Subgroups — C₂×#11720527 #13144313

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6226	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11720527	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63637	—	3	1	no	maximal
6	C₂×#63368	—	3	3	no	maximal
7	C₂²×C₃×C₄×C₉	—	2	1	no	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	—	2	1	no	—
10	C₂×#6226	—	2	1	no	—
11	C₂×#6226	—	2	2	no	—
12	C₂×#6226	C₄	1	4	yes	complemented
13	C₂×#6226	C₂²	1	2	yes	—
14	C₃×C₉×C₄⋊C₄	—	2	1	no	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
17	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
18	C₂²×#634	—	3	1	no	—
19	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
20	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂²×Dic₁₈	—	3	3	no	—
22	#63637	—	3	4	no	—
23	C₂×C₉×C₄⋊C₄	D₃	1	3	yes	—
24	C₂²×C₉⋊C₈	—	3	3	no	—
25	#63368	—	3	12	no	—
26	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	4	2	no	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₃⋊Dic₉	—	2	4	no	—
34	C₂×C₃⋊Dic₉	—	2	2	no	—
35	#6226	—	2	4	no	—
36	#6226	—	2	2	no	—
37	#6226	C₂×C₄	1	4	yes	complemented
38	C₃⋊(C₉⋊C₈)	—	2	2	no	—
39	C₂×#810	—	9	3	no	—
40	C₂×#810	—	9	1	no	—
41	C₂²×C₃²×C₄	—	2	1	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃⋊Dic₃	—	6	1	no	—
44	C₂×#634	—	3	4	no	—
45	C₂×#634	—	3	2	no	—
46	C₂×#634	—	6	2	no	—
47	C₂×#634	—	6	1	no	—
48	C₃²×C₄⋊C₄	—	2	1	no	—
49	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
51	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
52	C₂²×C₄×C₉	—	2	3	no	—
53	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
54	C₂²×Dic₉	—	6	3	no	—
55	C₂×Dic₁₈	—	3	12	no	—
56	C₂×Dic₁₈	—	3	6	no	—
57	C₂×Dic₁₈	—	6	6	no	—
58	C₂×Dic₁₈	—	6	3	no	—
59	C₉×C₄⋊C₄	—	2	3	no	—
60	C₉×C₄⋊C₄	C₂×D₃	1	6	yes	—
61	C₂×C₉⋊C₈	—	3	6	no	—
62	C₂×C₉⋊C₈	—	6	6	no	—
63	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
65	C₂²×C₃×C₉	Q₁₆	1	2	yes	—
66	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
67	C₃×C₄×C₉	—	4	2	no	—
68	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
69	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
70	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₃⋊Dic₉	—	2	4	no	—
72	C₂²×Dic₆	—	9	3	no	—
73	C₂²×Dic₆	—	9	1	no	—
74	C₂×C₃×C₄⋊C₄	D₉	1	3	yes	—
75	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
76	C₂²×C₃⋊C₈	—	9	3	no	—
77	C₂²×C₃⋊C₈	—	9	1	no	—
78	#810	—	9	12	no	—
79	#810	—	9	4	no	—
80	C₂³×C₃²	#67	1	1	yes	characteristic
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	4	2	no	—
83	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
84	C₂×C₃²×C₄	D₁₂	1	2	yes	—
85	C₂×C₃²×C₄	#67	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	—	6	4	no	—
88	C₂×C₃⋊Dic₃	—	6	2	no	—
89	#634	—	3	4	no	—
90	#634	—	6	4	no	—
91	#634	—	6	2	no	—
92	C₂³×C₉	#67	1	3	yes	—
93	C₃⋊(C₃⋊C₈)	—	6	2	no	—
94	C₂×C₄×C₉	—	2	6	no	—
95	C₂×C₄×C₉	—	4	6	no	—
96	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
97	C₂×C₄×C₉	D₁₂	1	6	yes	—
98	C₂×C₄×C₉	#67	1	3	yes	—
