Finite Group Databasebeta

← C3×#10590019 #13154869

Subgroups — C3×#10590019 #13154869

Order 1728.

234 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 38 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×#63809C211yesmaximal, characteristic
2C3×#63768C211yesmaximal, characteristic, complemented
3C3×#63544C211yesmaximal, characteristic, complemented
4#10590019C311yesmaximal, characteristic, direct product factor
5C3×#3796—21no—
6C3×C4⋊#157—21no—
7C3×C8×C3⋊D3C2211yescharacteristic
8C32×#204—21no—
9C3×#3718C411yescharacteristic, complemented
10C3×#3713C411yescharacteristic, complemented
11C32×C3⋊C16—21no—
12C3×#916—61no—
13#63809C2×C311yescharacteristic
14#63768C2×C311yescharacteristic, complemented
15C3×#798—61no—
16#63544C2×C311yescharacteristic, complemented
17C2×C3×#157—41no—
18C3×C4×C3⋊D3D411yescharacteristic
19C32×D12—21no—
20C32×Dic6—21no—
21C3×#626—41no—
22C3×D3×Dic3—41no—
23C33×C8D411yescharacteristic
24C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
25C3×#360—91nomaximal
26C3×#239—61no—
27C3×#237—61no—
28#3796—21no—
29C4⋊#157—21no—
30C32×SD16—61no—
31C8×C3⋊D3C22×C311yescharacteristic
32C3×#216—61no—
33C3×#213—61no—
34C3×#204—21no—
35C3×#204—21no—
36C3×#204—41no—
37C3×D3×C8—61no—
38C3×D3×C8—61no—
39#3718C3×C411yescharacteristic, complemented
40#3713C3×C411yescharacteristic, complemented
41C32×C16—61no—
42C3×C3⋊C16—21no—
43C3×C3⋊C16—21no—
44C3×C3⋊C16—41no—
45C2×C3×C3⋊D3D811yescharacteristic
46C2×C32×D3—41no—
47C3×#157—81no—
48C33×C4C22⋊C411yescharacteristic
49C3×C3⋊Dic3SD1611yescharacteristic
50C32×Dic3—41no—
51C3×#135—91no—
52#916—61no—
53C3×M32—91no—
54C3×#106—91no—
55#798—61no—
56C2×#157—41no—
57C32×Q8—61no—
58C32×D4—61no—
59C4×C3⋊D3C3×D411yescharacteristic
60C3×#67—121no—
61C2×C3×Dic3—121no—
62C3×D12—21no—
63C3×D12—21no—
64C3×D12—41no—
65C3×D12—121no—
66C3×C4×D3—61no—
67C3×C4×D3—61no—
68C3×C4×D3—121no—
69C3×Dic6—21no—
70C3×Dic6—21no—
71C3×Dic6—41no—
72#626—41no—
73D3×Dic3—41no—
74C32×C8—21no—
75C32×C8—21no—
76C32×C8—41no—
77C32×C8—41no—
78C32×C8C3×D411yescharacteristic
79C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
80C3×C3⋊C8—61no—
81C3×C3⋊C8—61no—
82#360—91noSylow
83C2×C33#10211yescharacteristic
84C3×C3⋊D3—21no—
85C32×D3—81no—
86C3×#41—181no—
87#239—61no—
88#237—61no—
89C3×Q16—91no—
90C3×SD16—61no—
91C3×SD16—61no—
92C3×SD16—121no—
93C3×D8—91no—
94C2×C3×C8—91no—
95C3×C4⋊C4—181no—
96#216—61no—
97#213—61no—
98#204—21no—
99D3×C8—61no—
100D3×C8—61no—
101C3×C16—61no—
102C3×C16—61no—
103C3×C16—121no—
104C3⋊C16—21no—
105C22×C32—121no—
106C2×C3⋊D3C3×D811yescharacteristic
107C2×C3×D3—41no—
108C2×C3×D3—41no—
109C2×C3×D3—61no—
110C2×C3×D3—61no—
111C2×C3×D3—81no—
112#157—81no—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4—41no—
116C32×C4—41no—
