Finite Group Databasebeta

← C3×#10590182 #13157107

Subgroups — C3×#10590182 #13157107

Order 1728.

424 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 92 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×C8×C3⋊D3C211yesmaximal, characteristic
3#10590182C311yesmaximal, characteristic, direct product factor
4C22×C3×#157—22no—
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C3×C4⋊#157—22no—
7C3×C4⋊#157C414yescomplemented
8C2×C33×C8C411yescharacteristic, complemented
9C2×C3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
10C3×C8×C3⋊D3—22no—
11C2×C4⋊#157C2×C312yes—
12C2×C8×C3⋊D3C2×C311yescharacteristic
13C2×C3×D3×C8—62no—
14C22×C3×C3⋊D3D411yescharacteristic
15C2×C3×#157—24no—
16C2×C3×#157—42no—
17C2×C3×#157—42no—
18C2×C33×C4C2×C411yescharacteristic
19C2×C3×C3⋊Dic3Q811yescharacteristic
20C3×C4×C3⋊D3C2×C412yes—
21C3×C4×C3⋊D3D412yes—
22C33×C8—21no—
23C3×C3⋊(C3⋊C8)—21no—
24C3×#340—91nomaximal
25C22×#157—22no—
26C2×C4×C3⋊D3C22×C311yescharacteristic
27C2×C3×C4×D3—62no—
28C4⋊#157—22no—
29C4⋊#157C3×C414yescomplemented
30C2×C32×C8—22no—
31C2×C32×C8—42no—
32C2×C32×C8C3×C411yescharacteristic, complemented
33C2×C3⋊(C3⋊C8)C3×C411yescharacteristic, complemented
34C8×C3⋊D3—22no—
35C2×C3×C3⋊C8—62no—
36C3×D3×C8—68no—
37C22×C33C4⋊C411yescharacteristic
38C2×C3×C3⋊D3C22⋊C412yes—
39C2×C3×C3⋊D3D811yescharacteristic
40C2×C3×C3⋊D3SD1612yes—
41C2×C3×C3⋊D3Q1611yescharacteristic
42C3×#157—44no—
43C33×C4C4211yescharacteristic
44C33×C4C22⋊C411yescharacteristic
45C3×C3⋊Dic3C4⋊C412yes—
46C22×C3×C8—91no—
47C2×C3×C4⋊C4—92no—
48C2×D3×C8—62no—
49C22×C3⋊D3C3×D411yescharacteristic
50C22×C3×D3—62no—
51C2×#157—24no—
52C2×#157—42no—
53C2×#157—42no—
54C2×C32×C4—22no—
55C2×C32×C4—42no—
56C2×C32×C4C2×C3×C411yescharacteristic
57C2×C3⋊Dic3C3×Q811yescharacteristic
58C4×C3⋊D3C2×C3×C412yes—
59C4×C3⋊D3C3×D412yes—
60C2×C3×Dic3—62no—
61C3×C4×D3—64no—
62C3×C4×D3—64no—
63C32×C8—21no—
64C32×C8—24no—
65C32×C8—44no—
66C3⋊(C3⋊C8)—21no—
67C3×C3⋊C8—64no—
68#340—91noSylow
69C2×C33#9511yescharacteristic
70C2×C33#10611yescharacteristic
71C2×C33#10711yescharacteristic
72C3×C3⋊D3#10212yes—
73C3×C3⋊D3#10312yes—
74C22×C3×C4—91no—
75C22×C3×C4—182no—
76C2×C4×D3—62no—
77C2×C3×C8—22no—
78C2×C3×C8—42no—
79C2×C3×C8—91no—
80C2×C3×C8—182no—
81C2×C3×C8#15711yescharacteristic, complemented
82C3×C4⋊C4—94no—
83C3×C4⋊C4—182no—
84C2×C3⋊C8—62no—
85D3×C8—68no—
86C22×C32—22no—
87C22×C32—42no—
88C22×C32C3×C4⋊C411yescharacteristic
89C2×C3⋊D3C3×C22⋊C412yes—
90C2×C3⋊D3C3×D811yescharacteristic
91C2×C3⋊D3C3×SD1612yes—
92C2×C3⋊D3C3×Q1611yescharacteristic
93C2×C3×D3—64no—
94C2×C3×D3—62no—
95C2×C3×D3—64no—
96C2×C3×D3—62no—
97#157—44no—
98C32×C4—22no—
99C32×C4—22no—
100C32×C4—42no—
101C32×C4—42no—
102C32×C4C3×C4211yescharacteristic
103C32×C4C3×C22⋊C411yescharacteristic
104C3⋊Dic3C3×C4⋊C412yes—
105C3×Dic3—64no—
