Subgroups — C₃×Q₈×D₃₆ #13152681

Order 1728.

910 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₉	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63459	C₂	1	6	yes	maximal, complemented
4	C₃×C₄×D₃₆	C₂	1	3	yes	maximal
5	C₃×#63431	C₂	1	3	yes	maximal, complemented
6	C₃×Q₈×D₁₂	—	3	1	no	maximal
7	Q₈×D₃₆	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₃×Q₈×D₉	—	2	8	no	—
10	C₂×C₃×D₃₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×D₉	C₂²	1	6	yes	—
12	C₂×C₃×Dic₁₈	C₂²	1	6	yes	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	3	yes	—
14	C₃×C₄²×C₉	C₂²	1	3	yes	—
15	C₃×(C₂×D₉)⋊C₄	C₂²	1	6	yes	—
16	C₃×C₄⋊Dic₉	C₂²	1	3	yes	—
17	C₃×C₄⋊Dic₉	C₂²	1	6	yes	—
18	C₂×C₃×D₃×Q₈	—	3	2	no	—
19	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
20	C₃×#897	—	3	6	no	—
21	C₃×C₄×D₁₂	—	3	3	no	—
22	C₃×#869	—	3	3	no	—
23	C₂×Q₈×D₉	C₂×C₃	1	2	yes	complemented
24	C₄×Q₈×C₉	—	2	1	no	—
25	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
26	#63459	C₂×C₃	1	6	yes	complemented
27	C₄×D₃₆	C₂×C₃	1	3	yes	—
28	#63431	C₂×C₃	1	3	yes	complemented
29	C₂²×C₃×D₉	C₂³	1	2	yes	—
30	C₃×Q₈×C₉	D₄	1	4	yes	complemented
31	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×C₃×Dic₉	C₂³	1	6	yes	—
35	C₃×D₃₆	Q₈	1	4	yes	direct product factor
36	C₃×C₄×D₉	—	2	12	no	—
37	C₃×Dic₁₈	—	2	12	no	—
38	C₃×D₄×Q₈	—	9	1	no	—
39	Q₈×D₁₂	—	3	1	no	—
40	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
41	C₃×D₃×Q₈	—	6	8	no	—
42	C₂×C₃×D₁₂	—	3	1	no	—
43	C₂×C₃×C₄×D₃	—	3	6	no	—
44	C₂×C₃×Dic₆	—	3	6	no	—
45	C₃²×C₄⋊C₄	C₂×D₃	1	3	yes	—
46	C₃²×C₄²	C₂×D₃	1	3	yes	—
47	C₃×(C₂×D₃)⋊C₄	—	3	6	no	—
48	C₃×C₄⋊Dic₃	—	3	6	no	—
49	C₃×C₄⋊Dic₃	—	3	3	no	—
50	C₂×Q₈×C₉	—	2	1	no	—
51	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
52	Q₈×D₉	—	2	8	no	—
53	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₄×D₉	C₂²×C₃	1	6	yes	—
55	C₂×Dic₁₈	C₂²×C₃	1	6	yes	—
56	C₉×C₄⋊C₄	—	2	3	no	—
57	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
58	C₄²×C₉	—	2	3	no	—
59	C₄²×C₉	C₂²×C₃	1	3	yes	—
60	(C₂×D₉)⋊C₄	C₂²×C₃	1	6	yes	—
61	C₄⋊Dic₉	C₂²×C₃	1	3	yes	—
62	C₄⋊Dic₉	C₂²×C₃	1	6	yes	—
63	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
64	C₂×C₃×D₉	—	2	2	no	—
65	C₂×C₃×D₉	—	2	2	no	—
66	C₂×C₃×D₉	C₂×Q₈	1	4	yes	complemented
67	C₃×C₄×C₉	—	2	3	no	—
68	C₃×C₄×C₉	C₂×D₄	1	6	yes	—
69	C₃×C₄×C₉	C₂×Q₈	1	2	yes	complemented
70	C₃×Dic₉	—	2	6	no	—
71	C₂²×C₃×Q₈	—	9	2	no	—
72	C₂×D₃×Q₈	—	3	2	no	—
73	C₃×C₄⋊Q₈	—	9	3	no	—
74	C₃×C₂²⋊Q₈	—	9	6	no	—
75	C₃×C₄×Q₈	—	2	1	no	—
76	C₃×C₄×Q₈	D₉	1	1	yes	characteristic, complemented
77	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
78	C₃×C₄×D₄	—	9	3	no	—
79	#897	—	3	6	no	—
80	C₄×D₁₂	—	3	3	no	—
81	#869	—	3	3	no	—
82	C₂²×C₃×D₃	—	3	2	no	—
83	C₃²×Q₈	D₁₂	1	4	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
85	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃×Dic₃	—	3	6	no	—
88	C₃×D₁₂	—	3	4	no	—
89	C₃×C₄×D₃	—	6	12	no	—
90	C₃×Dic₆	—	6	12	no	—
91	C₂²×D₉	C₂³×C₃	1	2	yes	—
92	Q₈×C₉	—	2	4	no	—
93	Q₈×C₉	C₃×D₄	1	4	yes	complemented
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	2	3	no	—
96	C₂×C₄×C₉	—	2	3	no	—