99	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
100	C₂×Dic₉	—	6	12	no	—
101	C₂×Dic₉	—	6	6	no	—
102	Dic₁₈	—	3	12	no	—
103	Dic₁₈	—	6	12	no	—
104	Dic₁₈	—	6	6	no	—
105	C₉⋊C₈	—	6	6	no	—
106	C₂×#103	—	27	1	no	Sylow
107	C₂×C₃×C₉	#103	1	4	yes	complemented
108	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	2	3	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
114	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
115	C₂²×Dic₃	—	18	3	no	—
116	C₂²×Dic₃	—	18	1	no	—
117	C₂×Dic₆	—	9	12	no	—
118	C₂×Dic₆	—	9	6	no	—
119	C₂×Dic₆	—	9	4	no	—
120	C₂×Dic₆	—	9	2	no	—
121	C₂×Dic₆	—	18	6	no	—
122	C₂×Dic₆	—	18	3	no	—
123	C₂×Dic₆	—	18	2	no	—
124	C₂×Dic₆	—	18	1	no	—
125	C₃×C₄⋊C₄	—	2	3	no	—
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	C₂×D₉	1	6	yes	—
128	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
129	C₂×C₃⋊C₈	—	9	6	no	—
130	C₂×C₃⋊C₈	—	9	2	no	—
131	C₂×C₃⋊C₈	—	18	6	no	—
132	C₂×C₃⋊C₈	—	18	2	no	—
133	C₂²×C₃²	(C₂×D₃)⋊C₄	1	2	yes	—
134	C₂²×C₃²	#214	1	2	yes	—
135	C₂²×C₃²	#216	1	2	yes	—
136	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
137	C₃²×C₄	—	4	2	no	—
138	C₃²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
139	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
140	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
141	C₃⋊Dic₃	—	6	4	no	—
142	C₂²×C₉	(C₂×D₃)⋊C₄	1	6	yes	—
143	C₂²×C₉	#214	1	6	yes	—
144	C₂²×C₉	#216	1	6	yes	—
145	C₂²×C₉	C₂×#67	1	3	yes	—
146	C₄×C₉	—	4	6	no	—
147	C₄×C₉	(C₂×D₃)⋊C₄	1	6	yes	—
148	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
149	C₄×C₉	C₂×D₁₂	1	3	yes	—
150	Dic₉	—	6	12	no	—
151	C₂²×Q₈	—	27	1	no	—
152	C₂²×C₈	—	27	1	no	—
153	C₂×C₄⋊C₄	C₃⋊D₉	1	1	yes	characteristic
154	#103	—	27	4	no	—
155	C₃×C₉	C₂×#103	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	#611	1	3	yes	—
157	C₂³×C₃	#638	1	1	yes	characteristic
158	C₂×C₃×C₄	—	2	6	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	4	6	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
163	C₂×C₃×C₄	D₃₆	1	6	yes	—
164	C₂×C₃×C₄	#611	1	3	yes	—
165	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
166	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
167	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
168	C₂×C₃×C₄	#638	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
170	C₂×Dic₃	—	18	12	no	—
171	C₂×Dic₃	—	18	6	no	—
172	C₂×Dic₃	—	18	4	no	—
173	C₂×Dic₃	—	18	2	no	—
174	Dic₆	—	9	12	no	—
175	Dic₆	—	9	4	no	—
176	Dic₆	—	18	12	no	—
177	Dic₆	—	18	6	no	—
178	Dic₆	—	18	4	no	—
179	Dic₆	—	18	2	no	—
180	C₃⋊C₈	—	18	6	no	—
181	C₃⋊C₈	—	18	2	no	—
182	C₂×C₃²	#810	1	4	yes	—
183	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂×C₃²	C₂×#214	1	1	yes	characteristic
185	C₂×C₃²	C₂×#216	1	1	yes	characteristic
186	C₂×C₉	#810	1	12	yes	complemented
187	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	3	yes	—
188	C₂×C₉	C₂×#214	1	3	yes	—
189	C₂×C₉	C₂×#216	1	3	yes	—
190	C₂×Q₈	—	27	4	no	—
191	C₂×Q₈	—	27	2	no	—
192	C₂×Q₈	—	54	2	no	—
