Finite Group Databasebeta

← Q8×C3⋊(C3⋊S4) #13181402

Subgroups — Q8×C3⋊(C3⋊S4) #13181402

Order 1728.

1544 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 1544 conjugacy classes of subgroups — too many to draw. Only its 217 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
1C32×Q8×A4C211yesmaximal, characteristic, complemented
2C4×C3⋊(C3⋊S4)C213yesmaximal
3#11724463C213yesmaximal, complemented
4Q8×C3⋊S4—312nomaximal
5Q8×#638—31nomaximal
6C2×C3⋊(C3⋊S4)C2211yescharacteristic
7Q8×C3⋊(C3⋊D3)—41nomaximal
8C32×C4×A4C2213yes—
9C3⋊(C3⋊SL(2,ℤ/4))C2213yes—
10C22×C32×Q8D311yescharacteristic, complemented
11C2×Q8×C3⋊D3—31no—
12C3×Q8×A4D3112yescomplemented
13C4×C3⋊S4—336no—
14#64259—336no—
15#64155—33no—
16Q8×C3⋊Dic3—31no—
17#64153—33no—
18C4×#638—33no—
19#64133—33no—
20C2×C32×A4C2311yescharacteristic
21C3⋊(C3⋊S4)Q812yesdirect product factor
22C33×Q8—41no—
23C4×C3⋊(C3⋊D3)—43no—
24#6308—43no—
25Q8×S4—99no—
26Q8×#67—94no—
27C2×C3⋊S4—312no—
28C2×C32×Q8—31no—
29C2×C32×Q8—33no—
30C22×C32×C4C2×D313yes—
31C2×#638—31no—
32Q8×C3⋊D3—63no—
33Q8×C3⋊D3—61no—
34Q8×C3⋊D3—1212no—
35C2×C4×C3⋊D3—33no—
36C2×#634—33no—
37C3×C4×A4C2×D3136yes—
38C3⋊SL(2,ℤ/4)—336no—
39#3757—33no—
40#3752—33no—
41C4⋊(C3⋊Dic3)—33no—
42#3750—36no—
43#3750—33no—
44C4×C3⋊Dic3—33no—
45C2×C3⋊(C3⋊D3)—41no—
46C32×A4C2×Q811yescharacteristic, complemented
47C33×C4—43no—
48C3⋊(C3⋊Dic3)—43no—
49C22×C3×Q8C3⋊D314yescomplemented
50C2×D3×Q8—94no—
51Q8×A4C3⋊D319yescomplemented
52C4×S4—927no—
53A4⋊Q8—927no—
54#946—912no—
55Q8×Dic3—94no—
56#944—912no—
57C4×#67—912no—
58#924—912no—
59C23×C32C22×D311yescharacteristic
60C22×C3⋊D3—31no—
61C2×C3×A4C22×D3112yes—
62C3⋊S4—324no—
63C32×Q8—33no—
64C32×Q8—412no—
65C32×Q8—61no—
66C32×Q8S411yescharacteristic, complemented
67C2×C32×C4—33no—
68C2×C32×C4—33no—
69#638—32no—
70#638—32no—
71C2×C3⋊Dic3—31no—
72C2×C3⋊Dic3—33no—
73C2×C3⋊Dic3—33no—
74C4×C3⋊D3—63no—
75C4×C3⋊D3—63no—
76C4×C3⋊D3—1236no—
77#634—63no—
78#634—63no—
79#634—1236no—
80D4×Q8—271noSylow
81C2×C33—41no—
82C3⋊(C3⋊D3)—42no—
83C2×S4—99no—
84C2×C3×Q8—34no—
85C2×C3×Q8—312no—
86C22×C3×C4C2×C3⋊D3112yes—
87C2×#67—94no—
88D3×Q8—1812no—
89D3×Q8—184no—
90D3×Q8—369no—
91C2×C4×D3—912no—
92C2×Dic6—912no—
93C4×A4C2×C3⋊D3127yes—
94SL(2,ℤ/4)—927no—
95#217—912no—
96(C2×D3)⋊C4—912no—
97C4⋊Dic3—912no—
98Dic3⋊C4—924no—
99Dic3⋊C4—912no—
100C4×Dic3—912no—
101C22×C32—31no—
102C22×C32—31no—
103C22×C32D3×Q811yescharacteristic, complemented
104C2×C3⋊D3—32no—
105C2×C3⋊D3—61no—
106C2×C3⋊D3—61no—
107C2×C3⋊D3—1212no—
108C3×A4D3×Q8112yescomplemented
109C32×C4—33no—
110C32×C4—436no—
111C32×C4C2×S413yes—
112C3⋊Dic3—32no—
113C3⋊Dic3—63no—
114C3⋊Dic3—63no—
115C3⋊Dic3—1236no—
116C22×Q8—271no—
117C22×Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
118C4⋊Q8—273no—
