Finite Group Databasebeta

← 3−1+2×B(4,𝔽2) #13145131

Subgroups — 3−1+2×B(4,𝔽2) #13145131

Order 1728.

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

normal subgroups only

This group has 792 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, complemented
23−1+2×#120C213yesmaximal, complemented
33−1+2×#99C213yesmaximal, complemented
4C32×B(4,𝔽2)C311yesmaximal, characteristic
5C9×B(4,𝔽2)C313yesmaximal, complemented
6C24×3−1+2C2211yescharacteristic, complemented
7#41×3−1+2—23no—
8C2×D4×3−1+2—23no—
9C2×D4×3−1+2—23no—
10C2×D4×3−1+2C2213yes—
11C22⋊C4×3−1+2—23no—
12C22⋊C4×3−1+2C2213yescomplemented
13C32×H(5,𝔽2)C2×C311yescharacteristic
14C32×#120C2×C313yes—
15C9×H(5,𝔽2)C2×C313yescomplemented
16C32×#99C2×C313yes—
17C9×#120C2×C319yescomplemented
18C9×#99C2×C319yescomplemented
19C23×3−1+2—21no—
20C23×3−1+2—23no—
21C23×3−1+2—42no—
22C23×3−1+2D413yescomplemented
23C23×3−1+2C2311yescharacteristic
24Q8×3−1+2—21no—
25D4×3−1+2—23no—
26D4×3−1+2—26no—
27D4×3−1+2—46no—
28C2×C4×3−1+2—23no—
29C2×C4×3−1+2—23no—
30C2×C4×3−1+2D413yescomplemented
31C3×B(4,𝔽2)—31no—
32C3×B(4,𝔽2)C3211yescharacteristic
33C24×C32C22×C311yescharacteristic
34C32×#41—23no—
35C2×C32×D4—23no—
36C2×C32×D4—23no—
37C2×C32×D4C22×C313yes—
38C24×C9C22×C313yescomplemented
39C32×C22⋊C4—23no—
40C32×C22⋊C4C22×C313yes—
41C9×#41—29no—
42C2×D4×C9—29no—
43C2×D4×C9—29no—
44C2×D4×C9C22×C319yes—
45C9×C22⋊C4—29no—
46C9×C22⋊C4C22×C319yescomplemented
47C22×3−1+2—26no—
48C22×3−1+2—23no—
49C22×3−1+2—41no—
50C22×3−1+2—43no—
51C22×3−1+2—43no—
52C22×3−1+2—41no—
53C22×3−1+2—81no—
54C22×3−1+2C2×D413yes—
55C4×3−1+2—23no—
56C4×3−1+2—43no—
57C3×H(5,𝔽2)—31no—
58C3×H(5,𝔽2)C2×C3211yescharacteristic
59C3×#120—33no—
60C3×#120C2×C3213yes—
61C3×#99—33no—
62C3×#99C2×C3213yes—
63C23×C32—21no—
64C23×C32—23no—
65C23×C32—42no—
66C23×C32C3×D413yes—
67C23×C32C23×C311yescharacteristic
68C32×Q8—21no—
69C32×D4—23no—
70C32×D4—26no—
71C32×D4—46no—
72C2×C32×C4—23no—
73C2×C32×C4—23no—
74C2×C32×C4C3×D413yes—
75C23×C9—23no—
76C23×C9—29no—
77C23×C9—46no—
78C23×C9C3×D419yescomplemented
79C23×C9C23×C313yes—
80Q8×C9—23no—
81D4×C9—29no—
82D4×C9—218no—
83D4×C9—418no—
84C2×C4×C9—29no—
85C2×C4×C9—29no—
86C2×C4×C9C3×D419yescomplemented
87B(4,𝔽2)3−1+211yescharacteristic, direct product factor, Sylow
88C2×3−1+2—23no—
89C2×3−1+2—42no—
90C2×3−1+2—43no—
91C2×3−1+2#12011yescharacteristic
92C24×C3—31no—
93C24×C3C22×C3211yescharacteristic
94C3×#41—23no—
95C3×#41—63no—
96C2×C3×D4—23no—
97C2×C3×D4—23no—
98C2×C3×D4—33no—
99C2×C3×D4—63no—
100C2×C3×D4—63no—
101C2×C3×D4C22×C3213yes—
102C3×C22⋊C4—23no—
103C3×C22⋊C4—33no—
104C3×C22⋊C4—63no—
105C3×C22⋊C4C22×C3213yes—
106C22×C32—26no—
107C22×C32—23no—
108C22×C32—41no—
109C22×C32—43no—
110C22×C32—43no—
111C22×C32—41no—
112C22×C32—81no—
113C22×C32C2×C3×D413yes—
114C32×C4—23no—
115C32×C4—43no—
116C22×C9—218no—
117C22×C9—29no—
118C22×C9—43no—
119C22×C9—49no—
120C22×C9—49no—
121C22×C9—43no—
122C22×C9—83no—
123C22×C9C2×C3×D419yes—
124C4×C9—29no—
