Subgroups — C₂×#11722642 #13164945

Order 1728.

2464 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	#11722642	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×AGL(1,ℤ/9)	C₂	1	2	yes	maximal, complemented
4	C₂×#3699⋊C₃	C₂	1	6	yes	maximal, complemented
5	C₂×#3697⋊C₃	C₂	1	3	yes	maximal, complemented
6	C₂²×3₋¹⁺²⋊Q₈	C₂	1	3	yes	maximal, complemented
7	C₂×C₃×#1010	—	3	1	no	maximal
8	C₂×#63716	C₃	1	1	yes	maximal, characteristic, complemented
9	#41×3₋¹⁺²	C₂²	1	8	yes	complemented
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂×D₄×3₋¹⁺²	C₂²	1	3	yes	—
12	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
13	C₂×#611⋊C₃	C₂²	1	6	yes	—
14	C₂²×3₋¹⁺²⋊C₄	C₂²	1	6	yes	—
15	Q₈×AGL(1,ℤ/9)	C₂²	1	16	yes	complemented
16	#3699⋊C₃	C₂²	1	48	yes	complemented
17	#3697⋊C₃	C₂²	1	24	yes	complemented
18	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
19	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	6	yes	—
20	C₂×3₋¹⁺²⋊Q₈	C₂²	1	3	yes	—
21	C₂×3₋¹⁺²⋊Q₈	C₂²	1	6	yes	—
22	C₂×3₋¹⁺²⋊Q₈	C₂²	1	24	yes	complemented
23	C₂×C₃²×#41	D₃	1	1	yes	characteristic
24	C₃×#1010	—	3	16	no	—
25	C₂×C₃×D₃×Q₈	—	3	2	no	—
26	C₂×C₃×#237	—	3	6	no	—
27	C₂×C₃×#235	—	3	3	no	—
28	C₂²×C₃×Dic₆	—	3	3	no	—
29	C₂×C₉×#41	—	2	1	no	—
30	C₂×C₉×#41	C₂×C₃	1	1	yes	characteristic, complemented
31	#63716	C₂×C₃	1	16	yes	complemented
32	C₂×Q₈×D₉	C₂×C₃	1	2	yes	complemented
33	C₂×#3699	C₂×C₃	1	6	yes	complemented
34	C₂×#3697	C₂×C₃	1	3	yes	complemented
35	C₂²×Dic₁₈	C₂×C₃	1	3	yes	complemented
36	C₂³×3₋¹⁺²	C₂³	1	3	yes	—
37	C₂²×AGL(1,ℤ/9)	C₂³	1	2	yes	—
38	Q₈×3₋¹⁺²	C₂³	1	4	yes	—
39	D₄×3₋¹⁺²	C₂³	1	12	yes	—
40	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
41	C₂×C₄×3₋¹⁺²	C₂³	1	3	yes	—
42	C₂×C₄×3₋¹⁺²	C₂³	1	12	yes	—
43	#611⋊C₃	C₂³	1	24	yes	—
44	C₂×3₋¹⁺²⋊C₄	C₂³	1	6	yes	—
45	C₂×3₋¹⁺²⋊C₄	C₂³	1	24	yes	—
46	C₄⋊AGL(1,ℤ/9)	C₂³	1	4	yes	—
47	C₄×AGL(1,ℤ/9)	C₂³	1	24	yes	—
48	3₋¹⁺²⋊Q₈	C₂³	1	12	yes	—
49	3₋¹⁺²⋊Q₈	C₂³	1	24	yes	—
50	C₂×C₃×2₋¹⁺⁴	—	9	1	no	—
51	C₂×#1010	—	3	1	no	—
52	C₃²×#41	C₂×D₃	1	8	yes	—
53	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×D₄	C₂×D₃	1	3	yes	—
55	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
56	C₂×C₃×#67	—	3	6	no	—
57	C₂²×C₃×Dic₃	—	3	6	no	—