119C22⋊Q8—273no—
120C22⋊Q8—273no—
121C4×Q8—271no—
122C4×D4—273no—
123C33—41noSylow
124C23×C3C22×C3⋊D314yes—
125C22×D3—94no—
126C2×A4C22×C3⋊D319yes—
127S4—918no—
128C3×Q8—312no—
129C3×Q8—49no—
130C3×Q8—64no—
131C3×Q8C3⋊S414yescomplemented
132C2×C3×C4—312no—
133C2×C3×C4—312no—
134#67—98no—
135#67—98no—
136C2×Dic3—94no—
137C2×Dic3—912no—
138C2×Dic3—912no—
139C4×D3—1812no—
140C4×D3—1812no—
141C4×D3—3627no—
142Dic6—1812no—
143Dic6—1812no—
144Dic6—3627no—
145C2×C32—31no—
146C2×C32—31no—
147C2×C32—412no—
148C2×C32C22×S411yescharacteristic
149C3⋊D3—62no—
150C3⋊D3—1224no—
151C2×Q8—31no—
152C2×Q8—33no—
153C2×Q8—273no—
154C2×Q8—543no—
155C2×Q8—541no—
156C2×D4—271no—
157C22×C4—273no—
158C22×C4C2×C3⋊(C3⋊D3)13yes—
159C4⋊C4—273no—
160C4⋊C4—273no—
161C4⋊C4—276no—
162C22⋊C4—273no—
163C22⋊C4—273no—
164C42—273no—
165C22×C3—34no—
166C22×C3—34no—
167C22×C3Q8×C3⋊D314yescomplemented
168C2×D3—98no—
169C2×D3—184no—
170C2×D3—184no—
171C2×D3—369no—
172A4Q8×C3⋊D319yescomplemented
173C3×C4—312no—
174C3×C4—427no—
175C3×C4C2×C3⋊S4112yes—
176Dic3—98no—
177Dic3—1812no—
178Dic3—1812no—
179Dic3—3627no—
180C32—412no—
181C32Q8×S411yescharacteristic, complemented
182C23—271no—
183C23C22×C3⋊(C3⋊D3)11yescharacteristic
184Q8—33no—
185Q8—61no—
186Q8—543no—
187Q8—543no—
188Q8C3⋊(C3⋊S4)11yescharacteristic, direct product factor
189D4—272no—
190D4—272no—
191C2×C4—33no—
192C2×C4—33no—
193C2×C4—271no—
194C2×C4—273no—
195C2×C4—273no—
196C2×C4—543no—
197C2×C4—543no—
198C2×C3—34no—
199C2×C3—34no—
200C2×C3—49no—
201C2×C3C22×C3⋊S414yes—
202D3—188no—
203D3—3618no—
204C22—31no—
205C22—31no—
206C22—272no—
207C22—541no—
208C22—541no—
209C22Q8×C3⋊(C3⋊D3)11yescharacteristic, complemented
210C4—33no—
211C4—272no—
212C4—543no—
213C4—543no—
214C4C2×C3⋊(C3⋊S4)13yes—
215C3—49no—
216C3Q8×C3⋊S414yescomplemented
217C2—31no—
218C2—31no—
219C2—542no—
220C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, FrattiniC2C22×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×A4C23
FittingC22×C32×Q8D3
socleC23×C32C22×D3
prime radicalC2×C3⋊(C3⋊S4)C22
hypercenterQ8C3⋊(C3⋊S4)
hypocenterC32×A4C2×Q8
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
3. #11724463C2semidirect
10. C22×C32×Q8D3semidirect
12. C3×Q8×A4D3semidirect
21. C3⋊(C3⋊S4)Q8direct (N × H)
46. C32×A4C2×Q8semidirect
49. C22×C3×Q8C3⋊D3semidirect
51. Q8×A4C3⋊D3semidirect
66. C32×Q8S4semidirect
103. C22×C32D3×Q8semidirect
108. C3×A4D3×Q8semidirect
117. C22×Q8C3⋊(C3⋊D3)semidirect
131. C3×Q8C3⋊S4semidirect
167. C22×C3Q8×C3⋊D3semidirect
172. A4Q8×C3⋊D3semidirect
181. C32Q8×S4semidirect
188. Q8C3⋊(C3⋊S4)direct (N × H)
209. C22Q8×C3⋊(C3⋊D3)semidirect
216. C3Q8×C3⋊S4semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×A42
C4×C3⋊(C3⋊S4)2
#117244632
Q8×C3⋊S43
Q8×#6383
Q8×C3⋊(C3⋊D3)4