Subgroups — C₂×#11723534 #13177266

Order 1728.

2108 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6293	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃²×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal
6	#11723534	C₂	1	8	yes	maximal, direct product factor
7	C₂×#64078	—	3	1	no	maximal
8	C₂×#63918	—	3	4	no	maximal
9	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂²×C₃×C₃⋊Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃²×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃²×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×#6293	—	2	8	no	—
15	C₂×#6293	C₂²	1	8	yes	—
16	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	4	yes	—
17	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
18	C₃²×C₄×Dic₃	C₂²	1	4	yes	—
19	Dic₃⋊(C₃⋊Dic₃)	C₂²	1	8	yes	—
20	C₂²×#634	—	3	2	no	—
21	C₂²×#627	—	3	8	no	—
22	C₂×C₃²×C₄²	—	3	1	no	—
23	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
24	C₂×C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
25	C₂×C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
26	C₂×C₄⋊(C₃⋊Dic₃)	—	6	4	no	—
27	C₂×C₄⋊(C₃⋊Dic₃)	D₃	1	1	yes	characteristic
28	#64078	—	3	8	no	—
29	C₂×C₃×C₄⋊Dic₃	—	3	4	no	—
30	C₂×C₃×C₄×Dic₃	D₃	1	4	yes	—
31	C₂×Dic₃⋊Dic₃	—	3	8	no	—
32	#63918	—	3	32	no	—
33	C₂³×C₃³	C₂³	1	1	yes	characteristic
34	C₂×C₃³×C₄	Q₈	1	4	yes	complemented
35	C₂×C₃³×C₄	C₂³	1	2	yes	—
36	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
37	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
38	C₂×C₃×C₃⋊Dic₃	—	2	4	no	—
39	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
40	C₂×C₃²×Dic₃	D₄	1	4	yes	—
41	C₂×C₃²×Dic₃	Q₈	1	4	yes	complemented
42	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
43	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
44	#6293	—	2	16	no	—
45	C₂×#888	—	9	1	no	—
46	C₂×#869	—	9	4	no	—
47	C₂²×C₃²×C₄	—	2	4	no	—
48	C₂²×C₃²×C₄	—	3	1	no	—
49	C₂²×C₃²×C₄	—	3	1	no	—
50	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
51	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃⋊Dic₃	—	3	2	no	—
53	C₂²×C₃⋊Dic₃	—	3	8	no	—
54	C₂²×C₃⋊Dic₃	—	6	8	no	—
55	C₂²×C₃⋊Dic₃	C₂×D₃	1	2	yes	—
56	C₂×#634	—	3	8	no	—
57	C₂×#634	—	6	8	no	—
58	C₂²×C₃×Dic₃	—	3	8	no	—
59	C₂²×C₃×Dic₃	C₂×D₃	1	4	yes	—
60	C₂²×C₃×Dic₃	C₂×D₃	1	4	yes	—
61	C₂×#627	—	3	32	no	—
62	C₂×#627	—	6	32	no	—
63	C₃²×C₄²	—	3	4	no	—
64	C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
65	C₄⋊(C₃⋊Dic₃)	—	3	16	no	—
66	C₄⋊(C₃⋊Dic₃)	—	3	8	no	—
67	C₄⋊(C₃⋊Dic₃)	—	6	16	no	—
68	C₄⋊(C₃⋊Dic₃)	C₂×D₃	1	4	yes	—
69	C₃×C₄⋊Dic₃	—	3	16	no	—
70	C₃×C₄×Dic₃	C₂×D₃	1	16	yes	—
71	Dic₃⋊Dic₃	—	3	32	no	—
72	C₂²×C₃³	C₂×D₄	1	2	yes	—
73	C₂²×C₃³	C₂×Q₈	1	2	yes	—
74	C₂²×C₃³	C₂×Q₈	1	2	yes	—
75	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
