Subgroups — C₃²×#5734 #13179111

Order 1728.

1688 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#947	C₂	1	4	yes	maximal, complemented
3	C₃²×#895	C₂	1	4	yes	maximal, complemented
4	C₃²×#892	C₂	1	4	yes	maximal, complemented
5	C₃²×C₄×D₁₂	C₂	1	2	yes	maximal, complemented
6	C₃²×#555	—	3	1	no	maximal
7	C₃×#5734	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃³×Q₈	C₂²	1	2	yes	—
9	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
10	C₂×C₃²×D₁₂	C₂²	1	4	yes	—
11	C₂×C₃²×C₄×D₃	C₂²	1	4	yes	—
12	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃²×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
15	C₃²×(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
16	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₃²×C₄×Dic₃	C₂²	1	4	yes	—
18	C₃²×C₄⋊Q₈	—	2	4	no	—
19	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
20	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	complemented
21	C₃²×#124	—	3	4	no	—
22	C₃²×#121	—	3	4	no	—
23	C₃²×C₄×D₄	—	3	2	no	—
24	C₃²×#117	—	3	4	no	—
25	C₃×#947	C₂×C₃	1	16	yes	complemented
26	C₃×#895	C₂×C₃	1	16	yes	complemented
27	C₃×#892	C₂×C₃	1	16	yes	complemented
28	C₃×C₄×D₁₂	C₂×C₃	1	8	yes	complemented
29	C₂²×C₃²×D₃	C₂³	1	4	yes	—
30	C₃³×Q₈	—	4	2	no	—
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	4	yes	—
33	C₂×C₃³×C₄	C₂³	1	2	yes	—
34	C₂×C₃²×Dic₃	C₂³	1	4	yes	—
35	C₃²×D₁₂	—	2	4	no	—
36	C₃²×D₁₂	—	4	4	no	—
37	C₃²×C₄×D₃	—	2	8	no	—
38	C₃×#555	—	3	4	no	—
39	#5734	C₃²	1	1	yes	characteristic, direct product factor
40	C₂×C₃²×Q₈	—	2	8	no	—
41	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
42	C₂×C₃²×Q₈	C₂²×C₃	1	8	yes	—
43	C₂×C₃²×D₄	—	3	2	no	—
44	C₂×C₃²×D₄	—	3	4	no	—
45	C₂²×C₃²×C₄	—	3	4	no	—
46	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
47	C₂×C₃×D₁₂	C₂²×C₃	1	16	yes	—
48	C₂×C₃×C₄×D₃	C₂²×C₃	1	16	yes	—
49	C₃²×C₄⋊C₄	—	2	16	no	—
50	C₃²×C₄⋊C₄	—	3	2	no	—
51	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
52	C₃²×C₄⋊C₄	C₂²×C₃	1	16	yes	—
53	C₃²×C₂²⋊C₄	—	3	8	no	—
54	C₃²×C₂²⋊C₄	—	3	4	no	—
55	C₃²×C₄²	—	2	4	no	—
56	C₃²×C₄²	—	3	4	no	—
57	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₃²×C₄²	C₂²×C₃	1	4	yes	—
59	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	32	yes	—
60	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	16	yes	—
61	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
62	C₃×C₄×Dic₃	C₂²×C₃	1	16	yes	—
63	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
64	C₂×C₃²×D₃	—	2	4	no	—
65	C₂×C₃²×D₃	—	2	4	no	—
66	C₂×C₃²×D₃	—	2	4	no	—
67	C₃³×C₄	—	2	1	no	—
68	C₃³×C₄	—	2	4	no	—
69	C₃³×C₄	#41	1	4	yes	—
70	C₃²×Dic₃	—	2	4	no	—
71	C₃×C₄⋊Q₈	—	2	4	no	—
72	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	complemented
73	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
74	C₃×#124	—	3	16	no	—
75	C₃×#121	—	3	16	no	—
76	C₃×C₄×D₄	—	3	8	no	—
77	C₃×#117	—	3	16	no	—
78	#947	C₂×C₃²	1	4	yes	complemented
79	#895	C₂×C₃²	1	4	yes	complemented
80	#892	C₂×C₃²	1	4	yes	complemented
81	C₄×D₁₂	C₂×C₃²	1	2	yes	complemented
82	C₂³×C₃²	—	3	4	no	—
83	C₂²×C₃×D₃	C₂³×C₃	1	16	yes	—
