Subgroups — C₂×#11722207 #13163872

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88855	C₂	1	1	yes	maximal, characteristic, complemented
4	#11722207	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63664	—	3	13	no	maximal
6	C₃³×#41	—	2	1	no	—
7	C₃³×#41	—	2	1	no	—
8	C₃³×#41	C₄	1	2	yes	complemented
9	C₃³×#41	C₂²	1	2	yes	—
10	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic, complemented
11	C₂×C₃³×D₄	—	2	1	no	—
12	C₂×C₃³×D₄	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₃³×C₄	—	2	1	no	—
14	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
16	C₄×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
17	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	complemented
18	#88855	—	2	1	no	—
19	#88855	C₂²	1	2	yes	complemented
20	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
21	C₂×C₃²×#41	D₃	1	13	yes	—
22	C₂×C₄×C₃⋊Dic₃	—	3	13	no	—
23	C₂×#3748	—	3	13	no	—
24	#63664	—	3	52	no	—
25	C₂³×C₃³	—	2	1	no	—
26	C₂³×C₃³	D₄	1	1	yes	characteristic
27	C₃³×Q₈	—	2	1	no	—
28	C₃³×Q₈	C₂×C₄	1	2	yes	complemented
29	C₃³×D₄	—	2	2	no	—
30	C₃³×D₄	—	2	2	no	—
31	C₃³×D₄	—	2	1	no	—
32	C₃³×D₄	C₂×C₄	1	2	yes	complemented
33	C₂×C₃³×C₄	—	2	1	no	—
34	C₂×C₃³×C₄	—	2	2	no	—
35	C₂×C₃³×C₄	—	2	2	no	—
36	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
40	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
41	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
43	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
44	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
45	C₂×#837	—	9	13	no	—
46	C₃²×#41	—	2	13	no	—
47	C₃²×#41	—	2	13	no	—
48	C₃²×#41	Dic₃	1	26	yes	complemented
49	C₃²×#41	C₂×D₃	1	26	yes	—
50	C₂×C₃²×Q₈	Dic₃	1	13	yes	complemented
51	C₂×C₃²×D₄	—	2	13	no	—
52	C₂×C₃²×D₄	Dic₃	1	13	yes	complemented
53	C₂²×C₃²×C₄	—	2	13	no	—
54	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
55	C₂²×C₃⋊Dic₃	—	6	13	no	—
56	C₄×C₃⋊Dic₃	—	3	26	no	—
57	C₄×C₃⋊Dic₃	—	6	13	no	—
58	#3748	—	3	26	no	—
59	#3748	—	6	13	no	—
60	C₂×C₃⋊(C₃⋊C₈)	—	6	13	no	—
61	C₂²×C₃³	—	2	2	no	—
62	C₂²×C₃³	—	2	2	no	—
63	C₂²×C₃³	—	4	2	no	—
64	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
67	C₃³×C₄	—	2	2	no	—
68	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
72	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
73	C₂×C₃×#41	C₃⋊D₃	1	13	yes	—
74	C₂×C₄×Dic₃	—	9	13	no	—
75	C₂×#208	—	9	13	no	—
76	#837	—	9	52	no	—
77	C₂³×C₃²	—	2	13	no	—
78	C₂³×C₃²	#67	1	13	yes	—
79	C₃²×Q₈	—	2	13	no	—
80	C₃²×Q₈	C₂×Dic₃	1	26	yes	complemented
81	C₃²×D₄	—	2	26	no	—
82	C₃²×D₄	—	2	26	no	—
83	C₃²×D₄	—	2	13	no	—
84	C₃²×D₄	C₂×Dic₃	1	26	yes	complemented
85	C₂×C₃²×C₄	—	2	13	no	—
86	C₂×C₃²×C₄	—	2	26	no	—
87	C₂×C₃²×C₄	—	2	26	no	—
88	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
89	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
90	C₂×C₃²×C₄	#67	1	13	yes	—
91	C₂×C₃²×C₄	#67	1	13	yes	—
92	C₂×C₃²×C₄	#67	1	13	yes	—
93	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
