Finite Group Databasebeta

← C2×#11723102 #13173391

Subgroups — C2×#11723102 #13173391

Order 1728.

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

normal subgroups only

This group has 616 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#11723102C214yesmaximal, direct product factor
4C2×#63777—31nomaximal
5C22×C33×C4C411yescharacteristic
6C22×C3×C3⋊Dic3C411yescharacteristic
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3⋊#622C418yescomplemented
9C2×C3⋊#622C2214yes—
10C3×C4×C3⋊Dic3C412yes—
11C3×C4×C3⋊Dic3C2212yes—
12C22×#622—32no—
13C2×C4×C3⋊Dic3D311yescharacteristic
14C2×C3×C4×Dic3—62no—
15#63777—34no—
16C23×C33C2×C411yescharacteristic
17C2×C33×C4—22no—
18C2×C33×C4C2×C412yes—
19C2×C3×C3⋊Dic3—22no—
20C2×C3×C3⋊Dic3C2×C412yes—
21C2×C3×C3⋊Dic3C2×C411yescharacteristic
22C2×C3×C3⋊Dic3C2×C414yes—
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C3⋊#622C2×C418yescomplemented
25C2×#803—91nomaximal
26C22×C32×C4—22no—
27C22×C32×C4—42no—
28C22×C32×C4Dic311yescharacteristic
29C22×C3⋊Dic3Dic311yescharacteristic
30C22×C3⋊Dic3C2×D311yescharacteristic
31C22×C3×Dic3—62no—
32C22×C3×Dic3—62no—
33C2×#622—38no—
34C2×#622—34no—
35C4×C3⋊Dic3Dic312yes—
36C4×C3⋊Dic3C2×D312yes—
37C3×C4×Dic3—64no—
38C3×C4×Dic3—64no—
39C22×C33C4212yes—
40C22×C33M1612yes—
41C22×C33M1612yes—
42C22×C33C22×C411yescharacteristic
43C33×C4—22no—
44C3×C3⋊Dic3—22no—
45C3×C3⋊Dic3C4212yes—
46C3×C3⋊Dic3C22×C412yes—
47C2×C3×C42—91no—
48C2×C4×Dic3—62no—
49C22×C3⋊C8—92no—
50#803—94no—
51C23×C32—22no—
52C23×C32—42no—
53C23×C32C2×Dic311yescharacteristic
54C2×C32×C4—22no—
55C2×C32×C4—28no—
56C2×C32×C4—24no—
57C2×C32×C4—48no—
58C2×C32×C4—44no—
59C2×C32×C4C2×Dic312yes—
60C2×C3⋊Dic3—22no—
61C2×C3⋊Dic3C4×D314yes—
62C2×C3⋊Dic3C2×Dic312yes—
63C2×C3⋊Dic3C2×Dic311yescharacteristic
64C2×C3⋊Dic3C22×D311yescharacteristic
65C2×C3×Dic3—62no—
66C2×C3×Dic3—64no—
67C2×C3×Dic3—62no—
68C2×C3×Dic3—68no—
69C2×C3×Dic3—68no—
70#622—38no—
71C2×#97—271noSylow
72C2×C33#9714yescomplemented
73C2×C33C2×C4211yescharacteristic
74C2×C33C2×M1611yescharacteristic
75C2×C33C2×M1611yescharacteristic
76C22×C3×C4—22no—
77C22×C3×C4—42no—
78C22×C3×C4—91no—
79C22×C3×C4—91no—
80C22×C3×C4#15711yescharacteristic
81C22×Dic3—62no—
82C22×Dic3—62no—
83C3×C42—92no—
84C3×C42—92no—
85C4×Dic3—64no—
86C4×Dic3—64no—
87C2×C3⋊C8—94no—
88C2×C3⋊C8—98no—
89C22×C32—22no—
90C22×C32—24no—
91C22×C32—24no—
92C22×C32—24no—
93C22×C32—44no—
94C22×C32—44no—
95C22×C32—44no—
96C22×C32—42no—
97C22×C32#20812yes—
98C22×C32#20812yes—
99C22×C32C4×Dic312yes—
100C22×C32C22×Dic311yescharacteristic
101C32×C4—22no—
102C32×C4—28no—
103C32×C4—48no—
104C3⋊Dic3—22no—
105C3⋊Dic3C4×Dic312yes—
106C3⋊Dic3C2×C4×D312yes—
107C3×Dic3—64no—
108C3×Dic3—64no—
109C3×Dic3—68no—