76	C₃³×C₄	C₂×Q₈	1	4	yes	complemented
77	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
78	C₃×C₃⋊Dic₃	—	2	4	no	—
79	C₃²×Dic₃	C₂×D₄	1	4	yes	—
80	C₃²×Dic₃	C₂×Q₈	1	4	yes	complemented
81	C₂²×Dic₆	—	9	8	no	—
82	C₂²×Dic₆	—	9	2	no	—
83	C₂×C₃×C₄⋊C₄	—	9	1	no	—
84	C₂×C₃×C₄²	—	3	4	no	—
85	C₂×C₄⋊Dic₃	—	3	4	no	—
86	C₂×C₄⋊Dic₃	—	9	8	no	—
87	C₂×C₄⋊Dic₃	—	9	4	no	—
88	C₂×C₄⋊Dic₃	—	9	1	no	—
89	C₂×C₄⋊Dic₃	—	18	4	no	—
90	C₂×Dic₃⋊C₄	—	9	2	no	—
91	C₂×C₄×Dic₃	C₃⋊D₃	1	1	yes	characteristic
92	#888	—	9	8	no	—
93	#869	—	9	32	no	—
94	C₂³×C₃²	—	2	4	no	—
95	C₂³×C₃²	C₂²×D₃	1	4	yes	—
96	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃²×C₄	—	2	16	no	—
98	C₂×C₃²×C₄	—	2	8	no	—
99	C₂×C₃²×C₄	—	3	4	no	—
100	C₂×C₃²×C₄	—	3	4	no	—
101	C₂×C₃²×C₄	—	3	2	no	—
102	C₂×C₃²×C₄	—	3	2	no	—
103	C₂×C₃²×C₄	Dic₆	1	16	yes	complemented
104	C₂×C₃²×C₄	Dic₆	1	4	yes	complemented
105	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
106	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
107	C₂×C₃⋊Dic₃	—	2	4	no	—
108	C₂×C₃⋊Dic₃	—	3	4	no	—
109	C₂×C₃⋊Dic₃	—	3	16	no	—
110	C₂×C₃⋊Dic₃	—	6	16	no	—
111	C₂×C₃⋊Dic₃	—	6	4	no	—
112	C₂×C₃⋊Dic₃	—	6	16	no	—
113	C₂×C₃⋊Dic₃	—	12	16	no	—
114	C₂×C₃⋊Dic₃	C₂²×D₃	1	4	yes	—
115	#634	—	6	16	no	—
116	C₂×C₃×Dic₃	—	3	16	no	—
117	C₂×C₃×Dic₃	—	6	16	no	—
118	C₂×C₃×Dic₃	Dic₆	1	16	yes	complemented
119	C₂×C₃×Dic₃	D₁₂	1	16	yes	—
120	C₂×C₃×Dic₃	C₂²×D₃	1	8	yes	—
121	C₂×C₃×Dic₃	C₂²×D₃	1	8	yes	—
122	#627	—	6	64	no	—
123	C₂×C₄⋊Q₈	—	27	1	no	Sylow
124	C₂×C₃³	C₄⋊Q₈	1	4	yes	complemented
125	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
128	C₂²×C₃×C₄	—	2	4	no	—
129	C₂²×C₃×C₄	—	3	4	no	—
130	C₂²×C₃×C₄	—	3	4	no	—
131	C₂²×C₃×C₄	—	9	2	no	—
132	C₂²×C₃×C₄	D₃²	1	4	yes	—
133	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
134	C₂²×Dic₃	—	3	8	no	—
135	C₂²×Dic₃	—	9	8	no	—
136	C₂²×Dic₃	—	9	2	no	—
137	C₂²×Dic₃	—	18	8	no	—
138	C₂²×Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
139	C₂²×Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
140	C₂×Dic₆	—	9	32	no	—
141	C₂×Dic₆	—	9	8	no	—
142	C₂×Dic₆	—	18	32	no	—
143	C₂×Dic₆	—	18	8	no	—
144	C₃×C₄⋊C₄	—	9	4	no	—
145	C₃×C₄²	—	3	16	no	—
146	C₄⋊Dic₃	—	3	16	no	—
147	C₄⋊Dic₃	—	9	32	no	—
148	C₄⋊Dic₃	—	9	16	no	—
149	C₄⋊Dic₃	—	9	4	no	—
150	C₄⋊Dic₃	—	18	16	no	—
151	Dic₃⋊C₄	—	9	8	no	—
152	C₄×Dic₃	C₂×C₃⋊D₃	1	4	yes	—
153	C₂²×C₃²	—	2	4	no	—
154	C₂²×C₃²	—	2	8	no	—
155	C₂²×C₃²	—	2	8	no	—
156	C₂²×C₃²	—	2	8	no	—
157	C₂²×C₃²	C₂×Dic₆	1	8	yes	—
158	C₂²×C₃²	C₂×Dic₆	1	8	yes	—
159	C₂²×C₃²	C₂×Dic₆	1	2	yes	—
160	C₂²×C₃²	C₂×D₁₂	1	8	yes	—
161	C₂²×C₃²	D₃×D₄	1	2	yes	—
162	C₂²×C₃²	D₃×Q₈	1	2	yes	—
163	C₂²×C₃²	C₂³×D₃	1	4	yes	—
164	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
165	C₃²×C₄	—	2	16	no	—
166	C₃²×C₄	—	3	4	no	—
