Subgroups — C₂×#11720279 #13139379

Order 1728.

482 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11720279	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63575	—	3	1	no	maximal
6	C₂×#63378	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	—	2	1	no	—
9	C₂²×C₃×Dic₉	—	2	1	no	—
10	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
11	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
12	C₃×C₄⋊Dic₉	—	2	1	no	—
13	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
14	C₂×C₉×C₃⋊C₈	C₄	1	4	yes	complemented
15	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
17	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
18	C₂²×C₃×C₃⋊C₈	D₃	1	1	yes	characteristic
19	#63575	—	3	4	no	—
20	C₂²×C₈×C₉	—	3	1	no	—
21	C₂×C₄⋊Dic₉	—	3	1	no	—
22	C₂×C₄⋊Dic₉	—	6	1	no	—
23	C₂×C₄⋊Dic₉	D₃	1	1	yes	characteristic
24	#63378	—	3	4	no	—
25	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	C₂×C₃⋊Dic₉	—	4	2	no	—
32	C₂×C₃×Dic₉	—	2	2	no	—
33	C₂×C₃×Dic₉	—	4	2	no	—
34	C₉×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
35	C₂×#818	—	9	1	no	—
36	C₂×#807	—	9	1	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃⋊Dic₃	—	6	1	no	—
39	C₂²×C₃×Dic₃	—	6	1	no	—
40	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
41	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
42	C₃×C₄⋊Dic₃	—	3	2	no	—
43	C₃×C₄⋊Dic₃	—	6	1	no	—
44	C₂×C₃×C₃⋊C₈	Dic₃	1	4	yes	—
45	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
46	C₂²×C₄×C₉	—	2	1	no	—
47	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
48	C₂²×Dic₉	—	2	1	no	—
49	C₂²×Dic₉	—	6	2	no	—
50	C₂²×Dic₉	—	6	1	no	—
51	C₂×C₈×C₉	—	3	4	no	—
52	C₂×C₈×C₉	—	3	2	no	—
53	C₄⋊Dic₉	—	2	1	no	—
54	C₄⋊Dic₉	—	3	2	no	—
55	C₄⋊Dic₉	—	6	2	no	—
56	C₄⋊Dic₉	—	6	2	no	—
57	C₄⋊Dic₉	—	6	1	no	—
58	C₄⋊Dic₉	C₂×D₃	1	2	yes	—
59	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
60	C₂²×C₃×C₉	D₈	1	2	yes	—
61	C₂²×C₃×C₉	Q₁₆	1	2	yes	—
62	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
63	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
64	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
66	C₃⋊Dic₉	—	4	2	no	—
67	C₃×Dic₉	—	4	2	no	—
68	C₂²×C₃×C₈	—	3	1	no	—
69	C₂×C₃×C₄⋊C₄	—	9	1	no	—
70	C₂×C₄⋊Dic₃	—	3	1	no	—
71	C₂×C₄⋊Dic₃	—	9	1	no	—
72	C₂×C₄⋊Dic₃	—	9	1	no	—
73	C₂×C₄⋊Dic₃	—	18	1	no	—
74	C₂²×C₃⋊C₈	D₉	1	1	yes	characteristic
75	#818	—	9	4	no	—
76	#807	—	9	4	no	—
77	C₂³×C₃²	D₁₂	1	1	yes	characteristic
78	C₂×C₃²×C₄	Dic₆	1	2	yes	—
79	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
81	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	—	6	2	no	—
83	C₂×C₃⋊Dic₃	—	12	2	no	—
84	C₂×C₃×Dic₃	—	6	2	no	—
85	C₂×C₃×Dic₃	—	12	2	no	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	#67	1	1	yes	characteristic
88	C₃×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	Dic₆	1	2	yes	—
94	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
95	C₂×C₄×C₉	#67	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
97	C₂×Dic₉	—	2	2	no	—
98	C₂×Dic₉	—	4	2	no	—
99	C₂×Dic₉	—	6	4	no	—
100	C₂×Dic₉	—	6	2	no	—
101	C₂×Dic₉	—	12	4	no	—
102	C₂×Dic₉	—	12	2	no	—
103	C₈×C₉	—	3	4	no	—
104	C₂×#107	—	27	1	no	Sylow
