Finite Group Databasebeta

← C3×#10593748 #13173219

Subgroups — C3×#10593748 #13173219

Order 1728.

464 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 136 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×#3788C211yesmaximal, characteristic
2C2×C3×#3787C211yesmaximal, characteristic
3C3×#64101C211yesmaximal, characteristic
4C3×#63777C212yesmaximal, complemented
5C3×C4×#622C212yesmaximal, complemented
6#10593748C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#622C2211yescharacteristic
9C2×C3×#622C2212yes—
10C2×C3×#622C2211yescharacteristic
11C3×#3788C414yescomplemented
12C3×#3787C414yescomplemented
13C33×C4⋊C4C411yescharacteristic
14C3×#3752C412yes—
15C3×C4⋊(C3⋊Dic3)C411yescharacteristic
16C3×C4×C3⋊Dic3C2212yes—
17C2×#3788C2×C311yescharacteristic
18C2×#3787C2×C311yescharacteristic
19#64101C2×C311yescharacteristic
20C3×#892—62no—
21#63777C2×C312yescomplemented
22C4×#622C2×C312yescomplemented
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C2×C33×C4C2×C412yes—
25C2×C33×C4C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2×C412yes—
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3×C4×C3⋊D3C2×C412yes—
29C3×C4×C3⋊D3C2×C412yes—
30C3×#622C2×C412yes—
31C3×#622C2×C412yes—
32C3×#622C2×C414yes—
33C3×#405—91nomaximal
34C2×C4×C3⋊D3C22×C311yescharacteristic
35C2×C3×C4×D3—62no—
36C2×#622C22×C311yescharacteristic
37C2×#622C22×C311yescharacteristic
38C2×#622C22×C312yes—
39#3788C3×C414yescomplemented
40#3787C3×C414yescomplemented
41C32×C4⋊C4—22no—
42C32×C4⋊C4—42no—
43C32×C4⋊C4C3×C411yescharacteristic
44#3752C3×C412yes—
45C4⋊(C3⋊Dic3)C3×C411yescharacteristic
46C4×C3⋊Dic3C22×C312yes—
47C3×(C2×D3)⋊C4—64no—
48C3×C4⋊Dic3—62no—
49C3×C4×Dic3—64no—
50C22×C33C22×C411yescharacteristic
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3C4212yes—
54C33×C4—22no—
55C33×C4C4212yes—
56C3×C3⋊Dic3—22no—
57C3×C3⋊Dic3C22×C411yescharacteristic
58C3×C3⋊Dic3C22×C411yescharacteristic
59C2×C3×M16—91no—
60C22×C3×C8—91no—
61C3×#117—91no—
62#892—62no—
63C3×#97—92no—
64C3×C4×C8—92no—
65C22×C3⋊D3C2×C3×C411yescharacteristic
66C22×C3×D3—62no—
67C2×C32×C4—22no—
68C2×C32×C4—24no—
69C2×C32×C4—42no—
70C2×C32×C4—44no—
71C2×C32×C4C2×C3×C412yes—
72C2×C32×C4C2×C3×C411yescharacteristic
73C2×C3⋊Dic3C2×C3×C412yes—
74C2×C3⋊Dic3C23×C311yescharacteristic
75C4×C3⋊D3C2×C3×C412yes—
76C4×C3⋊D3C2×C3×C412yes—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—64no—
79C3×C4×D3—64no—
80C3×C4×D3—64no—
81#622C2×C3×C412yes—
82#622C2×C3×C412yes—
83#622C2×C3×C414yes—
84#405—91noSylow
85C2×C33C2×C4211yescharacteristic
86C2×C33#13111yescharacteristic
87C2×C33#13111yescharacteristic
88C3×C3⋊D3—22no—
89C22×C3×C4—91no—
90C2×C4×D3—62no—
91C3×M16—94no—
92C2×C3×C8—91no—
93C2×C3×C8—92no—
94C2×C3×C8—91no—
95C2×C3×C8—94no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—42no—
98C3×C4⋊C4—91no—
99C3×C4⋊C4#15711yescharacteristic
100C3×C22⋊C4—92no—
101C3×C42—92no—
102(C2×D3)⋊C4—64no—
103C4⋊Dic3—62no—
104C4×Dic3—64no—
105C22×C32—22no—
106C22×C32—42no—
107C22×C32C22×C3×C411yescharacteristic
108C2×C3⋊D3—21no—
