Subgroups — C₃²×#5078 #13162514

Order 1728.

860 conjugacy classes of subgroups.

The subgroup lattice has 224 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
2	C₃²×#869	C₂	1	1	yes	maximal, characteristic
3	C₃²×#807	C₂	1	4	yes	maximal
4	C₃²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃²×#500	—	3	1	no	maximal
6	C₃×#5078	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃³×Q₈	—	2	1	no	—
8	C₂×C₃²×Dic₆	—	2	1	no	—
9	C₃³×C₄⋊C₄	—	2	1	no	—
10	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃²×C₄⋊Dic₃	—	2	1	no	—
13	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃²×C₄⋊Q₈	—	2	4	no	—
16	C₃²×C₄⋊Q₈	—	3	1	no	—
17	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
18	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	—
19	C₃×#869	C₂×C₃	1	4	yes	—
20	C₃²×#107	—	3	4	no	—
21	C₃²×C₄×C₈	—	3	1	no	—
22	C₃×#807	C₂×C₃	1	16	yes	—
23	C₃×C₄×C₃⋊C₈	C₂×C₃	1	4	yes	—
24	C₃³×Q₈	—	4	2	no	—
25	C₂×C₃³×C₄	—	2	2	no	—
26	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃²×Dic₃	—	2	2	no	—
30	C₃²×Dic₆	—	4	2	no	—
31	C₃²×C₃⋊C₈	Q₈	1	4	yes	complemented
32	C₃×#500	—	3	4	no	—
33	#5078	C₃²	1	1	yes	characteristic, direct product factor
34	C₂×C₃²×Q₈	—	2	8	no	—
35	C₂×C₃²×Q₈	—	2	1	no	—
36	C₂×C₃²×Q₈	—	2	4	no	—
37	C₂×C₃²×Q₈	—	6	1	no	—
38	C₂×C₃×Dic₆	—	2	4	no	—
39	C₂×C₃²×C₈	—	3	2	no	—
40	C₃²×C₄⋊C₄	—	2	8	no	—
41	C₃²×C₄⋊C₄	—	2	8	no	—
42	C₃²×C₄⋊C₄	—	2	1	no	—
43	C₃²×C₄⋊C₄	—	2	4	no	—
44	C₃²×C₄⋊C₄	—	3	2	no	—
45	C₃²×C₄⋊C₄	—	6	1	no	—
46	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
47	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
48	C₃²×C₄²	—	2	4	no	—
49	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₃²×C₄²	C₂²×C₃	1	4	yes	—
51	C₃×C₄⋊Dic₃	—	2	4	no	—
52	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
53	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
54	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
55	C₃³×C₄	—	4	2	no	—
56	C₃³×C₄	D₈	1	2	yes	—
57	C₃³×C₄	Q₁₆	1	2	yes	—
58	C₃³×C₄	C₂×Q₈	1	2	yes	—
59	C₃²×Dic₃	—	4	2	no	—
60	C₃×C₄⋊Q₈	—	2	4	no	—
61	C₃×C₄⋊Q₈	—	3	4	no	—
62	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	—
63	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
64	#869	C₂×C₃²	1	1	yes	characteristic
65	C₃×#107	—	3	16	no	—
66	C₃×C₄×C₈	—	3	4	no	—
67	#807	C₂×C₃²	1	4	yes	—
68	C₄×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic
69	C₃²×Q₈	—	4	16	no	—
70	C₃²×Q₈	—	4	2	no	—
71	C₃²×Q₈	—	4	8	no	—
72	C₃²×Q₈	—	12	2	no	—
73	C₂×C₃²×C₄	—	2	16	no	—
74	C₂×C₃²×C₄	—	2	2	no	—
75	C₂×C₃²×C₄	—	2	8	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	—	2	4	no	—
78	C₂×C₃²×C₄	—	2	4	no	—
79	C₂×C₃²×C₄	—	6	2	no	—
80	C₂×C₃²×C₄	#67	1	1	yes	characteristic
81	C₂×C₃²×C₄	#67	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
83	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
86	C₂×C₃×Dic₃	—	2	8	no	—
87	C₃×Dic₆	—	4	8	no	—
88	C₃²×C₈	—	3	4	no	—
89	C₃×C₃⋊C₈	C₃×Q₈	1	16	yes	complemented
90	#500	—	3	1	no	Sylow
91	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
93	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₃×Q₈	—	2	8	no	—
95	C₂×C₃×Q₈	—	2	4	no	—
96	C₂×C₃×Q₈	—	2	1	no	—
97	C₂×C₃×Q₈	—	6	4	no	—
98	C₂×Dic₆	—	2	1	no	—
99	C₂×C₃×C₈	—	3	8	no	—
100	C₃×C₄⋊C₄	—	2	8	no	—
101	C₃×C₄⋊C₄	—	2	4	no	—
102	C₃×C₄⋊C₄	—	2	1	no	—
103	C₃×C₄⋊C₄	—	2	8	no	—
104	C₃×C₄⋊C₄	—	3	8	no	—
105	C₃×C₄⋊C₄	—	6	4	no	—
106	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
107	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
108	C₃×C₄²	—	2	4	no	—
109	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
110	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
111	C₄⋊Dic₃	—	2	1	no	—
112	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
113	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
114	C₂²×C₃²	—	2	4	no	—
115	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
