Subgroups — C₂×#11720577 #13144816

Order 1728.

1050 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88551	C₂	1	2	yes	maximal, complemented
3	#11720577	C₂	1	4	yes	maximal, direct product factor
4	C₂×#63661	—	3	1	no	maximal
5	C₂×#63393	—	3	3	no	maximal
6	C₂⁴×C₃×C₉	C₄	1	1	yes	characteristic, complemented
7	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃×D₄×C₉	—	2	2	no	—
9	C₂×C₃×D₄×C₉	—	2	2	no	—
10	C₂×C₃×D₄×C₉	C₄	1	2	yes	complemented
11	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
12	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₃⋊Dic₉	—	2	2	no	—
14	#88551	—	2	2	no	—
15	#88551	C₂²	1	4	yes	complemented
16	C₂²×C₃²×D₄	D₃	1	1	yes	characteristic
17	C₂×#3757	—	3	2	no	—
18	C₂²×D₄×C₉	D₃	1	3	yes	—
19	#63661	—	3	4	no	—
20	C₂×#3676	—	3	6	no	—
21	#63393	—	3	12	no	—
22	C₂³×C₃×C₉	—	2	1	no	—
23	C₂³×C₃×C₉	—	2	1	no	—
24	C₂³×C₃×C₉	—	4	2	no	—
25	C₂³×C₃×C₉	—	4	2	no	—
26	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₃×C₉	C₂×C₄	1	2	yes	complemented
29	C₂³×C₃×C₉	D₄	1	2	yes	—
30	C₂³×C₃×C₉	D₄	1	2	yes	—
31	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
32	C₃×D₄×C₉	—	2	4	no	—
33	C₃×D₄×C₉	—	2	4	no	—
34	C₂×C₃×C₄×C₉	—	2	1	no	—
35	C₂×C₃×C₄×C₉	—	2	1	no	—
36	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	complemented
37	C₂×C₃⋊Dic₉	—	2	4	no	—
38	C₂×C₃⋊Dic₉	—	4	4	no	—
39	C₂×#834	—	9	3	no	—
40	C₂×#834	—	9	1	no	—
41	C₂⁴×C₃²	Dic₃	1	1	yes	characteristic
42	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃²×D₄	—	2	2	no	—
44	C₂×C₃²×D₄	—	2	2	no	—
45	C₂×C₃²×D₄	Dic₃	1	2	yes	—
46	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
47	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
48	C₂²×C₃⋊Dic₃	—	6	2	no	—
49	C₂⁴×C₉	Dic₃	1	3	yes	complemented
50	C₂⁴×C₉	C₂×D₃	1	3	yes	—
51	#3757	—	3	4	no	—
52	#3757	—	6	2	no	—
53	C₂×D₄×C₉	—	2	6	no	—
54	C₂×D₄×C₉	—	2	6	no	—
55	C₂×D₄×C₉	Dic₃	1	6	yes	complemented
56	C₂×D₄×C₉	C₂×D₃	1	6	yes	—
57	C₂²×C₄×C₉	Dic₃	1	3	yes	complemented
58	C₂²×Dic₉	—	6	6	no	—
59	#3676	—	3	12	no	—
60	#3676	—	6	6	no	—
61	C₂²×C₃×C₉	—	2	4	no	—
62	C₂²×C₃×C₉	—	2	2	no	—
63	C₂²×C₃×C₉	—	2	2	no	—
64	C₂²×C₃×C₉	—	4	1	no	—
65	C₂²×C₃×C₉	—	4	2	no	—
66	C₂²×C₃×C₉	—	4	1	no	—
67	C₂²×C₃×C₉	—	4	2	no	—
68	C₂²×C₃×C₉	—	4	2	no	—
69	C₂²×C₃×C₉	—	4	2	no	—
70	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
75	C₃×C₄×C₉	—	2	2	no	—
76	C₃⋊Dic₉	—	4	4	no	—
77	C₂²×C₃×D₄	D₉	1	3	yes	—
78	C₂²×C₃×D₄	C₃⋊D₃	1	1	yes	characteristic
79	C₂×#217	—	9	6	no	—
80	C₂×#217	—	9	2	no	—
81	#834	—	9	12	no	—
82	#834	—	9	4	no	—
83	C₂³×C₃²	—	2	1	no	—
84	C₂³×C₃²	—	2	1	no	—
85	C₂³×C₃²	—	4	2	no	—
86	C₂³×C₃²	—	4	2	no	—
87	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
88	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
89	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