105	C₂×C₃×C₉	#107	1	4	yes	complemented
106	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	18	1	no	—
111	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
112	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
113	C₂²×Dic₃	—	6	1	no	—
114	C₂²×Dic₃	—	18	2	no	—
115	C₂²×Dic₃	—	18	1	no	—
116	C₂²×Dic₃	—	18	1	no	—
117	C₂×C₃×C₈	—	3	4	no	—
118	C₂×C₃×C₈	—	3	2	no	—
119	C₃×C₄⋊C₄	—	9	2	no	—
120	C₃×C₄⋊C₄	—	18	1	no	—
121	C₄⋊Dic₃	—	3	2	no	—
122	C₄⋊Dic₃	—	6	1	no	—
123	C₄⋊Dic₃	—	9	2	no	—
124	C₄⋊Dic₃	—	9	2	no	—
125	C₄⋊Dic₃	—	18	2	no	—
126	C₄⋊Dic₃	—	18	2	no	—
127	C₄⋊Dic₃	—	18	1	no	—
128	C₄⋊Dic₃	—	18	1	no	—
129	C₂×C₃⋊C₈	Dic₉	1	4	yes	complemented
130	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
131	C₂²×C₃²	D₂₄	1	2	yes	—
132	C₂²×C₃²	Dic₁₂	1	2	yes	—
133	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
134	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
135	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
136	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
137	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
138	C₃⋊Dic₃	—	12	2	no	—
139	C₃×Dic₃	—	12	2	no	—
140	C₂²×C₉	—	2	2	no	—
141	C₂²×C₉	—	2	2	no	—
142	C₂²×C₉	—	2	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	Dic₃⋊C₄	1	2	yes	—
145	C₂²×C₉	#213	1	2	yes	—
146	C₂²×C₉	#216	1	2	yes	—
147	C₂²×C₉	C₂×#67	1	1	yes	characteristic
148	C₄×C₉	—	2	2	no	—
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	Dic₃⋊C₄	1	2	yes	—
152	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
153	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
154	Dic₉	—	4	2	no	—
155	Dic₉	—	12	4	no	—
156	Dic₉	—	12	2	no	—
157	C₂²×C₈	—	3	1	no	—
158	C₂×C₄⋊C₄	—	9	1	no	—
159	C₂×C₄⋊C₄	—	27	1	no	—
160	#107	—	27	4	no	—
161	C₃×C₉	C₂×#107	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₃	—	2	1	no	—
163	C₂³×C₃	D₃₆	1	1	yes	characteristic
164	C₂³×C₃	#626	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	18	2	no	—
170	C₂×C₃×C₄	—	36	2	no	—
171	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
172	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
174	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
175	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
176	C₂×C₃×C₄	#626	1	1	yes	characteristic
177	C₂×C₃×C₄	#627	1	2	yes	—
178	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
179	C₂×Dic₃	—	6	2	no	—
180	C₂×Dic₃	—	12	2	no	—
181	C₂×Dic₃	—	18	4	no	—
182	C₂×Dic₃	—	18	2	no	—
183	C₂×Dic₃	—	18	2	no	—
184	C₂×Dic₃	—	36	4	no	—
185	C₂×Dic₃	—	36	2	no	—
186	C₂×Dic₃	—	36	2	no	—
187	C₃×C₈	—	3	4	no	—
188	C₃⋊C₈	C₂×Dic₉	1	4	yes	complemented
189	C₂×C₃²	#818	1	4	yes	—
190	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
191	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
192	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	—	2	4	no	—
195	C₂×C₉	—	2	1	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	#807	1	4	yes	complemented
198	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
199	C₂×C₉	C₂×#213	1	1	yes	characteristic
200	C₂×C₉	C₂×#216	1	1	yes	characteristic
201	C₂²×C₄	—	18	1	no	—
202	C₂²×C₄	—	54	1	no	—
203	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