106C22×C8—91no—
107C2×C4⋊C4—92no—
108C33#34011yescharacteristic, complemented, Sylow
109C23×C3—91no—
110C22×D3—62no—
111C2×C3×C4—22no—
112C2×C3×C4—42no—
113C2×C3×C4—91no—
114C2×C3×C4—92no—
115C2×C3×C4—92no—
116C2×C3×C4—184no—
117C2×C3×C4—362no—
118C2×C3×C4—362no—
119C2×C3×C4C2×#15711yescharacteristic
120C2×Dic3—62no—
121C4×D3—64no—
122C4×D3—64no—
123C3×C8—24no—
124C3×C8—21no—
125C3×C8—44no—
126C3×C8—181no—
127C3⋊C8—64no—
128C2×C32—22no—
129C2×C32—22no—
130C2×C32—22no—
131C2×C32—42no—
132C2×C32—42no—
133C2×C32—42no—
134C2×C32C3×#9511yescharacteristic
135C2×C32C3×#10611yescharacteristic
136C2×C32C3×#10711yescharacteristic
137C3⋊D3C3×#10212yes—
138C3⋊D3C3×#10312yes—
139C3×D3—64no—
140C3×D3—64no—
141C22×C4—91no—
142C22×C4—182no—
143C2×C8—91no—
144C2×C8—182no—
145C2×C8C3×#15711yescharacteristic, complemented
146C4⋊C4—94no—
147C4⋊C4—182no—
148C22×C3—22no—
149C22×C3—42no—
150C22×C3—92no—
151C22×C3—91no—
152C22×C3—91no—
153C22×C3—92no—
154C22×C3C4⋊#15711yescharacteristic
155C2×D3—64no—
156C2×D3—62no—
157C2×D3—64no—
158C2×D3—62no—
159C3×C4—22no—
160C3×C4—22no—
161C3×C4—42no—
162C3×C4—42no—
163C3×C4—92no—
164C3×C4—364no—
165C3×C4C4×#15711yescharacteristic
166C3×C4C22⋊#15711yescharacteristic
167Dic3—64no—
168C32—22no—
169C32—42no—
170C32C3×#34011yescharacteristic, complemented
171C23—91no—
172C2×C4—91no—
173C2×C4—92no—
174C2×C4—92no—
175C2×C4—184no—
176C2×C4—362no—
177C2×C4—362no—
178C2×C4C2×C3×#15711yescharacteristic
179C8—21no—
180C8—181no—
181C2×C3—22no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—42no—
185C2×C3—42no—
186C2×C3—42no—
187C2×C3—92no—
188C2×C3—92no—
189C2×C3#6376811yescharacteristic
190C2×C3#6376911yescharacteristic
191C2×C3#6378111yescharacteristic
192D3—64no—
193D3—64no—
194C22—92no—
195C22—91no—
196C22—91no—
197C22—92no—
198C22C3×C4⋊#15711yescharacteristic
199C4—92no—
200C4—364no—
201C4C3×C4×#15711yescharacteristic
202C4C3×C22⋊#15711yescharacteristic
203C3—22no—
204C3—42no—
205C3#1059018211yescharacteristic, direct product factor
206C2—92no—
207C2—92no—
208C2C3×#6376811yescharacteristic
209C2C3×#6376911yescharacteristic
210C2C3×#6378111yescharacteristic
subgroupG/H
centerC22×C3C4⋊#157
commutatorC32×C4C3×C42
FrattiniC2×C4C2×C3×#157
FittingC2×C33×C8C4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C4×C3⋊D3C22×C3
hypercenterC2×C3×C8#157
hypocenterC32C3×#340
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. #10590182C3direct (N × H)
7. C3×C4⋊#157C4semidirect
8. C2×C33×C8C4semidirect
9. C2×C3×C3⋊(C3⋊C8)C4semidirect
29. C4⋊#157C3×C4semidirect
32. C2×C32×C8C3×C4semidirect
33. C2×C3⋊(C3⋊C8)C3×C4semidirect
81. C2×C3×C8#157semidirect
108. C33#340semidirect
145. C2×C8C3×#157semidirect
170. C32C3×#340semidirect
205. C3#10590182direct (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×C8×C3⋊D32
#105901823
C3×#3409