Finite Group Databasebeta

← C2×#11723104 #13173398

Subgroups — C2×#11723104 #13173398

Order 1728.

600 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 141 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⋊#622C212yesmaximal
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3#11723104C214yesmaximal, direct product factor
4C2×#63779—31nomaximal
5C22×C33×C4C411yescharacteristic
6C22×C3×C3⋊Dic3C411yescharacteristic
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3⋊#622—24no—
9C2×C3⋊#622C2214yes—
10C3×C4×C3⋊Dic3C412yes—
11C3×C4×C3⋊Dic3C2212yes—
12C22×#622—32no—
13C2×C4×C3⋊Dic3D311yescharacteristic
14C2×C3×C4×Dic3—62no—
15#63779—34no—
16C23×C33C2×C411yescharacteristic
17C2×C33×C4—22no—
18C2×C33×C4C2×C412yes—
19C2×C3×C3⋊Dic3C814yescomplemented
20C2×C3×C3⋊Dic3C2×C411yescharacteristic
21C2×C3×C3⋊Dic3C2×C412yes—
22C2×C3×C3⋊Dic3D412yes—
23C2×C3×C3⋊Dic3Q812yes—
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C3⋊#622—24no—
26C2×C4⋊(C3⋊C8)—91nomaximal
27C22×C32×C4—22no—
28C22×C32×C4—42no—
29C22×C32×C4Dic311yescharacteristic
30C22×C3⋊Dic3Dic311yescharacteristic
31C22×C3⋊Dic3C2×D311yescharacteristic
32C22×C3×Dic3—62no—
33C22×C3×Dic3—62no—
34C2×#622—34no—
35C2×#622—64no—
36C4×C3⋊Dic3Dic312yes—
37C4×C3⋊Dic3C2×D312yes—
38C3×C4×Dic3—64no—
39C3×C4×Dic3—64no—
40C22×C33C4⋊C412yes—
41C22×C33C2×C812yes—
42C22×C33M1612yes—
43C22×C33C22×C411yescharacteristic
44C33×C4—22no—
45C3×C3⋊Dic3C4⋊C412yes—
46C3×C3⋊Dic3C2×C814yescomplemented
47C3×C3⋊Dic3C2×D411yescharacteristic
48C3×C3⋊Dic3C2×Q811yescharacteristic
49C2×C3×C42—91no—
50C2×C4×Dic3—62no—
51C22×C3⋊C8—92no—
52C4⋊(C3⋊C8)—94no—
53C23×C32—22no—
54C23×C32—42no—
55C23×C32C2×Dic311yescharacteristic
56C2×C32×C4—22no—
57C2×C32×C4—28no—
58C2×C32×C4—24no—
59C2×C32×C4—48no—
60C2×C32×C4—44no—
61C2×C32×C4C2×Dic312yes—
62C2×C3⋊Dic3C3⋊C814yescomplemented
63C2×C3⋊Dic3Dic612yes—
64C2×C3⋊Dic3D1212yes—
65C2×C3⋊Dic3C2×Dic312yes—
66C2×C3⋊Dic3C2×Dic311yescharacteristic
67C2×C3⋊Dic3C22×D311yescharacteristic
68C2×C3×Dic3—62no—
69C2×C3×Dic3—64no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—68no—
72C2×C3×Dic3—64no—
73C2×C3×Dic3—64no—
74#622—64no—
75C2×C4⋊C8—271noSylow
76C2×C33C4⋊C814yescomplemented
77C2×C33C2×C4⋊C411yescharacteristic
78C2×C33C22×C811yescharacteristic
79C2×C33C2×M1611yescharacteristic
80C22×C3×C4—22no—
81C22×C3×C4—42no—
82C22×C3×C4—91no—
83C22×C3×C4—91no—
84C22×C3×C4#15711yescharacteristic
85C22×Dic3—62no—
86C22×Dic3—62no—
87C3×C42—92no—
88C3×C42—92no—
89C4×Dic3—64no—
90C4×Dic3—64no—
91C2×C3⋊C8—94no—
92C2×C3⋊C8—184no—
93C22×C32—22no—
94C22×C32—24no—
95C22×C32—24no—
96C22×C32—24no—
97C22×C32—44no—
98C22×C32—44no—
99C22×C32—44no—
100C22×C32—42no—
101C22×C32C2×C3⋊C812yes—
102C22×C32#20812yes—
103C22×C32C4⋊Dic312yes—
104C22×C32C22×Dic311yescharacteristic
105C32×C4—22no—
106C32×C4—28no—
107C32×C4—48no—
108C3⋊Dic3C2×C3⋊C814yescomplemented
109C3⋊Dic3C4⋊Dic312yes—
110C3⋊Dic3C2×Dic611yescharacteristic
111C3⋊Dic3C2×D1211yescharacteristic
112C3×Dic3—64no—
