Subgroups — C₂×#11723919 #13179606

Order 1728.

4508 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6308	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃³×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#88864	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88857	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88857	C₂	1	1	yes	maximal, characteristic, complemented
8	#11723919	C₂	1	8	yes	maximal, direct product factor
9	C₂×#64087	—	3	13	no	maximal
10	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
11	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
14	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
16	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
17	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
18	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
19	C₂×#6308	—	2	4	no	—
20	C₂×#6308	C₂²	1	4	yes	complemented
21	C₃³×C₂²⋊C₄	C₂²	1	4	yes	—
22	#88864	C₂²	1	4	yes	—
23	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	4	yes	complemented
24	#88857	C₂²	1	4	yes	complemented
25	#88857	C₂²	1	4	yes	complemented
26	C₂³×C₃⋊Dic₃	—	3	13	no	—
27	C₂²×#634	—	3	13	no	—
28	C₂×C₃²×C₂²⋊C₄	D₃	1	13	yes	—
29	C₂×#3757	—	3	13	no	—
30	C₂×C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
31	C₂×#3750	—	3	13	no	—
32	C₂×#3750	—	3	13	no	—
33	#64087	—	3	104	no	—
34	C₂³×C₃³	—	2	2	no	—
35	C₂³×C₃³	—	2	2	no	—
36	C₂³×C₃³	Q₈	1	4	yes	complemented
37	C₂³×C₃³	C₂³	1	2	yes	—
38	C₂³×C₃³	C₂³	1	1	yes	characteristic
39	C₂×C₃³×C₄	—	2	2	no	—
40	C₂×C₃³×C₄	—	2	2	no	—
41	C₂×C₃³×C₄	C₂³	1	2	yes	—
42	C₂×C₃³×C₄	C₂³	1	2	yes	—
43	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
44	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
45	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
46	C₂×C₃⋊(C₃⋊Dic₃)	—	2	8	no	—
47	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	4	yes	—
48	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
49	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
50	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
51	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
52	#6308	—	2	8	no	—
53	C₂×#878	—	9	13	no	—
54	C₂⁴×C₃²	C₂×D₃	1	13	yes	—
55	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
56	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
57	C₂²×C₃⋊Dic₃	—	3	13	no	—
58	C₂²×C₃⋊Dic₃	—	3	52	no	—
59	C₂²×C₃⋊Dic₃	—	3	13	no	—
60	C₂²×C₃⋊Dic₃	—	3	13	no	—
61	C₂²×C₃⋊Dic₃	—	3	13	no	—
62	C₂²×C₃⋊Dic₃	—	6	52	no	—
63	C₂×#634	—	3	52	no	—
64	C₂×#634	—	6	52	no	—
65	C₃²×C₂²⋊C₄	C₂×D₃	1	52	yes	—
66	#3757	—	3	52	no	—
67	C₄⋊(C₃⋊Dic₃)	—	3	52	no	—
68	#3750	—	3	52	no	—
69	#3750	—	3	52	no	—
70	C₂²×C₃³	—	2	8	no	—
71	C₂²×C₃³	—	2	2	no	—
72	C₂²×C₃³	—	2	2	no	—
73	C₂²×C₃³	C₂×D₄	1	2	yes	—
74	C₂²×C₃³	C₂×Q₈	1	2	yes	—
75	C₂²×C₃³	C₂×Q₈	1	4	yes	complemented
76	C₂²×C₃³	#41	1	2	yes	—
77	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
78	C₃³×C₄	—	2	2	no	—
79	C₃³×C₄	—	2	2	no	—
80	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
81	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
82	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
83	C₃⋊(C₃⋊Dic₃)	C₂×D₄	1	4	yes	—
84	C₂³×Dic₃	—	9	13	no	—
85	C₂²×Dic₆	—	9	13	no	—
86	C₂×C₃×C₂²⋊C₄	C₃⋊D₃	1	13	yes	—
