Subgroups — Q₈×C₉×D₁₂ #13152696

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

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₂×C₃×D₁₂	C₂²×C₃
hypercenter	C₄×Q₈×C₉	D₃
hypocenter	C₃	D₄×Q₈×C₉
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄×Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×Q₈×C₉	C₂	semidirect
2. C₃×C₄×Q₈×C₉	C₂	semidirect
3. C₉×#897	C₂	semidirect
5. C₉×#869	C₂	semidirect
27. C₄×Q₈×C₉	D₃	semidirect
30. C₃×Q₈×C₉	D₄	semidirect
35. C₉×D₁₂	Q₈	direct (N × H)
39. Q₈×D₁₂	C₉	direct (N × H)
66. C₂×D₃×C₉	C₂×Q₈	semidirect
69. C₃×C₄×C₉	C₂×Q₈	semidirect
72. C₂×D₃×Q₈	C₂×C₉	semidirect
76. C₃×C₄×Q₈	C₂×C₉	semidirect
79. #897	C₂×C₉	semidirect
81. #869	C₂×C₉	semidirect
94. Q₈×C₉	D₁₂	direct (N × H)
151. C₄×C₉	D₃×Q₈	semidirect
155. C₄×Q₈	D₃×C₉	semidirect
157. C₃×C₉	D₄×Q₈	semidirect
162. C₃×Q₈	D₄×C₉	semidirect
177. D₁₂	Q₈×C₉	direct (N × H)
209. C₂×D₃	C₂×Q₈×C₉	semidirect
217. C₃×C₄	C₂×Q₈×C₉	semidirect
223. C₉	Q₈×D₁₂	direct (N × H)
226. Q₈	C₉×D₁₂	direct (N × H)
251. C₄	D₃×Q₈×C₉	semidirect
253. C₃	D₄×Q₈×C₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₃×Q₈×C₉	2
C₃×C₄×Q₈×C₉	2
C₉×#897	2
C₄×C₉×D₁₂	2
C₉×#869	2
C₃×Q₈×D₁₂	3
D₄×Q₈×C₉	3

Subgroups — Q8×C9×D12 #13152696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₉×D₁₂ #13152696