Finite Group Databasebeta

← C32×#5936 #13181420

Subgroups — C32×#5936 #13181420

Order 1728.

1012 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×C32×C22⋊A4C211yesmaximal, characteristic
2C3×#5936C314yesmaximal, direct product factor
3C32×#379—31nomaximal
4C32×C22⋊A4C411yescharacteristic, complemented
5C32×SL(2,ℤ/4)—43nomaximal
6C25×C32D311yescharacteristic
7C2×C3×C22⋊A4—24no—
8C2×C3×C22⋊A4C2×C314yes—
9C2×C32×C22⋊C4—33no—
10C2×C32×C22⋊C4—33no—
11C2×C32×A4—43no—
12C2×C32×A4—81no—
13#5936C3211yescharacteristic, direct product factor
14C3×#379—34no—
15C24×C32—33no—
16C24×C32—33no—
17C24×C32—61no—
18C24×C32—61no—
19C24×C32Dic311yescharacteristic, complemented
20C3×C22⋊A4—24no—
21C3×C22⋊A4C3×C414yescomplemented
22C22×C32×C4—33no—
23C22×C32×C4—31no—
24C3×SL(2,ℤ/4)—412no—
25C32×C22⋊C4—63no—
26C32×C22⋊C4—63no—
27C32×C22⋊C4—63no—
28C32×C22⋊C4—63no—
29C32×A4—43no—
30C32×A4—81no—
31C32×Dic3—161no—
32C25×C3C3×D314yes—
33C2×C22⋊A4—24no—
34C2×C22⋊A4C2×C3211yescharacteristic
35C2×C3×C22⋊C4—312no—
36C2×C3×C22⋊C4—312no—
37C23×C32—21no—
38C23×C32—31no—
39C23×C32—33no—
40C23×C32—33no—
41C23×C32—33no—
42C23×C32—33no—
43C23×C32—33no—
44C23×C32—61no—
45C23×C32—61no—
46C23×C32—66no—
47C23×C32—63no—
48C23×C32—63no—
49C23×C32—63no—
50C23×C32S413yes—
51C2×C3×A4—412no—
52C2×C3×A4—84no—
53C2×C3×A4—88no—
54C2×C3×A4—812no—
55C2×C32×C4—66no—
56C2×C32×C4—63no—
57C2×C32×C4—63no—
58#379—31noSylow
59C2×C33—161no—
60C24×C3—312no—
61C24×C3—312no—
62C24×C3—64no—
63C24×C3—64no—
64C24×C3C3×Dic314yescomplemented
65C22⋊A4—24no—
66C22⋊A4C32×C411yescharacteristic, complemented
67C22×C3×C4—312no—
68C22×C3×C4—34no—
69SL(2,ℤ/4)—43no—
70C3×C22⋊C4—612no—
71C3×C22⋊C4—612no—
72C3×C22⋊C4—612no—
73C3×C22⋊C4—612no—
74C22×C32—21no—
75C22×C32—33no—
76C22×C32—33no—
77C22×C32—31no—
78C22×C32—33no—
79C22×C32—33no—
80C22×C32—33no—
81C22×C32—63no—
82C22×C32—66no—
83C22×C32—63no—
84C22×C32—63no—
85C22×C32—61no—
86C22×C32—61no—
87C22×C32SL(2,ℤ/4)13yescomplemented
88C3×A4—412no—
89C3×A4—84no—
90C3×A4—88no—
91C3×A4—812no—
92C32×C4—123no—
93C32×C4—121no—
94C3×Dic3—164no—
95C25C32×D311yescharacteristic
96C2×C22⋊C4—33no—
97C2×C22⋊C4—33no—
98C33—161noSylow
99C23×C3—24no—
100C23×C3—34no—
101C23×C3—312no—
102C23×C3—312no—
103C23×C3—312no—
104C23×C3—312no—
105C23×C3—312no—
106C23×C3—64no—
107C23×C3—64no—
108C23×C3—624no—
109C23×C3—612no—
110C23×C3—612no—
111C23×C3—612no—
112C23×C3C3×S4112yes—
113C2×A4—43no—
114C2×A4—81no—
115C2×A4—88no—
116C2×A4—812no—
117C2×C3×C4—624no—
118C2×C3×C4—612no—
119C2×C3×C4—612no—
120C2×C32—33no—
121C2×C32—33no—
122C2×C32—61no—
123C2×C32—61no—
124C2×C32—164no—
125C2×C32—324no—
126C2×C32#102011yescharacteristic
127C24—33no—
128C24—33no—
129C24—61no—
130C24—61no—
131C24C32×Dic311yescharacteristic, complemented
132C22×C4—33no—
133C22×C4—31no—
134C22⋊C4—63no—
135C22⋊C4—63no—
136C22⋊C4—63no—
137C22⋊C4—63no—
138C22×C3—24no—
139C22×C3—312no—
140C22×C3—312no—
141C22×C3—34no—
142C22×C3—312no—
143C22×C3—312no—
144C22×C3—312no—
145C22×C3—612no—
146C22×C3—624no—
147C22×C3—612no—
148C22×C3—612no—
149C22×C3—64no—
150C22×C3—64no—
151C22×C3C3×SL(2,ℤ/4)112yescomplemented
152A4—43no—
153A4—81no—
154A4—88no—
155A4—812no—
156C3×C4—1212no—
157C3×C4—124no—
158Dic3—161no—
159C32—164no—
160C32—324no—
161C32#593611yescharacteristic, direct product factor
162C23—21no—
163C23—31no—
164C23—33no—
165C23—33no—
166C23—33no—
167C23—33no—
168C23—33no—
169C23—61no—
170C23—61no—
171C23—66no—
172C23—63no—
173C23—63no—
174C23—63no—
175C23C32×S413yes—
176C2×C4—66no—
177C2×C4—63no—
178C2×C4—63no—
179C2×C3—312no—
180C2×C3—312no—
181C2×C3—64no—
182C2×C3—64no—
183C2×C3—161no—
184C2×C3—324no—
185C2×C3C3×#102014yes—
186C22—21no—
187C22—33no—
188C22—33no—
189C22—31no—
190C22—33no—
191C22—33no—
192C22—33no—
193C22—63no—
194C22—66no—
195C22—63no—
196C22—63no—
197C22—61no—
198C22—61no—
199C22C32×SL(2,ℤ/4)13yescomplemented
200C4—123no—
201C4—121no—
202C3—161no—
203C3—324no—
204C3C3×#593614yesdirect product factor
205C2—33no—
206C2—33no—
207C2—61no—
208C2—61no—
209C2C32×#102011yescharacteristic
subgroupG/H
center, hypercenterC2×C32#1020
commutator, hypocenterC22⋊A4C32×C4
FrattiniC2C32×#1020
Fitting, socleC25×C32D3
prime radicalC2×C32×C22⋊A4C2
cosocleC2×C22⋊A4C2×C32
LCSG ▷ C22⋊A4
UCSC1 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
2. C3×#5936C3direct (N × H)
4. C32×C22⋊A4C4semidirect
13. #5936C32direct (N × H)
19. C24×C32Dic3semidirect
21. C3×C22⋊A4C3×C4semidirect
64. C24×C3C3×Dic3semidirect
66. C22⋊A4C32×C4semidirect
87. C22×C32SL(2,ℤ/4)semidirect
131. C24C32×Dic3semidirect
151. C22×C3C3×SL(2,ℤ/4)semidirect
161. C32#5936direct (N × H)
199. C22C32×SL(2,ℤ/4)semidirect
204. C3C3×#5936direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊A42
C3×#59363
C32×#3793
C32×SL(2,ℤ/4)4