Subgroups — C₂×#11720576 #13144813

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×C₃×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720576	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63659	—	3	1	no	maximal
6	C₂×#63392	—	3	3	no	maximal
7	C₂⁴×C₃×C₉	—	2	1	no	—
8	C₂×C₃×D₄×C₉	—	2	1	no	—
9	C₂×C₃×D₄×C₉	—	2	2	no	—
10	C₂×C₃×D₄×C₉	C₄	1	4	yes	complemented
11	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
12	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊Dic₉	—	2	1	no	—
14	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
15	C₄⋊(C₃⋊Dic₉)	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×C₃²×D₄	D₃	1	1	yes	characteristic
19	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
20	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂²×D₄×C₉	D₃	1	3	yes	—
22	#63659	—	3	4	no	—
23	C₂×C₄⋊Dic₉	—	3	3	no	—
24	C₂²×C₉⋊C₈	—	3	3	no	—
25	#63392	—	3	12	no	—
26	C₂³×C₃×C₉	—	2	4	no	—
27	C₂³×C₃×C₉	—	2	2	no	—
28	C₂³×C₃×C₉	—	4	2	no	—
29	C₂³×C₃×C₉	—	4	2	no	—
30	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
31	C₃×D₄×C₉	—	2	4	no	—
32	C₃×D₄×C₉	—	2	2	no	—
33	C₃×D₄×C₉	C₂×C₄	1	4	yes	complemented
34	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊Dic₉	—	2	2	no	—
39	C₂×C₃⋊Dic₉	—	4	2	no	—
40	C₃⋊(C₉⋊C₈)	—	2	2	no	—
41	C₂×#832	—	9	3	no	—
42	C₂×#832	—	9	1	no	—
43	C₂⁴×C₃²	—	2	1	no	—
44	C₂×C₃²×D₄	—	2	1	no	—
45	C₂×C₃²×D₄	—	2	2	no	—
46	C₂×C₃²×D₄	Dic₃	1	4	yes	—
47	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
48	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃⋊Dic₃	—	6	1	no	—
50	C₂⁴×C₉	—	2	3	no	—
51	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
52	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
53	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
54	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
55	C₂×D₄×C₉	—	2	3	no	—
56	C₂×D₄×C₉	—	2	6	no	—
57	C₂×D₄×C₉	Dic₃	1	12	yes	complemented
58	C₂×D₄×C₉	C₂×D₃	1	6	yes	—
59	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
60	C₂²×Dic₉	—	6	3	no	—
61	C₄⋊Dic₉	—	3	6	no	—
62	C₄⋊Dic₉	—	6	3	no	—
63	C₂×C₉⋊C₈	—	3	6	no	—
64	C₂×C₉⋊C₈	—	6	6	no	—
65	C₂²×C₃×C₉	—	2	4	no	—
66	C₂²×C₃×C₉	—	4	8	no	—
67	C₂²×C₃×C₉	—	4	2	no	—
68	C₂²×C₃×C₉	—	4	2	no	—
69	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₃×C₉	D₈	1	2	yes	—
71	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
72	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
74	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
76	C₃⋊Dic₉	—	4	2	no	—
77	C₂²×C₃×D₄	D₉	1	3	yes	—
78	C₂²×C₃×D₄	C₃⋊D₃	1	1	yes	characteristic
79	C₂×C₄⋊Dic₃	—	9	3	no	—
80	C₂×C₄⋊Dic₃	—	9	1	no	—
81	C₂²×C₃⋊C₈	—	9	3	no	—
82	C₂²×C₃⋊C₈	—	9	1	no	—
83	#832	—	9	12	no	—
84	#832	—	9	4	no	—
85	C₂³×C₃²	—	2	4	no	—
86	C₂³×C₃²	—	2	2	no	—
87	C₂³×C₃²	—	4	2	no	—
88	C₂³×C₃²	—	4	2	no	—
89	C₂³×C₃²	#67	1	1	yes	characteristic
90	C₃²×D₄	—	2	4	no	—
91	C₃²×D₄	—	2	2	no	—
92	C₃²×D₄	C₂×Dic₃	1	4	yes	—
93	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
94	C₂×C₃²×C₄	#67	1	1	yes	characteristic
95	C₂×C₃²×C₄	#67	1	2	yes	—
96	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃⋊Dic₃	—	6	2	no	—
98	C₂×C₃⋊Dic₃	—	12	2	no	—
99	C₂³×C₉	—	2	12	no	—
100	C₂³×C₉	—	2	6	no	—
101	C₂³×C₉	—	4	6	no	—
102	C₂³×C₉	—	4	6	no	—
103	C₂³×C₉	#67	1	3	yes	—
104	C₃⋊(C₃⋊C₈)	—	6	2	no	—
105	D₄×C₉	—	2	12	no	—
106	D₄×C₉	—	2	6	no	—
