Subgroups — C₂×C₃²×#368 #11769209

Order 1152.

432 conjugacy classes of subgroups.

The subgroup lattice has 141 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#108	C₂	1	2	yes	maximal
3	C₃²×#368	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#368	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×M₁₆	—	2	2	no	—
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₁₆	C₄	1	2	yes	—
8	C₂×C₃²×C₁₆	C₄	1	2	yes	—
9	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
10	C₃²×#108	—	2	2	no	—
11	C₃²×#108	C₂²	1	4	yes	—
12	C₂²×C₃×C₁₆	C₂×C₃	1	4	yes	—
13	C₂×C₃×#108	C₂×C₃	1	8	yes	—
14	C₃×#368	C₂×C₃	1	16	yes	direct product factor
15	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
16	C₃²×M₁₆	—	2	4	no	—
17	C₃²×M₁₆	—	4	2	no	—
18	C₂×C₃²×C₈	—	4	2	no	—
19	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
22	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
23	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
24	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
25	C₃²×C₁₆	C₂×C₄	1	4	yes	—
26	C₂×#368	C₃²	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₃×M₁₆	—	2	8	no	—
28	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
29	C₂×C₃×C₁₆	C₃×C₄	1	8	yes	—
30	C₂×C₃×C₁₆	C₃×C₄	1	8	yes	—
31	C₂×C₃×C₁₆	C₂²×C₃	1	8	yes	—
32	C₃×#108	—	2	8	no	—
33	C₃×#108	C₂²×C₃	1	16	yes	—
34	C₂³×C₃²	Q₁₆	1	1	yes	characteristic
35	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
38	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
39	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
40	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
41	C₃²×C₈	—	4	4	no	—
42	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₁₆	C₂×C₃²	1	1	yes	characteristic
47	C₂×#108	C₂×C₃²	1	2	yes	—
48	#368	C₂×C₃²	1	4	yes	direct product factor
49	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
50	C₃×M₁₆	—	2	16	no	—
51	C₃×M₁₆	—	4	8	no	—
52	C₂×C₃×C₈	—	4	8	no	—
53	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
54	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
55	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
56	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
57	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
58	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
59	C₃×C₁₆	C₂×C₃×C₄	1	16	yes	—
60	C₂²×C₃²	—	2	2	no	—
61	C₂²×C₃²	#107	1	1	yes	characteristic
62	C₂²×C₃²	#107	1	1	yes	characteristic
63	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
64	C₃²×C₄	#107	1	1	yes	characteristic
65	C₃²×C₄	#107	1	1	yes	characteristic
66	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
68	C₂×M₁₆	—	2	2	no	—
69	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
70	C₂×C₁₆	C₃²×C₄	1	2	yes	—
71	C₂×C₁₆	C₃²×C₄	1	2	yes	—
72	C₂×C₁₆	C₂²×C₃²	1	2	yes	—
73	#108	—	2	2	no	—
74	#108	C₂²×C₃²	1	4	yes	—
75	C₂³×C₃	C₃×Q₁₆	1	4	yes	—
76	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
77	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
78	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
79	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
80	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
81	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
82	C₃×C₈	—	4	16	no	—
83	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
84	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
85	C₃×C₈	C₂²×C₃×C₄	1	4	yes	—
86	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
87	C₂×C₃²	—	2	1	no	—
88	C₂×C₃²	—	2	1	no	—
89	C₂×C₃²	#368	1	2	yes	direct product factor
90	C₂×C₃²	C₂×#107	1	1	yes	characteristic
91	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
92	M₁₆	—	2	4	no	—
93	M₁₆	—	4	2	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
96	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
97	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
98	C₂×C₈	C₃²×Q₈	1	1	yes	characteristic
99	C₂×C₈	C₃²×Q₈	1	1	yes	characteristic
100	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
101	C₁₆	C₂×C₃²×C₄	1	4	yes	—
102	C₂²×C₃	—	2	8	no	—
103	C₂²×C₃	C₃×#107	1	4	yes	—
104	C₂²×C₃	C₃×#107	1	4	yes	—
105	C₂²×C₃	C₂×C₃×Q₁₆	1	4	yes	—
106	C₃×C₄	C₃×#107	1	4	yes	—
107	C₃×C₄	C₃×#107	1	4	yes	—
108	C₃×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
109	C₃×C₄	C₂×C₃×D₈	1	4	yes	—
110	C₃²	C₂×#368	1	1	yes	characteristic, direct product factor, Sylow
111	C₂³	C₃²×Q₁₆	1	1	yes	characteristic
112	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄	C₃²×D₈	1	1	yes	characteristic
115	C₂×C₄	C₃²×D₈	1	1	yes	characteristic
116	C₂×C₄	C₃²×Q₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
118	C₈	—	4	4	no	—
119	C₈	C₃²×C₄⋊C₄	1	1	yes	characteristic
120	C₈	C₃²×C₄⋊C₄	1	1	yes	characteristic
121	C₈	C₂²×C₃²×C₄	1	1	yes	characteristic
122	C₈	C₂×C₃²×Q₈	1	1	yes	characteristic
123	C₂×C₃	—	2	4	no	—
124	C₂×C₃	—	2	4	no	—
125	C₂×C₃	C₃×#368	1	8	yes	direct product factor
126	C₂×C₃	C₂×C₃×#107	1	4	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	C₃²×#107	1	1	yes	characteristic
129	C₂²	C₃²×#107	1	1	yes	characteristic
130	C₂²	C₂×C₃²×Q₁₆	1	1	yes	characteristic
131	C₄	C₃²×#107	1	1	yes	characteristic
132	C₄	C₃²×#107	1	1	yes	characteristic
133	C₄	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
134	C₄	C₂×C₃²×D₈	1	1	yes	characteristic
135	C₃	C₂×C₃×#368	1	4	yes	direct product factor
136	C₂	—	2	1	no	—
137	C₂	—	2	1	no	—
138	C₂	C₃²×#368	1	2	yes	direct product factor
139	C₂	C₂×C₃²×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	D₈
commutator	C₈	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂²×C₃²	#107
prime radical	C₂³×C₃²	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#368	C₂	direct (N × H)
4. C₂×C₃×#368	C₃	direct (N × H)
14. C₃×#368	C₂×C₃	direct (N × H)
26. C₂×#368	C₃²	direct (N × H)
48. #368	C₂×C₃²	direct (N × H)
89. C₂×C₃²	#368	direct (N × H)
110. C₃²	C₂×#368	direct (N × H)
125. C₂×C₃	C₃×#368	direct (N × H)
135. C₃	C₂×C₃×#368	direct (N × H)
138. C₂	C₃²×#368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₁₆	2
C₂×C₃²×#108	2
C₃²×#368	2
C₂×C₃×#368	3

Subgroups — C2×C32×#368 #11769209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#368 #11769209