94	C₂×C₃⋊Dic₃	—	6	52	no	—
95	C₂×C₃⋊Dic₃	—	6	26	no	—
96	C₃⋊(C₃⋊C₈)	—	6	26	no	—
97	C₂×#104	—	27	1	no	Sylow
98	C₂×C₃³	—	2	1	no	—
99	C₂×C₃³	—	2	1	no	—
100	C₂×C₃³	—	4	2	no	—
101	C₂×C₃³	#104	1	2	yes	complemented
102	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₃×#41	—	2	13	no	—
104	C₃×#41	—	2	13	no	—
105	C₃×#41	C₃⋊Dic₃	1	26	yes	complemented
106	C₃×#41	C₂×C₃⋊D₃	1	26	yes	—
107	C₂×C₃×Q₈	C₃⋊Dic₃	1	13	yes	complemented
108	C₂×C₃×D₄	—	2	13	no	—
109	C₂×C₃×D₄	C₃⋊Dic₃	1	13	yes	complemented
110	C₂²×C₃×C₄	—	2	13	no	—
111	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
112	C₂²×Dic₃	—	18	13	no	—
113	C₄×Dic₃	—	9	26	no	—
114	C₄×Dic₃	—	18	13	no	—
115	#208	—	9	26	no	—
116	#208	—	18	13	no	—
117	C₂×C₃⋊C₈	—	18	13	no	—
118	C₂²×C₃²	—	2	26	no	—
119	C₂²×C₃²	—	2	26	no	—
120	C₂²×C₃²	—	4	26	no	—
121	C₂²×C₃²	#217	1	13	yes	—
122	C₂²×C₃²	#217	1	13	yes	—
123	C₂²×C₃²	C₂×#67	1	13	yes	—
124	C₃²×C₄	—	2	26	no	—
125	C₃²×C₄	#217	1	13	yes	—
126	C₃²×C₄	#217	1	13	yes	—
127	C₃²×C₄	C₂²×Dic₃	1	13	yes	—
128	C₃²×C₄	C₂×#67	1	13	yes	—
129	C₃⋊Dic₃	—	6	52	no	—
130	C₂×#41	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
131	C₂×M₁₆	—	27	1	no	—
132	C₂×C₄²	—	27	1	no	—
133	#104	—	27	4	no	—
134	C₃³	C₂×#104	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₃	—	2	13	no	—
136	C₂³×C₃	#638	1	13	yes	—
137	C₃×Q₈	—	2	13	no	—
138	C₃×Q₈	C₂×C₃⋊Dic₃	1	26	yes	complemented
139	C₃×D₄	—	2	26	no	—
140	C₃×D₄	—	2	26	no	—
141	C₃×D₄	—	2	13	no	—
142	C₃×D₄	C₂×C₃⋊Dic₃	1	26	yes	complemented
143	C₂×C₃×C₄	—	2	13	no	—
144	C₂×C₃×C₄	—	2	26	no	—
145	C₂×C₃×C₄	—	2	26	no	—
146	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
147	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
148	C₂×C₃×C₄	#638	1	13	yes	—
149	C₂×C₃×C₄	#638	1	13	yes	—
150	C₂×C₃×C₄	#638	1	13	yes	—
151	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
152	C₂×Dic₃	—	18	52	no	—
153	C₂×Dic₃	—	18	26	no	—
154	C₃⋊C₈	—	18	26	no	—
155	C₂×C₃²	—	2	13	no	—
156	C₂×C₃²	—	2	13	no	—
157	C₂×C₃²	—	4	26	no	—
158	C₂×C₃²	#837	1	26	yes	complemented
159	C₂×C₃²	C₂×#217	1	13	yes	—
160	#41	—	2	1	no	—
161	#41	—	2	1	no	—
162	#41	C₃⋊(C₃⋊Dic₃)	1	2	yes	complemented
163	#41	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
164	C₂×Q₈	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, complemented
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, complemented
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	54	1	no	—
169	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
170	M₁₆	—	27	2	no	—
171	M₁₆	—	54	1	no	—
172	C₂×C₈	—	54	1	no	—
173	C₄²	—	27	2	no	—
174	C₄²	—	54	1	no	—
175	C₂²×C₃	—	2	26	no	—
176	C₂²×C₃	—	2	26	no	—
177	C₂²×C₃	—	4	26	no	—
178	C₂²×C₃	#3757	1	13	yes	—
179	C₂²×C₃	#3757	1	13	yes	—
180	C₂²×C₃	C₂×#638	1	13	yes	—
181	C₃×C₄	—	2	26	no	—
182	C₃×C₄	#3757	1	13	yes	—
183	C₃×C₄	#3757	1	13	yes	—
184	C₃×C₄	C₂²×C₃⋊Dic₃	1	13	yes	—
185	C₃×C₄	C₂×#638	1	13	yes	—
186	Dic₃	—	18	52	no	—