84	C₃²×Q₈	—	4	2	no	—
85	C₃²×Q₈	—	4	8	no	—
86	C₃²×Q₈	—	4	16	no	—
87	C₃²×D₄	—	6	4	no	—
88	C₃²×D₄	—	12	4	no	—
89	C₂×C₃²×C₄	—	2	8	no	—
90	C₂×C₃²×C₄	—	2	4	no	—
91	C₂×C₃²×C₄	—	2	16	no	—
92	C₂×C₃²×C₄	—	3	4	no	—
93	C₂×C₃²×C₄	—	6	8	no	—
94	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
95	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
97	C₂×C₃²×C₄	C₂³×C₃	1	16	yes	—
98	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
99	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
100	C₂×C₃×Dic₃	C₂³×C₃	1	16	yes	—
101	C₃×D₁₂	—	2	16	no	—
102	C₃×D₁₂	—	4	16	no	—
103	C₃×C₄×D₃	—	2	32	no	—
104	#555	—	3	1	no	Sylow
105	C₂×C₃³	C₂×#41	1	2	yes	—
106	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
107	C₃²×D₃	—	4	4	no	—
108	C₂×C₃×Q₈	—	2	8	no	—
109	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
110	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
111	C₂×C₃×D₄	—	3	8	no	—
112	C₂×C₃×D₄	—	3	16	no	—
113	C₂²×C₃×C₄	—	3	16	no	—
114	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
115	C₂×D₁₂	C₂²×C₃²	1	4	yes	—
116	C₂×C₄×D₃	C₂²×C₃²	1	4	yes	—
117	C₃×C₄⋊C₄	—	2	16	no	—
118	C₃×C₄⋊C₄	—	3	8	no	—
119	C₃×C₄⋊C₄	C₂×C₃×D₃	1	16	yes	—
120	C₃×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
121	C₃×C₂²⋊C₄	—	3	32	no	—
122	C₃×C₂²⋊C₄	—	3	16	no	—
123	C₃×C₄²	—	2	4	no	—
124	C₃×C₄²	—	3	16	no	—
125	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
126	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
127	(C₂×D₃)⋊C₄	C₂²×C₃²	1	4	yes	—
128	(C₂×D₃)⋊C₄	C₂²×C₃²	1	8	yes	—
129	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
130	C₄×Dic₃	C₂²×C₃²	1	4	yes	—
131	C₂²×C₃²	—	2	4	no	—
132	C₂²×C₃²	—	6	4	no	—
133	C₂²×C₃²	—	6	4	no	—
134	C₂²×C₃²	—	6	4	no	—
135	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
136	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
137	C₂×C₃×D₃	—	2	16	no	—
138	C₂×C₃×D₃	—	2	16	no	—
139	C₂×C₃×D₃	—	2	16	no	—
140	C₃²×C₄	—	2	8	no	—
141	C₃²×C₄	—	2	4	no	—
142	C₃²×C₄	—	2	1	no	—
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	—	2	16	no	—
145	C₃²×C₄	—	2	16	no	—
146	C₃²×C₄	—	4	16	no	—
147	C₃²×C₄	—	6	4	no	—
148	C₃²×C₄	#239	1	4	yes	—
149	C₃²×C₄	C₃×#41	1	16	yes	—
150	C₃×Dic₃	—	2	16	no	—
151	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
152	#124	—	3	4	no	—
153	#121	—	3	4	no	—
154	C₄×D₄	—	3	2	no	—
155	#117	—	3	4	no	—
156	C₃³	#555	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₃	—	3	16	no	—
158	C₂²×D₃	C₂³×C₃²	1	4	yes	—
159	C₃×Q₈	—	4	8	no	—
160	C₃×Q₈	—	4	2	no	—
161	C₃×Q₈	—	4	16	no	—
162	C₃×D₄	—	6	16	no	—
163	C₃×D₄	—	12	16	no	—
164	C₂×C₃×C₄	—	2	8	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	16	no	—
167	C₂×C₃×C₄	—	3	16	no	—
168	C₂×C₃×C₄	—	6	32	no	—
169	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
170	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
171	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
172	C₂×C₃×C₄	C₂³×C₃²	1	4	yes	—
173	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
175	C₂×Dic₃	C₂³×C₃²	1	4	yes	—
176	D₁₂	—	2	4	no	—
177	D₁₂	—	4	4	no	—