97	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
98	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
99	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
100	C₂×Dic₉	C₂³×C₃	1	6	yes	—
101	D₃₆	C₃×Q₈	1	4	yes	direct product factor
102	C₄×D₉	—	2	12	no	—
103	Dic₁₈	—	2	12	no	—
104	D₄×Q₈	—	9	1	no	Sylow
105	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
108	C₃×D₉	—	2	4	no	—
109	C₂×C₃×Q₈	—	2	1	no	—
110	C₂×C₃×Q₈	—	9	6	no	—
111	C₂×C₃×Q₈	—	18	8	no	—
112	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
113	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
114	C₂×C₃×D₄	—	9	1	no	—
115	C₂²×C₃×C₄	—	9	6	no	—
116	D₃×Q₈	—	6	8	no	—
117	C₂×D₁₂	—	3	1	no	—
118	C₂×C₄×D₃	—	3	6	no	—
119	C₂×Dic₆	—	3	6	no	—
120	C₃×C₄⋊C₄	—	2	3	no	—
121	C₃×C₄⋊C₄	—	9	3	no	—
122	C₃×C₄⋊C₄	—	9	6	no	—
123	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
124	C₃×C₄⋊C₄	C₂×C₃×D₃	1	3	yes	—
125	C₃×C₂²⋊C₄	—	9	6	no	—
126	C₃×C₄²	—	2	3	no	—
127	C₃×C₄²	C₂×D₉	1	3	yes	—
128	C₃×C₄²	C₂×C₃×D₃	1	3	yes	—
129	(C₂×D₃)⋊C₄	—	3	6	no	—
130	C₄⋊Dic₃	—	3	3	no	—
131	C₄⋊Dic₃	—	3	6	no	—
132	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
133	C₂×C₃×D₃	—	3	4	no	—
134	C₂×C₃×D₃	—	6	2	no	—
135	C₂×C₃×D₃	—	6	2	no	—
136	C₃²×C₄	—	2	3	no	—
137	C₃²×C₄	C₂×D₁₂	1	6	yes	—
138	C₃²×C₄	D₃×Q₈	1	2	yes	—
139	C₃×Dic₃	—	6	6	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
142	C₂×D₉	—	2	2	no	—
143	C₂×D₉	—	2	2	no	—
144	C₂×D₉	C₂×C₃×Q₈	1	4	yes	complemented
145	C₄×C₉	—	2	3	no	—
146	C₄×C₉	—	2	6	no	—
147	C₄×C₉	—	2	2	no	—
148	C₄×C₉	—	4	3	no	—
149	C₄×C₉	C₂×C₃×D₄	1	6	yes	—
150	C₄×C₉	C₂×C₃×Q₈	1	2	yes	complemented
151	Dic₉	—	2	6	no	—
152	C₂²×Q₈	—	9	2	no	—
153	C₄⋊Q₈	—	9	3	no	—
154	C₂²⋊Q₈	—	9	6	no	—
155	C₄×Q₈	C₃×D₉	1	1	yes	characteristic, complemented
156	C₄×D₄	—	9	3	no	—
157	C₃×C₉	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₃	—	9	2	no	—
159	C₂²×D₃	—	3	2	no	—
160	C₃×Q₈	—	2	4	no	—
161	C₃×Q₈	—	18	12	no	—
162	C₃×Q₈	D₃₆	1	4	yes	direct product factor
163	C₃×Q₈	C₃×D₁₂	1	4	yes	—
164	C₃×D₄	—	9	4	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	3	no	—
167	C₂×C₃×C₄	—	2	3	no	—
168	C₂×C₃×C₄	—	9	6	no	—
169	C₂×C₃×C₄	—	18	12	no	—
170	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
171	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
172	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
174	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
175	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
176	C₂×Dic₃	—	3	6	no	—
177	D₁₂	—	3	4	no	—
178	C₄×D₃	—	6	12	no	—
179	Dic₆	—	6	12	no	—
180	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
181	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
182	C₂×C₃²	#1010	1	1	yes	characteristic
183	C₃×D₃	—	6	4	no	—
184	C₂×C₉	—	2	1	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	2	1	no	—
187	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
188	C₂×C₉	C₂²×C₃×Q₈	1	1	yes	characteristic
189	C₂×C₉	C₃×2₋¹⁺⁴	1	1	yes	characteristic
190	D₉	—	2	4	no	—
191	C₂×Q₈	—	9	6	no	—
192	C₂×Q₈	—	18	8	no	—
193	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
194	C₂×D₄	—	9	1	no	—
195	C₂²×C₄	—	9	6	no	—
196	C₄⋊C₄	—	9	3	no	—
