Subgroups — 3₋¹⁺²×#382 #13144906

Order 1728.

616 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×3₋¹⁺²	C₂	1	2	yes	maximal
2	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#95	C₂	1	4	yes	maximal
4	C₃²×#382	C₃	1	1	yes	maximal, characteristic
5	C₉×#382	C₃	1	3	yes	maximal, complemented
6	C₂²×C₄×3₋¹⁺²	C₂²	1	4	yes	—
7	C₂²×C₄×3₋¹⁺²	C₂²	1	2	yes	—
8	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×3₋¹⁺²	—	2	4	no	—
10	C₄²×3₋¹⁺²	C₄	1	4	yes	complemented
11	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	2	yes	—
12	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
13	C₃²×#95	C₂×C₃	1	4	yes	—
14	C₂×C₉×C₄⋊C₄	C₂×C₃	1	6	yes	—
15	C₂×C₄²×C₉	C₂×C₃	1	3	yes	—
16	C₉×#95	C₂×C₃	1	12	yes	—
17	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
18	C₂×C₄×3₋¹⁺²	—	2	4	no	—
19	C₂×C₄×3₋¹⁺²	—	2	4	no	—
20	C₂×C₄×3₋¹⁺²	—	2	8	no	—
21	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
22	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
23	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
24	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
25	C₃×#382	—	3	1	no	—
26	C₃×#382	C₃²	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
28	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
29	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₃²×C₄⋊C₄	—	2	4	no	—
31	C₃²×C₄²	C₃×C₄	1	4	yes	—
32	C₂²×C₄×C₉	C₂²×C₃	1	12	yes	—
33	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
34	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
35	C₉×C₄⋊C₄	—	2	12	no	—
36	C₄²×C₉	C₃×C₄	1	12	yes	complemented
37	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
38	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
39	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
40	C₂²×3₋¹⁺²	#41	1	4	yes	—
41	C₄×3₋¹⁺²	—	2	4	no	—
42	C₄×3₋¹⁺²	—	4	4	no	—
43	C₄×3₋¹⁺²	C₄⋊C₄	1	4	yes	—
44	C₂×C₃×C₄⋊C₄	—	3	2	no	—
45	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	2	yes	—
46	C₂×C₃×C₄²	—	3	1	no	—
47	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
48	C₃×#95	—	3	4	no	—
49	C₃×#95	C₂×C₃²	1	4	yes	—
50	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
51	C₂×C₃²×C₄	—	2	4	no	—
52	C₂×C₃²×C₄	—	2	4	no	—
53	C₂×C₃²×C₄	—	2	8	no	—
54	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
55	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
56	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
57	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
58	C₂³×C₉	C₂³×C₃	1	3	yes	—
59	C₂×C₄×C₉	—	2	12	no	—
60	C₂×C₄×C₉	—	2	12	no	—
61	C₂×C₄×C₉	—	2	24	no	—
62	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
63	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
64	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
65	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
66	#382	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
67	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×3₋¹⁺²	#117	1	2	yes	—
69	C₂×3₋¹⁺²	#124	1	2	yes	—
70	C₂×3₋¹⁺²	#125	1	2	yes	—
71	C₂²×C₃×C₄	—	3	1	no	—
72	C₂²×C₃×C₄	—	3	2	no	—
73	C₂²×C₃×C₄	—	3	4	no	—
74	C₂²×C₃×C₄	C₂²×C₃²	1	4	yes	—
75	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
76	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
77	C₃×C₄⋊C₄	—	2	4	no	—
78	C₃×C₄⋊C₄	—	6	4	no	—
79	C₃×C₄²	—	3	4	no	—
80	C₃×C₄²	C₃²×C₄	1	4	yes	—
81	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
82	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
83	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
84	C₂²×C₃²	C₃×#41	1	4	yes	—
85	C₃²×C₄	—	2	4	no	—
86	C₃²×C₄	—	4	4	no	—
87	C₃²×C₄	C₃×C₄⋊C₄	1	4	yes	—
