Finite Group Databasebeta

← C2×#11722576 #13164858

Subgroups — C2×#11722576 #13164858

Order 1728.

1108 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1108 conjugacy classes of subgroups — too many to draw. Only its 169 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
1#11722576C214yesmaximal, direct product factor
2C2×#88691C213yesmaximal, complemented
3C2×C32×H(5,𝔽2)C311yesmaximal, characteristic
4C2×#63700C313yesmaximal, complemented
5C2×Q8×3−1+2—41nomaximal
6#88691C2216yescomplemented
7C2×#6204C2211yescharacteristic, complemented
8C32×H(5,𝔽2)—34no—
9C32×H(5,𝔽2)C2×C314yes—
10C2×C32×#41—31no—
11C2×C32×#41C2×C313yes—
12C22×C32×D4—33no—
13#63700C2×C3112yescomplemented
14C2×#41⋊C9C2×C319yescomplemented
15Q8×3−1+2—44no—
16C2×C4×3−1+2—43no—
17#6204C2311yescharacteristic, complemented
18C2×C3×H(5,𝔽2)—31no—
19C2×C3×H(5,𝔽2)C3211yescharacteristic
20C24×C32—32no—
21C32×#41—34no—
22C32×#41—34no—
23C32×#41—36no—
24C32×#41C22×C316yes—
25C2×C32×Q8A411yescharacteristic
26C2×C32×Q8C22×C311yescharacteristic
27C2×C32×D4—312no—
28C2×C32×D4—33no—
29C2×C32×D4—312no—
30C2×C32×D4—33no—
31C22×C32×C4—33no—
32C2×Q8×C9—43no—
33#41⋊C9C22×C3118yescomplemented
34C2×Q8⋊C9C22×C313yescomplemented
35C22×3−1+2—41no—
36C4×3−1+2—46no—
37C3×H(5,𝔽2)—34no—
38C3×H(5,𝔽2)—312no—
39C3×H(5,𝔽2)—34no—
40C3×H(5,𝔽2)C2×C3214yes—
41C2×C3×#41—33no—
42C2×C3×#41—31no—
43C2×C3×#41—33no—
44C2×C3×#41C2×C3213yes—
45C22×C3×D4—39no—
46C22×C3×D4—33no—
47C23×C32—38no—
48C23×C32—33no—
49C23×C32—124no—
50C32×Q8—31no—
51C32×Q8C2×A414yes—
52C32×Q8C23×C311yescharacteristic
53C32×D4—36no—
54C32×D4—36no—
55C32×D4—312no—
56C2×C32×C4—31no—
57C2×C32×C4—36no—
58C2×C32×C4—36no—
59C2×C32×C4C2×A413yes—
60Q8×C9—412no—
61C2×C4×C9—49no—
62Q8⋊C9C23×C313yescomplemented
63C2×H(5,𝔽2)3−1+211yescharacteristic, complemented, Sylow
64C2×3−1+2—42no—
65C2×3−1+2—41no—
66C24×C3—36no—
67C24×C3—32no—
68C3×#41—312no—
69C3×#41—312no—
70C3×#41—318no—
71C3×#41—34no—
72C3×#41—34no—
73C3×#41—36no—
74C3×#41—36no—
75C3×#41C22×C3216yes—
76C2×C3×Q8—31no—
77C2×C3×Q8—31no—
78C2×C3×Q8C3×A411yescharacteristic
79C2×C3×Q8C22×C3211yescharacteristic
80C2×C3×D4—33no—
81C2×C3×D4—39no—
82C2×C3×D4—39no—
83C2×C3×D4—336no—
84C2×C3×D4—312no—
85C2×C3×D4—336no—
86C2×C3×D4—312no—
87C2×C3×D4—33no—
88C22×C3×C4—39no—
89C22×C3×C4—33no—
90C22×C32—36no—
91C22×C32—66no—
92C22×C32—128no—
93C22×C32C22×A411yescharacteristic
94C32×C4—31no—
95C32×C4—31no—
96C32×C4C22×A416yes—
97C22×C9—43no—
98C4×C9—418no—
99H(5,𝔽2)—34no—
100H(5,𝔽2)C2×3−1+214yescomplemented
101C2×#41—31no—
102C2×#41C2×3−1+213yes—
103C22×D4—33no—