110C22×C8—272no—
111C2×C42—91no—
112#97—274no—
113C33C2×#9711yescharacteristic, complemented, Sylow
114C23×C3—22no—
115C23×C3—42no—
116C23×C3C2×#15711yescharacteristic
117C2×C3×C4—28no—
118C2×C3×C4—22no—
119C2×C3×C4—24no—
120C2×C3×C4—48no—
121C2×C3×C4—44no—
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—94no—
125C2×C3×C4—92no—
126C2×C3×C4—182no—
127C2×C3×C4C2×#15712yes—
128C2×Dic3—68no—
129C2×Dic3—62no—
130C2×Dic3—64no—
131C2×Dic3—62no—
132C2×Dic3—68no—
133C3⋊C8—98no—
134C2×C32—22no—
135C2×C32—22no—
136C2×C32—28no—
137C2×C32—22no—
138C2×C32—48no—
139C2×C32—42no—
140C2×C32—42no—
141C2×C32—42no—
142C2×C32#80314yescomplemented
143C2×C32C2×#20811yescharacteristic
144C2×C32C2×#20811yescharacteristic
145C2×C32C2×C4×Dic311yescharacteristic
146C22×C4—91no—
147C22×C4—91no—
148C22×C4C3⋊#15711yescharacteristic
149C2×C8—274no—
150C2×C8—278no—
151C42—92no—
152C42—92no—
153C22×C3—22no—
154C22×C3—24no—
155C22×C3—24no—
156C22×C3—24no—
157C22×C3—42no—
158C22×C3—44no—
159C22×C3—44no—
160C22×C3—44no—
161C22×C3#378812yes—
162C22×C3C4×#15712yes—
163C22×C3#379212yes—
164C22×C3C22×#15711yescharacteristic
165C3×C4—28no—
166C3×C4—22no—
167C3×C4—48no—
168C3×C4—92no—
169C3×C4—92no—
170C3×C4—182no—
171Dic3—64no—
172Dic3—64no—
173Dic3—68no—
174C32—22no—
175C32—42no—
176C32C2×#80311yescharacteristic, complemented
177C23C2×C3⋊#15711yescharacteristic
178C2×C4—22no—
179C2×C4—91no—
180C2×C4—91no—
181C2×C4—94no—
182C2×C4—92no—
183C2×C4—182no—
184C2×C4C2×C3⋊#15712yes—
185C8—278no—
186C2×C3—28no—
187C2×C3—22no—
188C2×C3—22no—
189C2×C3—22no—
190C2×C3—48no—
191C2×C3—42no—
192C2×C3—42no—
193C2×C3—42no—
194C2×C3#6377714yescomplemented
195C2×C3C2×#378811yescharacteristic
196C2×C3C2×C4×#15711yescharacteristic
197C2×C3C2×#379211yescharacteristic
198C22#8898912yes—
199C22C4×C3⋊#15712yes—
200C22#8899312yes—
201C22C22×C3⋊#15711yescharacteristic
202C4—22no—
203C4—92no—
204C4—92no—
205C4—182no—
206C3—22no—
207C3—42no—
208C3C2×#6377711yescharacteristic, complemented
209C2#1172310214yesdirect product factor
210C2C2×#8898911yescharacteristic
211C2C2×C4×C3⋊#15711yescharacteristic
212C2C2×#8899311yescharacteristic
subgroupG/H
centerC23C2×C3⋊#157
commutatorC2×C33C2×C42
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socle, prime radicalC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33C2×#97
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11723102C2direct (N × H)
8. C2×C3⋊#622C4semidirect
24. C3⋊#622C2×C4semidirect
72. C2×C33#97semidirect
113. C33C2×#97semidirect
142. C2×C32#803semidirect
176. C32C2×#803semidirect
194. C2×C3#63777semidirect
208. C3C2×#63777semidirect
209. C2#11723102direct (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
#117231022
C2×#637773
C2×#8039