90	C₂³×C₃²	#67	1	2	yes	—
91	C₂³×C₃²	#67	1	2	yes	—
92	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
93	C₃²×D₄	—	2	4	no	—
94	C₃²×D₄	—	2	4	no	—
95	C₂×C₃²×C₄	—	2	1	no	—
96	C₂×C₃²×C₄	—	2	1	no	—
97	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
98	C₂×C₃⋊Dic₃	—	6	4	no	—
99	C₂×C₃⋊Dic₃	—	12	4	no	—
100	C₂³×C₉	—	2	3	no	—
101	C₂³×C₉	—	2	3	no	—
102	C₂³×C₉	—	4	6	no	—
103	C₂³×C₉	—	4	6	no	—
104	C₂³×C₉	C₂×Dic₃	1	3	yes	—
105	C₂³×C₉	C₂×Dic₃	1	3	yes	—
106	C₂³×C₉	C₂×Dic₃	1	6	yes	complemented
107	C₂³×C₉	#67	1	6	yes	—
108	C₂³×C₉	#67	1	6	yes	—
109	C₂³×C₉	C₂²×D₃	1	3	yes	—
110	D₄×C₉	—	2	12	no	—
111	D₄×C₉	—	2	12	no	—
112	C₂×C₄×C₉	—	2	3	no	—
113	C₂×C₄×C₉	—	2	3	no	—
114	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	complemented
115	C₂×Dic₉	—	6	12	no	—
116	C₂×Dic₉	—	12	12	no	—
117	C₂×#99	—	27	1	no	Sylow
118	C₂×C₃×C₉	—	2	2	no	—
119	C₂×C₃×C₉	—	2	1	no	—
120	C₂×C₃×C₉	—	2	1	no	—
121	C₂×C₃×C₉	—	2	2	no	—
122	C₂×C₃×C₉	—	4	2	no	—
123	C₂×C₃×C₉	#99	1	2	yes	complemented
124	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂⁴×C₃	Dic₉	1	3	yes	complemented
126	C₂⁴×C₃	C₂×D₉	1	3	yes	—
127	C₂⁴×C₃	C₃⋊Dic₃	1	1	yes	characteristic
128	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
129	C₂×C₃×D₄	—	2	6	no	—
130	C₂×C₃×D₄	—	2	6	no	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	—	2	2	no	—
133	C₂×C₃×D₄	Dic₉	1	6	yes	complemented
134	C₂×C₃×D₄	C₂×D₉	1	6	yes	—
135	C₂×C₃×D₄	C₃⋊Dic₃	1	2	yes	—
136	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
137	C₂²×C₃×C₄	Dic₉	1	3	yes	complemented
138	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
139	C₂²×Dic₃	—	18	6	no	—
140	C₂²×Dic₃	—	18	2	no	—
141	#217	—	9	12	no	—
142	#217	—	9	4	no	—
143	#217	—	18	6	no	—
144	#217	—	18	2	no	—
145	C₂²×C₃²	—	2	4	no	—
146	C₂²×C₃²	—	2	2	no	—
147	C₂²×C₃²	—	2	2	no	—
148	C₂²×C₃²	—	4	1	no	—
149	C₂²×C₃²	—	4	2	no	—
150	C₂²×C₃²	—	4	1	no	—
151	C₂²×C₃²	—	4	2	no	—
152	C₂²×C₃²	—	4	2	no	—
153	C₂²×C₃²	—	4	2	no	—
154	C₂²×C₃²	#217	1	1	yes	characteristic
155	C₂²×C₃²	#217	1	2	yes	—
156	C₂²×C₃²	#217	1	1	yes	characteristic
157	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
158	C₂²×C₃²	C₂×#67	1	2	yes	—
159	C₃²×C₄	—	2	2	no	—
160	C₃⋊Dic₃	—	12	4	no	—
161	C₂²×C₉	—	2	12	no	—
162	C₂²×C₉	—	2	6	no	—
163	C₂²×C₉	—	2	6	no	—
164	C₂²×C₉	—	4	3	no	—
165	C₂²×C₉	—	4	6	no	—
166	C₂²×C₉	—	4	3	no	—
167	C₂²×C₉	—	4	6	no	—
168	C₂²×C₉	—	4	6	no	—
169	C₂²×C₉	—	4	6	no	—
170	C₂²×C₉	#217	1	3	yes	—
171	C₂²×C₉	#217	1	6	yes	—
172	C₂²×C₉	#217	1	3	yes	—
173	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
174	C₂²×C₉	C₂×#67	1	6	yes	—
175	C₄×C₉	—	2	6	no	—
176	Dic₉	—	12	12	no	—
177	C₂²×D₄	C₃⋊D₉	1	1	yes	characteristic
178	C₂×C₂²⋊C₄	—	27	2	no	—
179	#99	—	27	4	no	—
