Finite Group Databasebeta

← C3×#10593553 #13167630

Subgroups — C3×#10593553 #13167630

Order 1728.

442 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 146 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×C3×C4⋊#157C212yesmaximal
2C2×C3×C4×#157C211yesmaximal, characteristic
3C2×C3×#3777C214yesmaximal
4#10593553C311yesmaximal, characteristic, direct product factor
5C22×C3×#157C2214yes—
6C22×C3×#157C2212yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C3×C4⋊#157—24no—
9C3×C4×#157C414yescomplemented
10C3×#3777—28no—
11C2×C4⋊#157C2×C312yes—
12C2×C4×#157C2×C311yescharacteristic
13C2×#3777C2×C314yes—
14C22×C3×C3⋊D3C2311yescharacteristic
15C2×C3×#157—28no—
16C2×C3×#157—24no—
17C2×C3×#157C2×C414yes—
18C2×C3×#157D414yes—
19C2×C3×#157Q814yes—
20C2×C33×C4Q811yescharacteristic
21C2×C3×C3⋊Dic3Q811yescharacteristic
22C3×C4×C3⋊D3C2×C412yes—
23C3×C4×C3⋊D3D412yes—
24C3×#384—91nomaximal
25C22×#157C22×C314yes—
26C22×#157C22×C312yes—
27C2×C4×C3⋊D3C22×C311yescharacteristic
28C2×C3×C4×D3—121no—
29C4⋊#157—24no—
30C4×#157C3×C414yescomplemented
31#3777—28no—
32C22×C33C2×Q811yescharacteristic
33C2×C3×C3⋊D3C22×C411yescharacteristic
34C2×C3×C3⋊D3C2×D412yes—
35C2×C3×C3⋊D3C2×D411yescharacteristic
36C2×C3×C3⋊D3C2×Q812yes—
37C3×#157—44no—
38C3×#157C4⋊C418yescomplemented
39C33×C4C4⋊C412yescomplemented
40C3×C3⋊Dic3C4⋊C412yescomplemented
41C2×C3×C4⋊C4—94no—
42C2×C3×C4⋊C4—92no—
43C2×C3×C42—91no—
44C22×C3⋊D3C23×C311yescharacteristic
45C22×C3×D3—121no—
46C2×#157—28no—
47C2×#157—24no—
48C2×#157C2×C3×C414yes—
49C2×#157C3×D414yes—
50C2×#157C3×Q814yes—
51C2×C32×C4—41no—
52C2×C32×C4—81no—
53C2×C32×C4C3×Q811yescharacteristic
54C2×C3⋊Dic3C3×Q811yescharacteristic
55C4×C3⋊D3C2×C3×C412yes—
56C4×C3⋊D3C3×D412yes—
57C2×C3×Dic3—121no—
58C3×C4×D3—122no—
59C3×C4×D3—122no—
60#384—91noSylow
61C2×C33C2×C4⋊C411yescharacteristic
62C2×C33C4⋊Q812yes—
63C3×C3⋊D3C2×C4⋊C412yes—
64C3×C3⋊D3#12711yescharacteristic
65C3×C3⋊D3C4⋊Q811yescharacteristic
66C22×C3×C4—92no—
67C22×C3×C4—91no—
68C22×C3×C4—94no—
69C2×C4×D3—121no—
70C3×C4⋊C4—188no—
71C3×C4⋊C4—184no—
72C3×C42—94no—
73C22×C32—41no—
74C22×C32—81no—
75C22×C32C2×C3×Q811yescharacteristic
76C2×C3⋊D3C22×C3×C411yescharacteristic
77C2×C3⋊D3C2×C3×D412yes—
78C2×C3⋊D3C2×C3×D411yescharacteristic
79C2×C3⋊D3C2×C3×Q812yes—
80C2×C3×D3—121no—
81C2×C3×D3—121no—
82C2×C3×D3—122no—
83C2×C3×D3—122no—
84#157—44no—
85#157C3×C4⋊C418yescomplemented
86C32×C4—42no—
87C32×C4—82no—
88C32×C4C3×C4⋊C412yescomplemented
89C3⋊Dic3C3×C4⋊C412yescomplemented
90C3×Dic3—122no—
91C2×C4⋊C4—92no—
92C2×C4⋊C4—94no—
