescharacteristic
30C2×C32×D4C22×C311yescharacteristic
31C22×C32×C4C3×C411yescharacteristic
32C32×M16—21no—
33C32×C4⋊C4—21no—
34C32×C22⋊C4—21no—
35C32×C22⋊C4—21no—
36C32×C22⋊C4C3×C411yescharacteristic
37C2×D4×C9C22×C313yes—
38C22×C4×C9C3×C413yescomplemented
39C9×M16—23no—
40C9×C4⋊C4—23no—
41C9×C22⋊C4—23no—
42C9×C22⋊C4—23no—
43C9×C22⋊C4C3×C413yescomplemented
44C22×3−1+2—21no—
45C22×3−1+2—21no—
46C22×3−1+2—41no—
47C22×3−1+2—41no—
48C22×3−1+2C22⋊C411yescharacteristic
49C4×3−1+2—21no—
50C4×3−1+2—41no—
51C4×3−1+2—41no—
52C4×3−1+2—81no—
53C3×#123—31no—
54C3×#123C2×C3211yescharacteristic
55C3×#100—31no—
56C3×#100C2×C3211yescharacteristic
57C3×#99—31no—
58C3×#99C2×C3211yescharacteristic
59C23×C32C2×C3×C411yescharacteristic
60C23×C32C3×D411yescharacteristic
61C32×D4—41no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4—41no—
65C2×C32×C4—41no—
66C2×C32×C4C3×D411yescharacteristic
67C23×C9C2×C3×C413yes—
68C23×C9C3×D413yes—
69C32×C8—41no—
70D4×C9—43no—
71C2×C4×C9—23no—
72C2×C4×C9—23no—
73C2×C4×C9—43no—
74C2×C4×C9—43no—
75C2×C4×C9C3×D413yes—
76C8×C9—43no—
77#3523−1+211yescharacteristic, direct product factor, Sylow
78C2×3−1+2—21no—
79C2×3−1+2—41no—
80C2×3−1+2—41no—
81C2×3−1+2#9911yescharacteristic
82C2×C3×D4—31no—
83C2×C3×D4C22×C3211yescharacteristic
84C22×C3×C4—31no—
85C22×C3×C4C32×C411yescharacteristic
86C3×M16—21no—
87C3×M16—61no—
88C3×C4⋊C4—21no—
89C3×C4⋊C4—61no—
90C3×C22⋊C4—21no—
91C3×C22⋊C4—21no—
92C3×C22⋊C4—31no—
93C3×C22⋊C4—61no—
94C3×C22⋊C4—61no—
95C3×C22⋊C4C32×C411yescharacteristic
96C22×C32—21no—
97C22×C32—21no—
98C22×C32—41no—
99C22×C32—41no—
100C22×C32C3×C22⋊C411yescharacteristic
101C32×C4—21no—
102C32×C4—41no—
103C32×C4—41no—
104C32×C4—81no—
105C22×C9—23no—
106C22×C9—23no—
107C22×C9—43no—
108C22×C9—43no—
109C22×C9C3×C22⋊C413yes—
110C4×C9—23no—
111C4×C9—43no—
112C4×C9—43no—
113C4×C9—83no—
114#123C2×3−1+211yescharacteristic
115#100C2×3−1+211yescharacteristic
116#99C2×3−1+211yescharacteristic
1173−1+2#35211yescharacteristic, direct product factor, Sylow
118C23×C3—31no—
119C23×C3—31no—
120C23×C3C2×C32×C411yescharacteristic
121C23×C3C32×D411yescharacteristic
122C3×D4—41no—
123C3×D4—121no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—31no—
127C2×C3×C4—41no—
128C2×C3×C4—41no—
129C2×C3×C4—61no—
130C2×C3×C4—61no—
131C2×C3×C4—121no—
132C2×C3×C4—121no—
133C2×C3×C4C32×D411yescharacteristic
134C3×C8—41no—
135C3×C8—121no—
136C2×C32—21no—
137C2×C32—41no—
138C2×C32—41no—
139C2×C32C3×#9911yescharacteristic
140C2×C9—23no—
141C2×C9—43no—
142C2×C9—43no—
143C2×C9C3×#9913yes—
144C2×D4C22×3−1+211yescharacteristic
145C22×C4C4×3−1+211yescharacteristic, complemented
146M16—21no—
147C4⋊C4—21no—
148C22⋊C4—21no—
149C22⋊C4—21no—
150C22⋊C4C4×3−1+211yescharacteristic, complemented
151C22×C3—21no—
152C22×C3—21no—
153C22×C3—31no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—61no—
157C22×C3—61no—
158C22×C3—121no—
159C22×C3—121no—
160C22×C3C32×C22⋊C411yescharacteristic
161C3×C4—21no—
162C3×C4—41no—
163C3×C4—41no—
164C3×C4—61no—
165C3×C4—81no—
166C3×C4—121no—
167C3×C4—121no—
168C3×C4—241no—
169C32C3×#35211yescharacteristic
170C9C3×#35213yescomplemented
171C23C2×C4×3−1+211yescharacteristic
172C23D4×3−1+211yescharacteristic
173D4—41no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4D4×3−1+211yescharacteristic
179C8—41no—
180C2×C3—21no—
181C2×C3—31no—
182C2×C3—41no—
183C2×C3—41no—
184C2×C3—61no—
185C2×C3—121no—
186C2×C3—121no—
187C2×C3C32×#9911yescharacteristic
188C22—21no—
189C22—21no—
190C22—41no—
191C22—41no—
192C22C22⋊C4×3−1+211yescharacteristic
193C4—21no—
194C4—41no—
195C4—41no—
196C4—81no—
197C3—31no—
198C3C32×#35211yescharacteristic
199C2—21no—
200C2—41no—
201C2—41no—
202C23−1+2×#9911yescharacteristic
subgroupG/H
center, socleC2×C3C32×#99
commutatorC23×C3C2×C32×C4
Frattini, cosocleC2×C3×D4C22×C32
prime radicalC2×C32×D4C22×C3
LCSG ▷ C23×C3 ▷ 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
5. C9×#352C3semidirect
7. C22×C4×3−1+2C4semidirect
12. C22⋊C4×3−1+2C4semidirect
38. C22×C4×C9C3×C4semidirect
43. C9×C22⋊C4C3×C4semidirect
77. #3523−1+2direct (N × H)
117. 3−1+2#352direct (N × H)
145. C22×C4C4×3−1+2semidirect
150. C22⋊C4C4×3−1+2semidirect
170. C9C3×#352semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1232
3−1+2×#1002
3−1+2×#992
C32×#3523
C9×#3523