Finite Group Databasebeta

← C2×#11723098 #13173387

Subgroups — C2×#11723098 #13173387

Order 1728.

440 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2#11723098C214yesmaximal, direct product factor
3C2×C3⋊#3708C212yesmaximal, complemented
4C2×#63773—31nomaximal
5C22×C33×C4C411yescharacteristic
6C2×C3×C3⋊(C3⋊C8)C412yes—
7C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9C2×C3×C3⋊(C3⋊C8)C2212yes—
10C3⋊#3708C2214yescomplemented
11C22×C3⋊(C3⋊C8)D311yescharacteristic
12C22×C3×C3⋊C8—62no—
13#63773—34no—
14C2×#3708—32no—
15C23×C33C811yescharacteristic
16C2×C33×C4C812yes—
17C2×C33×C4C811yescharacteristic
18C2×C33×C4C2×C412yes—
19C2×C33×C4C2×C411yescharacteristic
20C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
21C3×C3⋊(C3⋊C8)C2×C412yes—
22C3×C3⋊(C3⋊C8)C2311yescharacteristic
23C2×#812—91nomaximal
24C22×C32×C4—22no—
25C22×C32×C4—42no—
26C22×C32×C4Dic311yescharacteristic
27C2×C3⋊(C3⋊C8)Dic312yes—
28C2×C3⋊(C3⋊C8)Dic311yescharacteristic
29C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
30C2×C3⋊(C3⋊C8)C2×D312yes—
31C2×C3×C3⋊C8—64no—
32C2×C3×C3⋊C8—62no—
33C2×C3×C3⋊C8—62no—
34C2×C3×C3⋊C8—64no—
35#3708—34no—
36C22×C33—22no—
37C22×C33C2×C812yes—
38C22×C33C2×C811yescharacteristic
39C33×C4C2×C811yescharacteristic
40C33×C4C2×C812yes—
41C33×C4C22×C411yescharacteristic
42C22×C3×C8—91no—
43C22×C3⋊C8—62no—
44#812—94no—
45C2×C3⋊C16—92no—
46C23×C32—22no—
47C23×C32—42no—
48C23×C32C3⋊C811yescharacteristic
49C2×C32×C4—24no—
50C2×C32×C4—22no—
51C2×C32×C4—24no—
52C2×C32×C4—22no—
53C2×C32×C4—44no—
54C2×C32×C4—42no—
55C2×C32×C4—44no—
56C2×C32×C4—42no—
57C2×C32×C4C3⋊C812yes—
58C2×C32×C4C3⋊C811yescharacteristic
59C2×C32×C4C2×Dic312yes—
60C2×C32×C4C2×Dic311yescharacteristic
61C3⋊(C3⋊C8)C2×Dic311yescharacteristic
62C3⋊(C3⋊C8)C2×Dic312yes—
63C3⋊(C3⋊C8)C22×D311yescharacteristic
64C3×C3⋊C8—62no—
65C3×C3⋊C8—64no—
66C3×C3⋊C8—62no—
67C2×M32—271noSylow
68C2×C33—22no—
69C2×C33M3212yescomplemented
70C2×C33C22×C811yescharacteristic
71C22×C3×C4—22no—
72C22×C3×C4—42no—
73C22×C3×C4#15711yescharacteristic
74C2×C3×C8—91no—
75C2×C3×C8—92no—
76C2×C3×C8—92no—
77C2×C3×C8—91no—
78C2×C3⋊C8—64no—
79C2×C3⋊C8—62no—
80C2×C3⋊C8—64no—
81C2×C3⋊C8—62no—
82C3⋊C16—94no—
83C22×C32—22no—
84C22×C32—28no—
85C22×C32—24no—
86C22×C32—22no—
87C22×C32—48no—
88C22×C32—44no—
89C22×C32—42no—
90C22×C32C2×C3⋊C812yes—
91C22×C32C2×C3⋊C811yescharacteristic
92C32×C4—22no—
93C32×C4—24no—
94C32×C4—22no—
95C32×C4—44no—
96C32×C4—42no—
97C32×C4—42no—
98C32×C4C2×C3⋊C812yes—
99C32×C4C2×C3⋊C811yescharacteristic
100C32×C4C22×Dic311yescharacteristic
101C22×C8—91no—