167	C₃²×C₄	—	3	4	no	—
168	C₃²×C₄	C₂×Dic₆	1	16	yes	complemented
169	C₃²×C₄	C₂×Dic₆	1	4	yes	complemented
170	C₃⋊Dic₃	—	2	4	no	—
171	C₃⋊Dic₃	—	6	4	no	—
172	C₃⋊Dic₃	—	6	16	no	—
173	C₃⋊Dic₃	—	12	16	no	—
174	C₃×Dic₃	—	6	16	no	—
175	C₃×Dic₃	C₂×Dic₆	1	16	yes	complemented
176	C₃×Dic₃	C₂×D₁₂	1	16	yes	—
177	C₂²×Q₈	—	27	2	no	—
178	C₄⋊Q₈	—	27	8	no	—
179	C₂×C₄⋊C₄	—	9	1	no	—
180	C₂×C₄⋊C₄	—	27	2	no	—
181	C₂×C₄⋊C₄	—	27	1	no	—
182	C₂×C₄²	—	3	1	no	—
183	C₃³	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₃	—	2	4	no	—
185	C₂³×C₃	C₂×D₃²	1	4	yes	—
186	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
187	C₂×C₃×C₄	—	2	16	no	—
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	3	8	no	—
190	C₂×C₃×C₄	—	3	8	no	—
191	C₂×C₃×C₄	—	3	16	no	—
192	C₂×C₃×C₄	—	3	16	no	—
193	C₂×C₃×C₄	—	9	4	no	—
194	C₂×C₃×C₄	—	18	4	no	—
195	C₂×C₃×C₄	#627	1	16	yes	complemented
196	C₂×C₃×C₄	#634	1	4	yes	complemented
197	C₂×C₃×C₄	C₂×D₃²	1	8	yes	—
198	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
199	C₂×Dic₃	—	3	16	no	—
200	C₂×Dic₃	—	6	16	no	—
201	C₂×Dic₃	—	9	16	no	—
202	C₂×Dic₃	—	9	4	no	—
203	C₂×Dic₃	—	18	16	no	—
204	C₂×Dic₃	—	18	16	no	—
205	C₂×Dic₃	—	18	4	no	—
206	C₂×Dic₃	—	36	16	no	—
207	C₂×Dic₃	#634	1	4	yes	complemented
208	C₂×Dic₃	C₄⋊(C₃⋊D₃)	1	4	yes	—
209	C₂×Dic₃	C₂²×C₃⋊D₃	1	2	yes	—
210	C₂×Dic₃	C₂²×C₃⋊D₃	1	2	yes	—
211	Dic₆	—	18	64	no	—
212	Dic₆	—	18	16	no	—
213	C₂×C₃²	—	2	4	no	—
214	C₂×C₃²	—	2	4	no	—
215	C₂×C₃²	—	2	4	no	—
216	C₂×C₃²	—	2	16	no	—
217	C₂×C₃²	#869	1	16	yes	complemented
218	C₂×C₃²	#888	1	4	yes	complemented
219	C₂×C₃²	C₂²×Dic₆	1	4	yes	—
220	C₂×C₃²	C₂²×Dic₆	1	4	yes	—
221	C₂×C₃²	C₂²×Dic₆	1	1	yes	characteristic
222	C₂×C₃²	C₂²×D₁₂	1	4	yes	—
223	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
224	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
225	C₂×Q₈	—	27	8	no	—
226	C₂×Q₈	—	54	8	no	—
227	C₂²×C₄	—	3	1	no	—
228	C₂²×C₄	—	3	1	no	—
229	C₂²×C₄	—	9	2	no	—
230	C₂²×C₄	—	27	2	no	—
231	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
232	C₄⋊C₄	—	9	4	no	—
233	C₄⋊C₄	—	27	8	no	—
234	C₄⋊C₄	—	27	4	no	—
235	C₄²	—	3	4	no	—
236	C₂²×C₃	—	2	4	no	—
237	C₂²×C₃	—	2	8	no	—
238	C₂²×C₃	—	2	8	no	—
239	C₂²×C₃	—	2	8	no	—
240	C₂²×C₃	D₃×Dic₆	1	8	yes	—
241	C₂²×C₃	D₃×D₁₂	1	8	yes	—
242	C₂²×C₃	C₂×#627	1	8	yes	—
243	C₂²×C₃	C₂×#634	1	2	yes	—
244	C₂²×C₃	C₂×#634	1	2	yes	—
245	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
246	C₂²×C₃	C₂²×D₃²	1	4	yes	—
247	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
248	C₃×C₄	—	2	16	no	—
249	C₃×C₄	—	3	16	no	—
250	C₃×C₄	—	3	16	no	—
251	C₃×C₄	—	18	4	no	—
252	C₃×C₄	C₂×#627	1	16	yes	complemented
253	C₃×C₄	C₂×#634	1	4	yes	complemented
254	Dic₃	—	6	16	no	—
255	Dic₃	—	18	16	no	—
256	Dic₃	—	18	4	no	—
257	Dic₃	—	36	16	no	—
