Finite Group Databasebeta

← C3×#10590018 #13154868

Subgroups — C3×#10590018 #13154868

Order 1728.

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

normal subgroups only

This group has 232 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×#63770C211yesmaximal, characteristic, complemented
3C3×#63542C211yesmaximal, characteristic, complemented
4#10590018C311yesmaximal, characteristic, direct product factor
5C3×#3796—21no—
6C3×#3788—21no—
7C3×C8×C3⋊D3C2211yescharacteristic
8C32×#204—21no—
9C3×#3718C411yescharacteristic
10C3×#3713C411yescharacteristic
11C32×C3⋊C16—21no—
12C3×#916—61no—
13#63809C2×C311yescharacteristic
14#63770C2×C311yescharacteristic, complemented
15C3×D3×C16—61no—
16#63542C2×C311yescharacteristic, complemented
17C3×C4×C3⋊D3D411yescharacteristic
18C32×D12—21no—
19C32×Dic6—21no—
20C3×#626—41no—
21C3×D3×Dic3—41no—
22C3×#622—41no—
23C33×C8D411yescharacteristic
24C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
25C3×#359—91nomaximal
26C3×#239—61no—
27C3×#237—61no—
28#3796—21no—
29#3788—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
40#3713C3×C411yescharacteristic
41C32×C16—61no—
42C3×C3⋊C16—21no—
43C3×C3⋊C16—21no—
44C3×C3⋊C16—41no—
45C2×C3×C3⋊D3SD1611yescharacteristic
46C2×C32×D3—41no—
47C33×C4C22⋊C411yescharacteristic
48C3×C3⋊Dic3D811yescharacteristic
49C32×Dic3—41no—
50C3×#135—91no—
51#916—61no—
52C2×C3×C16—91no—
53C3×#108—91no—
54D3×C16—61no—
55C32×Q8—61no—
56C32×D4—61no—
57C4×C3⋊D3C3×D411yescharacteristic
58C3×#67—121no—
59C2×C3×Dic3—121no—
60C3×D12—21no—
61C3×D12—21no—
62C3×D12—41no—
63C3×D12—121no—
64C3×C4×D3—61no—
65C3×C4×D3—61no—
66C3×C4×D3—121no—
67C3×Dic6—21no—
68C3×Dic6—21no—
69C3×Dic6—41no—
70#626—41no—
71D3×Dic3—41no—
72#622—41no—
73C32×C8—21no—
74C32×C8—21no—
75C32×C8—41no—
76C32×C8—41no—
77C32×C8C3×D411yescharacteristic
78C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
79C3×C3⋊C8—61no—
80C3×C3⋊C8—61no—
81#359—91noSylow
82C2×C33#10211yescharacteristic
83C3×C3⋊D3—21no—
84C32×D3—81no—
85C3×#41—181no—
86#239—61no—
87#237—61no—
88C3×Q16—91no—
89C3×SD16—61no—
90C3×SD16—61no—
91C3×SD16—121no—
92C3×D8—91no—
93C3×M16—181no—
94C2×C3×C8—91no—
95#216—61no—
96#213—61no—
97#204—21no—
98D3×C8—61no—
99D3×C8—61no—
100C3×C16—61no—
101C3×C16—61no—
102C3×C16—121no—
103C3⋊C16—21no—
104C22×C32—121no—
105C2×C3⋊D3C3×SD1611yescharacteristic
106C2×C3×D3—41no—
107C2×C3×D3—41no—
108C2×C3×D3—61no—
109C2×C3×D3—61no—
110C2×C3×D3—81no—
111C32×C4—21no—
112C32×C4—21no—
113C32×C4—41no—
114C32×C4—41no—
115C32×C4—121no—
116C32×C4C3×C22⋊C411yescharacteristic
117C3⋊Dic3C3×D811yescharacteristic
118C3×Dic3—41no—
119C3×Dic3—41no—
120C3×Dic3—61no—
121C3×Dic3—61no—
122C3×Dic3—81no—
123#135—91no—
124C2×C16—91no—
125#108—91no—
126C33#35911yescharacteristic, complemented, Sylow
127C3×Q8—61no—
128C3×Q8—61no—
129C3×Q8—121no—
130C3×D4—61no—
131C3×D4—61no—
132C3×D4—121no—
133C3×D4—361no—
134C2×C3×C4—91no—
135C2×C3×C4—361no—
136#67—121no—
137C2×Dic3—121no—
138D12—21no—
139D12—121no—
140C4×D3—61no—
141C4×D3—61no—
142C4×D3—121no—
143Dic6—21no—
144C3×C8—21no—
145C3×C8—21no—
146C3×C8—41no—
147C3×C8—41no—
148C3×C8—91no—
149C3×C8—361no—
150C3×C8GO+(4,𝔽2)11yescharacteristic
151C3⋊C8—61no—
152C3⋊C8—61no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32—41no—
156C2×C32—41no—
157C2×C32—241no—
158C2×C32C3×#10211yescharacteristic
159C3⋊D3—21no—
160C3×D3—62no—
161C3×D3—81no—
162C3×D3—81no—
163C3×D3—121no—
164C3×D3—161no—
165#41—181no—
166Q16—91no—
167SD16—61no—
168D8—91no—
169M16—181no—
170C2×C8—91no—
171C16—61no—
172C22×C3—91no—
173C22×C3—121no—
174C22×C3—121no—
175C22×C3—241no—
176C2×D3—41no—
177C2×D3—61no—
178C2×D3—61no—
179C3×C4—21no—
180C3×C4—21no—
181C3×C4—41no—
182C3×C4—41no—
183C3×C4—91no—
184C3×C4—121no—
185C3×C4—121no—
186C3×C4—241no—
187C3×C4D32⋊C411yescharacteristic
188Dic3—41no—
189Dic3—61no—
190Dic3—61no—
191C32—21no—
192C32—21no—
193C32—41no—
194C32—41no—
195C32C3×#35911yescharacteristic, complemented
196Q8—61no—
197D4—61no—
198D4—361no—
199C2×C4—91no—
200C2×C4—361no—
201C8—91no—
202C8—361no—
203C8C3×GO+(4,𝔽2)11yescharacteristic
204C2×C3—21no—
205C2×C3—21no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—181no—
209C2×C3—241no—
210C2×C3—241no—
211C2×C3—481no—
212C2×C3#6372911yescharacteristic
213D3—62no—
214D3—81no—
215D3—121no—
216C22—91no—
217C22—121no—
218C4—91no—
219C4—121no—
220C4C3×D32⋊C411yescharacteristic
221C3—21no—
222C3—21no—
223C3—41no—
224C3—41no—
225C3#1059001811yescharacteristic, direct product factor
226C2—181no—
227C2—241no—
228C2C3×#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×#359
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×#63770C2semidirect
3. C3×#63542C2semidirect
4. #10590018C3direct (N × H)
14. #63770C2×C3semidirect
16. #63542C2×C3semidirect
126. C33#359semidirect
195. C32C3×#359semidirect
225. C3#10590018direct (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×#637702
C3×#635422
#105900183
C3×#3599