Subgroups — C₃×D₃×Q₈² #11903714

Order 1152.

2330 conjugacy classes of subgroups.

The subgroup lattice has 233 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×C₄⋊Q₈	C₂	1	9	yes	maximal
3	C₃×#5722	C₂	1	9	yes	maximal, complemented
4	C₃×C₄×D₃×Q₈	C₂	1	6	yes	maximal
5	C₃×Q₈×Dic₆	C₂	1	6	yes	maximal, complemented
6	C₂×C₃×Q₈²	—	3	1	no	maximal
7	D₃×Q₈²	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₃×Q₈	C₂²	1	6	yes	—
9	C₂×C₃×D₃×Q₈	C₂²	1	2	yes	—
10	C₃²×C₄⋊Q₈	C₂²	1	9	yes	—
11	C₃²×C₄×Q₈	C₂²	1	6	yes	—
12	C₃×Q₈×Dic₃	C₂²	1	6	yes	—
13	C₃×#944	C₂²	1	18	yes	—
14	C₃×D₃×C₄⋊C₄	C₂²	1	18	yes	—
15	C₃×#888	C₂²	1	36	yes	—
16	C₃×#887	C₂²	1	18	yes	—
17	C₃×C₄²×D₃	C₂²	1	9	yes	—
18	C₃×#869	C₂²	1	9	yes	—
19	C₃×C₄×Dic₆	C₂²	1	18	yes	—
20	C₃×Q₈²	—	2	1	no	—
21	C₃×Q₈²	—	3	9	no	—
22	C₃×Q₈²	—	3	6	no	—
23	C₃×Q₈²	D₃	1	1	yes	characteristic, direct product factor
24	C₃×Q₈²	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×C₃×C₄⋊Q₈	—	3	9	no	—
26	C₂×C₃×C₄×Q₈	—	3	6	no	—
27	D₃×C₄⋊Q₈	C₂×C₃	1	9	yes	—
28	#5722	C₂×C₃	1	9	yes	complemented
29	C₄×D₃×Q₈	C₂×C₃	1	6	yes	—
30	Q₈×Dic₆	C₂×C₃	1	6	yes	complemented
31	C₂×C₃²×Q₈	C₂³	1	6	yes	—
32	C₂×C₃²×Q₈	C₂³	1	2	yes	—
33	C₃×D₃×Q₈	—	4	12	no	—
34	C₃×D₃×Q₈	Q₈	1	16	yes	direct product factor
35	C₂×C₃×C₄×D₃	C₂³	1	9	yes	—
36	C₂×C₃×C₄×D₃	C₂³	1	6	yes	—
37	C₂×C₃×Dic₆	C₂³	1	18	yes	—
38	C₂×C₃×Dic₆	C₂³	1	6	yes	—
39	C₃²×C₄⋊C₄	C₂³	1	18	yes	—
40	C₃²×C₄²	C₂³	1	9	yes	—
41	C₃×C₄⋊Dic₃	C₂³	1	18	yes	—
42	C₃×Dic₃⋊C₄	C₂³	1	36	yes	—
43	C₃×C₄×Dic₃	C₂³	1	9	yes	—
44	C₃×C₄×Dic₃	C₂³	1	18	yes	—
45	C₂×Q₈²	—	3	1	no	Sylow
46	C₂²×C₃×Q₈	—	3	2	no	—
47	C₂²×C₃×Q₈	—	3	6	no	—
48	C₂×D₃×Q₈	C₂²×C₃	1	6	yes	—
49	C₂×D₃×Q₈	C₂²×C₃	1	2	yes	—
50	C₃×C₄⋊Q₈	—	2	9	no	—
51	C₃×C₄⋊Q₈	—	3	18	no	—
52	C₃×C₄⋊Q₈	—	3	36	no	—
53	C₃×C₄⋊Q₈	—	3	9	no	—
54	C₃×C₄⋊Q₈	C₂×D₃	1	9	yes	—
55	C₃×C₄⋊Q₈	C₂²×C₃	1	9	yes	—
56	C₃×C₄×Q₈	—	2	6	no	—
57	C₃×C₄×Q₈	—	3	6	no	—
58	C₃×C₄×Q₈	—	3	18	no	—
59	C₃×C₄×Q₈	—	3	18	no	—
60	C₃×C₄×Q₈	C₂×D₃	1	6	yes	—
61	C₃×C₄×Q₈	C₂²×C₃	1	6	yes	—
62	C₂×C₃×C₄⋊C₄	—	3	18	no	—
63	C₂×C₃×C₄²	—	3	9	no	—
64	Q₈×Dic₃	C₂²×C₃	1	6	yes	—
65	#944	C₂²×C₃	1	18	yes	—
66	D₃×C₄⋊C₄	C₂²×C₃	1	18	yes	—
67	#888	C₂²×C₃	1	36	yes	—
68	#887	C₂²×C₃	1	18	yes	—
69	C₄²×D₃	C₂²×C₃	1	9	yes	—
70	#869	C₂²×C₃	1	9	yes	—
71	C₄×Dic₆	C₂²×C₃	1	18	yes	—
72	C₂²×C₃×D₃	C₂⁴	1	1	yes	characteristic
73	C₃²×Q₈	—	4	6	no	—
74	C₃²×Q₈	C₂×Q₈	1	8	yes	complemented
75	C₂×C₃²×C₄	C₂⁴	1	9	yes	—
