Subgroups — C₂×#11720528 #13144314

Order 1728.

1010 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₃⋊D₉)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720528	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63636	—	3	1	no	maximal
6	C₂×#63369	—	3	3	no	maximal
7	C₂³×C₃⋊D₉	—	2	1	no	—
8	C₂²×C₃×C₄×C₉	—	2	1	no	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊D₉)	—	2	1	no	—
11	C₂×C₄⋊(C₃⋊D₉)	—	2	2	no	—
12	C₂×C₄⋊(C₃⋊D₉)	C₄	1	4	yes	complemented
13	C₂×C₄⋊(C₃⋊D₉)	C₂²	1	2	yes	—
14	C₃×C₉×C₄⋊C₄	—	2	1	no	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	complemented
16	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
17	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	complemented
18	C₂²×C₄⋊(C₃⋊D₃)	—	3	1	no	—
19	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
20	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂²×D₃₆	—	3	3	no	—
22	#63636	—	3	4	no	—
23	C₂×C₉×C₄⋊C₄	D₃	1	3	yes	complemented
24	C₂²×C₉⋊C₈	—	3	3	no	—
25	#63369	—	3	12	no	—
26	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₃⋊D₉	—	2	4	no	—
28	C₂²×C₃⋊D₉	—	2	2	no	—
29	C₂²×C₃⋊D₉	—	4	2	no	—
30	C₂²×C₃⋊D₉	—	4	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	—	4	2	no	—
33	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₄⋊(C₃⋊D₉)	—	2	4	no	—
38	C₄⋊(C₃⋊D₉)	—	2	2	no	—
39	C₄⋊(C₃⋊D₉)	C₂×C₄	1	4	yes	complemented
40	C₃⋊(C₉⋊C₈)	—	2	2	no	—
41	C₂×#809	—	9	3	no	—
42	C₂×#809	—	9	1	no	—
43	C₂³×C₃⋊D₃	—	6	1	no	—
44	C₂²×C₃²×C₄	—	2	1	no	—
45	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄⋊(C₃⋊D₃)	—	3	4	no	—
47	C₂×C₄⋊(C₃⋊D₃)	—	3	2	no	—
48	C₂×C₄⋊(C₃⋊D₃)	—	6	2	no	—
49	C₂×C₄⋊(C₃⋊D₃)	—	6	1	no	—
50	C₂³×D₉	—	6	3	no	—
51	C₃²×C₄⋊C₄	—	2	1	no	—
52	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
53	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
54	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
55	C₂²×C₄×C₉	—	2	3	no	—
56	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
57	C₂×D₃₆	—	3	12	no	—
58	C₂×D₃₆	—	3	6	no	—
59	C₂×D₃₆	—	6	6	no	—
60	C₂×D₃₆	—	6	3	no	—
61	C₉×C₄⋊C₄	—	2	3	no	—
62	C₉×C₄⋊C₄	C₂×D₃	1	6	yes	complemented
63	C₂×C₉⋊C₈	—	3	6	no	—
64	C₂×C₉⋊C₈	—	6	6	no	—
65	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₃×C₉	D₈	1	2	yes	—
67	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
68	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃⋊D₉	—	2	4	no	—
70	C₂×C₃⋊D₉	—	4	8	no	—
71	C₂×C₃⋊D₉	—	4	2	no	—
72	C₂×C₃⋊D₉	—	4	2	no	—
73	C₃×C₄×C₉	—	4	2	no	—
74	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
75	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×D₁₂	—	9	3	no	—