178	C₄×D₃	—	2	8	no	—
179	C₂×C₃²	—	2	4	no	—
180	C₂×C₃²	—	2	8	no	—
181	C₂×C₃²	—	12	4	no	—
182	C₂×C₃²	#1004	1	1	yes	characteristic
183	C₂×C₃²	C₂×#239	1	2	yes	—
184	C₂×C₃²	C₂×C₃×#41	1	8	yes	—
185	C₂×C₃²	C₃×H(5,𝔽₂)	1	4	yes	—
186	C₃×D₃	—	4	16	no	—
187	C₂×Q₈	C₂×C₃²×D₃	1	2	yes	—
188	C₂×D₄	—	3	2	no	—
189	C₂×D₄	—	3	4	no	—
190	C₂²×C₄	—	3	4	no	—
191	C₄⋊C₄	—	3	2	no	—
192	C₄⋊C₄	C₂×C₃²×D₃	1	4	yes	—
193	C₂²⋊C₄	—	3	8	no	—
194	C₂²⋊C₄	—	3	4	no	—
195	C₄²	—	3	4	no	—
196	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	6	16	no	—
199	C₂²×C₃	—	6	16	no	—
200	C₂²×C₃	—	6	16	no	—
201	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
202	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
203	C₂×D₃	—	2	4	no	—
204	C₂×D₃	—	2	4	no	—
205	C₂×D₃	—	2	4	no	—
206	C₃×C₄	—	2	8	no	—
207	C₃×C₄	—	2	16	no	—
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	16	no	—
212	C₃×C₄	—	4	16	no	—
213	C₃×C₄	—	6	16	no	—
214	C₃×C₄	C₃×#239	1	16	yes	—
215	C₃×C₄	C₃²×#41	1	4	yes	—
216	Dic₃	—	2	4	no	—
217	C₃²	—	2	4	no	—
218	C₃²	#5734	1	1	yes	characteristic, direct product factor
219	C₃²	C₃×#555	1	4	yes	complemented
220	C₂³	—	3	4	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	6	4	no	—
223	D₄	—	12	4	no	—
224	C₂×C₄	—	3	4	no	—
225	C₂×C₄	—	6	8	no	—
226	C₂×C₄	C₂²×C₃²×D₃	1	4	yes	—
227	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	—	2	8	no	—
231	C₂×C₃	—	12	16	no	—
232	C₂×C₃	C₃×#1004	1	4	yes	—
233	C₂×C₃	C₂×C₃×#239	1	8	yes	—
234	C₂×C₃	C₂×C₃²×#41	1	2	yes	—
235	C₂×C₃	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
236	D₃	—	4	4	no	—
237	C₂²	—	6	4	no	—
238	C₂²	—	6	4	no	—
239	C₂²	—	6	4	no	—
240	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	1	no	—
243	C₄	—	6	4	no	—
244	C₄	C₃²×#239	1	4	yes	—
245	C₃	—	2	4	no	—
246	C₃	C₃×#5734	1	4	yes	direct product factor
247	C₃	C₃²×#555	1	1	yes	characteristic, complemented
248	C₂	—	12	4	no	—
249	C₂	C₃²×#1004	1	1	yes	characteristic
250	C₂	C₂×C₃²×#239	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃²×C₄⋊Q₈	D₃
hypocenter	C₃	C₃²×#555
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄⋊Q₈	C₂	semidirect
2. C₃²×#947	C₂	semidirect
3. C₃²×#895	C₂	semidirect
4. C₃²×#892	C₂	semidirect
5. C₃²×C₄×D₁₂	C₂	semidirect
7. C₃×#5734	C₃	direct (N × H)
19. C₃²×C₄⋊Q₈	D₃	semidirect
20. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
25. C₃×#947	C₂×C₃	semidirect
26. C₃×#895	C₂×C₃	semidirect
27. C₃×#892	C₂×C₃	semidirect
28. C₃×C₄×D₁₂	C₂×C₃	semidirect
39. #5734	C₃²	direct (N × H)
72. C₃×C₄⋊Q₈	C₃×D₃	semidirect
73. C₃×C₄⋊Q₈	C₂×C₃²	semidirect
78. #947	C₂×C₃²	semidirect
79. #895	C₂×C₃²	semidirect
80. #892	C₂×C₃²	semidirect
81. C₄×D₁₂	C₂×C₃²	semidirect
151. C₄⋊Q₈	C₃²×D₃	semidirect
156. C₃³	#555	semidirect
218. C₃²	#5734	direct (N × H)
219. C₃²	C₃×#555	semidirect
246. C₃	C₃×#5734	direct (N × H)
247. C₃	C₃²×#555	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₃²×#947	2
C₃²×#895	2
C₃²×#892	2
C₃²×C₄×D₁₂	2
C₃²×#555	3
C₃×#5734	3

Subgroups — C32×#5734 #13179111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5734 #13179111