113C3×Dic3—68no—
114C3×Dic3—62no—
115C3×Dic3—62no—
116C22×C8—272no—
117C2×C42—91no—
118C4⋊C8—274no—
119C33C2×C4⋊C811yescharacteristic, complemented, Sylow
120C23×C3—22no—
121C23×C3—42no—
122C23×C3C2×#15711yescharacteristic
123C2×C3×C4—28no—
124C2×C3×C4—22no—
125C2×C3×C4—24no—
126C2×C3×C4—48no—
127C2×C3×C4—44no—
128C2×C3×C4—91no—
129C2×C3×C4—92no—
130C2×C3×C4—91no—
131C2×C3×C4—94no—
132C2×C3×C4—92no—
133C2×C3×C4—92no—
134C2×C3×C4C2×#15712yes—
135C2×Dic3—68no—
136C2×Dic3—64no—
137C2×Dic3—64no—
138C2×Dic3—64no—
139C2×Dic3—62no—
140C2×Dic3—62no—
141C3⋊C8—184no—
142C2×C32—22no—
143C2×C32—22no—
144C2×C32—28no—
145C2×C32—22no—
146C2×C32—48no—
147C2×C32—42no—
148C2×C32—42no—
149C2×C32—42no—
150C2×C32C4⋊(C3⋊C8)14yescomplemented
151C2×C32C22×C3⋊C811yescharacteristic
152C2×C32C2×#20811yescharacteristic
153C2×C32C2×C4⋊Dic311yescharacteristic
154C22×C4—91no—
155C22×C4—91no—
156C22×C4C3⋊#15711yescharacteristic
157C2×C8—274no—
158C2×C8—544no—
159C42—92no—
160C42—92no—
161C22×C3—22no—
162C22×C3—24no—
163C22×C3—24no—
164C22×C3—24no—
165C22×C3—42no—
166C22×C3—44no—
167C22×C3—44no—
168C22×C3—44no—
169C22×C3#378712yes—
170C22×C3C4⋊#15712yes—
171C22×C3#379212yes—
172C22×C3C22×#15711yescharacteristic
173C3×C4—28no—
174C3×C4—22no—
175C3×C4—48no—
176C3×C4—94no—
177C3×C4—91no—
178C3×C4—91no—
179C3×C4—92no—
180Dic3—68no—
181Dic3—64no—
182Dic3—62no—
183Dic3—62no—
184C32—22no—
185C32—42no—
186C32C2×C4⋊(C3⋊C8)11yescharacteristic, complemented
187C23C2×C3⋊#15711yescharacteristic
188C2×C4—22no—
189C2×C4—91no—
190C2×C4—92no—
191C2×C4—91no—
192C2×C4—94no—
193C2×C4—92no—
194C2×C4—92no—
195C2×C4C2×C3⋊#15712yes—
196C8—544no—
197C2×C3—28no—
198C2×C3—22no—
199C2×C3—22no—
200C2×C3—22no—
201C2×C3—48no—
202C2×C3—42no—
203C2×C3—42no—
204C2×C3—42no—
205C2×C3#6377914yescomplemented
206C2×C3C2×#378711yescharacteristic
207C2×C3C2×C4⋊#15711yescharacteristic
208C2×C3C2×#379211yescharacteristic
209C22#8898812yes—
210C22C4⋊(C3⋊#157)12yes—
211C22#8899312yes—
212C22C22×C3⋊#15711yescharacteristic
213C4—22no—
214C4—94no—
215C4—91no—
216C4—91no—
217C4—92no—
218C3—22no—
219C3—42no—
220C3C2×#6377911yescharacteristic, complemented
221C2#1172310414yesdirect product factor
222C2C2×#8898811yescharacteristic
223C2C2×C4⋊(C3⋊#157)11yescharacteristic
224C2C2×#8899311yescharacteristic
subgroupG/H
centerC23C2×C3⋊#157
commutatorC2×C33C22×C8
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socle, prime radicalC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33C2×C4⋊C8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11723104C2direct (N × H)
19. C2×C3×C3⋊Dic3C8semidirect
46. C3×C3⋊Dic3C2×C8semidirect
62. C2×C3⋊Dic3C3⋊C8semidirect
76. C2×C33C4⋊C8semidirect
108. C3⋊Dic3C2×C3⋊C8semidirect
119. C33C2×C4⋊C8semidirect
150. C2×C32C4⋊(C3⋊C8)semidirect
186. C32C2×C4⋊(C3⋊C8)semidirect
205. C2×C3#63779semidirect
220. C3C2×#63779semidirect
221. C2#11723104direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C2×C3×C4×C3⋊Dic32
#117231042
C2×#637793
C2×C4⋊(C3⋊C8)9