Subgroups — C₂×#11720581 #13144832

Order 1728.

730 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	2	yes	maximal
3	#11720581	C₂	1	4	yes	maximal, direct product factor
4	C₂×#63663	—	3	1	no	maximal
5	C₂×#63394	—	3	3	no	maximal
6	C₂×C₃×Q₈×C₉	—	2	2	no	—
7	C₂×C₃×Q₈×C₉	—	2	2	no	—
8	C₂×C₃×Q₈×C₉	C₄	1	2	yes	—
9	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	#88532	—	2	2	no	—
13	#88532	C₂²	1	4	yes	—
14	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
15	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
16	C₂×#3748	—	3	2	no	—
17	C₂²×Q₈×C₉	D₃	1	3	yes	—
18	#63663	—	3	4	no	—
19	C₂×#3667	—	3	6	no	—
20	#63394	—	3	12	no	—
21	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
22	C₃×Q₈×C₉	—	2	4	no	—
23	C₃×Q₈×C₉	—	2	4	no	—
24	C₂×C₃×C₄×C₉	—	2	2	no	—
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₃⋊(C₉⋊C₈)	—	2	4	no	—
32	C₂×#836	—	9	3	no	—
33	C₂×#836	—	9	1	no	—
34	C₂×C₃²×Q₈	—	2	2	no	—
35	C₂×C₃²×Q₈	—	2	2	no	—
36	C₂×C₃²×Q₈	Dic₃	1	2	yes	—
37	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
38	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	#3748	—	3	4	no	—
41	#3748	—	6	2	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
43	C₂×Q₈×C₉	—	2	6	no	—
44	C₂×Q₈×C₉	—	2	6	no	—
45	C₂×Q₈×C₉	Dic₃	1	6	yes	—
46	C₂×Q₈×C₉	C₂×D₃	1	6	yes	—
47	C₂²×C₄×C₉	Dic₃	1	6	yes	—
48	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
49	#3667	—	3	12	no	—
50	#3667	—	6	6	no	—
51	C₂×C₉⋊C₈	—	6	6	no	—
52	C₂²×C₃×C₉	—	2	2	no	—
53	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₄×C₉	—	2	4	no	—
57	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
58	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
59	C₂²×C₃×Q₈	D₉	1	3	yes	—
60	C₂²×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
61	C₂×#208	—	9	6	no	—
62	C₂×#208	—	9	2	no	—
63	#836	—	9	12	no	—
64	#836	—	9	4	no	—
65	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
66	C₃²×Q₈	—	2	4	no	—
67	C₃²×Q₈	—	2	4	no	—
68	C₂×C₃²×C₄	—	2	2	no	—
69	C₂×C₃²×C₄	—	2	2	no	—
70	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
71	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	#67	1	2	yes	—
73	C₂×C₃²×C₄	#67	1	2	yes	—
74	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂³×C₉	C₂×Dic₃	1	3	yes	—
76	C₃⋊(C₃⋊C₈)	—	6	4	no	—
77	Q₈×C₉	—	2	12	no	—
78	Q₈×C₉	—	2	12	no	—
79	C₂×C₄×C₉	—	2	6	no	—
80	C₂×C₄×C₉	—	2	6	no	—
81	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
82	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
83	C₂×C₄×C₉	#67	1	6	yes	—
84	C₂×C₄×C₉	#67	1	6	yes	—
85	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
86	C₉⋊C₈	—	6	12	no	—
87	C₂×#101	—	27	1	no	Sylow
88	C₂×C₃×C₉	—	2	1	no	—
89	C₂×C₃×C₉	—	2	1	no	—
90	C₂×C₃×C₉	#101	1	2	yes	complemented
91	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃×Q₈	—	2	6	no	—
93	C₂×C₃×Q₈	—	2	6	no	—
94	C₂×C₃×Q₈	—	2	2	no	—
95	C₂×C₃×Q₈	—	2	2	no	—
96	C₂×C₃×Q₈	Dic₉	1	6	yes	—
97	C₂×C₃×Q₈	C₂×D₉	1	6	yes	—
98	C₂×C₃×Q₈	C₃⋊Dic₃	1	2	yes	—
99	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
100	C₂²×C₃×C₄	Dic₉	1	6	yes	—
101	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
102	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
103	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
104	#208	—	9	12	no	—
105	#208	—	9	4	no	—
106	#208	—	18	6	no	—
107	#208	—	18	2	no	—
108	C₂×C₃⋊C₈	—	18	6	no	—
109	C₂×C₃⋊C₈	—	18	2	no	—
110	C₂²×C₃²	—	2	2	no	—
111	C₂²×C₃²	#217	1	1	yes	characteristic
112	C₂²×C₃²	#217	1	1	yes	characteristic
113	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
114	C₃²×C₄	—	2	4	no	—
115	C₃²×C₄	#217	1	2	yes	—
116	C₃²×C₄	C₂×#67	1	2	yes	—
117	C₂²×C₉	—	2	6	no	—
118	C₂²×C₉	#217	1	3	yes	—
119	C₂²×C₉	#217	1	3	yes	—
120	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