87	C₂×#217	—	9	13	no	—
88	C₂×C₄⋊Dic₃	—	9	13	no	—
89	C₂×Dic₃⋊C₄	—	9	13	no	—
90	C₂×Dic₃⋊C₄	—	9	13	no	—
91	#878	—	9	104	no	—
92	C₂³×C₃²	—	2	26	no	—
93	C₂³×C₃²	—	2	26	no	—
94	C₂³×C₃²	Dic₆	1	52	yes	complemented
95	C₂³×C₃²	C₂²×D₃	1	26	yes	—
96	C₂³×C₃²	C₂²×D₃	1	13	yes	—
97	C₂×C₃²×C₄	—	2	26	no	—
98	C₂×C₃²×C₄	—	2	26	no	—
99	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
100	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
101	C₂×C₃⋊Dic₃	—	3	52	no	—
102	C₂×C₃⋊Dic₃	—	3	26	no	—
103	C₂×C₃⋊Dic₃	—	3	26	no	—
104	C₂×C₃⋊Dic₃	—	3	26	no	—
105	C₂×C₃⋊Dic₃	—	3	26	no	—
106	C₂×C₃⋊Dic₃	—	6	104	no	—
107	C₂×C₃⋊Dic₃	—	6	26	no	—
108	C₂×C₃⋊Dic₃	—	6	26	no	—
109	C₂×C₃⋊Dic₃	—	6	26	no	—
110	#634	—	6	104	no	—
111	C₂×C₂²⋊Q₈	—	27	1	no	Sylow
112	C₂×C₃³	—	2	4	no	—
113	C₂×C₃³	C₂²⋊Q₈	1	4	yes	complemented
114	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₃³	C₂×#41	1	1	yes	characteristic
117	C₂⁴×C₃	C₂×C₃⋊D₃	1	13	yes	—
118	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
119	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
120	C₂²×Dic₃	—	9	13	no	—
121	C₂²×Dic₃	—	9	52	no	—
122	C₂²×Dic₃	—	9	13	no	—
123	C₂²×Dic₃	—	9	13	no	—
124	C₂²×Dic₃	—	9	13	no	—
125	C₂²×Dic₃	—	18	52	no	—
126	C₂×Dic₆	—	9	52	no	—
127	C₂×Dic₆	—	18	52	no	—
128	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	52	yes	—
129	#217	—	9	52	no	—
130	C₄⋊Dic₃	—	9	52	no	—
131	Dic₃⋊C₄	—	9	52	no	—
132	Dic₃⋊C₄	—	9	52	no	—
133	C₂²×C₃²	—	2	104	no	—
134	C₂²×C₃²	—	2	26	no	—
135	C₂²×C₃²	—	2	26	no	—
136	C₂²×C₃²	C₂×Dic₆	1	26	yes	—
137	C₂²×C₃²	C₂×Dic₆	1	52	yes	complemented
138	C₂²×C₃²	D₃×D₄	1	26	yes	—
139	C₂²×C₃²	#237	1	26	yes	—
140	C₂²×C₃²	C₂³×D₃	1	13	yes	—
141	C₃²×C₄	—	2	26	no	—
142	C₃²×C₄	—	2	26	no	—
143	C₃⋊Dic₃	—	3	52	no	—
144	C₃⋊Dic₃	—	6	26	no	—
145	C₃⋊Dic₃	—	6	26	no	—
146	C₃⋊Dic₃	—	6	26	no	—
147	C₂²×Q₈	—	27	1	no	—
148	C₂³×C₄	—	27	1	no	—
149	C₂²⋊Q₈	—	27	8	no	—
150	C₂×C₄⋊C₄	—	27	1	no	—
151	C₂×C₄⋊C₄	—	27	1	no	—
152	C₂×C₄⋊C₄	—	27	1	no	—
153	C₂×C₂²⋊C₄	—	27	1	no	—
154	C₂×C₂²⋊C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
155	C₃³	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	2	26	no	—
157	C₂³×C₃	—	2	26	no	—
158	C₂³×C₃	#634	1	52	yes	complemented
159	C₂³×C₃	C₂²×C₃⋊D₃	1	26	yes	—
160	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
161	C₂×C₃×C₄	—	2	26	no	—
162	C₂×C₃×C₄	—	2	26	no	—
163	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
164	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
165	C₂×Dic₃	—	9	26	no	—
166	C₂×Dic₃	—	9	52	no	—
167	C₂×Dic₃	—	9	26	no	—
168	C₂×Dic₃	—	9	26	no	—
169	C₂×Dic₃	—	9	26	no	—
170	C₂×Dic₃	—	18	104	no	—
171	C₂×Dic₃	—	18	26	no	—
172	C₂×Dic₃	—	18	26	no	—
173	C₂×Dic₃	—	18	26	no	—
174	Dic₆	—	18	104	no	—
175	C₂×C₃²	—	2	52	no	—
176	C₂×C₃²	#878	1	52	yes	complemented
177	C₂×C₃²	C₂²×Dic₆	1	13	yes	—
178	C₂×C₃²	C₂×D₃×D₄	1	13	yes	—
179	C₂×C₃²	C₂×#237	1	13	yes	—
180	C₂⁴	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
181	C₂×Q₈	—	27	4	no	—
182	C₂×Q₈	—	54	4	no	—
183	C₂²×C₄	—	27	1	no	—
184	C₂²×C₄	—	27	4	no	—
185	C₂²×C₄	—	27	1	no	—
186	C₂²×C₄	—	27	1	no	—
187	C₂²×C₄	—	27	1	no	—
188	C₂²×C₄	—	54	4	no	—
189	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
191	C₄⋊C₄	—	27	4	no	—
