Subgroups — C₂×C₄×Dic₃₆ #11859930

Order 1152.

732 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₈	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#63378	C₂	1	1	yes	maximal, characteristic
5	C₂×#63376	C₂	1	2	yes	maximal
6	C₄×Dic₃₆	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄×Dic₁₂	—	3	1	no	maximal
8	C₂²×Dic₁₈	C₂²	1	2	yes	—
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₂×Dic₉⋊C₄	—	2	2	no	—
14	C₂×C₄×Dic₉	—	2	2	no	—
15	C₂×Dic₃₆	C₄	1	8	yes	direct product factor
16	C₂×Dic₃₆	C₂²	1	4	yes	—
17	C₄×Dic₁₈	—	2	4	no	—
18	C₄×Dic₁₈	C₂²	1	8	yes	—
19	C₄×C₈×C₉	C₂²	1	4	yes	—
20	#63378	C₂²	1	4	yes	—
21	#63376	C₂²	1	8	yes	—
22	C₂²×Dic₁₂	—	3	1	no	—
23	C₂×C₄×Dic₆	—	3	2	no	—
24	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
25	C₂×#818	—	3	1	no	—
26	C₂×#816	—	3	2	no	—
27	C₄×Dic₁₂	—	3	8	no	—
28	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₉	—	2	2	no	—
32	C₂²×Dic₉	—	2	2	no	—
33	C₂×Dic₁₈	—	2	2	no	—
34	C₂×Dic₁₈	—	2	4	no	—
35	C₂×Dic₁₈	C₂×C₄	1	8	yes	—
36	C₂×Dic₁₈	C₂³	1	4	yes	—
37	C₂×C₈×C₉	—	2	2	no	—
38	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₉	C₂³	1	2	yes	—
40	C₂×C₈×C₉	C₂³	1	2	yes	—
41	C₄²×C₉	D₄	1	2	yes	—
42	C₄²×C₉	C₂³	1	2	yes	—
43	C₄⋊Dic₉	—	2	2	no	—
44	C₄⋊Dic₉	C₂³	1	4	yes	—
45	Dic₉⋊C₄	—	2	8	no	—
46	C₄×Dic₉	—	2	8	no	—
47	Dic₃₆	C₂×C₄	1	16	yes	direct product factor
48	C₂×C₄×Q₁₆	—	9	1	no	Sylow
49	C₂²×Dic₆	—	3	2	no	—
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₂×C₄⋊Dic₃	—	3	2	no	—
54	C₂×Dic₃⋊C₄	—	6	2	no	—
55	C₂×C₄×Dic₃	—	6	2	no	—
56	C₂×Dic₁₂	—	3	8	no	—
57	C₂×Dic₁₂	—	3	4	no	—
58	C₄×Dic₆	—	3	8	no	—
59	C₄×Dic₆	—	6	4	no	—
60	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
61	#818	—	3	4	no	—
62	#816	—	3	8	no	—
63	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	Q₁₆	1	4	yes	complemented
66	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₉	#41	1	2	yes	—
71	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₉	—	2	8	no	—
73	C₂×Dic₉	—	2	4	no	—
74	C₂×Dic₉	—	2	4	no	—
75	C₂×Dic₉	—	4	4	no	—
76	Dic₁₈	—	2	8	no	—
77	Dic₁₈	—	2	4	no	—
78	Dic₁₈	C₂²×C₄	1	8	yes	—
79	C₈×C₉	—	2	2	no	—
80	C₈×C₉	C₂²×C₄	1	4	yes	—
81	C₂²×Q₁₆	—	9	1	no	—
82	C₂×C₄×Q₈	—	9	2	no	—
83	C₄×Q₁₆	—	9	8	no	—
84	C₂×#107	—	9	1	no	—
85	C₂×#103	—	9	2	no	—
86	C₂×C₄×C₈	D₉	1	1	yes	characteristic
87	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
88	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
90	C₂²×Dic₃	—	6	2	no	—
91	C₂²×Dic₃	—	6	2	no	—
92	C₂×Dic₆	—	3	8	no	—
93	C₂×Dic₆	—	3	4	no	—
94	C₂×Dic₆	—	6	2	no	—
95	C₂×Dic₆	—	6	4	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
98	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
99	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
100	C₃×C₄²	D₁₂	1	2	yes	—
101	C₃×C₄²	C₂²×D₃	1	2	yes	—
102	C₄⋊Dic₃	—	3	4	no	—
103	C₄⋊Dic₃	—	6	2	no	—
104	Dic₃⋊C₄	—	6	8	no	—
105	C₄×Dic₃	—	6	8	no	—
106	Dic₁₂	—	3	16	no	—
107	C₂²×C₉	C₄×D₄	1	2	yes	—
108	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
109	C₂²×C₉	#135	1	2	yes	—
110	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₉	—	2	2	no	—
112	C₄×C₉	C₄×D₄	1	2	yes	—
113	C₄×C₉	C₂×Q₁₆	1	4	yes	complemented
114	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
115	C₄×C₉	C₂×#41	1	1	yes	characteristic
116	Dic₉	—	2	8	no	—
117	Dic₉	—	4	4	no	—
118	C₂²×Q₈	—	9	2	no	—
119	C₂×Q₁₆	—	9	8	no	—
120	C₂×Q₁₆	—	9	4	no	—
121	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
122	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
123	C₄×Q₈	—	9	8	no	—
124	C₄×Q₈	—	18	4	no	—
125	C₂×C₄⋊C₄	—	9	2	no	—
126	C₂×C₄⋊C₄	—	18	2	no	—
127	C₂×C₄²	—	18	2	no	—
128	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
129	#107	—	9	4	no	—
130	#103	—	9	8	no	—
131	C₄×C₈	C₂×D₉	1	4	yes	—