76	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
77	C₂×C₃×Dic₃	C₂⁴	1	9	yes	—
78	C₂×C₃×Dic₃	C₂⁴	1	6	yes	—
79	C₃×C₄×D₃	—	2	18	no	—
80	C₃×C₄×D₃	C₂×Q₈	1	24	yes	—
81	C₃×Dic₆	—	4	18	no	—
82	C₃×Dic₆	C₂×Q₈	1	24	yes	complemented
83	Q₈²	—	3	9	no	—
84	Q₈²	—	3	6	no	—
85	Q₈²	C₃×D₃	1	1	yes	characteristic, direct product factor
86	C₂×C₄⋊Q₈	—	3	9	no	—
87	C₂×C₄×Q₈	—	3	6	no	—
88	C₂×C₃×Q₈	—	2	2	no	—
89	C₂×C₃×Q₈	—	2	6	no	—
90	C₂×C₃×Q₈	—	3	16	no	—
91	C₂×C₃×Q₈	—	3	6	no	—
92	C₂×C₃×Q₈	—	3	18	no	—
93	C₂×C₃×Q₈	—	12	12	no	—
94	C₂×C₃×Q₈	C₂²×D₃	1	6	yes	—
95	C₂×C₃×Q₈	C₂²×D₃	1	2	yes	—
96	C₂×C₃×Q₈	C₂³×C₃	1	6	yes	—
97	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
98	C₂²×C₃×C₄	—	3	6	no	—
99	C₂²×C₃×C₄	—	3	9	no	—
100	D₃×Q₈	—	4	12	no	—
101	D₃×Q₈	C₃×Q₈	1	16	yes	direct product factor
102	C₂×C₄×D₃	C₂³×C₃	1	9	yes	—
103	C₂×C₄×D₃	C₂³×C₃	1	6	yes	—
104	C₂×Dic₆	C₂³×C₃	1	18	yes	—
105	C₂×Dic₆	C₂³×C₃	1	6	yes	—
106	C₃×C₄⋊C₄	—	2	18	no	—
107	C₃×C₄⋊C₄	—	3	36	no	—
108	C₃×C₄⋊C₄	—	3	18	no	—
109	C₃×C₄⋊C₄	C₂²×D₃	1	18	yes	—
110	C₃×C₄⋊C₄	C₂³×C₃	1	18	yes	—
111	C₃×C₄²	—	2	9	no	—
112	C₃×C₄²	—	3	9	no	—
113	C₃×C₄²	—	3	18	no	—
114	C₃×C₄²	C₂²×D₃	1	9	yes	—
115	C₃×C₄²	C₂³×C₃	1	9	yes	—
116	C₄⋊Dic₃	C₂³×C₃	1	18	yes	—
117	Dic₃⋊C₄	C₂³×C₃	1	36	yes	—
118	C₄×Dic₃	C₂³×C₃	1	18	yes	—
119	C₄×Dic₃	C₂³×C₃	1	9	yes	—
120	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
121	C₂×C₃×D₃	C₂²×Q₈	1	4	yes	—
122	C₂×C₃×D₃	H(5,𝔽₂)	1	2	yes	—
123	C₃²×C₄	—	2	9	no	—
124	C₃²×C₄	C₂²×Q₈	1	12	yes	—
125	C₃×Dic₃	—	2	9	no	—
126	C₃×Dic₃	C₂²×Q₈	1	12	yes	—
127	C₂²×Q₈	—	3	2	no	—
128	C₂²×Q₈	—	3	6	no	—
129	C₄⋊Q₈	—	3	18	no	—
130	C₄⋊Q₈	—	3	36	no	—
131	C₄⋊Q₈	—	3	9	no	—
132	C₄⋊Q₈	C₂×C₃×D₃	1	9	yes	—
133	C₄×Q₈	—	3	6	no	—
134	C₄×Q₈	—	3	18	no	—
135	C₄×Q₈	—	3	18	no	—
136	C₄×Q₈	C₂×C₃×D₃	1	6	yes	—
137	C₂×C₄⋊C₄	—	3	18	no	—
138	C₂×C₄²	—	3	9	no	—
139	C₂³×C₃	—	3	1	no	—
140	C₂²×D₃	C₂⁴×C₃	1	1	yes	characteristic
141	C₃×Q₈	—	2	8	no	—
142	C₃×Q₈	—	3	24	no	—
143	C₃×Q₈	—	4	6	no	—
144	C₃×Q₈	—	4	6	no	—
145	C₃×Q₈	—	8	6	no	—
146	C₃×Q₈	—	12	18	no	—
147	C₃×Q₈	D₃×Q₈	1	8	yes	direct product factor
148	C₃×Q₈	C₂×C₃×Q₈	1	8	yes	complemented
149	C₂×C₃×C₄	—	2	6	no	—
150	C₂×C₃×C₄	—	2	9	no	—
151	C₂×C₃×C₄	—	3	6	no	—
152	C₂×C₃×C₄	—	3	24	no	—
153	C₂×C₃×C₄	—	3	9	no	—
154	C₂×C₃×C₄	—	6	18	no	—
155	C₂×C₃×C₄	C₂³×D₃	1	9	yes	—
156	C₂×C₃×C₄	C₂³×D₃	1	6	yes	—
157	C₂×C₃×C₄	C₂⁴×C₃	1	9	yes	—
158	C₂×C₃×C₄	C₂⁴×C₃	1	6	yes	—
159	C₂×Dic₃	C₂⁴×C₃	1	9	yes	—