258	Dic₃	C₂×#634	1	4	yes	complemented
259	Dic₃	C₂×C₄⋊(C₃⋊D₃)	1	4	yes	—
260	C₃²	—	2	4	no	—
261	C₃²	C₂×#869	1	4	yes	complemented
262	C₃²	C₂×#888	1	1	yes	characteristic, complemented
263	C₂³	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
264	Q₈	—	54	16	no	—
265	C₂×C₄	—	3	2	no	—
266	C₂×C₄	—	3	2	no	—
267	C₂×C₄	—	3	4	no	—
268	C₂×C₄	—	3	4	no	—
269	C₂×C₄	—	9	4	no	—
270	C₂×C₄	—	18	4	no	—
271	C₂×C₄	—	27	4	no	—
272	C₂×C₄	—	54	4	no	—
273	C₂×C₄	#6293	1	4	yes	complemented
274	C₂×C₄	C₂×D₃×C₃⋊D₃	1	2	yes	—
275	C₂×C₃	—	2	16	no	—
276	C₂×C₃	—	2	4	no	—
277	C₂×C₃	—	2	4	no	—
278	C₂×C₃	—	2	4	no	—
279	C₂×C₃	#63918	1	16	yes	complemented
280	C₂×C₃	#64078	1	4	yes	complemented
281	C₂×C₃	C₂×D₃×Dic₆	1	4	yes	—
282	C₂×C₃	C₂×D₃×D₁₂	1	4	yes	—
283	C₂×C₃	C₂²×#627	1	4	yes	—
284	C₂×C₃	C₂²×#634	1	1	yes	characteristic
285	C₂×C₃	C₂²×#634	1	1	yes	characteristic
286	C₂×C₃	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
287	C₂²	D₃×#634	1	2	yes	—
288	C₂²	D₃×C₄⋊(C₃⋊D₃)	1	2	yes	—
289	C₂²	C₂×#6293	1	2	yes	—
290	C₂²	C₂²×D₃×C₃⋊D₃	1	1	yes	characteristic
291	C₄	—	3	4	no	—
292	C₄	—	3	4	no	—
293	C₄	—	18	4	no	—
294	C₄	—	54	4	no	—
295	C₄	C₂×#6293	1	4	yes	complemented
296	C₃	—	2	4	no	—
297	C₃	C₂×#63918	1	4	yes	complemented
298	C₃	C₂×#64078	1	1	yes	characteristic, complemented
299	C₂	#11723534	1	4	yes	direct product factor
300	C₂	C₂×D₃×#634	1	1	yes	characteristic
301	C₂	C₂×D₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
302	C₂	C₂²×#6293	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
6. #11723534	C₂	direct (N × H)
34. C₂×C₃³×C₄	Q₈	semidirect
41. C₂×C₃²×Dic₃	Q₈	semidirect
76. C₃³×C₄	C₂×Q₈	semidirect
80. C₃²×Dic₃	C₂×Q₈	semidirect
103. C₂×C₃²×C₄	Dic₆	semidirect
104. C₂×C₃²×C₄	Dic₆	semidirect
118. C₂×C₃×Dic₃	Dic₆	semidirect
124. C₂×C₃³	C₄⋊Q₈	semidirect
168. C₃²×C₄	C₂×Dic₆	semidirect
169. C₃²×C₄	C₂×Dic₆	semidirect
175. C₃×Dic₃	C₂×Dic₆	semidirect
183. C₃³	C₂×C₄⋊Q₈	semidirect
195. C₂×C₃×C₄	#627	semidirect
196. C₂×C₃×C₄	#634	semidirect
207. C₂×Dic₃	#634	semidirect
217. C₂×C₃²	#869	semidirect
218. C₂×C₃²	#888	semidirect
252. C₃×C₄	C₂×#627	semidirect
253. C₃×C₄	C₂×#634	semidirect
258. Dic₃	C₂×#634	semidirect
261. C₃²	C₂×#869	semidirect
262. C₃²	C₂×#888	semidirect
273. C₂×C₄	#6293	semidirect
279. C₂×C₃	#63918	semidirect
280. C₂×C₃	#64078	semidirect
295. C₄	C₂×#6293	semidirect
297. C₃	C₂×#63918	semidirect
298. C₃	C₂×#64078	semidirect
299. C₂	#11723534	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#6293	2
C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	2
C₂×C₃×C₄⋊(C₃⋊Dic₃)	2
C₂×C₃²×C₄×Dic₃	2
C₂×Dic₃⋊(C₃⋊Dic₃)	2
#11723534	2
C₂×#64078	3
C₂×#63918	3

Subgroups — C2×#11723534 #13177266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11723534 #13177266