Finite Group Databasebeta

← C3×#10593756 #13173226

Subgroups — C3×#10593756 #13173226

Order 1728.

380 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C3×#64096C211yesmaximal, characteristic
2C3×#63777C212yesmaximal
3C3×C4⋊#622C211yesmaximal, characteristic
4C3×C4⋊#622C212yesmaximal
5C3×C4×#622C211yesmaximal, characteristic
6#10593756C311yesmaximal, characteristic, direct product factor
7C2×C3×#634C411yescharacteristic
8C2×C3×#622C2211yescharacteristic
9C2×C3×#622C2212yes—
10C2×C3×#622C2211yescharacteristic
11C33×C4⋊C4C411yescharacteristic
12C3×#3750C412yes—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C3×C4×C3⋊Dic3C2212yes—
15#64096C2×C311yescharacteristic
16C3×#887—62no—
17#63777C2×C312yes—
18C4⋊#622C2×C311yescharacteristic
19C4⋊#622C2×C312yes—
20C4×#622C2×C311yescharacteristic
21C2×C33×C4C2×C412yes—
22C2×C33×C4C2×C411yescharacteristic
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3C2×C412yes—
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×#634—22no—
27C3×#622—21no—
28C3×#622—22no—
29C3×#622Q812yescomplemented
30C3×#446—91nomaximal
31C2×#634C3×C411yescharacteristic
32C2×C3×Dic6—62no—
33C2×#622C22×C311yescharacteristic
34C2×#622C22×C312yes—
35C2×#622C22×C311yescharacteristic
36C32×C4⋊C4—22no—
37C32×C4⋊C4—42no—
38C32×C4⋊C4C3×C411yescharacteristic
39#3750C3×C412yes—
40C4×C3⋊Dic3C22×C311yescharacteristic
41C4×C3⋊Dic3C22×C312yes—
42C3×Dic3⋊C4—64no—
43C3×C4×Dic3—64no—
44C3×C4×Dic3—62no—
45C22×C33C22×C411yescharacteristic
46C33×C4—22no—
47C33×C4M1612yes—
48C3×C3⋊Dic3—22no—
49C3×C3⋊Dic3—21no—
50C3×C3⋊Dic3C2×Q811yescharacteristic
51C3×C3⋊Dic3#4111yescharacteristic
52C3×C4×Q8—91no—
53#887—62no—
54C3×C4⋊C8—91no—
55C3×C4⋊C8—92no—
56C3×#97—92no—
57C3×C4×C8—91no—
58C2×C32×C4—22no—
59C2×C32×C4—24no—
60C2×C32×C4—42no—
61C2×C32×C4—44no—
62C2×C32×C4C2×C3×C412yes—
63C2×C32×C4C2×C3×C411yescharacteristic
64C2×C3⋊Dic3C2×C3×C411yescharacteristic
65C2×C3⋊Dic3C2×C3×C412yes—
66C2×C3⋊Dic3C23×C311yescharacteristic
67#634—22no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—64no—
70C2×C3×Dic3—62no—
71C3×Dic6—68no—
72#622—21no—
73#622—22no—
74#622C3×Q812yescomplemented
75#446—91noSylow
76C2×C33C4×Q811yescharacteristic
77C2×C33C2×M1611yescharacteristic
78C2×C33#13111yescharacteristic
79C2×C3×Q8—91no—
80C2×Dic6—62no—
81C2×C3×C8—91no—
82C2×C3×C8—91no—
83C2×C3×C8—92no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—92no—
87C3×C4⋊C4#15711yescharacteristic
88C3×C42—91no—
89C3×C42—92no—
90Dic3⋊C4—64no—
91C4×Dic3—62no—
92C4×Dic3—64no—
93C22×C32—22no—
94C22×C32—42no—
95C22×C32C22×C3×C411yescharacteristic
96C32×C4—24no—
97C32×C4—22no—
98C32×C4—44no—
99C32×C4—44no—
100C32×C4—84no—
101C32×C4C3×M1612yes—
102C3⋊Dic3—22no—
103C3⋊Dic3—21no—
104C3⋊Dic3C2×C3×Q811yescharacteristic
105C3⋊Dic3C3×#4111yescharacteristic
106C3×Dic3—62no—
107C3×Dic3—62no—
108C3×Dic3—68no—
109C3×Dic3—122no—
110C4×Q8—91no—
111C4⋊C8—91no—
112C4⋊C8—92no—
113#97—92no—
114C4×C8—91no—
115C33#44611yescharacteristic, complemented, Sylow
116C3×Q8—182no—
117C2×C3×C4—22no—
118C2×C3×C4—24no—
119C2×C3×C4—42no—
120C2×C3×C4—44no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—92no—
124C2×C3×C4C2×#15712yes—
125C2×C3×C4C2×#15711yescharacteristic
126C2×Dic3—64no—
127C2×Dic3—62no—
128C2×Dic3—62no—
129Dic6—68no—
130C3×C8—92no—
131C3×C8—181no—
132C3×C8—182no—
133C2×C32—22no—
134C2×C32—22no—
135C2×C32—22no—
136C2×C32—42no—
137C2×C32—42no—
138C2×C32—42no—
139C2×C32C3×C4×Q811yescharacteristic
140C2×C32C2×C3×M1611yescharacteristic
141C2×C32C3×#13111yescharacteristic
142C2×Q8—91no—
143C2×C8—91no—
144C2×C8—91no—
145C2×C8—92no—
146C4⋊C4—92no—
147C4⋊C4C3×#15711yescharacteristic
148C42—91no—
149C42—92no—
150C22×C3—22no—
151C22×C3—42no—
152C22×C3C22×#15711yescharacteristic
153C3×C4—22no—
154C3×C4—24no—
155C3×C4—44no—
156C3×C4—44no—
157C3×C4—84no—
158C3×C4—91no—
159C3×C4—91no—
160C3×C4—182no—
161C3×C4—181no—
162C3×C4#378812yes—
163Dic3—62no—
164Dic3—62no—
165Dic3—68no—
166Dic3—122no—
167C32—22no—
168C32—42no—
169C32C3×#44611yescharacteristic, complemented
170Q8—182no—
171C2×C4—91no—
172C2×C4—91no—
173C2×C4—92no—
174C2×C4C2×C3×#15712yes—
175C2×C4C2×C3×#15711yescharacteristic
176C8—92no—
177C8—181no—
178C8—182no—
179C2×C3—22no—
180C2×C3—22no—
181C2×C3—22no—
182C2×C3—42no—
183C2×C3—42no—
184C2×C3—42no—
185C2×C3C2×#378811yescharacteristic
186C2×C3#6428711yescharacteristic
187C2×C3Q8×#15711yescharacteristic
188C22C22×C3×#15711yescharacteristic
189C4—22no—
190C4—91no—
191C4—91no—
192C4—182no—
193C4—181no—
194C4C3×#378812yes—
195C3—22no—
196C3—42no—
197C3#1059375611yescharacteristic, direct product factor
198C2C2×C3×#378811yescharacteristic
199C2C3×#6428711yescharacteristic
200C2C3×Q8×#15711yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC22×C32C22×C3×C4
FrattiniC22C22×C3×#157
FittingC33×C4⋊C4C4
socle, prime radicalC22×C33C22×C4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×#446
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #10593756C3direct (N × H)
29. C3×#622Q8semidirect
74. #622C3×Q8semidirect
115. C33#446semidirect
169. C32C3×#446semidirect
197. C3#10593756direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#640962
C3×#637772
C3×C4⋊#6222
C3×C4×#6222
#105937563
C3×#4469