Subgroups — #67702

Order 384.

188 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#4935	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#1296	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	—	2	1	no	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₃	—	2	1	no	—
8	C₂×C₄⋊Dic₃	—	2	1	no	—
9	C₂×C₄⋊Dic₃	C₄	1	2	yes	—
10	C₂²×C₃⋊C₈	—	2	1	no	—
11	C₃×C₄⋊C₈	—	2	1	no	—
12	C₃×C₄⋊C₈	C₄	1	2	yes	complemented
13	C₄⋊(C₃⋊C₈)	—	2	1	no	—
14	C₄⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
15	C₂×C₄⋊C₈	—	3	1	no	—
16	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic
17	#384	—	3	1	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
21	C₂²×Dic₃	—	2	2	no	—
22	C₂×C₃×C₈	—	2	2	no	—
23	C₂×C₃×C₈	—	4	1	no	—
24	C₂×C₃×C₈	—	4	1	no	—
25	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²	D₄	1	1	yes	characteristic
28	C₃×C₄²	Q₈	1	1	yes	characteristic
29	C₄⋊Dic₃	—	4	1	no	—
30	C₄⋊Dic₃	—	4	1	no	—
31	C₄⋊Dic₃	—	4	1	no	—
32	C₄⋊Dic₃	—	4	1	no	—
33	C₄⋊Dic₃	—	4	2	no	—
34	C₂×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃⋊C₈	—	4	1	no	—
36	C₂×C₃⋊C₈	—	4	1	no	—
37	C₂²×C₈	—	2	1	no	—
38	C₂²×C₈	—	6	1	no	—
39	C₂×C₄⋊C₄	—	3	2	no	—
40	C₂×C₄⋊C₄	—	6	1	no	—
41	C₂×C₄⋊C₄	—	6	1	no	—
42	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₄⋊C₈	—	2	1	no	—
44	C₄⋊C₈	—	3	2	no	—
45	C₄⋊C₈	—	6	1	no	—
46	C₄⋊C₈	Dic₃	1	2	yes	complemented
47	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₄	—	2	1	no	—
49	C₂×C₃×C₄	—	2	1	no	—
50	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₃×C₄	D₈	1	1	yes	characteristic
57	C₂×C₃×C₄	D₈	1	1	yes	characteristic
58	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
59	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
60	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
61	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
62	C₂×C₃×C₄	Q₁₆	1	1	yes	characteristic
63	C₂×C₃×C₄	Q₁₆	1	1	yes	characteristic
64	C₂×Dic₃	—	4	2	no	—
65	C₂×Dic₃	—	4	2	no	—
66	C₂×Dic₃	—	4	2	no	—
67	C₃×C₈	—	4	2	no	—
68	C₃⋊C₈	—	4	2	no	—
69	C₂²×C₄	—	6	2	no	—
70	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
71	C₂²×C₄	D₁₂	1	1	yes	characteristic
72	C₂²×C₄	#67	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	—	6	2	no	—
77	C₂×C₈	—	12	1	no	—
78	C₂×C₈	—	12	1	no	—
79	C₄⋊C₄	—	12	2	no	—
80	C₄⋊C₄	—	12	1	no	—
81	C₄⋊C₄	—	12	1	no	—
82	C₄⋊C₄	—	12	1	no	—
83	C₄⋊C₄	—	12	1	no	—
84	C₄²	Dic₆	1	1	yes	characteristic
85	C₄²	C₄×D₃	1	1	yes	characteristic
86	C₄²	C₂×Dic₃	1	1	yes	characteristic
87	C₄²	#67	1	1	yes	characteristic
88	C₂²×C₃	#95	1	1	yes	characteristic
89	C₂²×C₃	#100	1	1	yes	characteristic
90	C₂²×C₃	#101	1	1	yes	characteristic
91	C₂²×C₃	#102	1	1	yes	characteristic
92	C₂²×C₃	#102	1	1	yes	characteristic
93	C₂²×C₃	#103	1	1	yes	characteristic
94	C₂²×C₃	#103	1	1	yes	characteristic
95	C₃×C₄	—	2	1	no	—
96	C₃×C₄	—	2	1	no	—
97	C₃×C₄	#102	1	1	yes	characteristic
98	C₃×C₄	#102	1	1	yes	characteristic
99	C₃×C₄	#103	1	1	yes	characteristic
100	C₃×C₄	#103	1	1	yes	characteristic
101	C₃×C₄	#106	1	1	yes	characteristic
102	C₃×C₄	#106	1	1	yes	characteristic
103	C₃×C₄	#107	1	1	yes	characteristic
104	C₃×C₄	#107	1	1	yes	characteristic
105	Dic₃	—	8	2	no	—
106	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	12	2	no	—
110	C₂×C₄	—	12	2	no	—
111	C₂×C₄	—	12	2	no	—
112	C₂×C₄	#204	1	1	yes	characteristic
113	C₂×C₄	#204	1	1	yes	characteristic
114	C₂×C₄	D₂₄	1	1	yes	characteristic
115	C₂×C₄	Dic₁₂	1	1	yes	characteristic
116	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
117	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
118	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
119	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	#213	1	1	yes	characteristic
122	C₂×C₄	#214	1	1	yes	characteristic
123	C₂×C₄	#215	1	1	yes	characteristic
124	C₂×C₄	#216	1	1	yes	characteristic
125	C₂×C₄	#217	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₈	—	12	2	no	—
128	C₂×C₃	#331	1	1	yes	characteristic
129	C₂×C₃	#332	1	1	yes	characteristic
130	C₂×C₃	#332	1	1	yes	characteristic
131	C₂×C₃	#333	1	1	yes	characteristic
132	C₂×C₃	#340	1	1	yes	characteristic
133	C₂×C₃	#340	1	1	yes	characteristic
134	C₂×C₃	#343	1	1	yes	characteristic
135	C₂²	#809	1	1	yes	characteristic
136	C₂²	#810	1	1	yes	characteristic
137	C₂²	#816	1	1	yes	characteristic
138	C₂²	#821	1	1	yes	characteristic
139	C₂²	#823	1	1	yes	characteristic
140	C₂²	#824	1	1	yes	characteristic
141	C₂²	C₃⋊#95	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	24	2	no	—
145	C₄	#807	1	1	yes	characteristic
146	C₄	#808	1	1	yes	characteristic
147	C₄	#816	1	1	yes	characteristic
148	C₄	#817	1	1	yes	characteristic
149	C₄	#818	1	1	yes	characteristic
150	C₄	#821	1	1	yes	characteristic
151	C₄	#832	1	1	yes	characteristic
152	C₄	#835	1	1	yes	characteristic
153	C₃	#1296	1	1	yes	characteristic, complemented, Sylow
154	C₂	#4481	1	1	yes	characteristic
155	C₂	#4482	1	1	yes	characteristic
156	C₂	#4485	1	1	yes	characteristic
157	C₂	#4486	1	1	yes	characteristic
158	C₂	#4529	1	1	yes	characteristic
159	C₂	#4551	1	1	yes	characteristic
160	C₂	#4554	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₄²
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₂×C₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#1296
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
12. C₃×C₄⋊C₈	C₄	semidirect
14. C₄⋊(C₃⋊C₈)	C₄	semidirect
46. C₄⋊C₈	Dic₃	semidirect
153. C₃	#1296	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊C₈	2
#4935	2
C₂×C₄⋊(C₃⋊C₈)	2
#1296	3