116	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
117	C₃²×C₄	—	2	8	no	—
118	C₃²×C₄	—	2	8	no	—
119	C₃²×C₄	—	2	8	no	—
120	C₃²×C₄	—	4	16	no	—
121	C₃²×C₄	—	4	2	no	—
122	C₃²×C₄	—	4	8	no	—
123	C₃²×C₄	—	12	2	no	—
124	C₃²×C₄	#213	1	2	yes	—
125	C₃²×C₄	#216	1	2	yes	—
126	C₃²×C₄	C₃×D₈	1	8	yes	—
127	C₃²×C₄	C₃×Q₁₆	1	8	yes	—
128	C₃²×C₄	D₃×Q₈	1	2	yes	—
129	C₃²×C₄	C₂×C₃×Q₈	1	8	yes	—
130	C₃×Dic₃	—	4	8	no	—
131	C₄⋊Q₈	—	3	1	no	—
132	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic
133	#107	—	3	4	no	—
134	C₄×C₈	—	3	1	no	—
135	C₃³	#500	1	1	yes	characteristic, complemented, Sylow
136	C₃×Q₈	—	4	16	no	—
137	C₃×Q₈	—	4	8	no	—
138	C₃×Q₈	—	4	2	no	—
139	C₃×Q₈	—	12	8	no	—
140	C₂×C₃×C₄	—	2	16	no	—
141	C₂×C₃×C₄	—	2	8	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	6	8	no	—
147	C₂×C₃×C₄	C₃×#67	1	4	yes	—
148	C₂×C₃×C₄	C₃×#67	1	4	yes	—
149	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
152	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
153	C₂×Dic₃	—	2	2	no	—
154	Dic₆	—	4	2	no	—
155	C₃×C₈	—	3	16	no	—
156	C₃⋊C₈	C₃²×Q₈	1	4	yes	complemented
157	C₂×C₃²	—	2	4	no	—
158	C₂×C₃²	—	2	4	no	—
159	C₂×C₃²	—	2	4	no	—
160	C₂×C₃²	C₂×#213	1	1	yes	characteristic
161	C₂×C₃²	C₂×#216	1	1	yes	characteristic
162	C₂×C₃²	#944	1	1	yes	characteristic
163	C₂×C₃²	C₃×C₄⋊Q₈	1	4	yes	—
164	C₂×C₃²	C₂×C₃×D₈	1	4	yes	—
165	C₂×C₃²	C₂×C₃×Q₁₆	1	4	yes	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	6	1	no	—
168	C₂×C₈	—	3	2	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	3	2	no	—
171	C₄⋊C₄	—	6	1	no	—
172	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
173	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
174	C₂²×C₃	—	2	4	no	—
175	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
176	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
177	C₃×C₄	—	2	8	no	—
178	C₃×C₄	—	2	8	no	—
179	C₃×C₄	—	2	8	no	—
180	C₃×C₄	—	4	16	no	—
181	C₃×C₄	—	4	8	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	—	12	8	no	—
184	C₃×C₄	C₃×#213	1	8	yes	—
185	C₃×C₄	C₃×#216	1	8	yes	—
186	C₃×C₄	C₃²×D₈	1	2	yes	—
187	C₃×C₄	C₃²×Q₁₆	1	2	yes	—
188	C₃×C₄	C₃×D₃×Q₈	1	8	yes	—
189	C₃×C₄	C₂×C₃²×Q₈	1	2	yes	—
190	Dic₃	—	4	2	no	—
191	C₃²	—	2	4	no	—
192	C₃²	#5078	1	1	yes	characteristic, direct product factor
193	C₃²	C₃×#500	1	4	yes	complemented
194	Q₈	—	4	2	no	—
195	Q₈	—	12	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	6	2	no	—
198	C₂×C₄	C₃²×#67	1	1	yes	characteristic
199	C₂×C₄	C₃²×#67	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
201	C₈	—	3	4	no	—
202	C₂×C₃	—	2	4	no	—
203	C₂×C₃	—	2	4	no	—
204	C₂×C₃	—	2	4	no	—
205	C₂×C₃	C₂×C₃×#213	1	4	yes	—
206	C₂×C₃	C₂×C₃×#216	1	4	yes	—
207	C₂×C₃	C₃×#944	1	4	yes	—
208	C₂×C₃	C₃²×C₄⋊Q₈	1	1	yes	characteristic
209	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
210	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
211	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
212	C₄	—	4	2	no	—
213	C₄	—	12	2	no	—
214	C₄	C₃²×#213	1	2	yes	—
215	C₄	C₃²×#216	1	2	yes	—
216	C₄	C₃²×D₃×Q₈	1	2	yes	—
217	C₃	—	2	4	no	—
218	C₃	C₃×#5078	1	4	yes	direct product factor
219	C₃	C₃²×#500	1	1	yes	characteristic, complemented
220	C₂	C₂×C₃²×#213	1	1	yes	characteristic
221	C₂	C₂×C₃²×#216	1	1	yes	characteristic
222	C₂	C₃²×#944	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₃²×C₄⋊Q₈	D₃
hypocenter	C₃	C₃²×#500
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. C₃×#5078	C₃	direct (N × H)
31. C₃²×C₃⋊C₈	Q₈	semidirect
33. #5078	C₃²	direct (N × H)
89. C₃×C₃⋊C₈	C₃×Q₈	semidirect
135. C₃³	#500	semidirect
156. C₃⋊C₈	C₃²×Q₈	semidirect
192. C₃²	#5078	direct (N × H)
193. C₃²	C₃×#500	semidirect
218. C₃	C₃×#5078	direct (N × H)
219. C₃	C₃²×#500	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×#869	2
C₃²×#807	2
C₃²×C₄×C₃⋊C₈	2
C₃²×#500	3
C₃×#5078	3

Subgroups — C32×#5078 #13162514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5078 #13162514