107	D₄×C₉	C₂×Dic₃	1	12	yes	complemented
108	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
109	C₂×C₄×C₉	#67	1	3	yes	—
110	C₂×C₄×C₉	#67	1	6	yes	—
111	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
112	C₂×Dic₉	—	6	6	no	—
113	C₂×Dic₉	—	12	6	no	—
114	C₉⋊C₈	—	6	6	no	—
115	C₂×#102	—	27	1	no	Sylow
116	C₂×C₃×C₉	—	4	4	no	—
117	C₂×C₃×C₉	#102	1	4	yes	complemented
118	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂⁴×C₃	—	2	3	no	—
122	C₂⁴×C₃	—	2	1	no	—
123	C₂×C₃×D₄	—	2	3	no	—
124	C₂×C₃×D₄	—	2	6	no	—
125	C₂×C₃×D₄	—	2	1	no	—
126	C₂×C₃×D₄	—	2	2	no	—
127	C₂×C₃×D₄	Dic₉	1	12	yes	complemented
128	C₂×C₃×D₄	C₂×D₉	1	6	yes	—
129	C₂×C₃×D₄	C₃⋊Dic₃	1	4	yes	—
130	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
131	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
132	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
133	C₂²×Dic₃	—	18	3	no	—
134	C₂²×Dic₃	—	18	1	no	—
135	C₄⋊Dic₃	—	9	6	no	—
136	C₄⋊Dic₃	—	9	2	no	—
137	C₄⋊Dic₃	—	18	3	no	—
138	C₄⋊Dic₃	—	18	1	no	—
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	—	2	4	no	—
144	C₂²×C₃²	—	4	8	no	—
145	C₂²×C₃²	—	4	2	no	—
146	C₂²×C₃²	—	4	2	no	—
147	C₂²×C₃²	#213	1	2	yes	—
148	C₂²×C₃²	#214	1	2	yes	—
149	C₂²×C₃²	#217	1	2	yes	—
150	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
151	C₃²×C₄	#217	1	2	yes	—
152	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
153	C₃²×C₄	C₂×#67	1	1	yes	characteristic
154	C₃⋊Dic₃	—	12	2	no	—
155	C₂²×C₉	—	2	12	no	—
156	C₂²×C₉	—	4	24	no	—
157	C₂²×C₉	—	4	6	no	—
158	C₂²×C₉	—	4	6	no	—
159	C₂²×C₉	#213	1	6	yes	—
160	C₂²×C₉	#214	1	6	yes	—
161	C₂²×C₉	#217	1	6	yes	—
162	C₂²×C₉	C₂×#67	1	3	yes	—
163	C₄×C₉	#217	1	6	yes	—
164	C₄×C₉	C₂²×Dic₃	1	3	yes	—
165	C₄×C₉	C₂×#67	1	3	yes	—
166	Dic₉	—	12	6	no	—
167	C₂²×D₄	C₃⋊D₉	1	1	yes	characteristic
168	C₂²×C₈	—	27	1	no	—
169	C₂×C₄⋊C₄	—	27	1	no	—
170	#102	—	27	4	no	—
171	C₃×C₉	C₂×#102	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₃	—	2	12	no	—
173	C₂³×C₃	—	2	6	no	—
174	C₂³×C₃	—	2	4	no	—
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	—	4	6	no	—
177	C₂³×C₃	—	4	6	no	—
178	C₂³×C₃	—	4	2	no	—
179	C₂³×C₃	—	4	2	no	—
180	C₂³×C₃	#611	1	3	yes	—
181	C₂³×C₃	#638	1	1	yes	characteristic
182	C₃×D₄	—	2	12	no	—
183	C₃×D₄	—	2	6	no	—
184	C₃×D₄	—	2	4	no	—
185	C₃×D₄	—	2	2	no	—
186	C₃×D₄	C₂×Dic₉	1	12	yes	complemented
187	C₃×D₄	C₂×C₃⋊Dic₃	1	4	yes	—
188	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
189	C₂×C₃×C₄	#611	1	3	yes	—
190	C₂×C₃×C₄	#611	1	6	yes	—
191	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
192	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
193	C₂×C₃×C₄	#638	1	1	yes	characteristic
194	C₂×C₃×C₄	#638	1	2	yes	—
195	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
196	C₂×Dic₃	—	18	6	no	—
197	C₂×Dic₃	—	18	2	no	—
198	C₂×Dic₃	—	36	6	no	—
199	C₂×Dic₃	—	36	2	no	—
200	C₃⋊C₈	—	18	6	no	—
201	C₃⋊C₈	—	18	2	no	—
202	C₂×C₃²	—	4	4	no	—
203	C₂×C₃²	#832	1	4	yes	—
204	C₂×C₃²	C₂×#213	1	1	yes	characteristic
205	C₂×C₃²	C₂×#214	1	1	yes	characteristic
206	C₂×C₃²	C₂×#217	1	1	yes	characteristic
207	C₂×C₉	—	4	12	no	—
208	C₂×C₉	#832	1	12	yes	complemented
209	C₂×C₉	C₂×#213	1	3	yes	—
210	C₂×C₉	C₂×#214	1	3	yes	—
211	C₂×C₉	C₂×#217	1	3	yes	—
212	C₂⁴	—	2	1	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	C₃⋊Dic₉	1	4	yes	complemented
216	C₂×D₄	C₂×C₃⋊D₉	1	2	yes	—