58	C₃×D₃×Q₈	—	3	16	no	—
59	C₃×#237	—	3	48	no	—
60	C₃×#235	—	3	24	no	—
61	C₂×C₃×D₁₂	—	3	1	no	—
62	C₂×C₃×C₄×D₃	—	3	6	no	—
63	C₂×C₃×Dic₆	—	3	3	no	—
64	C₂×C₃×Dic₆	—	3	6	no	—
65	C₂×C₃×Dic₆	—	3	24	no	—
66	C₉×#41	—	2	8	no	—
67	C₉×#41	C₂²×C₃	1	8	yes	complemented
68	C₂×Q₈×C₉	—	2	1	no	—
69	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
70	C₂×D₄×C₉	—	2	3	no	—
71	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
72	C₂²×C₄×C₉	—	2	3	no	—
73	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
74	C₂×#611	C₂²×C₃	1	6	yes	—
75	C₂²×Dic₉	C₂²×C₃	1	6	yes	—
76	Q₈×D₉	C₂²×C₃	1	16	yes	complemented
77	#3699	C₂²×C₃	1	48	yes	complemented
78	#3697	C₂²×C₃	1	24	yes	complemented
79	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₄×D₉	C₂²×C₃	1	6	yes	—
81	C₂×Dic₁₈	C₂²×C₃	1	3	yes	—
82	C₂×Dic₁₈	C₂²×C₃	1	6	yes	—
83	C₂×Dic₁₈	C₂²×C₃	1	24	yes	complemented
84	C₂²×3₋¹⁺²	—	2	6	no	—
85	C₂²×3₋¹⁺²	C₂⁴	1	6	yes	—
86	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
87	C₂×AGL(1,ℤ/9)	—	2	4	no	—
88	C₂×AGL(1,ℤ/9)	C₂⁴	1	4	yes	—
89	C₄×3₋¹⁺²	C₂⁴	1	2	yes	—
90	C₄×3₋¹⁺²	C₂⁴	1	6	yes	—
91	3₋¹⁺²⋊C₄	C₂⁴	1	12	yes	—
92	C₃×2₋¹⁺⁴	—	9	16	no	—
93	C₂×C₃×#41	—	3	1	no	—
94	C₂×C₃×#41	—	9	3	no	—
95	C₂×C₃×#41	—	9	6	no	—
96	C₂×C₃×#41	C₃×D₃	1	1	yes	characteristic
97	C₂²×C₃×Q₈	—	9	3	no	—
98	C₂²×C₃×Q₈	—	9	2	no	—
99	#1010	—	3	16	no	—
100	C₂×D₃×Q₈	—	3	2	no	—
101	C₂×#237	—	3	6	no	—
102	C₂×#235	—	3	3	no	—
103	C₂²×Dic₆	—	3	3	no	—
104	C₂³×C₃²	C₂²×D₃	1	3	yes	—
105	C₂²×C₃×D₃	—	3	2	no	—
106	C₃²×Q₈	C₂²×D₃	1	4	yes	—
107	C₃²×D₄	C₂²×D₃	1	12	yes	—
108	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
110	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	—
111	C₃×#67	—	3	24	no	—
112	C₂×C₃×Dic₃	—	3	6	no	—
113	C₂×C₃×Dic₃	—	3	24	no	—
114	C₃×D₁₂	—	3	4	no	—
115	C₃×C₄×D₃	—	3	24	no	—
116	C₃×Dic₆	—	3	12	no	—
117	C₃×Dic₆	—	3	24	no	—
118	C₂³×C₉	—	2	3	no	—
119	C₂³×C₉	C₂³×C₃	1	3	yes	—
120	C₂²×D₉	C₂³×C₃	1	2	yes	—
121	Q₈×C₉	—	2	4	no	—
122	Q₈×C₉	C₂³×C₃	1	4	yes	—
123	D₄×C₉	—	2	12	no	—
124	D₄×C₉	C₂³×C₃	1	12	yes	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	2	3	no	—
127	C₂×C₄×C₉	—	2	12	no	—
128	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
129	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