102M32—274no—
103C2×C16—272no—
104C33C2×M3211yescharacteristic, complemented, Sylow
105C23×C3—22no—
106C23×C3—42no—
107C23×C3#62211yescharacteristic
108C2×C3×C4—22no—
109C2×C3×C4—24no—
110C2×C3×C4—24no—
111C2×C3×C4—22no—
112C2×C3×C4—42no—
113C2×C3×C4—44no—
114C2×C3×C4—44no—
115C2×C3×C4—42no—
116C2×C3×C4#62212yes—
117C2×C3×C4#62211yescharacteristic
118C2×C3×C4C2×#15712yes—
119C2×C3×C4C2×#15711yescharacteristic
120C3×C8—91no—
121C3×C8—91no—
122C3×C8—92no—
123C3⋊C8—64no—
124C3⋊C8—62no—
125C3⋊C8—62no—
126C2×C32—22no—
127C2×C32—28no—
128C2×C32—24no—
129C2×C32—22no—
130C2×C32—48no—
131C2×C32—44no—
132C2×C32—42no—
133C2×C32#81212yescomplemented
134C2×C32C22×C3⋊C811yescharacteristic
135C22×C4C3⋊#15711yescharacteristic
136C2×C8—92no—
137C2×C8—92no—
138C2×C8—91no—
139C2×C8—91no—
140C16—274no—
141C22×C3—28no—
142C22×C3—22no—
143C22×C3—24no—
144C22×C3—22no—
145C22×C3—48no—
146C22×C3—44no—
147C22×C3—42no—
148C22×C3C2×#62212yes—
149C22×C3C2×#62211yescharacteristic
150C3×C4—22no—
151C3×C4—24no—
152C3×C4—22no—
153C3×C4—42no—
154C3×C4—44no—
155C3×C4—42no—
156C3×C4C2×#62212yes—
157C3×C4C2×#62211yescharacteristic
158C3×C4C22×#15711yescharacteristic
159C32—22no—
160C32—42no—
161C32C2×#81211yescharacteristic, complemented
162C23C3⋊#62211yescharacteristic
163C2×C4C3⋊#62211yescharacteristic
164C2×C4C3⋊#62212yes—
165C2×C4C2×C3⋊#15712yes—
166C2×C4C2×C3⋊#15711yescharacteristic
167C8—91no—
168C8—92no—
169C8—91no—
170C2×C3—28no—
171C2×C3—22no—
172C2×C3—24no—
173C2×C3—22no—
174C2×C3—48no—
175C2×C3—44no—
176C2×C3—42no—
177C2×C3#6377312yescomplemented
178C2×C3C22×#62211yescharacteristic
179C22—22no—
180C22C2×C3⋊#62212yes—
181C22C2×C3⋊#62211yescharacteristic
182C4C2×C3⋊#62212yes—
183C4C2×C3⋊#62211yescharacteristic
184C4C22×C3⋊#15711yescharacteristic
185C3—22no—
186C3—42no—
187C3C2×#6377311yescharacteristic, complemented
188C2—22no—
189C2#1172309812yesdirect product factor
190C2C22×C3⋊#62211yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊#157
commutatorC2×C33C22×C8
FrattiniC4C22×C3⋊#157
FittingC22×C33×C4C4
socleC22×C33C2×C8
prime radicalC23×C33C8
cosocleC3×C3⋊(C3⋊C8)C23
hypercenterC22×C4C3⋊#157
hypocenterC33C2×M32
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11723098C2direct (N × H)
3. C2×C3⋊#3708C2semidirect
10. C3⋊#3708C22semidirect
69. C2×C33M32semidirect
104. C33C2×M32semidirect
133. C2×C32#812semidirect
161. C32C2×#812semidirect
177. C2×C3#63773semidirect
187. C3C2×#63773semidirect
189. C2#11723098direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊C8)2
#117230982
C2×C3⋊#37082
C2×#637733
C2×#8129