160	C₂×Dic₃	C₂⁴×C₃	1	6	yes	—
161	C₄×D₃	—	2	18	no	—
162	C₄×D₃	C₂×C₃×Q₈	1	24	yes	—
163	Dic₆	—	4	18	no	—
164	Dic₆	C₂×C₃×Q₈	1	24	yes	complemented
165	C₂×C₃²	C₂³×Q₈	1	2	yes	—
166	C₂×C₃²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
167	C₃×D₃	Q₈²	1	4	yes	direct product factor
168	C₂×Q₈	—	3	16	no	—
169	C₂×Q₈	—	3	6	no	—
170	C₂×Q₈	—	3	18	no	—
171	C₂×Q₈	—	12	12	no	—
172	C₂×Q₈	C₂²×C₃×D₃	1	6	yes	—
173	C₂×Q₈	C₂²×C₃×D₃	1	2	yes	—
174	C₂²×C₄	—	3	6	no	—
175	C₂²×C₄	—	3	9	no	—
176	C₄⋊C₄	—	3	36	no	—
177	C₄⋊C₄	—	3	18	no	—
178	C₄⋊C₄	C₂²×C₃×D₃	1	18	yes	—
179	C₄²	—	3	9	no	—
180	C₄²	—	3	18	no	—
181	C₄²	C₂²×C₃×D₃	1	9	yes	—
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	3	4	no	—
184	C₂²×C₃	—	3	2	no	—
185	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
186	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
187	C₂×D₃	C₂²×C₃×Q₈	1	4	yes	—
188	C₂×D₃	C₃×H(5,𝔽₂)	1	2	yes	—
189	C₃×C₄	—	2	9	no	—
190	C₃×C₄	—	2	9	no	—
191	C₃×C₄	—	2	12	no	—
192	C₃×C₄	—	3	12	no	—
193	C₃×C₄	—	4	9	no	—
194	C₃×C₄	—	6	9	no	—
195	C₃×C₄	C₂×D₃×Q₈	1	12	yes	—
196	C₃×C₄	C₂²×C₃×Q₈	1	12	yes	—
197	Dic₃	—	2	9	no	—
198	Dic₃	C₂²×C₃×Q₈	1	12	yes	—
199	C₃²	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
200	C₂³	—	3	1	no	—
201	Q₈	—	3	24	no	—
202	Q₈	—	4	6	no	—
203	Q₈	—	12	18	no	—
204	Q₈	C₃×D₃×Q₈	1	8	yes	direct product factor
205	C₂×C₄	—	3	6	no	—
206	C₂×C₄	—	3	24	no	—
207	C₂×C₄	—	3	9	no	—
208	C₂×C₄	—	6	18	no	—
209	C₂×C₄	C₂³×C₃×D₃	1	9	yes	—
210	C₂×C₄	C₂³×C₃×D₃	1	6	yes	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	3	4	no	—
214	C₂×C₃	C₂²×D₃×Q₈	1	2	yes	—
215	C₂×C₃	D₃×H(5,𝔽₂)	1	1	yes	characteristic
216	C₂×C₃	C₂³×C₃×Q₈	1	2	yes	—
217	C₂×C₃	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
218	D₃	C₃×Q₈²	1	4	yes	direct product factor
219	C₂²	—	3	4	no	—
220	C₂²	—	3	2	no	—
221	C₂²	C₂⁴×C₃×D₃	1	1	yes	characteristic
222	C₄	—	2	9	no	—
223	C₄	—	3	12	no	—
224	C₄	—	6	9	no	—
225	C₄	C₂×C₃×D₃×Q₈	1	12	yes	—
226	C₃	—	2	1	no	—
227	C₃	D₃×Q₈²	1	1	yes	characteristic, direct product factor
228	C₃	C₂×C₃×Q₈²	1	1	yes	characteristic, complemented
229	C₂	—	3	4	no	—
230	C₂	C₂²×C₃×D₃×Q₈	1	2	yes	—
231	C₂	C₃×D₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂⁴×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₃
Fitting	C₃²×Q₈²	C₂
socle	C₂²×C₃²	C₂⁵
prime radical	C₂²×C₃×D₃	C₂⁴
hypercenter	C₃×Q₈²	D₃
hypocenter	C₃	C₂×C₃×Q₈²
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×Q₈²