1043−1+2—41noSylow
105C23×C3—39no—
106C23×C3—324no—
107C23×C3—38no—
108C23×C3—33no—
109C23×C3—1212no—
110C23×C3—124no—
111C3×Q8—31no—
112C3×Q8—33no—
113C3×Q8—34no—
114C3×Q8—31no—
115C3×Q8C2×C3×A414yes—
116C3×Q8C23×C3211yescharacteristic
117C3×D4—36no—
118C3×D4—318no—
119C3×D4—318no—
120C3×D4—36no—
121C3×D4—336no—
122C3×D4—312no—
123C2×C3×C4—33no—
124C2×C3×C4—318no—
125C2×C3×C4—318no—
126C2×C3×C4—31no—
127C2×C3×C4—36no—
128C2×C3×C4—36no—
129C2×C3×C4—33no—
130C2×C3×C4C2×C3×A413yes—
131C2×C32—66no—
132C2×C32#99412yes—
133C2×C32C23×A411yescharacteristic
134C2×C9—46no—
135C2×C9—43no—
136C24—32no—
137#41—34no—
138#41—34no—
139#41—36no—
140#41C22×3−1+216yes—
141C2×Q8#109411yescharacteristic
142C2×Q8C22×3−1+211yescharacteristic
143C2×D4—33no—
144C2×D4—312no—
145C2×D4—312no—
146C2×D4—33no—
147C22×C4—33no—
148C22×C3—318no—
149C22×C3—36no—
150C22×C3—31no—
151C22×C3—618no—
152C22×C3—66no—
153C22×C3—1224no—
154C22×C3—128no—
155C22×C3C22×C3×A411yescharacteristic
156C3×C4—33no—
157C3×C4—33no—
158C3×C4—31no—
159C3×C4—31no—
160C3×C4—36no—
161C3×C4C22×C3×A416yes—
162C32C2×#99411yescharacteristic
163C9—43no—
164C23—38no—
165C23—33no—
166C23—124no—
167Q8—31no—
168Q8C2×#109414yes—
169Q8C23×3−1+211yescharacteristic
170D4—36no—
171D4—36no—
172D4—312no—
173C2×C4—31no—
174C2×C4—36no—
175C2×C4—36no—
176C2×C4C2×#109413yes—
177C2×C3—31no—
178C2×C3—32no—
179C2×C3—618no—
180C2×C3—66no—
181C2×C3C3×#99412yes—
182C2×C3C23×C3×A411yescharacteristic
183C22—36no—
184C22—66no—
185C22—128no—
186C22C22×#109411yescharacteristic
187C4—31no—
188C4—31no—
189C4C22×#109416yes—
190C3—31no—
191C3C2×C3×#99411yescharacteristic
192C2—66no—
193C2#1172257612yesdirect product factor
194C2C23×#109411yescharacteristic
subgroupG/H
center, socleC22×C3C22×C3×A4
commutator, cosocleC3×Q8C23×C32
FrattiniC2×C3C23×C3×A4
Fitting, prime radicalC2×C32×H(5,𝔽2)C3
hypercenterC2×C32×Q8A4
hypocenterQ8C23×3−1+2
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C32×Q8

Complements

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

NH ≅ G/Ntype
1. #11722576C2direct (N × H)
2. C2×#88691C2semidirect
4. C2×#63700C3semidirect
6. #88691C22semidirect
7. C2×#6204C22semidirect
13. #63700C2×C3semidirect
14. C2×#41⋊C9C2×C3semidirect
17. #6204C23semidirect
33. #41⋊C9C22×C3semidirect
34. C2×Q8⋊C9C22×C3semidirect
62. Q8⋊C9C23×C3semidirect
63. C2×H(5,𝔽2)3−1+2semidirect
100. H(5,𝔽2)C2×3−1+2semidirect
193. C2#11722576direct (N × H)

Maximal subgroups

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

subgroupindex
#117225762
C2×#886912
C2×C32×H(5,𝔽2)3
C2×#637003
C2×Q8×3−1+24