78	C₂²×D₁₂	—	9	1	no	—
79	C₂×C₃×C₄⋊C₄	D₉	1	3	yes	complemented
80	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
81	C₂²×C₃⋊C₈	—	9	3	no	—
82	C₂²×C₃⋊C₈	—	9	1	no	—
83	#809	—	9	12	no	—
84	#809	—	9	4	no	—
85	C₂³×C₃²	#67	1	1	yes	characteristic
86	C₂²×C₃⋊D₃	—	6	4	no	—
87	C₂²×C₃⋊D₃	—	6	2	no	—
88	C₂²×C₃⋊D₃	—	12	2	no	—
89	C₂²×C₃⋊D₃	—	12	2	no	—
90	C₂×C₃²×C₄	—	2	2	no	—
91	C₂×C₃²×C₄	—	4	2	no	—
92	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
93	C₂×C₃²×C₄	D₁₂	1	2	yes	—
94	C₂×C₃²×C₄	#67	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₄⋊(C₃⋊D₃)	—	3	4	no	—
97	C₄⋊(C₃⋊D₃)	—	6	4	no	—
98	C₄⋊(C₃⋊D₃)	—	6	2	no	—
99	C₂³×C₉	#67	1	3	yes	—
100	C₂²×D₉	—	6	12	no	—
101	C₂²×D₉	—	6	6	no	—
102	C₂²×D₉	—	12	6	no	—
103	C₂²×D₉	—	12	6	no	—
104	C₃⋊(C₃⋊C₈)	—	6	2	no	—
105	C₂×C₄×C₉	—	2	6	no	—
106	C₂×C₄×C₉	—	4	6	no	—
107	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
108	C₂×C₄×C₉	D₁₂	1	6	yes	—
109	C₂×C₄×C₉	#67	1	3	yes	—
110	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
111	D₃₆	—	3	12	no	—
112	D₃₆	—	6	12	no	—
113	D₃₆	—	6	6	no	—
114	C₉⋊C₈	—	6	6	no	—
115	C₂×#102	—	27	1	no	Sylow
116	C₂×C₃×C₉	#102	1	4	yes	complemented
117	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₃⋊D₉	—	4	4	no	—
121	C₂³×D₃	—	18	3	no	—
122	C₂³×D₃	—	18	1	no	—
123	C₂²×C₃×C₄	—	2	3	no	—
124	C₂²×C₃×C₄	—	2	1	no	—
125	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
126	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
127	C₂×D₁₂	—	9	12	no	—
128	C₂×D₁₂	—	9	6	no	—
129	C₂×D₁₂	—	9	4	no	—
130	C₂×D₁₂	—	9	2	no	—
131	C₂×D₁₂	—	18	6	no	—
132	C₂×D₁₂	—	18	3	no	—
133	C₂×D₁₂	—	18	2	no	—
134	C₂×D₁₂	—	18	1	no	—
135	C₃×C₄⋊C₄	—	2	3	no	—
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	C₂×D₉	1	6	yes	complemented
138	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
139	C₂×C₃⋊C₈	—	9	6	no	—
140	C₂×C₃⋊C₈	—	9	2	no	—
141	C₂×C₃⋊C₈	—	18	6	no	—
142	C₂×C₃⋊C₈	—	18	2	no	—
143	C₂²×C₃²	(C₂×D₃)⋊C₄	1	2	yes	—
144	C₂²×C₃²	#213	1	2	yes	—
145	C₂²×C₃²	#215	1	2	yes	—
146	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
147	C₂×C₃⋊D₃	—	6	4	no	—
148	C₂×C₃⋊D₃	—	12	8	no	—
149	C₂×C₃⋊D₃	—	12	2	no	—
150	C₂×C₃⋊D₃	—	12	2	no	—
151	C₃²×C₄	—	4	2	no	—
152	C₃²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
153	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
154	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
155	C₂²×C₉	(C₂×D₃)⋊C₄	1	6	yes	—
156	C₂²×C₉	#213	1	6	yes	—
157	C₂²×C₉	#215	1	6	yes	—
158	C₂²×C₉	C₂×#67	1	3	yes	—
159	C₂×D₉	—	6	12	no	—
160	C₂×D₉	—	12	24	no	—
161	C₂×D₉	—	12	6	no	—
162	C₂×D₉	—	12	6	no	—
163	C₄×C₉	—	4	6	no	—
164	C₄×C₉	(C₂×D₃)⋊C₄	1	6	yes	—