130	C₂×C₄×C₉	C₂³×C₃	1	12	yes	—
131	#611	C₂³×C₃	1	24	yes	—
132	C₂×Dic₉	C₂³×C₃	1	6	yes	—
133	C₂×Dic₉	C₂³×C₃	1	24	yes	—
134	D₃₆	C₂³×C₃	1	4	yes	—
135	C₄×D₉	C₂³×C₃	1	24	yes	—
136	Dic₁₈	C₂³×C₃	1	12	yes	—
137	Dic₁₈	C₂³×C₃	1	24	yes	—
138	C₂×2₋¹⁺⁴	—	9	1	no	Sylow
139	C₂×3₋¹⁺²	—	2	6	no	—
140	C₂×3₋¹⁺²	2₋¹⁺⁴	1	2	yes	complemented
141	C₂×3₋¹⁺²	C₂⁵	1	1	yes	characteristic
142	AGL(1,ℤ/9)	—	2	4	no	—
143	C₃×#41	—	3	8	no	—
144	C₃×#41	—	9	24	no	—
145	C₃×#41	—	9	48	no	—
146	C₃×#41	C₂×C₃×D₃	1	8	yes	—
147	C₂×C₃×Q₈	—	3	1	no	—
148	C₂×C₃×Q₈	—	9	3	no	—
149	C₂×C₃×Q₈	—	9	6	no	—
150	C₂×C₃×Q₈	—	9	24	no	—
151	C₂×C₃×Q₈	—	9	16	no	—
152	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
153	C₂×C₃×D₄	—	3	3	no	—
154	C₂×C₃×D₄	—	9	1	no	—
155	C₂×C₃×D₄	—	9	6	no	—
156	C₂×C₃×D₄	C₂×C₃×D₃	1	3	yes	—
157	C₂²×C₃×C₄	—	3	3	no	—
158	C₂²×C₃×C₄	—	9	6	no	—
159	C₂²×C₃×C₄	—	9	6	no	—
160	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
161	C₂×#67	—	3	6	no	—
162	C₂²×Dic₃	—	3	6	no	—
163	D₃×Q₈	—	3	16	no	—
164	#237	—	3	48	no	—
165	#235	—	3	24	no	—
166	C₂×D₁₂	—	3	1	no	—
167	C₂×C₄×D₃	—	3	6	no	—
168	C₂×Dic₆	—	3	3	no	—
169	C₂×Dic₆	—	3	6	no	—
170	C₂×Dic₆	—	3	24	no	—
171	C₂²×C₃²	—	2	6	no	—
172	C₂²×C₃²	C₂³×D₃	1	6	yes	—
173	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
174	C₂×C₃×D₃	—	3	4	no	—
175	C₂×C₃×D₃	—	6	4	no	—
176	C₃²×C₄	C₂³×D₃	1	2	yes	—
177	C₃²×C₄	C₂³×D₃	1	6	yes	—
178	C₃×Dic₃	—	3	12	no	—
179	C₂²×C₉	—	2	6	no	—
180	C₂²×C₉	—	2	1	no	—
181	C₂²×C₉	—	2	6	no	—
182	C₂²×C₉	—	4	6	no	—
183	C₂²×C₉	C₂⁴×C₃	1	6	yes	—
184	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
185	C₂×D₉	—	2	4	no	—
186	C₂×D₉	C₂⁴×C₃	1	4	yes	—
187	C₄×C₉	—	2	2	no	—
188	C₄×C₉	—	2	6	no	—
189	C₄×C₉	C₂⁴×C₃	1	2	yes	—
190	C₄×C₉	C₂⁴×C₃	1	6	yes	—
191	Dic₉	C₂⁴×C₃	1	12	yes	—
192	2₋¹⁺⁴	—	9	16	no	—
193	C₂×#41	—	9	6	no	—
194	C₂×#41	—	9	3	no	—
195	C₂×#41	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
196	C₂²×Q₈	—	9	2	no	—
197	C₂²×Q₈	—	9	3	no	—
198	3₋¹⁺²	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	3	3	no	—
200	C₂³×C₃	—	9	2	no	—
201	C₂³×C₃	C₂²×C₃×D₃	1	3	yes	—
202	C₂²×D₃	—	3	2	no	—
203	C₃×Q₈	—	3	4	no	—
204	C₃×Q₈	—	9	12	no	—
205	C₃×Q₈	—	9	24	no	—
206	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
