Finite Group Databasebeta

← 3−1+2×#458 #13145134

Subgroups — 3−1+2×#458 #13145134

Order 1728.

640 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 189 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×H(5,𝔽2)C211yesmaximal, characteristic
23−1+2×#123C213yesmaximal, complemented
33−1+2×#99C213yesmaximal, complemented
4C32×#458C311yesmaximal, characteristic
5C9×#458C313yesmaximal, complemented
6#41×3−1+2—23no—
7C2×D4×3−1+2—23no—
8C2×D4×3−1+2C2213yes—
9C22×C4×3−1+2C2211yescharacteristic, complemented
10C4⋊C4×3−1+2—23no—
11C22⋊C4×3−1+2—23no—
12C22⋊C4×3−1+2C2213yescomplemented
13C32×H(5,𝔽2)C2×C311yescharacteristic
14C32×#123C2×C313yes—
15C9×H(5,𝔽2)C2×C313yes—
16C32×#99C2×C313yes—
17C9×#123C2×C319yescomplemented
18C9×#99C2×C319yescomplemented
19C23×3−1+2—21no—
20C23×3−1+2D413yes—
21C23×3−1+2C2311yescharacteristic
22Q8×3−1+2—21no—
23D4×3−1+2—23no—
24D4×3−1+2—26no—
25C2×C4×3−1+2—23no—
26C2×C4×3−1+2—23no—
27C2×C4×3−1+2—23no—
28C2×C4×3−1+2D413yes—
29C3×#458—31no—
30C3×#458C3211yescharacteristic
31C32×#41—23no—
32C2×C32×D4—23no—
33C2×C32×D4C22×C313yes—
34C22×C32×C4C22×C311yescharacteristic
35C32×C4⋊C4—23no—
36C32×C22⋊C4—23no—
37C32×C22⋊C4C22×C313yes—
38C9×#41—29no—
39C2×D4×C9—29no—
40C2×D4×C9C22×C319yes—
41C22×C4×C9C22×C313yescomplemented
42C9×C4⋊C4—29no—
43C9×C22⋊C4—29no—
44C9×C22⋊C4C22×C319yescomplemented
45C22×3−1+2—23no—
46C22×3−1+2—41no—
47C22×3−1+2—43no—
48C22×3−1+2—81no—
49C22×3−1+2C2×D413yes—
50C4×3−1+2—23no—
51C4×3−1+2—43no—
52C4×3−1+2—41no—
53C3×H(5,𝔽2)—31no—
54C3×H(5,𝔽2)C2×C3211yescharacteristic
55C3×#123—33no—
56C3×#123C2×C3213yes—
57C3×#99—33no—
58C3×#99C2×C3213yes—
59C23×C32—21no—
60C23×C32C3×D413yes—
61C23×C32C23×C311yescharacteristic
62C32×Q8—21no—
63C32×D4—23no—
64C32×D4—26no—
65C2×C32×C4—23no—
66C2×C32×C4—23no—
67C2×C32×C4—23no—
68C2×C32×C4C3×D413yes—
69C23×C9—23no—
70C23×C9C3×D419yes—
71C23×C9C23×C313yes—
72Q8×C9—23no—
73D4×C9—29no—
74D4×C9—218no—
75C2×C4×C9—29no—
76C2×C4×C9—29no—
77C2×C4×C9—29no—
78C2×C4×C9C3×D419yes—
79#4583−1+211yescharacteristic, direct product factor, Sylow
80C2×3−1+2—23no—
81C2×3−1+2—43no—
82C2×3−1+2#12011yescharacteristic
83C3×#41—23no—
84C3×#41—63no—
85C2×C3×D4—23no—
86C2×C3×D4—33no—
87C2×C3×D4—63no—
88C2×C3×D4C22×C3213yes—
89C22×C3×C4—31no—
90C22×C3×C4C22×C3211yescharacteristic
91C3×C4⋊C4—23no—
92C3×C4⋊C4—63no—
93C3×C22⋊C4—23no—
94C3×C22⋊C4—33no—
95C3×C22⋊C4—63no—
96C3×C22⋊C4C22×C3213yes—
97C22×C32—23no—
98C22×C32—41no—
99C22×C32—43no—
100C22×C32—81no—
101C22×C32C2×C3×D413yes—
102C32×C4—23no—
103C32×C4—43no—
104C32×C4—41no—
105C22×C9—29no—
106C22×C9—43no—
107C22×C9—49no—