Complements

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

N	H ≅ G/N	type
1. C₃²×Q₈²	C₂	semidirect
3. C₃×#5722	C₂	semidirect
5. C₃×Q₈×Dic₆	C₂	semidirect
7. D₃×Q₈²	C₃	direct (N × H)
23. C₃×Q₈²	D₃	direct (N × H)
24. C₃×Q₈²	C₂×C₃	semidirect
28. #5722	C₂×C₃	semidirect
30. Q₈×Dic₆	C₂×C₃	semidirect
34. C₃×D₃×Q₈	Q₈	direct (N × H)
74. C₃²×Q₈	C₂×Q₈	semidirect
82. C₃×Dic₆	C₂×Q₈	semidirect
85. Q₈²	C₃×D₃	direct (N × H)
101. D₃×Q₈	C₃×Q₈	direct (N × H)
147. C₃×Q₈	D₃×Q₈	direct (N × H)
148. C₃×Q₈	C₂×C₃×Q₈	semidirect
164. Dic₆	C₂×C₃×Q₈	semidirect
167. C₃×D₃	Q₈²	direct (N × H)
199. C₃²	C₂×Q₈²	semidirect
204. Q₈	C₃×D₃×Q₈	direct (N × H)
218. D₃	C₃×Q₈²	direct (N × H)
227. C₃	D₃×Q₈²	direct (N × H)
228. C₃	C₂×C₃×Q₈²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×Q₈²	2
C₃×D₃×C₄⋊Q₈	2
C₃×#5722	2
C₃×C₄×D₃×Q₈	2
C₃×Q₈×Dic₆	2
C₂×C₃×Q₈²	3
D₃×Q₈²	3

Subgroups — C3×D3×Q82 #11903714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×Q₈² #11903714