109C2×C3⋊D3—21no—
110C2×C3⋊D3C3×C4212yes—
111C2×C3×D3—64no—
112C2×C3×D3—122no—
113C2×C3×D3—122no—
114C32×C4—24no—
115C32×C4—22no—
116C32×C4—44no—
117C32×C4—44no—
118C32×C4—84no—
119C32×C4C3×C4212yes—
120C3⋊Dic3—22no—
121C3⋊Dic3C22×C3×C411yescharacteristic
122C3⋊Dic3C22×C3×C411yescharacteristic
123C3×Dic3—62no—
124C3×Dic3—62no—
125C3×Dic3—124no—
126C2×M16—91no—
127C22×C8—91no—
128#117—91no—
129#97—92no—
130C4×C8—92no—
131C33#40511yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—62no—
134C2×C3×C4—22no—
135C2×C3×C4—24no—
136C2×C3×C4—42no—
137C2×C3×C4—44no—
138C2×C3×C4—91no—
139C2×C3×C4—92no—
140C2×C3×C4—92no—
141C2×C3×C4—92no—
142C2×C3×C4C2×#15712yes—
143C2×C3×C4C2×#15711yescharacteristic
144C2×Dic3—62no—
145C2×Dic3—64no—
146C4×D3—64no—
147C4×D3—64no—
148C3×C8—92no—
149C3×C8—92no—
150C3×C8—94no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—42no—
155C2×C32—42no—
156C2×C32—42no—
157C2×C32C2×C3×C4211yescharacteristic
158C2×C32C3×#13111yescharacteristic
159C2×C32C3×#13111yescharacteristic
160C3⋊D3—22no—
161C3×D3—124no—
162C22×C4—91no—
163M16—94no—
164C2×C8—91no—
165C2×C8—92no—
166C2×C8—91no—
167C2×C8—94no—
168C4⋊C4—91no—
169C4⋊C4C3×#15711yescharacteristic
170C22⋊C4—92no—
171C42—92no—
172C22×C3—22no—
173C22×C3—42no—
174C22×C3—92no—
175C22×C3—181no—
176C22×C3—181no—
177C22×C3C22×#15711yescharacteristic
178C2×D3—64no—
179C2×D3—122no—
180C2×D3—122no—
181C3×C4—22no—
182C3×C4—24no—
183C3×C4—44no—
184C3×C4—44no—
185C3×C4—84no—
186C3×C4—91no—
187C3×C4—91no—
188C3×C4—182no—
189C3×C4C4×#15712yes—
190Dic3—62no—
191Dic3—62no—
192Dic3—124no—
193C32—22no—
194C32—42no—
195C32C3×#40511yescharacteristic, complemented
196C23—91no—
197C2×C4—91no—
198C2×C4—92no—
199C2×C4—92no—
200C2×C4—92no—
201C2×C4C2×C3×#15712yes—
202C2×C4C2×C3×#15711yescharacteristic
203C8—92no—
204C8—92no—
205C8—94no—
206C2×C3—22no—
207C2×C3—22no—
208C2×C3—22no—
209C2×C3—42no—
210C2×C3—42no—
211C2×C3—42no—
212C2×C3—182no—
213C2×C3C2×C4×#15711yescharacteristic
214C2×C3#6428711yescharacteristic
215C2×C3#6428911yescharacteristic
216D3—124no—
217C22—92no—
218C22—181no—
219C22—181no—
220C22C22×C3×#15711yescharacteristic
221C4—22no—
222C4—91no—
223C4—91no—
224C4—182no—
225C4C3×C4×#15712yes—
226C3—22no—
227C3—42no—
228C3#1059374811yescharacteristic, direct product factor
229C2—182no—
230C2C2×C3×C4×#15711yescharacteristic
231C2C3×#6428711yescharacteristic
232C2C3×#6428911yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC2×C32C2×C3×C42
FrattiniC22C22×C3×#157
FittingC33×C4⋊C4C4
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×#405
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#63777C2semidirect
5. C3×C4×#622C2semidirect
6. #10593748C3direct (N × H)
11. C3×#3788C4semidirect
12. C3×#3787C4semidirect
21. #63777C2×C3semidirect
22. C4×#622C2×C3semidirect
39. #3788C3×C4semidirect
40. #3787C3×C4semidirect
131. C33#405semidirect
195. C32C3×#405semidirect
228. C3#10593748direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37882
C2×C3×#37872
C3×#641012
C3×#637772
C3×C4×#6222
#105937483
C3×#4059