165	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
166	C₄×C₉	C₂×D₁₂	1	3	yes	—
167	C₂²×D₄	—	27	1	no	—
168	C₂²×C₈	—	27	1	no	—
169	C₂×C₄⋊C₄	C₃⋊D₉	1	1	yes	characteristic, complemented
170	#102	—	27	4	no	—
171	C₃×C₉	C₂×#102	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₃	#611	1	3	yes	—
173	C₂³×C₃	#638	1	1	yes	characteristic
174	C₂²×D₃	—	18	12	no	—
175	C₂²×D₃	—	18	6	no	—
176	C₂²×D₃	—	18	4	no	—
177	C₂²×D₃	—	18	2	no	—
178	C₂²×D₃	—	36	6	no	—
179	C₂²×D₃	—	36	6	no	—
180	C₂²×D₃	—	36	2	no	—
181	C₂²×D₃	—	36	2	no	—
182	C₂×C₃×C₄	—	2	6	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	4	6	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
187	C₂×C₃×C₄	D₃₆	1	6	yes	—
188	C₂×C₃×C₄	#611	1	3	yes	—
189	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
190	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
191	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
192	C₂×C₃×C₄	#638	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
194	D₁₂	—	9	12	no	—
195	D₁₂	—	9	4	no	—
196	D₁₂	—	18	12	no	—
197	D₁₂	—	18	6	no	—
198	D₁₂	—	18	4	no	—
199	D₁₂	—	18	2	no	—
200	C₃⋊C₈	—	18	6	no	—
201	C₃⋊C₈	—	18	2	no	—
202	C₂×C₃²	#809	1	4	yes	—
203	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₂×C₃²	C₂×#213	1	1	yes	characteristic
205	C₂×C₃²	C₂×#215	1	1	yes	characteristic
206	C₃⋊D₃	—	12	4	no	—
207	C₂×C₉	#809	1	12	yes	complemented
208	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	3	yes	—
209	C₂×C₉	C₂×#213	1	3	yes	—
210	C₂×C₉	C₂×#215	1	3	yes	—
211	D₉	—	12	12	no	—
212	C₂⁴	—	54	1	no	—
213	C₂×D₄	—	27	4	no	—
214	C₂×D₄	—	27	2	no	—
215	C₂×D₄	—	54	2	no	—
216	C₂×D₄	—	54	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
219	C₂×C₈	—	27	2	no	—
220	C₂×C₈	—	54	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	C₂×C₃⋊D₉	1	2	yes	complemented
223	C₂²×C₃	(C₂×D₉)⋊C₄	1	6	yes	—
224	C₂²×C₃	#3673	1	6	yes	—
225	C₂²×C₃	#3675	1	6	yes	—
226	C₂²×C₃	C₂×#611	1	3	yes	—
227	C₂²×C₃	#3752	1	2	yes	—
228	C₂²×C₃	#3753	1	2	yes	—
229	C₂²×C₃	#3755	1	2	yes	—
230	C₂²×C₃	C₂×#638	1	1	yes	characteristic
231	C₂×D₃	—	18	12	no	—
232	C₂×D₃	—	18	4	no	—
233	C₂×D₃	—	36	6	no	—
234	C₂×D₃	—	36	6	no	—
235	C₂×D₃	—	36	24	no	—
236	C₂×D₃	—	36	2	no	—
237	C₂×D₃	—	36	8	no	—
238	C₂×D₃	—	36	2	no	—
239	C₃×C₄	—	4	6	no	—
240	C₃×C₄	—	4	2	no	—
241	C₃×C₄	(C₂×D₉)⋊C₄	1	6	yes	—
242	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
243	C₃×C₄	C₂×D₃₆	1	3	yes	—
244	C₃×C₄	#3752	1	2	yes	—
245	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
246	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
247	C₃²	C₂×#809	1	1	yes	characteristic