93C2×C42—91no—
94C33#38411yescharacteristic, complemented, Sylow
95C23×C3—91no—
96C22×D3—121no—
97C2×C3×C4—41no—
98C2×C3×C4—81no—
99C2×C3×C4—94no—
100C2×C3×C4—94no—
101C2×C3×C4—94no—
102C2×C3×C4—91no—
103C2×C3×C4—92no—
104C2×C3×C4—92no—
105C2×C3×C4—188no—
106C2×C3×C4—184no—
107C2×C3×C4PSU(3,𝔽2)11yescharacteristic
108C2×Dic3—121no—
109C4×D3—122no—
110C4×D3—122no—
111C2×C32—42no—
112C2×C32—41no—
113C2×C32—81no—
114C2×C32—82no—
115C2×C32C2×C3×C4⋊C411yescharacteristic
116C2×C32C3×C4⋊Q812yes—
117C3⋊D3C2×C3×C4⋊C412yes—
118C3⋊D3C3×#12711yescharacteristic
119C3⋊D3C3×C4⋊Q811yescharacteristic
120C3×D3—122no—
121C3×D3—121no—
122C3×D3—121no—
123C22×C4—92no—
124C22×C4—91no—
125C22×C4—94no—
126C4⋊C4—188no—
127C4⋊C4—184no—
128C42—94no—
129C22×C3—41no—
130C22×C3—81no—
131C22×C3—92no—
132C22×C3—92no—
133C22×C3—91no—
134C22×C3—91no—
135C22×C3C2×PSU(3,𝔽2)11yescharacteristic
136C2×D3—121no—
137C2×D3—121no—
138C2×D3—122no—
139C2×D3—122no—
140C3×C4—42no—
141C3×C4—82no—
142C3×C4—98no—
143C3×C4—92no—
144C3×C4—364no—
145C3×C4#377712yescomplemented
146Dic3—122no—
147C32—41no—
148C32—81no—
149C32C3×#38411yescharacteristic, complemented
150C23—91no—
151C2×C4—94no—
152C2×C4—94no—
153C2×C4—94no—
154C2×C4—91no—
155C2×C4—92no—
156C2×C4—92no—
157C2×C4—188no—
158C2×C4—184no—
159C2×C4C3×PSU(3,𝔽2)11yescharacteristic
160C2×C3—42no—
161C2×C3—41no—
162C2×C3—82no—
163C2×C3—81no—
164C2×C3—92no—
165C2×C3—91no—
166C2×C3—91no—
167C2×C3C4⋊PSU(3,𝔽2)12yes—
168C2×C3C2×#377711yescharacteristic
169D3—122no—
170D3—121no—
171D3—121no—
172C22—92no—
173C22—92no—
174C22—91no—
175C22—91no—
176C22C2×C3×PSU(3,𝔽2)11yescharacteristic
177C4—98no—
178C4—92no—
179C4—364no—
180C4C3×#377712yescomplemented
181C3—41no—
182C3—81no—
183C3#1059355311yescharacteristic, direct product factor
184C2—92no—
185C2—91no—
186C2—91no—
187C2C3×C4⋊PSU(3,𝔽2)12yes—
188C2C2×C3×#377711yescharacteristic
subgroupG/H
centerC22×C3C2×PSU(3,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC22×C33C2×Q8
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#384
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. #10593553C3direct (N × H)
9. C3×C4×#157C4semidirect
30. C4×#157C3×C4semidirect
38. C3×#157C4⋊C4semidirect
39. C33×C4C4⋊C4semidirect
40. C3×C3⋊Dic3C4⋊C4semidirect
85. #157C3×C4⋊C4semidirect
88. C32×C4C3×C4⋊C4semidirect
89. C3⋊Dic3C3×C4⋊C4semidirect
94. C33#384semidirect
145. C3×C4#3777semidirect
149. C32C3×#384semidirect
180. C4C3×#3777semidirect
183. C3#10593553direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×C3×C4×#1572
C2×C3×#37772
#105935533
C3×#3849