125C4×C9—49no—
126H(5,𝔽2)C2×3−1+211yescharacteristic, complemented
127#120C2×3−1+213yescomplemented
128#99C2×3−1+213yescomplemented
1293−1+2B(4,𝔽2)11yescharacteristic, direct product factor, Sylow
130C23×C3—21no—
131C23×C3—23no—
132C23×C3—33no—
133C23×C3—31no—
134C23×C3—42no—
135C23×C3—63no—
136C23×C3—61no—
137C23×C3—122no—
138C23×C3C32×D413yes—
139C23×C3C23×C3211yescharacteristic
140C3×Q8—21no—
141C3×Q8—61no—
142C3×D4—23no—
143C3×D4—26no—
144C3×D4—46no—
145C3×D4—66no—
146C3×D4—63no—
147C3×D4—126no—
148C2×C3×C4—23no—
149C2×C3×C4—23no—
150C2×C3×C4—33no—
151C2×C3×C4—63no—
152C2×C3×C4—63no—
153C2×C3×C4C32×D413yes—
154C2×C32—23no—
155C2×C32—42no—
156C2×C32—43no—
157C2×C32C3×#12011yescharacteristic
158C2×C9—29no—
159C2×C9—46no—
160C2×C9—49no—
161C2×C9C3×#12013yes—
162C24C22×3−1+211yescharacteristic, complemented
163#41—23no—
164C2×D4—23no—
165C2×D4—23no—
166C2×D4C22×3−1+213yes—
167C22⋊C4—23no—
168C22⋊C4C22×3−1+213yescomplemented
169C22×C3—26no—
170C22×C3—23no—
171C22×C3—33no—
172C22×C3—41no—
173C22×C3—43no—
174C22×C3—43no—
175C22×C3—41no—
176C22×C3—63no—
177C22×C3—66no—
178C22×C3—81no—
179C22×C3—121no—
180C22×C3—121no—
181C22×C3—123no—
182C22×C3—123no—
183C22×C3—241no—
184C22×C3C2×C32×D413yes—
185C3×C4—23no—
186C3×C4—43no—
187C3×C4—63no—
188C3×C4—123no—
189C32C3×B(4,𝔽2)11yescharacteristic
190C9C3×B(4,𝔽2)13yescomplemented
191C23—21no—
192C23—23no—
193C23—42no—
194C23D4×3−1+213yescomplemented
195C23C23×3−1+211yescharacteristic
196Q8—21no—
197D4—23no—
198D4—26no—
199D4—46no—
200C2×C4—23no—
201C2×C4—23no—
202C2×C4D4×3−1+213yescomplemented
203C2×C3—23no—
204C2×C3—31no—
205C2×C3—42no—
206C2×C3—43no—
207C2×C3—63no—
208C2×C3—122no—
209C2×C3—123no—
210C2×C3C32×#12011yescharacteristic
211C22—26no—
212C22—23no—
213C22—41no—
214C22—43no—
215C22—43no—
216C22—41no—
217C22—81no—
218C22C2×D4×3−1+213yes—
219C4—23no—
220C4—43no—
221C3—31no—
222C3C32×B(4,𝔽2)11yescharacteristic
223C2—23no—
224C2—42no—
225C2—43no—
226C23−1+2×#12011yescharacteristic
subgroupG/H
center, socleC2×C3C32×#120
commutator, Frattini, cosocleC23×C3C23×C32
prime radicalC32×B(4,𝔽2)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
1. 3−1+2×H(5,𝔽2)C2semidirect
2. 3−1+2×#120C2semidirect
3. 3−1+2×#99C2semidirect
5. C9×B(4,𝔽2)C3semidirect
6. C24×3−1+2C22semidirect
12. C22⋊C4×3−1+2C22semidirect
15. C9×H(5,𝔽2)C2×C3semidirect
17. C9×#120C2×C3semidirect
18. C9×#99C2×C3semidirect
22. C23×3−1+2D4semidirect
30. C2×C4×3−1+2D4semidirect
38. C24×C9C22×C3semidirect
46. C9×C22⋊C4C22×C3semidirect
78. C23×C9C3×D4semidirect
86. C2×C4×C9C3×D4semidirect
87. B(4,𝔽2)3−1+2direct (N × H)
126. H(5,𝔽2)C2×3−1+2semidirect
127. #120C2×3−1+2semidirect
128. #99C2×3−1+2semidirect
129. 3−1+2B(4,𝔽2)direct (N × H)
162. C24C22×3−1+2semidirect
168. C22⋊C4C22×3−1+2semidirect
190. C9C3×B(4,𝔽2)semidirect
194. C23D4×3−1+2semidirect
202. C2×C4D4×3−1+2semidirect

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×#1202
3−1+2×#992
C32×B(4,𝔽2)3
C9×B(4,𝔽2)3