207	C₃×D₄	—	3	12	no	—
208	C₃×D₄	—	9	4	no	—
209	C₃×D₄	—	9	24	no	—
210	C₃×D₄	C₂²×C₃×D₃	1	12	yes	—
211	C₂×C₃×C₄	—	3	1	no	—
212	C₂×C₃×C₄	—	3	3	no	—
213	C₂×C₃×C₄	—	3	12	no	—
214	C₂×C₃×C₄	—	9	24	no	—
215	C₂×C₃×C₄	—	9	6	no	—
216	C₂×C₃×C₄	—	9	24	no	—
217	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
219	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
220	#67	—	3	24	no	—
221	C₂×Dic₃	—	3	6	no	—
222	C₂×Dic₃	—	3	24	no	—
223	D₁₂	—	3	4	no	—
224	C₄×D₃	—	3	24	no	—
225	Dic₆	—	3	12	no	—
226	Dic₆	—	3	24	no	—
227	C₂×C₃²	—	2	6	no	—
228	C₂×C₃²	#1010	1	2	yes	—
229	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
230	C₃×D₃	—	6	4	no	—
231	C₂×C₉	—	2	6	no	—
232	C₂×C₉	—	2	2	no	—
233	C₂×C₉	—	2	1	no	—
234	C₂×C₉	—	4	6	no	—
235	C₂×C₉	C₃×2₋¹⁺⁴	1	2	yes	complemented
236	C₂×C₉	C₂⁵×C₃	1	1	yes	characteristic
237	D₉	—	2	4	no	—
238	#41	—	9	48	no	—
239	#41	—	9	24	no	—
240	#41	C₂×AGL(1,ℤ/9)	1	8	yes	complemented
241	C₂×Q₈	—	9	16	no	—
242	C₂×Q₈	—	9	6	no	—
243	C₂×Q₈	—	9	3	no	—
244	C₂×Q₈	—	9	24	no	—
245	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
246	C₂×D₄	—	9	6	no	—
247	C₂×D₄	—	9	1	no	—
248	C₂×D₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
249	C₂²×C₄	—	9	6	no	—
250	C₂²×C₄	—	9	6	no	—
251	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
252	C₂²×C₃	—	2	6	no	—
253	C₂²×C₃	—	3	6	no	—
254	C₂²×C₃	—	3	1	no	—
255	C₂²×C₃	—	6	6	no	—
256	C₂²×C₃	—	9	4	no	—
257	C₂²×C₃	—	18	4	no	—
258	C₂²×C₃	C₂³×C₃×D₃	1	6	yes	—
259	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
260	C₂×D₃	—	3	4	no	—
261	C₂×D₃	—	6	4	no	—
262	C₃×C₄	—	3	2	no	—
263	C₃×C₄	—	3	6	no	—
264	C₃×C₄	—	9	12	no	—
265	C₃×C₄	C₂³×C₃×D₃	1	2	yes	—
266	C₃×C₄	C₂³×C₃×D₃	1	6	yes	—
267	Dic₃	—	3	12	no	—
268	C₃²	C₂×#1010	1	1	yes	characteristic
269	C₉	—	2	1	no	—
270	C₉	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic, complemented
271	C₂³	—	9	2	no	—
272	C₂³	C₂²×AGL(1,ℤ/9)	1	3	yes	—
273	Q₈	—	9	24	no	—
274	Q₈	—	9	12	no	—
275	Q₈	C₂²×AGL(1,ℤ/9)	1	4	yes	—
276	D₄	—	9	24	no	—
277	D₄	—	9	4	no	—
278	D₄	C₂²×AGL(1,ℤ/9)	1	12	yes	—
279	C₂×C₄	—	9	24	no	—
280	C₂×C₄	—	9	6	no	—
281	C₂×C₄	—	9	24	no	—
282	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	12	yes	—