197	C₄⋊C₄	—	9	6	no	—
198	C₄⋊C₄	C₂×C₃×D₉	1	3	yes	—
199	C₂²⋊C₄	—	9	6	no	—
200	C₄²	C₂×C₃×D₉	1	3	yes	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	9	4	no	—
203	C₂²×C₃	—	18	2	no	—
204	C₂²×C₃	—	18	2	no	—
205	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
206	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
207	C₂×D₃	—	3	4	no	—
208	C₂×D₃	—	6	2	no	—
209	C₂×D₃	—	6	2	no	—
210	C₃×C₄	—	2	3	no	—
211	C₃×C₄	—	2	3	no	—
212	C₃×C₄	—	2	6	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	4	3	no	—
215	C₃×C₄	—	18	6	no	—
216	C₃×C₄	C₂×D₃₆	1	6	yes	—
217	C₃×C₄	Q₈×D₉	1	2	yes	complemented
218	C₃×C₄	C₂×C₃×D₁₂	1	6	yes	—
219	C₃×C₄	C₃×D₃×Q₈	1	2	yes	—
220	Dic₃	—	6	6	no	—
221	C₃²	Q₈×D₁₂	1	1	yes	characteristic
222	C₉	—	2	1	no	—
223	C₉	C₃×D₄×Q₈	1	1	yes	characteristic, complemented
224	C₂³	—	9	2	no	—
225	Q₈	—	18	12	no	—
226	Q₈	C₃×D₃₆	1	4	yes	direct product factor
227	D₄	—	9	4	no	—
228	C₂×C₄	—	9	6	no	—
229	C₂×C₄	—	18	12	no	—
230	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
231	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
232	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	18	4	no	—
237	C₂×C₃	C₂²×D₃₆	1	1	yes	characteristic
238	C₂×C₃	C₂×Q₈×D₉	1	1	yes	characteristic
239	C₂×C₃	#63716	1	1	yes	characteristic
240	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
241	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
242	C₂×C₃	C₃×#1010	1	1	yes	characteristic
243	D₃	—	6	4	no	—
244	C₂²	—	9	4	no	—
245	C₂²	—	18	2	no	—
246	C₂²	—	18	2	no	—
247	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
248	C₄	—	2	3	no	—
249	C₄	—	18	6	no	—
250	C₄	C₂×C₃×D₃₆	1	6	yes	—
251	C₄	C₃×Q₈×D₉	1	2	yes	complemented
252	C₃	—	2	1	no	—
253	C₃	Q₈×D₃₆	1	1	yes	characteristic, direct product factor
254	C₃	C₃×Q₈×D₁₂	1	1	yes	characteristic
255	C₂	—	18	4	no	—
256	C₂	C₂²×C₃×D₃₆	1	1	yes	characteristic
257	C₂	C₂×C₃×Q₈×D₉	1	1	yes	characteristic
258	C₂	C₃×#63716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle	C₂²×C₃²	C₂³×D₃
prime radical	C₂×C₃×D₃₆	C₂²
hypercenter	C₃×C₄×Q₈	D₉
hypocenter	C₉	C₃×D₄×Q₈
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₈×D₉	C₂	semidirect
2. C₃×C₄×Q₈×C₉	C₂	semidirect
3. C₃×#63459	C₂	semidirect
5. C₃×#63431	C₂	semidirect
7. Q₈×D₃₆	C₃	direct (N × H)
23. C₂×Q₈×D₉	C₂×C₃	semidirect
25. C₄×Q₈×C₉	C₂×C₃	semidirect
26. #63459	C₂×C₃	semidirect
28. #63431	C₂×C₃	semidirect
30. C₃×Q₈×C₉	D₄	semidirect
35. C₃×D₃₆	Q₈	direct (N × H)
66. C₂×C₃×D₉	C₂×Q₈	semidirect
69. C₃×C₄×C₉	C₂×Q₈	semidirect
76. C₃×C₄×Q₈	D₉	semidirect
93. Q₈×C₉	C₃×D₄	semidirect
101. D₃₆	C₃×Q₈	direct (N × H)
144. C₂×D₉	C₂×C₃×Q₈	semidirect
150. C₄×C₉	C₂×C₃×Q₈	semidirect
155. C₄×Q₈	C₃×D₉	semidirect
157. C₃×C₉	D₄×Q₈	semidirect
162. C₃×Q₈	D₃₆	direct (N × H)
217. C₃×C₄	Q₈×D₉	semidirect
223. C₉	C₃×D₄×Q₈	semidirect
226. Q₈	C₃×D₃₆	direct (N × H)
251. C₄	C₃×Q₈×D₉	semidirect
253. C₃	Q₈×D₃₆	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×C₃×Q₈×D₉	2
C₃×C₄×Q₈×C₉	2
C₃×#63459	2
C₃×C₄×D₃₆	2
C₃×#63431	2
C₃×Q₈×D₁₂	3
Q₈×D₃₆	3

Subgroups — C3×Q8×D36 #13152681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×D₃₆ #13152681