88	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
89	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
90	C₂²×C₉	C₂×C₃×Q₈	1	3	yes	—
91	C₂²×C₉	C₃×#41	1	12	yes	—
92	C₄×C₉	—	2	12	no	—
93	C₄×C₉	—	4	12	no	—
94	C₄×C₉	C₃×C₄⋊C₄	1	12	yes	—
95	C₂×C₄⋊C₄	C₂×3₋¹⁺²	1	2	yes	—
96	C₂×C₄²	C₂×3₋¹⁺²	1	1	yes	characteristic
97	#95	C₂×3₋¹⁺²	1	4	yes	—
98	3₋¹⁺²	#382	1	1	yes	characteristic, direct product factor, Sylow
99	C₂³×C₃	—	3	1	no	—
100	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
101	C₂×C₃×C₄	—	2	4	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	8	no	—
104	C₂×C₃×C₄	—	3	2	no	—
105	C₂×C₃×C₄	—	3	4	no	—
106	C₂×C₃×C₄	—	3	2	no	—
107	C₂×C₃×C₄	—	3	2	no	—
108	C₂×C₃×C₄	—	6	4	no	—
109	C₂×C₃×C₄	—	6	4	no	—
110	C₂×C₃×C₄	—	6	8	no	—
111	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
112	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
113	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
114	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
115	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₃²	C₃×#117	1	2	yes	—
117	C₂×C₃²	C₃×#124	1	2	yes	—
118	C₂×C₃²	C₃×#125	1	2	yes	—
119	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
120	C₂×C₉	C₃×#117	1	6	yes	—
121	C₂×C₉	C₃×#124	1	6	yes	—
122	C₂×C₉	C₃×#125	1	6	yes	—
123	C₂²×C₄	C₂²×3₋¹⁺²	1	4	yes	—
124	C₂²×C₄	C₂²×3₋¹⁺²	1	2	yes	—
125	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
126	C₄⋊C₄	—	2	4	no	—
127	C₄²	C₄×3₋¹⁺²	1	4	yes	complemented
128	C₂²×C₃	—	3	1	no	—
129	C₂²×C₃	—	3	1	no	—
130	C₂²×C₃	—	3	1	no	—
131	C₂²×C₃	—	3	4	no	—
132	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
133	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
134	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
135	C₂²×C₃	C₃²×#41	1	4	yes	—
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	—	3	4	no	—
138	C₃×C₄	—	4	4	no	—
139	C₃×C₄	—	6	4	no	—
140	C₃×C₄	—	12	4	no	—
141	C₃×C₄	C₃²×C₄⋊C₄	1	4	yes	—
142	C₃²	C₃×#382	1	1	yes	characteristic
143	C₉	C₃×#382	1	3	yes	complemented
144	C₂³	C₂³×3₋¹⁺²	1	1	yes	characteristic
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	C₂×C₄×3₋¹⁺²	1	4	yes	—
149	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
150	C₂×C₄	D₄×3₋¹⁺²	1	2	yes	—
151	C₂×C₄	Q₈×3₋¹⁺²	1	2	yes	—
152	C₂×C₃	—	3	1	no	—
153	C₂×C₃	—	3	2	no	—
154	C₂×C₃	—	3	2	no	—
155	C₂×C₃	—	3	2	no	—
156	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₃	C₃²×#117	1	2	yes	—
158	C₂×C₃	C₃²×#124	1	2	yes	—
159	C₂×C₃	C₃²×#125	1	2	yes	—
160	C₂²	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
161	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
162	C₂²	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
163	C₂²	#41×3₋¹⁺²	1	4	yes	—
164	C₄	—	2	4	no	—
165	C₄	—	4	4	no	—
166	C₄	C₄⋊C₄×3₋¹⁺²	1	4	yes	—
167	C₃	—	3	1	no	—
168	C₃	C₃²×#382	1	1	yes	characteristic
169	C₂	C₂×C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
170	C₂	3₋¹⁺²×#117	1	2	yes	—
171	C₂	3₋¹⁺²×#124	1	2	yes	—
172	C₂	3₋¹⁺²×#125	1	2	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³×C₃	C₂³×C₃²
commutator	C₂²×C₃	C₂²×C₃²×C₄
prime radical	C₂³×C₃²	C₂³×C₃
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₉×#382	C₃	semidirect
10. C₄²×3₋¹⁺²	C₄	semidirect
36. C₄²×C₉	C₃×C₄	semidirect
66. #382	3₋¹⁺²	direct (N × H)
98. 3₋¹⁺²	#382	direct (N × H)
127. C₄²	C₄×3₋¹⁺²	semidirect
143. C₉	C₃×#382	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×3₋¹⁺²	2
C₂×C₄²×3₋¹⁺²	2
3₋¹⁺²×#95	2
C₃²×#382	3
C₉×#382	3

Subgroups — 3−1+2×#382 #13144906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#382 #13144906