Subgroups — #82062

Order 384.

656 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	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×#926	C₂	1	2	yes	maximal
3	#5212	C₂	1	4	yes	maximal
4	#4936	C₂	1	2	yes	maximal, complemented
5	#4935	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₃	C₂	1	4	yes	maximal, complemented
7	#2285	—	3	1	no	maximal, Sylow
8	C₂³×C₃×C₄	C₂²	1	2	yes	—
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²	C₄	1	8	yes	complemented
11	C₂×C₃×C₄²	C₂²	1	2	yes	—
12	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×#217	C₂²	1	4	yes	—
15	#926	—	2	8	no	—
16	C₂×C₄⋊Dic₃	C₂²	1	8	yes	—
17	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
18	C₂×C₄×Dic₃	C₂²	1	4	yes	—
19	C₃⋊#95	C₂²	1	8	yes	—
20	C₂×#117	—	3	2	no	—
21	C₂²×C₄²	D₃	1	1	yes	characteristic
22	#384	—	3	2	no	—
23	#382	—	3	2	no	—
24	#380	—	3	4	no	—
25	C₄×C₄⋊C₄	—	3	4	no	—
26	C₂⁴×C₃	C₂³	1	1	yes	characteristic
27	C₂²×C₃×C₄	—	2	8	no	—
28	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
29	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
30	C₂²×C₃×C₄	D₄	1	4	yes	—
31	C₂²×C₃×C₄	Q₈	1	4	yes	—
32	C₂²×C₃×C₄	C₂³	1	2	yes	—
33	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄	C₂³	1	2	yes	—
35	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂²×Dic₃	C₂³	1	8	yes	—
37	C₃×C₄²	C₂×C₄	1	4	yes	—
38	C₃×C₄²	C₂×C₄	1	8	yes	—
39	C₃×C₄²	C₂×C₄	1	4	yes	—
40	#217	—	2	8	no	—
41	C₄⋊Dic₃	—	2	8	no	—
42	C₄⋊Dic₃	—	2	16	no	—
43	C₄×Dic₃	—	2	8	no	—
44	C₂³×C₄	C₂×D₃	1	2	yes	—
45	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
46	#117	—	6	8	no	—
47	C₂×C₄⋊C₄	—	3	4	no	—
48	C₂×C₄⋊C₄	—	3	8	no	—
49	C₂×C₂²⋊C₄	—	3	4	no	—
50	C₂×C₄²	—	3	4	no	—
51	C₂×C₄²	Dic₃	1	8	yes	complemented
52	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₂×C₄²	C₂×D₃	1	2	yes	—
55	#95	—	3	8	no	—
56	C₂³×C₃	—	2	4	no	—
57	C₂³×C₃	C₂²×C₄	1	2	yes	—
58	C₂³×C₃	C₂×D₄	1	2	yes	—
59	C₂³×C₃	C₂×Q₈	1	2	yes	—
60	C₂³×C₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₃×C₄	—	2	4	no	—
62	C₂×C₃×C₄	—	2	16	no	—
63	C₂×C₃×C₄	C₄⋊C₄	1	16	yes	complemented
64	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
65	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
66	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
67	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
68	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
69	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
70	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
71	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
72	C₂×C₃×C₄	#41	1	8	yes	—
73	C₂×Dic₃	—	2	8	no	—
74	C₂×Dic₃	—	2	16	no	—
75	C₂⁴	C₂²×D₃	1	1	yes	characteristic
76	C₂²×C₄	—	2	8	no	—
77	C₂²×C₄	—	3	8	no	—
78	C₂²×C₄	Dic₆	1	4	yes	—
79	C₂²×C₄	D₁₂	1	4	yes	—
80	C₂²×C₄	C₂×Dic₃	1	8	yes	—
81	C₂²×C₄	C₂×Dic₃	1	4	yes	—
82	C₂²×C₄	C₂²×D₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₄⋊C₄	—	6	8	no	—
87	C₄⋊C₄	—	6	16	no	—
88	C₂²⋊C₄	—	6	8	no	—
89	C₄²	—	6	8	no	—
90	C₄²	C₂×Dic₃	1	4	yes	—
91	C₄²	C₂×Dic₃	1	8	yes	—
92	C₄²	C₂×Dic₃	1	4	yes	—
93	C₂²×C₃	—	2	4	no	—
94	C₂²×C₃	—	2	4	no	—
95	C₂²×C₃	—	2	4	no	—
96	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
99	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₃	C₂×#41	1	4	yes	—
101	C₃×C₄	—	2	4	no	—
102	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
103	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
104	C₃×C₄	#117	1	8	yes	complemented
105	Dic₃	—	4	8	no	—
106	C₂³	—	2	4	no	—
107	C₂³	C₂×Dic₆	1	2	yes	—
108	C₂³	C₂×D₁₂	1	2	yes	—
109	C₂³	C₂²×Dic₃	1	2	yes	—
110	C₂³	C₂³×D₃	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	16	no	—
113	C₂×C₄	—	6	8	no	—
114	C₂×C₄	—	6	16	no	—
115	C₂×C₄	C₄⋊Dic₃	1	16	yes	complemented
116	C₂×C₄	C₂×Dic₆	1	2	yes	—
117	C₂×C₄	C₂×Dic₆	1	2	yes	—
118	C₂×C₄	C₂×D₁₂	1	2	yes	—
119	C₂×C₄	C₂×D₁₂	1	2	yes	—
120	C₂×C₄	#235	1	8	yes	—
121	C₂×C₄	C₂²×Dic₃	1	4	yes	—
122	C₂×C₄	C₂²×Dic₃	1	2	yes	—
123	C₂×C₄	C₂²×Dic₃	1	2	yes	—
124	C₂×C₄	C₂²×Dic₃	1	4	yes	—
125	C₂×C₃	—	2	4	no	—
126	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₃	C₂×#117	1	2	yes	—
128	C₂×C₃	#532	1	2	yes	—
129	C₂×C₃	#533	1	2	yes	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	4	no	—
132	C₂²	—	2	4	no	—
133	C₂²	C₂×C₄⋊Dic₃	1	4	yes	—
134	C₂²	C₂²×Dic₆	1	1	yes	characteristic
135	C₂²	C₂²×D₁₂	1	1	yes	characteristic
136	C₂²	C₂×#235	1	4	yes	—
137	C₂²	C₂³×Dic₃	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	12	8	no	—
140	C₄	C₂×C₄⋊Dic₃	1	4	yes	—
141	C₄	C₂×C₄⋊Dic₃	1	4	yes	—
142	C₄	#926	1	8	yes	complemented
143	C₃	#2285	1	1	yes	characteristic, complemented, Sylow
144	C₂	—	2	4	no	—
145	C₂	#5470	1	2	yes	—
146	C₂	#5477	1	2	yes	—
147	C₂	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
148	C₂	C₂×#926	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄
Frattini	C₂³	C₂³×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	#2285
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
4. #4936	C₂	semidirect
5. #4935	C₂	semidirect
6. C₄×C₄⋊Dic₃	C₂	semidirect
10. C₂×C₃×C₄²	C₄	semidirect
51. C₂×C₄²	Dic₃	semidirect
63. C₂×C₃×C₄	C₄⋊C₄	semidirect
104. C₃×C₄	#117	semidirect
115. C₂×C₄	C₄⋊Dic₃	semidirect
142. C₄	#926	semidirect
143. C₃	#2285	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄²	2
C₂×#926	2
#5212	2
#4936	2
#4935	2
C₄×C₄⋊Dic₃	2
#2285	3