204	C₂×C₈	—	3	4	no	—
205	C₂×C₈	—	3	2	no	—
206	C₄⋊C₄	—	9	2	no	—
207	C₄⋊C₄	—	18	1	no	—
208	C₄⋊C₄	—	27	2	no	—
209	C₄⋊C₄	—	54	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	Dic₃₆	1	2	yes	—
215	C₂²×C₃	D₇₂	1	2	yes	—
216	C₂²×C₃	C₄⋊Dic₉	1	2	yes	—
217	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
218	C₂²×C₃	#3714	1	2	yes	—
219	C₂²×C₃	#3719	1	2	yes	—
220	C₂²×C₃	Dic₃⋊Dic₃	1	2	yes	—
221	C₂²×C₃	C₂×#626	1	1	yes	characteristic
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	36	2	no	—
226	C₃×C₄	C₄⋊Dic₉	1	2	yes	—
227	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
228	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
229	C₃×C₄	Dic₃⋊Dic₃	1	2	yes	—
230	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
231	C₃×C₄	C₂×#627	1	1	yes	characteristic
232	Dic₃	—	12	2	no	—
233	Dic₃	—	36	4	no	—
234	Dic₃	—	36	2	no	—
235	Dic₃	—	36	2	no	—
236	C₃²	C₂×#818	1	1	yes	characteristic
237	C₉	—	2	1	no	—
238	C₉	C₂×#807	1	1	yes	characteristic, complemented
239	C₂³	#6192	1	1	yes	characteristic
240	C₂×C₄	—	18	2	no	—
241	C₂×C₄	—	36	2	no	—
242	C₂×C₄	—	54	2	no	—
243	C₂×C₄	—	108	2	no	—
244	C₂×C₄	#6189	1	2	yes	—
245	C₂×C₄	#6192	1	1	yes	characteristic
246	C₂×C₄	D₃×Dic₉	1	2	yes	—
247	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
248	C₈	—	3	4	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	4	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	#63378	1	4	yes	complemented
254	C₂×C₃	C₂×Dic₃₆	1	1	yes	characteristic
255	C₂×C₃	C₂×D₇₂	1	1	yes	characteristic
256	C₂×C₃	C₂×C₄⋊Dic₉	1	1	yes	characteristic
257	C₂×C₃	#63575	1	4	yes	—
258	C₂×C₃	C₂×#3714	1	1	yes	characteristic
259	C₂×C₃	C₂×#3719	1	1	yes	characteristic
260	C₂×C₃	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
261	C₂²	#88417	1	2	yes	—
262	C₂²	#88418	1	2	yes	—
263	C₂²	Dic₃⋊Dic₉	1	2	yes	—
264	C₂²	C₂×#6192	1	1	yes	characteristic
265	C₄	—	36	2	no	—
266	C₄	—	108	2	no	—
267	C₄	Dic₃⋊Dic₉	1	2	yes	—
268	C₄	C₂×#6189	1	1	yes	characteristic
269	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
270	C₃	—	2	1	no	—
271	C₃	C₂×#63378	1	1	yes	characteristic, complemented
272	C₃	C₂×#63575	1	1	yes	characteristic
273	C₂	#11720279	1	4	yes	direct product factor
274	C₂	C₂×#88417	1	1	yes	characteristic
275	C₂	C₂×#88418	1	1	yes	characteristic
276	C₂	C₂×Dic₃⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6192
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	D₁₂
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₂×#107
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720279	C₂	direct (N × H)
14. C₂×C₉×C₃⋊C₈	C₄	semidirect
34. C₉×C₃⋊C₈	C₂×C₄	semidirect
105. C₂×C₃×C₉	#107	semidirect
129. C₂×C₃⋊C₈	Dic₉	semidirect
161. C₃×C₉	C₂×#107	semidirect
188. C₃⋊C₈	C₂×Dic₉	semidirect
197. C₂×C₉	#807	semidirect
238. C₉	C₂×#807	semidirect
253. C₂×C₃	#63378	semidirect
271. C₃	C₂×#63378	semidirect
273. C₂	#11720279	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×C₄⋊(C₃⋊Dic₉)	2
C₂×C₃×C₄⋊Dic₉	2
C₂²×C₉×C₃⋊C₈	2
#11720279	2
C₂×#63575	3
C₂×#63378	3

Subgroups — C2×#11720279 #13139379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720279 #13139379