217	C₂²×C₄	—	54	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	—	27	2	no	—
222	C₄⋊C₄	—	54	1	no	—
223	C₂²×C₃	—	2	12	no	—
224	C₂²×C₃	—	2	4	no	—
225	C₂²×C₃	—	4	6	no	—
226	C₂²×C₃	—	4	6	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	—	4	2	no	—
229	C₂²×C₃	—	4	24	no	—
230	C₂²×C₃	—	4	8	no	—
231	C₂²×C₃	#3672	1	6	yes	—
232	C₂²×C₃	#3673	1	6	yes	—
233	C₂²×C₃	#3676	1	6	yes	—
234	C₂²×C₃	C₂×#611	1	3	yes	—
235	C₂²×C₃	#3753	1	2	yes	—
236	C₂²×C₃	#3754	1	2	yes	—
237	C₂²×C₃	#3757	1	2	yes	—
238	C₂²×C₃	C₂×#638	1	1	yes	characteristic
239	C₃×C₄	#3676	1	6	yes	—
240	C₃×C₄	C₂²×Dic₉	1	3	yes	—
241	C₃×C₄	C₂×#611	1	3	yes	—
242	C₃×C₄	#3757	1	2	yes	—
243	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
244	C₃×C₄	C₂×#638	1	1	yes	characteristic
245	Dic₃	—	36	6	no	—
246	Dic₃	—	36	2	no	—
247	C₃²	C₂×#832	1	1	yes	characteristic
248	C₉	C₂×#832	1	3	yes	complemented
249	C₂³	—	2	2	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	#6233	1	1	yes	characteristic
254	D₄	—	2	2	no	—
255	D₄	—	2	4	no	—
256	D₄	C₂×C₃⋊Dic₉	1	4	yes	complemented
257	C₂×C₄	—	54	2	no	—
258	C₂×C₄	—	108	2	no	—
259	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
260	C₂×C₄	#6233	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	—	4	4	no	—
265	C₂×C₃	—	4	12	no	—
266	C₂×C₃	#63392	1	12	yes	complemented
267	C₂×C₃	C₂×#3672	1	3	yes	—
268	C₂×C₃	C₂×#3673	1	3	yes	—
269	C₂×C₃	C₂×#3676	1	3	yes	—
270	C₂×C₃	#63659	1	4	yes	—
271	C₂×C₃	C₂×#3753	1	1	yes	characteristic
272	C₂×C₃	C₂×#3754	1	1	yes	characteristic
273	C₂×C₃	C₂×#3757	1	1	yes	characteristic
274	C₂²	—	2	4	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	8	no	—
278	C₂²	#88543	1	2	yes	—
279	C₂²	#88544	1	2	yes	—
280	C₂²	#88551	1	2	yes	—
281	C₂²	C₂×#6233	1	1	yes	characteristic
282	C₄	—	108	2	no	—
283	C₄	#88551	1	2	yes	—
284	C₄	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
285	C₄	C₂×#6233	1	1	yes	characteristic
286	C₃	C₂×#63392	1	3	yes	complemented
287	C₃	C₂×#63659	1	1	yes	characteristic
288	C₂	—	4	4	no	—
289	C₂	#11720576	1	4	yes	direct product factor
290	C₂	C₂×#88543	1	1	yes	characteristic
291	C₂	C₂×#88544	1	1	yes	characteristic
292	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6233
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂²×C₃×D₄×C₉	C₂
socle	C₂³×C₃²	#67
prime radical	C₂²×C₃²×D₄	D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×D₄	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#102
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₃⋊Dic₉)	C₂	semidirect
3. C₂²×C₃⋊(C₉⋊C₈)	C₂	semidirect
4. #11720576	C₂	direct (N × H)
10. C₂×C₃×D₄×C₉	C₄	semidirect
15. C₄⋊(C₃⋊Dic₉)	C₂²	semidirect
17. C₂×C₃⋊(C₉⋊C₈)	C₂²	semidirect
33. C₃×D₄×C₉	C₂×C₄	semidirect
57. C₂×D₄×C₉	Dic₃	semidirect
107. D₄×C₉	C₂×Dic₃	semidirect
117. C₂×C₃×C₉	#102	semidirect
127. C₂×C₃×D₄	Dic₉	semidirect
171. C₃×C₉	C₂×#102	semidirect
186. C₃×D₄	C₂×Dic₉	semidirect
208. C₂×C₉	#832	semidirect
215. C₂×D₄	C₃⋊Dic₉	semidirect
248. C₉	C₂×#832	semidirect
256. D₄	C₂×C₃⋊Dic₉	semidirect
266. C₂×C₃	#63392	semidirect
286. C₃	C₂×#63392	semidirect
289. C₂	#11720576	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×C₄⋊(C₃⋊Dic₉)	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720576	2
C₂×#63659	3
C₂×#63392	3

Subgroups — C2×#11720576 #13144813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720576 #13144813