187	C₃²	C₂×#837	1	13	yes	complemented
188	C₂³	—	2	1	no	—
189	C₂³	#6312	1	1	yes	characteristic
190	Q₈	—	2	1	no	—
191	Q₈	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	complemented
192	D₄	—	2	2	no	—
193	D₄	—	2	2	no	—
194	D₄	—	2	1	no	—
195	D₄	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	complemented
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	54	4	no	—
200	C₂×C₄	—	54	2	no	—
201	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
203	C₂×C₄	#6312	1	1	yes	characteristic
204	C₂×C₄	#6312	1	1	yes	characteristic
205	C₂×C₄	#6312	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
207	C₈	—	54	2	no	—
208	C₂×C₃	—	2	13	no	—
209	C₂×C₃	—	2	13	no	—
210	C₂×C₃	—	4	26	no	—
211	C₂×C₃	#63664	1	26	yes	complemented
212	C₂×C₃	C₂×#3757	1	13	yes	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	#88864	1	1	yes	characteristic
217	C₂²	#88864	1	1	yes	characteristic
218	C₂²	C₂×#6312	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	54	4	no	—
221	C₄	#88864	1	1	yes	characteristic
222	C₄	#88864	1	1	yes	characteristic
223	C₄	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
224	C₄	C₂×#6312	1	1	yes	characteristic
225	C₃	C₂×#63664	1	13	yes	complemented
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	4	2	no	—
229	C₂	#11722207	1	2	yes	direct product factor
230	C₂	C₂×#88864	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6312
commutator	C₃³×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting, prime radical	C₂×C₃³×#41	C₂
socle	C₂²×C₃³	C₂²⋊C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂×#41	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×#104
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
3. C₂×#88855	C₂	semidirect
4. #11722207	C₂	direct (N × H)
8. C₃³×#41	C₄	semidirect
10. C₂×C₃³×Q₈	C₄	semidirect
12. C₂×C₃³×D₄	C₄	semidirect
17. C₄×C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
19. #88855	C₂²	semidirect
28. C₃³×Q₈	C₂×C₄	semidirect
32. C₃³×D₄	C₂×C₄	semidirect
48. C₃²×#41	Dic₃	semidirect
50. C₂×C₃²×Q₈	Dic₃	semidirect
52. C₂×C₃²×D₄	Dic₃	semidirect
80. C₃²×Q₈	C₂×Dic₃	semidirect
84. C₃²×D₄	C₂×Dic₃	semidirect
101. C₂×C₃³	#104	semidirect
105. C₃×#41	C₃⋊Dic₃	semidirect
107. C₂×C₃×Q₈	C₃⋊Dic₃	semidirect
109. C₂×C₃×D₄	C₃⋊Dic₃	semidirect
134. C₃³	C₂×#104	semidirect
138. C₃×Q₈	C₂×C₃⋊Dic₃	semidirect
142. C₃×D₄	C₂×C₃⋊Dic₃	semidirect
158. C₂×C₃²	#837	semidirect
162. #41	C₃⋊(C₃⋊Dic₃)	semidirect
164. C₂×Q₈	C₃⋊(C₃⋊Dic₃)	semidirect
166. C₂×D₄	C₃⋊(C₃⋊Dic₃)	semidirect
187. C₃²	C₂×#837	semidirect
191. Q₈	C₂×C₃⋊(C₃⋊Dic₃)	semidirect
195. D₄	C₂×C₃⋊(C₃⋊Dic₃)	semidirect
211. C₂×C₃	#63664	semidirect
225. C₃	C₂×#63664	semidirect
229. C₂	#11722207	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃³×#41	2
C₂×C₄×C₃⋊(C₃⋊Dic₃)	2
C₂×#88855	2
#11722207	2
C₂×#63664	3

Subgroups — C2×#11722207 #13163872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11722207 #13163872