283	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
284	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
285	C₂×C₃	—	2	6	no	—
286	C₂×C₃	—	3	1	no	—
287	C₂×C₃	—	3	2	no	—
288	C₂×C₃	—	6	6	no	—
289	C₂×C₃	—	18	4	no	—
290	C₂×C₃	C₃×#1010	1	2	yes	—
291	C₂×C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
292	D₃	—	6	4	no	—
293	C₂²	—	2	6	no	—
294	C₂²	—	9	4	no	—
295	C₂²	—	18	4	no	—
296	C₂²	C₂³×AGL(1,ℤ/9)	1	6	yes	—
297	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
298	C₄	—	9	12	no	—
299	C₄	C₂³×AGL(1,ℤ/9)	1	6	yes	—
300	C₄	C₂³×AGL(1,ℤ/9)	1	2	yes	—
301	C₃	—	3	1	no	—
302	C₃	C₂×C₃×#1010	1	1	yes	characteristic
303	C₂	—	2	6	no	—
304	C₂	—	18	4	no	—
305	C₂	#11722642	1	2	yes	direct product factor
306	C₂	C₂⁴×AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₉	C₂⁵×C₃
Frattini	C₂×C₃	C₂⁴×C₃×D₃
Fitting	C₂×#41×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂³×C₃×D₃
hypercenter	C₂×#41	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×2₋¹⁺⁴
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#41×3₋¹⁺²	C₂	semidirect
2. #11722642	C₂	direct (N × H)
3. C₂×Q₈×AGL(1,ℤ/9)	C₂	semidirect
4. C₂×#3699⋊C₃	C₂	semidirect
5. C₂×#3697⋊C₃	C₂	semidirect
6. C₂²×3₋¹⁺²⋊Q₈	C₂	semidirect
8. C₂×#63716	C₃	semidirect
9. #41×3₋¹⁺²	C₂²	semidirect
15. Q₈×AGL(1,ℤ/9)	C₂²	semidirect
16. #3699⋊C₃	C₂²	semidirect
17. #3697⋊C₃	C₂²	semidirect
22. C₂×3₋¹⁺²⋊Q₈	C₂²	semidirect
30. C₂×C₉×#41	C₂×C₃	semidirect
31. #63716	C₂×C₃	semidirect
32. C₂×Q₈×D₉	C₂×C₃	semidirect
33. C₂×#3699	C₂×C₃	semidirect
34. C₂×#3697	C₂×C₃	semidirect
35. C₂²×Dic₁₈	C₂×C₃	semidirect
67. C₉×#41	C₂²×C₃	semidirect
76. Q₈×D₉	C₂²×C₃	semidirect
77. #3699	C₂²×C₃	semidirect
78. #3697	C₂²×C₃	semidirect
83. C₂×Dic₁₈	C₂²×C₃	semidirect
140. C₂×3₋¹⁺²	2₋¹⁺⁴	semidirect
195. C₂×#41	AGL(1,ℤ/9)	semidirect
198. 3₋¹⁺²	C₂×2₋¹⁺⁴	semidirect
235. C₂×C₉	C₃×2₋¹⁺⁴	semidirect
240. #41	C₂×AGL(1,ℤ/9)	semidirect
270. C₉	C₂×C₃×2₋¹⁺⁴	semidirect
305. C₂	#11722642	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×#41×3₋¹⁺²	2
#11722642	2
C₂×Q₈×AGL(1,ℤ/9)	2
C₂×#3699⋊C₃	2
C₂×#3697⋊C₃	2
C₂²×3₋¹⁺²⋊Q₈	2
C₂×C₃×#1010	3
C₂×#63716	3

Subgroups — C2×#11722642 #13164945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11722642 #13164945