132	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	Dic₁₂	1	4	yes	—
135	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
136	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
137	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
139	C₂×C₃×C₄	#235	1	2	yes	—
140	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
141	C₂×Dic₃	—	6	8	no	—
142	C₂×Dic₃	—	6	4	no	—
143	C₂×Dic₃	—	6	4	no	—
144	C₂×Dic₃	—	12	4	no	—
145	Dic₆	—	3	8	no	—
146	Dic₆	—	6	8	no	—
147	Dic₆	—	6	4	no	—
148	C₃×C₈	—	2	2	no	—
149	C₃×C₈	C₂×C₄×D₃	1	4	yes	—
150	C₂×C₉	C₄×Q₁₆	1	4	yes	complemented
151	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
153	C₂×C₉	C₂×#135	1	1	yes	characteristic
154	C₂×Q₈	—	9	8	no	—
155	C₂×Q₈	—	9	4	no	—
156	C₂×Q₈	—	18	2	no	—
157	C₂×Q₈	—	18	4	no	—
158	C₂²×C₄	—	18	2	no	—
159	C₂²×C₄	—	18	2	no	—
160	C₂²×C₄	D₃₆	1	1	yes	characteristic
161	C₂²×C₄	D₃₆	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
163	Q₁₆	—	9	16	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	C₄×D₉	1	4	yes	—
166	C₂×C₈	C₂²×D₉	1	2	yes	—
167	C₂×C₈	C₂²×D₉	1	2	yes	—
168	C₄⋊C₄	—	9	4	no	—
169	C₄⋊C₄	—	18	2	no	—
170	C₄⋊C₄	—	18	8	no	—
171	C₄²	—	18	8	no	—
172	C₄²	D₃₆	1	2	yes	—
173	C₄²	C₂²×D₉	1	2	yes	—
174	C₂²×C₃	C₄×D₁₂	1	2	yes	—
175	C₂²×C₃	#904	1	2	yes	—
176	C₂²×C₃	C₂×Dic₁₂	1	2	yes	—
177	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	C₄×D₁₂	1	2	yes	—
180	C₃×C₄	C₂×Dic₁₂	1	4	yes	—
181	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
182	C₃×C₄	C₂×#235	1	1	yes	characteristic
183	Dic₃	—	6	8	no	—
184	Dic₃	—	12	4	no	—
185	C₉	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
186	C₂³	C₂×D₃₆	1	1	yes	characteristic
187	Q₈	—	9	8	no	—
188	Q₈	—	18	8	no	—
189	Q₈	—	18	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	18	8	no	—
192	C₂×C₄	—	18	4	no	—
193	C₂×C₄	—	18	4	no	—
194	C₂×C₄	—	36	4	no	—
195	C₂×C₄	Dic₃₆	1	4	yes	direct product factor
196	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
197	C₂×C₄	C₂×D₃₆	1	2	yes	—
198	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
199	C₂×C₄	C₂×D₃₆	1	2	yes	—
200	C₂×C₄	#3697	1	2	yes	—
201	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	C₂×C₄×D₉	1	4	yes	—
204	C₂×C₃	C₄×Dic₁₂	1	4	yes	—
205	C₂×C₃	C₂×C₄×D₁₂	1	1	yes	characteristic
206	C₂×C₃	C₂×#904	1	1	yes	characteristic
207	C₂×C₃	C₂²×Dic₁₂	1	1	yes	characteristic
208	C₂²	C₄×D₃₆	1	2	yes	—
209	C₂²	C₂×Dic₃₆	1	2	yes	—
210	C₂²	#63467	1	2	yes	—
211	C₂²	C₂²×D₃₆	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	18	8	no	—
214	C₄	—	36	4	no	—
215	C₄	C₄×D₃₆	1	2	yes	—
216	C₄	C₂×Dic₃₆	1	4	yes	direct product factor
217	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
218	C₄	C₂×#3697	1	1	yes	characteristic
219	C₃	C₂×C₄×Dic₁₂	1	1	yes	characteristic
220	C₂	C₄×Dic₃₆	1	4	yes	direct product factor
221	C₂	C₂×C₄×D₃₆	1	1	yes	characteristic
222	C₂	C₂²×Dic₃₆	1	1	yes	characteristic
223	C₂	C₂×#63467	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₃₆
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₄×C₈×C₉	C₂
socle, prime radical	C₂³×C₃	C₂×D₁₂
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₉
hypocenter	C₉	C₂×C₄×Q₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₄×Dic₃₆	C₂	direct (N × H)
15. C₂×Dic₃₆	C₄	direct (N × H)
47. Dic₃₆	C₂×C₄	direct (N × H)
65. C₂×C₄×C₉	Q₁₆	semidirect
113. C₄×C₉	C₂×Q₁₆	semidirect
150. C₂×C₉	C₄×Q₁₆	semidirect
185. C₉	C₂×C₄×Q₁₆	semidirect
195. C₂×C₄	Dic₃₆	direct (N × H)
216. C₄	C₂×Dic₃₆	direct (N × H)
220. C₂	C₄×Dic₃₆	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×Dic₃₆	2
C₂×C₄×Dic₁₈	2
C₂×C₄×C₈×C₉	2
C₂×#63378	2
C₂×#63376	2
C₄×Dic₃₆	2
C₂×C₄×Dic₁₂	3

Subgroups — C2×C4×Dic36 #11859930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₃₆ #11859930