193	C₂×Q₈	—	54	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	54	1	no	—
196	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
197	C₂×C₈	—	27	2	no	—
198	C₂×C₈	—	54	2	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂×C₃⋊D₉	1	2	yes	—
201	C₂²×C₃	(C₂×D₉)⋊C₄	1	6	yes	—
202	C₂²×C₃	#3672	1	6	yes	—
203	C₂²×C₃	#3674	1	6	yes	—
204	C₂²×C₃	C₂×#611	1	3	yes	—
205	C₂²×C₃	#3752	1	2	yes	—
206	C₂²×C₃	#3754	1	2	yes	—
207	C₂²×C₃	#3756	1	2	yes	—
208	C₂²×C₃	C₂×#638	1	1	yes	characteristic
209	C₃×C₄	—	4	6	no	—
210	C₃×C₄	—	4	2	no	—
211	C₃×C₄	(C₂×D₉)⋊C₄	1	6	yes	—
212	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
213	C₃×C₄	C₂×D₃₆	1	3	yes	—
214	C₃×C₄	#3752	1	2	yes	—
215	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
216	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
217	Dic₃	—	18	12	no	—
218	Dic₃	—	18	4	no	—
219	C₃²	C₂×#810	1	1	yes	characteristic
220	C₉	C₂×#810	1	3	yes	complemented
221	C₂³	#6233	1	1	yes	characteristic
222	Q₈	—	27	4	no	—
223	Q₈	—	54	2	no	—
224	Q₈	—	54	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	54	2	no	—
228	C₂×C₄	—	54	4	no	—
229	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
230	C₂×C₄	C₄⋊(C₃⋊D₉)	1	2	yes	—
231	C₂×C₄	#6233	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
233	C₈	—	54	2	no	—
234	C₂×C₃	#63368	1	12	yes	complemented
235	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	3	yes	—
236	C₂×C₃	C₂×#3672	1	3	yes	—
237	C₂×C₃	C₂×#3674	1	3	yes	—
238	C₂×C₃	#63637	1	4	yes	—
239	C₂×C₃	C₂×#3752	1	1	yes	characteristic
240	C₂×C₃	C₂×#3754	1	1	yes	characteristic
241	C₂×C₃	C₂×#3756	1	1	yes	characteristic
242	C₂²	#88536	1	2	yes	—
243	C₂²	#88543	1	2	yes	—
244	C₂²	#88547	1	2	yes	—
245	C₂²	C₂×#6233	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	54	4	no	—
248	C₄	#88536	1	2	yes	—
249	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
250	C₄	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
251	C₃	C₂×#63368	1	3	yes	complemented
252	C₃	C₂×#63637	1	1	yes	characteristic
253	C₂	#11720527	1	4	yes	direct product factor
254	C₂	C₂×#88536	1	1	yes	characteristic
255	C₂	C₂×#88543	1	1	yes	characteristic
256	C₂	C₂×#88547	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, prime radical	C₂³×C₃²	#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊C₄	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#103
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
4. #11720527	C₂	direct (N × H)
12. C₂×#6226	C₄	semidirect
37. #6226	C₂×C₄	semidirect
107. C₂×C₃×C₉	#103	semidirect
155. C₃×C₉	C₂×#103	semidirect
186. C₂×C₉	#810	semidirect
220. C₉	C₂×#810	semidirect
234. C₂×C₃	#63368	semidirect
251. C₃	C₂×#63368	semidirect
253. C₂	#11720527	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×#6226	2
C₂×C₃×C₉×C₄⋊C₄	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720527	2
C₂×#63637	3
C₂×#63368	3

Subgroups — C2×#11720527 #13144313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720527 #13144313