192	C₄⋊C₄	—	27	4	no	—
193	C₄⋊C₄	—	27	4	no	—
194	C₂²⋊C₄	—	27	4	no	—
195	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	4	yes	—
196	C₂²×C₃	—	2	104	no	—
197	C₂²×C₃	—	2	26	no	—
198	C₂²×C₃	—	2	26	no	—
199	C₂²×C₃	C₂×#634	1	26	yes	—
200	C₂²×C₃	C₂×#634	1	52	yes	complemented
201	C₂²×C₃	D₄×C₃⋊D₃	1	26	yes	—
202	C₂²×C₃	#3830	1	26	yes	—
203	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
204	C₃×C₄	—	2	26	no	—
205	C₃×C₄	—	2	26	no	—
206	Dic₃	—	9	52	no	—
207	Dic₃	—	18	26	no	—
208	Dic₃	—	18	26	no	—
209	Dic₃	—	18	26	no	—
210	C₃²	C₂×#878	1	13	yes	complemented
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#6308	1	4	yes	complemented
214	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
215	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
216	Q₈	—	54	8	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	27	2	no	—
220	C₂×C₄	—	27	4	no	—
221	C₂×C₄	—	27	2	no	—
222	C₂×C₄	—	27	2	no	—
223	C₂×C₄	—	27	2	no	—
224	C₂×C₄	—	54	8	no	—
225	C₂×C₄	—	54	2	no	—
226	C₂×C₄	—	54	2	no	—
227	C₂×C₄	—	54	2	no	—
228	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
229	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
230	C₂×C₃	—	2	52	no	—
231	C₂×C₃	#64087	1	52	yes	complemented
232	C₂×C₃	C₂²×#634	1	13	yes	—
233	C₂×C₃	C₂×D₄×C₃⋊D₃	1	13	yes	—
234	C₂×C₃	C₂×#3830	1	13	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	8	no	—
238	C₂²	C₂×#6308	1	2	yes	—
239	C₂²	C₂×#6308	1	4	yes	complemented
240	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
241	C₂²	#89078	1	2	yes	—
242	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	27	4	no	—
246	C₄	—	54	2	no	—
247	C₄	—	54	2	no	—
248	C₄	—	54	2	no	—
249	C₃	C₂×#64087	1	13	yes	complemented
250	C₂	—	2	4	no	—
251	C₂	#11723919	1	4	yes	direct product factor
252	C₂	C₂²×#6308	1	1	yes	characteristic
253	C₂	C₂×D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
254	C₂	C₂×#89078	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂²×C₃³	C₂⁴
Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×C₂²⋊C₄	C₂
socle	C₂³×C₃³	C₂³
prime radical	C₂⁴×C₃³	C₂²
hypercenter	C₂×C₂²⋊C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#6308	C₂	semidirect
5. C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	semidirect
6. C₂×#88857	C₂	semidirect
7. C₂×#88857	C₂	semidirect
8. #11723919	C₂	direct (N × H)
20. C₂×#6308	C₂²	semidirect
23. C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	semidirect
24. #88857	C₂²	semidirect
25. #88857	C₂²	semidirect
36. C₂³×C₃³	Q₈	semidirect
75. C₂²×C₃³	C₂×Q₈	semidirect
94. C₂³×C₃²	Dic₆	semidirect
113. C₂×C₃³	C₂²⋊Q₈	semidirect
137. C₂²×C₃²	C₂×Dic₆	semidirect
155. C₃³	C₂×C₂²⋊Q₈	semidirect
158. C₂³×C₃	#634	semidirect
176. C₂×C₃²	#878	semidirect
200. C₂²×C₃	C₂×#634	semidirect
210. C₃²	C₂×#878	semidirect
213. C₂³	#6308	semidirect
231. C₂×C₃	#64087	semidirect
239. C₂²	C₂×#6308	semidirect
249. C₃	C₂×#64087	semidirect
251. C₂	#11723919	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃⋊(C₃⋊Dic₃)	2
C₂²×#6308	2
C₂×C₃³×C₂²⋊C₄	2
C₂×#88864	2
C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	2
C₂×#88857	2
#11723919	2
C₂×#64087	3

Subgroups — C2×#11723919 #13179606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11723919 #13179606