121	C₄×C₉	—	2	12	no	—
122	C₄×C₉	#217	1	6	yes	—
123	C₄×C₉	C₂×#67	1	6	yes	—
124	C₂²×Q₈	C₃⋊D₉	1	1	yes	characteristic
125	C₂×M₁₆	—	27	2	no	—
126	#101	—	27	4	no	—
127	C₃×C₉	C₂×#101	1	1	yes	characteristic, complemented, Sylow
128	C₂³×C₃	C₂×Dic₉	1	3	yes	—
129	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
130	C₃×Q₈	—	2	12	no	—
131	C₃×Q₈	—	2	12	no	—
132	C₃×Q₈	—	2	4	no	—
133	C₃×Q₈	—	2	4	no	—
134	C₂×C₃×C₄	—	2	6	no	—
135	C₂×C₃×C₄	—	2	6	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	C₂×Dic₉	1	12	yes	—
139	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
140	C₂×C₃×C₄	#611	1	6	yes	—
141	C₂×C₃×C₄	#611	1	6	yes	—
142	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
143	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
144	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
145	C₂×C₃×C₄	#638	1	2	yes	—
146	C₂×C₃×C₄	#638	1	2	yes	—
147	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
148	C₃⋊C₈	—	18	12	no	—
149	C₃⋊C₈	—	18	4	no	—
150	C₂×C₃²	—	2	1	no	—
151	C₂×C₃²	—	2	1	no	—
152	C₂×C₃²	#836	1	2	yes	—
153	C₂×C₃²	C₂×#217	1	1	yes	characteristic
154	C₂×C₉	—	2	3	no	—
155	C₂×C₉	—	2	3	no	—
156	C₂×C₉	#836	1	6	yes	complemented
157	C₂×C₉	C₂×#217	1	3	yes	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₃⋊Dic₉	1	2	yes	—
161	C₂×Q₈	C₂×C₃⋊D₉	1	2	yes	—
162	C₂²×C₄	C₃⋊Dic₉	1	2	yes	—
163	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
164	M₁₆	—	27	4	no	—
165	M₁₆	—	54	2	no	—
166	C₂×C₈	—	54	2	no	—
167	C₂²×C₃	—	2	6	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	#3676	1	3	yes	—
170	C₂²×C₃	#3676	1	3	yes	—
171	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
172	C₂²×C₃	#3757	1	1	yes	characteristic
173	C₂²×C₃	#3757	1	1	yes	characteristic
174	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
175	C₃×C₄	—	2	12	no	—
176	C₃×C₄	—	2	4	no	—
177	C₃×C₄	#3676	1	6	yes	—
178	C₃×C₄	C₂×#611	1	6	yes	—
179	C₃×C₄	#3757	1	2	yes	—
180	C₃×C₄	C₂×#638	1	2	yes	—
181	C₃²	C₂×#836	1	1	yes	characteristic
182	C₉	C₂×#836	1	3	yes	complemented
183	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
184	Q₈	—	2	4	no	—
185	Q₈	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	C₂×C₃⋊Dic₉	1	4	yes	—
189	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
190	C₂×C₄	#6233	1	2	yes	—
191	C₂×C₄	#6233	1	2	yes	—
192	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
193	C₈	—	54	4	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	1	no	—
196	C₂×C₃	—	2	3	no	—
197	C₂×C₃	—	2	3	no	—
198	C₂×C₃	#63394	1	6	yes	complemented
199	C₂×C₃	C₂×#3676	1	3	yes	—
200	C₂×C₃	#63663	1	2	yes	—
201	C₂×C₃	C₂×#3757	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	#88551	1	1	yes	characteristic
204	C₂²	#88551	1	1	yes	characteristic
205	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
206	C₄	—	2	4	no	—
207	C₄	#88551	1	2	yes	—
208	C₄	C₂×#6233	1	2	yes	—
209	C₃	C₂×#63394	1	3	yes	complemented
210	C₃	C₂×#63663	1	1	yes	characteristic
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	#11720581	1	2	yes	direct product factor
214	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#88551
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle	C₂²×C₃²	#217
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#101
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
3. #11720581	C₂	direct (N × H)
90. C₂×C₃×C₉	#101	semidirect
127. C₃×C₉	C₂×#101	semidirect
156. C₂×C₉	#836	semidirect
182. C₉	C₂×#836	semidirect
198. C₂×C₃	#63394	semidirect
209. C₃	C₂×#63394	semidirect
213. C₂	#11720581	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×Q₈×C₉	2
C₂×#88532	2
#11720581	2
C₂×#63663	3
C₂×#63394	3

Subgroups — C2×#11720581 #13144832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720581 #13144832