Subgroups — C₃²×#1834 #11768883

Order 1152.

744 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×#384	C₂	1	2	yes	maximal
3	C₃²×#340	C₂	1	4	yes	maximal
4	C₃×#1834	C₃	1	4	yes	maximal, direct product factor
5	C₂²×C₃²×C₈	C₂²	1	2	yes	—
6	C₂×C₃²×C₄⋊C₄	—	2	2	no	—
7	C₂×C₃²×C₄⋊C₄	—	2	2	no	—
8	C₂×C₃²×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₃²×C₄×C₈	C₄	1	4	yes	complemented
11	C₂×C₃×C₄×C₈	C₂×C₃	1	4	yes	—
12	C₃×#384	C₂×C₃	1	8	yes	—
13	C₃×#340	C₂×C₃	1	16	yes	—
14	C₂²×C₃²×C₄	—	2	4	no	—
15	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
16	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
17	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
18	C₂×C₃²×C₈	—	2	4	no	—
19	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
20	C₃²×C₄⋊C₄	—	2	8	no	—
21	C₃²×C₄⋊C₄	—	4	4	no	—
22	C₃²×C₄⋊C₄	—	4	4	no	—
23	C₃²×C₄²	C₂×C₄	1	2	yes	—
24	C₃²×C₄²	Q₈	1	2	yes	—
25	#1834	C₃²	1	1	yes	characteristic, direct product factor, Sylow
26	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
27	C₂×C₃×C₄⋊C₄	—	2	8	no	—
28	C₂×C₃×C₄⋊C₄	—	2	8	no	—
29	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	16	yes	—
30	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
31	C₃×C₄×C₈	C₃×C₄	1	16	yes	complemented
32	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃²×C₄	—	4	4	no	—
34	C₂×C₃²×C₄	—	4	4	no	—
35	C₂×C₃²×C₄	—	4	4	no	—
36	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
37	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
38	C₂×C₃²×C₄	D₈	1	2	yes	—
39	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
40	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
41	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₃²×C₄	#41	1	4	yes	—
44	C₃²×C₈	—	2	4	no	—
45	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
46	#384	C₂×C₃²	1	2	yes	—
47	#340	C₂×C₃²	1	4	yes	—
48	C₂²×C₃×C₄	—	2	16	no	—
49	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
50	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
51	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
52	C₂×C₃×C₈	—	2	16	no	—
53	C₂×C₃×C₈	C₂×C₃×C₄	1	16	yes	—
54	C₃×C₄⋊C₄	—	2	32	no	—
55	C₃×C₄⋊C₄	—	4	16	no	—
56	C₃×C₄⋊C₄	—	4	16	no	—
57	C₃×C₄²	C₂×C₃×C₄	1	8	yes	—
58	C₃×C₄²	C₃×Q₈	1	8	yes	—
59	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₃²	#124	1	2	yes	—
61	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
62	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
63	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
64	C₃²×C₄	—	8	4	no	—
65	C₃²×C₄	#106	1	4	yes	—
66	C₃²×C₄	#107	1	4	yes	—
67	C₃²×C₄	#117	1	2	yes	—
68	C₃²×C₄	#125	1	2	yes	—
69	C₂²×C₈	C₂²×C₃²	1	2	yes	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
73	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
74	C₄×C₈	C₃²×C₄	1	4	yes	complemented
75	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
76	C₂×C₃×C₄	—	4	16	no	—
77	C₂×C₃×C₄	—	4	16	no	—
78	C₂×C₃×C₄	—	4	16	no	—
79	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
80	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
81	C₂×C₃×C₄	C₃×D₈	1	8	yes	—
82	C₂×C₃×C₄	C₃×SD₁₆	1	16	yes	—
83	C₂×C₃×C₄	C₃×Q₁₆	1	8	yes	—
84	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
85	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
86	C₂×C₃×C₄	C₃×#41	1	16	yes	—
87	C₃×C₈	—	2	16	no	—
88	C₂×C₃²	#382	1	1	yes	characteristic
89	C₂×C₃²	C₂×#106	1	1	yes	characteristic
90	C₂×C₃²	C₂×#107	1	1	yes	characteristic
91	C₂×C₃²	#486	1	2	yes	—
92	C₂×C₃²	#487	1	2	yes	—
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
95	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₂×C₃²×C₄	1	4	yes	—
99	C₄⋊C₄	—	2	8	no	—
100	C₄⋊C₄	—	4	4	no	—
101	C₄⋊C₄	—	4	4	no	—
102	C₄²	C₂×C₃²×C₄	1	2	yes	—
103	C₄²	C₃²×Q₈	1	2	yes	—
104	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
105	C₂²×C₃	C₃×#124	1	8	yes	—
106	C₂²×C₃	C₂×C₃×D₈	1	4	yes	—
107	C₂²×C₃	C₂×C₃×SD₁₆	1	8	yes	—
108	C₂²×C₃	C₂×C₃×Q₁₆	1	4	yes	—
109	C₃×C₄	—	8	16	no	—
110	C₃×C₄	C₃×#106	1	16	yes	—
111	C₃×C₄	C₃×#107	1	16	yes	—
112	C₃×C₄	C₃×#117	1	8	yes	—
113	C₃×C₄	C₃×#125	1	8	yes	—
114	C₃²	#1834	1	1	yes	characteristic, direct product factor, Sylow
115	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	4	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
120	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
121	C₂×C₄	C₃²×D₈	1	2	yes	—
122	C₂×C₄	C₃²×SD₁₆	1	4	yes	—
123	C₂×C₄	C₃²×Q₁₆	1	2	yes	—
124	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
125	C₂×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
126	C₂×C₄	C₃²×#41	1	4	yes	—
127	C₈	—	2	4	no	—
128	C₂×C₃	C₃×#382	1	4	yes	—
129	C₂×C₃	C₂×C₃×#106	1	4	yes	—
130	C₂×C₃	C₂×C₃×#107	1	4	yes	—
131	C₂×C₃	C₃×#486	1	8	yes	—
132	C₂×C₃	C₃×#487	1	8	yes	—
133	C₂²	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²	C₃²×#124	1	2	yes	—
135	C₂²	C₂×C₃²×D₈	1	1	yes	characteristic
136	C₂²	C₂×C₃²×SD₁₆	1	2	yes	—
137	C₂²	C₂×C₃²×Q₁₆	1	1	yes	characteristic
138	C₄	—	8	4	no	—
139	C₄	C₃²×#106	1	4	yes	—
140	C₄	C₃²×#107	1	4	yes	—
141	C₄	C₃²×#117	1	2	yes	—
142	C₄	C₃²×#125	1	2	yes	—
143	C₃	C₃×#1834	1	4	yes	direct product factor
144	C₂	C₃²×#382	1	1	yes	characteristic
145	C₂	C₂×C₃²×#106	1	1	yes	characteristic
146	C₂	C₂×C₃²×#107	1	1	yes	characteristic
147	C₂	C₃²×#486	1	2	yes	—
148	C₂	C₃²×#487	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃²	C₂×D₄
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ C₂×C₃²×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#1834	C₃	direct (N × H)
10. C₃²×C₄×C₈	C₄	semidirect
25. #1834	C₃²	direct (N × H)
31. C₃×C₄×C₈	C₃×C₄	semidirect
74. C₄×C₈	C₃²×C₄	semidirect
114. C₃²	#1834	direct (N × H)
143. C₃	C₃×#1834	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₃²×#384	2
C₃²×#340	2
C₃×#1834	3

Subgroups — C32×#1834 #11768883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1834 #11768883