180	C₃×C₉	C₂×#99	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₃	—	2	3	no	—
182	C₂³×C₃	—	2	3	no	—
183	C₂³×C₃	—	2	1	no	—
184	C₂³×C₃	—	2	1	no	—
185	C₂³×C₃	—	4	2	no	—
186	C₂³×C₃	—	4	6	no	—
187	C₂³×C₃	—	4	2	no	—
188	C₂³×C₃	—	4	6	no	—
189	C₂³×C₃	C₂×Dic₉	1	3	yes	—
190	C₂³×C₃	C₂×Dic₉	1	3	yes	—
191	C₂³×C₃	C₂×Dic₉	1	6	yes	complemented
192	C₂³×C₃	#611	1	6	yes	—
193	C₂³×C₃	#611	1	6	yes	—
194	C₂³×C₃	C₂²×D₉	1	3	yes	—
195	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
196	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
197	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
198	C₂³×C₃	#638	1	2	yes	—
199	C₂³×C₃	#638	1	2	yes	—
200	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
201	C₃×D₄	—	2	12	no	—
202	C₃×D₄	—	2	12	no	—
203	C₃×D₄	—	2	4	no	—
204	C₃×D₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	3	no	—
206	C₂×C₃×C₄	—	2	3	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	complemented
210	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
211	C₂×Dic₃	—	18	12	no	—
212	C₂×Dic₃	—	18	4	no	—
213	C₂×Dic₃	—	36	4	no	—
214	C₂×Dic₃	—	36	12	no	—
215	C₂×C₃²	—	2	2	no	—
216	C₂×C₃²	—	2	1	no	—
217	C₂×C₃²	—	2	1	no	—
218	C₂×C₃²	—	2	2	no	—
219	C₂×C₃²	—	4	2	no	—
220	C₂×C₃²	#834	1	2	yes	—
221	C₂×C₃²	C₂×#217	1	1	yes	characteristic
222	C₂×C₉	—	2	6	no	—
223	C₂×C₉	—	2	3	no	—
224	C₂×C₉	—	2	3	no	—
225	C₂×C₉	—	2	6	no	—
226	C₂×C₉	—	4	6	no	—
227	C₂×C₉	#834	1	6	yes	complemented
228	C₂×C₉	C₂×#217	1	3	yes	—
229	C₂⁴	C₃⋊Dic₉	1	1	yes	characteristic, complemented
230	C₂⁴	C₂×C₃⋊D₉	1	1	yes	characteristic
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	C₃⋊Dic₉	1	2	yes	complemented
234	C₂×D₄	C₂×C₃⋊D₉	1	2	yes	—
235	C₂²×C₄	—	54	2	no	—
236	C₂²×C₄	C₃⋊Dic₉	1	1	yes	characteristic, complemented
237	C₂²⋊C₄	—	27	4	no	—
238	C₂²⋊C₄	—	54	2	no	—
239	C₂²×C₃	—	2	12	no	—
240	C₂²×C₃	—	2	6	no	—
241	C₂²×C₃	—	2	4	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	6	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	4	3	no	—
246	C₂²×C₃	—	4	6	no	—
247	C₂²×C₃	—	4	3	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	2	no	—
250	C₂²×C₃	—	4	1	no	—
251	C₂²×C₃	—	4	6	no	—
252	C₂²×C₃	—	4	2	no	—
253	C₂²×C₃	—	4	2	no	—
254	C₂²×C₃	—	4	6	no	—
255	C₂²×C₃	—	4	6	no	—
256	C₂²×C₃	—	4	2	no	—
257	C₂²×C₃	#3676	1	3	yes	—
258	C₂²×C₃	#3676	1	6	yes	—
259	C₂²×C₃	#3676	1	3	yes	—
260	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
261	C₂²×C₃	C₂×#611	1	6	yes	—
262	C₂²×C₃	#3757	1	1	yes	characteristic
263	C₂²×C₃	#3757	1	2	yes	—
264	C₂²×C₃	#3757	1	1	yes	characteristic
265	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
266	C₂²×C₃	C₂×#638	1	2	yes	—
267	C₃×C₄	—	2	6	no	—
268	C₃×C₄	—	2	2	no	—
269	Dic₃	—	36	4	no	—
270	Dic₃	—	36	12	no	—
271	C₃²	C₂×#834	1	1	yes	characteristic
272	C₉	C₂×#834	1	3	yes	complemented