248	C₉	C₂×#809	1	3	yes	complemented
249	C₂³	—	54	2	no	—
250	C₂³	—	54	4	no	—
251	C₂³	—	108	2	no	—
252	C₂³	—	108	2	no	—
253	C₂³	#6233	1	1	yes	characteristic
254	D₄	—	27	4	no	—
255	D₄	—	54	2	no	—
256	D₄	—	54	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
260	C₂×C₄	C₄⋊(C₃⋊D₉)	1	2	yes	—
261	C₂×C₄	#6233	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
263	C₈	—	54	2	no	—
264	C₂×C₃	#63369	1	12	yes	complemented
265	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	3	yes	—
266	C₂×C₃	C₂×#3673	1	3	yes	—
267	C₂×C₃	C₂×#3675	1	3	yes	—
268	C₂×C₃	#63636	1	4	yes	—
269	C₂×C₃	C₂×#3752	1	1	yes	characteristic
270	C₂×C₃	C₂×#3753	1	1	yes	characteristic
271	C₂×C₃	C₂×#3755	1	1	yes	characteristic
272	D₃	—	36	4	no	—
273	D₃	—	36	12	no	—
274	C₂²	—	54	4	no	—
275	C₂²	—	108	8	no	—
276	C₂²	—	108	2	no	—
277	C₂²	—	108	2	no	—
278	C₂²	#88536	1	2	yes	—
279	C₂²	#88544	1	2	yes	—
280	C₂²	#88548	1	2	yes	—
281	C₂²	C₂×#6233	1	1	yes	characteristic
282	C₄	—	4	2	no	—
283	C₄	#88536	1	2	yes	—
284	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
285	C₄	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
286	C₃	C₂×#63369	1	3	yes	complemented
287	C₃	C₂×#63636	1	1	yes	characteristic
288	C₂	—	108	4	no	—
289	C₂	#11720528	1	4	yes	direct product factor
290	C₂	C₂×#88536	1	1	yes	characteristic
291	C₂	C₂×#88544	1	1	yes	characteristic
292	C₂	C₂×#88548	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6233
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle	C₂³×C₃²	#67
prime radical	C₂²×C₄⋊(C₃⋊D₉)	C₂
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊C₄	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#102
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₉×C₄⋊C₄	C₂	semidirect
3. C₂²×C₃⋊(C₉⋊C₈)	C₂	semidirect
4. #11720528	C₂	direct (N × H)
12. C₂×C₄⋊(C₃⋊D₉)	C₄	semidirect
15. C₃×C₉×C₄⋊C₄	C₂²	semidirect
17. C₂×C₃⋊(C₉⋊C₈)	C₂²	semidirect
23. C₂×C₉×C₄⋊C₄	D₃	semidirect
39. C₄⋊(C₃⋊D₉)	C₂×C₄	semidirect
62. C₉×C₄⋊C₄	C₂×D₃	semidirect
79. C₂×C₃×C₄⋊C₄	D₉	semidirect
116. C₂×C₃×C₉	#102	semidirect
137. C₃×C₄⋊C₄	C₂×D₉	semidirect
169. C₂×C₄⋊C₄	C₃⋊D₉	semidirect
171. C₃×C₉	C₂×#102	semidirect
207. C₂×C₉	#809	semidirect
222. C₄⋊C₄	C₂×C₃⋊D₉	semidirect
248. C₉	C₂×#809	semidirect
264. C₂×C₃	#63369	semidirect
286. C₃	C₂×#63369	semidirect
289. C₂	#11720528	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×C₄⋊(C₃⋊D₉)	2
C₂×C₃×C₉×C₄⋊C₄	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720528	2
C₂×#63636	3
C₂×#63369	3

Subgroups — C2×#11720528 #13144314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720528 #13144314