117C32×C4—121no—
118C32×C4C3×C22⋊C411yescharacteristic
119C3⋊Dic3C3×SD1611yescharacteristic
120C3×Dic3—41no—
121C3×Dic3—41no—
122C3×Dic3—61no—
123C3×Dic3—61no—
124C3×Dic3—81no—
125#135—91no—
126M32—91no—
127#106—91no—
128C33#36011yescharacteristic, complemented, Sylow
129C3×Q8—61no—
130C3×Q8—61no—
131C3×Q8—121no—
132C3×D4—61no—
133C3×D4—61no—
134C3×D4—121no—
135C3×D4—361no—
136C2×C3×C4—91no—
137C2×C3×C4—361no—
138C2×C3×C4—361no—
139#67—121no—
140C2×Dic3—121no—
141D12—21no—
142D12—121no—
143C4×D3—61no—
144C4×D3—61no—
145C4×D3—121no—
146Dic6—21no—
147C3×C8—21no—
148C3×C8—21no—
149C3×C8—41no—
150C3×C8—41no—
151C3×C8—91no—
152C3×C8GO+(4,𝔽2)11yescharacteristic
153C3⋊C8—61no—
154C3⋊C8—61no—
155C2×C32—21no—
156C2×C32—21no—
157C2×C32—41no—
158C2×C32—41no—
159C2×C32—241no—
160C2×C32C3×#10211yescharacteristic
161C3⋊D3—21no—
162C3×D3—81no—
163C3×D3—81no—
164C3×D3—121no—
165C3×D3—121no—
166C3×D3—161no—
167#41—181no—
168Q16—91no—
169SD16—61no—
170D8—91no—
171C2×C8—91no—
172C4⋊C4—181no—
173C16—61no—
174C22×C3—91no—
175C22×C3—121no—
176C22×C3—121no—
177C22×C3—241no—
178C2×D3—41no—
179C2×D3—61no—
180C2×D3—61no—
181C3×C4—21no—
182C3×C4—21no—
183C3×C4—41no—
184C3×C4—41no—
185C3×C4—91no—
186C3×C4—121no—
187C3×C4—121no—
188C3×C4—241no—
189C3×C4—721no—
190C3×C4D32⋊C411yescharacteristic
191Dic3—41no—
192Dic3—61no—
193Dic3—61no—
194C32—21no—
195C32—21no—
196C32—41no—
197C32—41no—
198C32C3×#36011yescharacteristic, complemented
199Q8—61no—
200D4—61no—
201D4—361no—
202C2×C4—91no—
203C2×C4—361no—
204C2×C4—361no—
205C8—91no—
206C8C3×GO+(4,𝔽2)11yescharacteristic
207C2×C3—21no—
208C2×C3—21no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—181no—
212C2×C3—241no—
213C2×C3—241no—
214C2×C3—481no—
215C2×C3#6372911yescharacteristic
216D3—81no—
217D3—121no—
218D3—121no—
219C22—91no—
220C22—121no—
221C4—91no—
222C4—121no—
223C4—721no—
224C4C3×D32⋊C411yescharacteristic
225C3—21no—
226C3—21no—
227C3—41no—
228C3—41no—
229C3#1059001911yescharacteristic, direct product factor
230C2—181no—
231C2—241no—
232C2C3×#6372911yescharacteristic
subgroupG/H
centerC2×C3#63729
commutatorC3⋊(C3⋊C8)C2×C3×C4
FrattiniC8C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#102
prime radicalC3×#63809C2
cosocleC8×C3⋊D3C22×C3
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#360
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63768C2semidirect
3. C3×#63544C2semidirect
4. #10590019C3direct (N × H)
9. C3×#3718C4semidirect
10. C3×#3713C4semidirect
14. #63768C2×C3semidirect
16. #63544C2×C3semidirect
39. #3718C3×C4semidirect
40. #3713C3×C4semidirect
128. C33#360semidirect
198. C32C3×#360semidirect
229. C3#10590019direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638092
C3×#637682
C3×#635442
#105900193
C3×#3609