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
278	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
279	C₂³	C₂×C₃⋊Dic₉	1	2	yes	complemented
280	C₂³	#6233	1	2	yes	—
281	C₂³	#6233	1	2	yes	—
282	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
283	D₄	—	2	4	no	—
284	D₄	—	2	4	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	54	4	no	—
288	C₂×C₄	—	108	4	no	—
289	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	complemented
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	2	6	no	—
295	C₂×C₃	—	2	3	no	—
296	C₂×C₃	—	2	3	no	—
297	C₂×C₃	—	2	6	no	—
298	C₂×C₃	—	4	2	no	—
299	C₂×C₃	—	4	6	no	—
300	C₂×C₃	#63393	1	6	yes	complemented
301	C₂×C₃	C₂×#3676	1	3	yes	—
302	C₂×C₃	#63661	1	2	yes	—
303	C₂×C₃	C₂×#3757	1	1	yes	characteristic
304	C₂²	—	2	4	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	#88551	1	1	yes	characteristic
314	C₂²	#88551	1	2	yes	—
315	C₂²	#88551	1	1	yes	characteristic
316	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
317	C₂²	C₂×#6233	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	108	4	no	—
320	C₃	C₂×#63393	1	3	yes	complemented
321	C₃	C₂×#63661	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	4	2	no	—
327	C₂	#11720577	1	2	yes	direct product factor
328	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#88551
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂²×C₃×D₄×C₉	C₂
socle	C₂²×C₃²	#217
prime radical	C₂²×C₃²×D₄	D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂²×D₄	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#99
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#88551	C₂	semidirect
3. #11720577	C₂	direct (N × H)
6. C₂⁴×C₃×C₉	C₄	semidirect
10. C₂×C₃×D₄×C₉	C₄	semidirect
12. C₂²×C₃×C₄×C₉	C₄	semidirect
15. #88551	C₂²	semidirect
28. C₂³×C₃×C₉	C₂×C₄	semidirect
36. C₂×C₃×C₄×C₉	C₂×C₄	semidirect
49. C₂⁴×C₉	Dic₃	semidirect
55. C₂×D₄×C₉	Dic₃	semidirect
57. C₂²×C₄×C₉	Dic₃	semidirect
106. C₂³×C₉	C₂×Dic₃	semidirect
114. C₂×C₄×C₉	C₂×Dic₃	semidirect
123. C₂×C₃×C₉	#99	semidirect
125. C₂⁴×C₃	Dic₉	semidirect
133. C₂×C₃×D₄	Dic₉	semidirect
137. C₂²×C₃×C₄	Dic₉	semidirect
180. C₃×C₉	C₂×#99	semidirect
191. C₂³×C₃	C₂×Dic₉	semidirect
209. C₂×C₃×C₄	C₂×Dic₉	semidirect
227. C₂×C₉	#834	semidirect
229. C₂⁴	C₃⋊Dic₉	semidirect
233. C₂×D₄	C₃⋊Dic₉	semidirect
236. C₂²×C₄	C₃⋊Dic₉	semidirect
272. C₉	C₂×#834	semidirect
279. C₂³	C₂×C₃⋊Dic₉	semidirect
289. C₂×C₄	C₂×C₃⋊Dic₉	semidirect
300. C₂×C₃	#63393	semidirect
320. C₃	C₂×#63393	semidirect
327. C₂	#11720577	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×C₃×D₄×C₉	2
C₂×#88551	2
#11720577	2
C₂×#63661	3
C₂×#63393	3

Subgroups — C2×#11720577 #13144816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720577 #13144816