108C22×C9—83no—
109C22×C9C2×C3×D419yes—
110C4×C9—29no—
111C4×C9—49no—
112C4×C9—43no—
113H(5,𝔽2)C2×3−1+211yescharacteristic
114#123C2×3−1+213yescomplemented
115#99C2×3−1+213yescomplemented
1163−1+2#45811yescharacteristic, direct product factor, Sylow
117C23×C3—21no—
118C23×C3—33no—
119C23×C3—31no—
120C23×C3—61no—
121C23×C3C32×D413yes—
122C23×C3C23×C3211yescharacteristic
123C3×Q8—21no—
124C3×Q8—61no—
125C3×D4—23no—
126C3×D4—26no—
127C3×D4—66no—
128C3×D4—63no—
129C2×C3×C4—23no—
130C2×C3×C4—23no—
131C2×C3×C4—23no—
132C2×C3×C4—33no—
133C2×C3×C4—63no—
134C2×C3×C4—63no—
135C2×C3×C4—63no—
136C2×C3×C4C32×D413yes—
137C2×C32—23no—
138C2×C32—43no—
139C2×C32C3×#12011yescharacteristic
140C2×C9—29no—
141C2×C9—49no—
142C2×C9C3×#12013yes—
143#41—23no—
144C2×D4—23no—
145C2×D4C22×3−1+213yes—
146C22×C4C22×3−1+211yescharacteristic, complemented
147C4⋊C4—23no—
148C22⋊C4—23no—
149C22⋊C4C22×3−1+213yescomplemented
150C22×C3—23no—
151C22×C3—33no—
152C22×C3—41no—
153C22×C3—43no—
154C22×C3—63no—
155C22×C3—81no—
156C22×C3—121no—
157C22×C3—123no—
158C22×C3—241no—
159C22×C3C2×C32×D413yes—
160C3×C4—23no—
161C3×C4—43no—
162C3×C4—41no—
163C3×C4—63no—
164C3×C4—121no—
165C3×C4—123no—
166C32C3×#45811yescharacteristic
167C9C3×#45813yescomplemented
168C23—21no—
169C23D4×3−1+213yes—
170C23C23×3−1+211yescharacteristic
171Q8—21no—
172D4—23no—
173D4—26no—
174C2×C4—23no—
175C2×C4—23no—
176C2×C4—23no—
177C2×C4D4×3−1+213yes—
178C2×C3—23no—
179C2×C3—31no—
180C2×C3—43no—
181C2×C3—63no—
182C2×C3—123no—
183C2×C3C32×#12011yescharacteristic
184C22—23no—
185C22—41no—
186C22—43no—
187C22—81no—
188C22C2×D4×3−1+213yes—
189C4—23no—
190C4—43no—
191C4—41no—
192C3—31no—
193C3C32×#45811yescharacteristic
194C2—23no—
195C2—43no—
196C23−1+2×#12011yescharacteristic
subgroupG/H
center, socleC2×C3C32×#120
commutator, Frattini, cosocleC23×C3C23×C32
prime radicalC32×H(5,𝔽2)C2×C3
LCSG ▷ C23×C3 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C23×3−1+2 ◁ G

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#123C2semidirect
3. 3−1+2×#99C2semidirect
5. C9×#458C3semidirect
9. C22×C4×3−1+2C22semidirect
12. C22⋊C4×3−1+2C22semidirect
17. C9×#123C2×C3semidirect
18. C9×#99C2×C3semidirect
41. C22×C4×C9C22×C3semidirect
44. C9×C22⋊C4C22×C3semidirect
79. #4583−1+2direct (N × H)
114. #123C2×3−1+2semidirect
115. #99C2×3−1+2semidirect
116. 3−1+2#458direct (N × H)
146. C22×C4C22×3−1+2semidirect
149. C22⋊C4C22×3−1+2semidirect
167. C9C3×#458semidirect

Maximal subgroups

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

subgroupindex
3−1+2×H(5,𝔽2)2
3−1+2×#1232
3−1+2×#992
C32×#4583
C9×#4583