Subgroups — #71980

Order 384.

416 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#926	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4550	C₂	1	4	yes	maximal, complemented
5	#1729	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃×M₁₆	—	2	2	no	—
8	C₂×C₃×M₁₆	—	2	2	no	—
9	C₂×C₃×M₁₆	C₄	1	4	yes	complemented
10	C₂²×C₃×C₈	C₂²	1	2	yes	—
11	C₂×#217	C₄	1	2	yes	—
12	#926	—	2	2	no	—
13	#926	—	2	2	no	—
14	C₂×C₄⋊Dic₃	C₄	1	2	yes	—
15	C₂×C₄×Dic₃	C₂²	1	2	yes	—
16	C₂²×C₃⋊C₈	—	2	2	no	—
17	C₂²×C₃⋊C₈	—	2	2	no	—
18	C₂²×C₃⋊C₈	C₄	1	4	yes	complemented
19	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
20	C₂²×M₁₆	D₃	1	1	yes	characteristic
21	C₂³×C₈	—	3	1	no	—
22	C₂×#117	—	3	1	no	—
23	#336	—	3	4	no	—
24	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄	D₄	1	2	yes	—
29	C₂²×C₃×C₄	D₄	1	2	yes	—
30	C₂²×C₃×C₄	D₄	1	2	yes	—
31	C₂²×C₃×C₄	Q₈	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃	C₂×C₄	1	4	yes	—
34	C₃×M₁₆	—	2	8	no	—
35	C₂×C₃×C₈	—	2	2	no	—
36	C₂×C₃×C₈	—	2	2	no	—
37	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
38	#217	—	2	4	no	—
39	C₄⋊Dic₃	—	2	4	no	—
40	C₄×Dic₃	—	2	2	no	—
41	C₄×Dic₃	—	2	2	no	—
42	C₂×C₃⋊C₈	—	2	8	no	—
43	C₂×C₃⋊C₈	—	2	2	no	—
44	C₂×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
46	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
47	C₂×M₁₆	—	2	2	no	—
48	C₂×M₁₆	—	2	2	no	—
49	C₂×M₁₆	Dic₃	1	4	yes	complemented
50	C₂²×C₈	—	3	4	no	—
51	C₂²×C₈	—	3	2	no	—
52	C₂²×C₈	—	6	2	no	—
53	C₂²×C₈	—	6	2	no	—
54	C₂²×C₈	C₂×D₃	1	2	yes	—
55	#117	—	6	2	no	—
56	#117	—	6	2	no	—
57	C₂×C₄⋊C₄	—	3	2	no	—
58	C₂×C₂²⋊C₄	—	3	2	no	—
59	C₂×C₄²	—	3	2	no	—
60	C₂³×C₃	—	2	4	no	—
61	C₂³×C₃	C₄²	1	2	yes	—
62	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
63	C₂³×C₃	C₄⋊C₄	1	2	yes	—
64	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄	C₄²	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	C₂²⋊C₄	1	4	yes	—
69	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
70	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
71	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
72	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂×Dic₃	—	2	4	no	—
79	C₂×Dic₃	—	2	8	no	—
80	C₃×C₈	—	2	4	no	—
81	C₃⋊C₈	—	2	4	no	—
82	C₂⁴	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₄	—	3	4	no	—
84	C₂²×C₄	Dic₆	1	2	yes	—
85	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₃	1	2	yes	—
87	C₂²×C₄	D₁₂	1	2	yes	—
88	C₂²×C₄	C₂×Dic₃	1	2	yes	—
89	C₂²×C₄	#67	1	2	yes	—
90	C₂²×C₄	#67	1	2	yes	—
91	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
92	M₁₆	—	2	8	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	3	4	no	—
96	C₂×C₈	—	6	8	no	—
97	C₂×C₈	—	6	2	no	—
98	C₂×C₈	—	6	2	no	—
99	C₂×C₈	C₂×Dic₃	1	4	yes	—
100	C₄⋊C₄	—	6	4	no	—
101	C₂²⋊C₄	—	6	4	no	—
102	C₄²	—	6	2	no	—
103	C₄²	—	6	2	no	—
104	C₂²×C₃	—	2	4	no	—
105	C₂²×C₃	—	2	4	no	—
106	C₂²×C₃	—	2	4	no	—
107	C₂²×C₃	#95	1	4	yes	—
108	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
109	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₃	#131	1	4	yes	—
112	C₃×C₄	#95	1	4	yes	—
113	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
117	Dic₃	—	4	4	no	—
118	C₂³	—	2	4	no	—
119	C₂³	C₄×Dic₃	1	2	yes	—
120	C₂³	Dic₃⋊C₄	1	2	yes	—
121	C₂³	(C₂×D₃)⋊C₄	1	2	yes	—
122	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
123	C₂×C₄	—	6	8	no	—
124	C₂×C₄	—	6	4	no	—
125	C₂×C₄	C₄×Dic₃	1	2	yes	—
126	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
127	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
128	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
129	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
130	C₂×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
131	C₂×C₄	#217	1	4	yes	—
132	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
134	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
135	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
136	C₂×C₄	C₂×#67	1	1	yes	characteristic
137	C₂×C₄	C₂×#67	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	6	4	no	—
140	C₂×C₃	—	2	4	no	—
141	C₂×C₃	C₂×#95	1	1	yes	characteristic
142	C₂×C₃	#405	1	2	yes	—
143	C₂×C₃	#408	1	2	yes	—
144	C₂×C₃	#424	1	2	yes	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	C₃⋊#95	1	4	yes	—
149	C₂²	#907	1	4	yes	—
150	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
151	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
152	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
153	C₄	—	12	4	no	—
154	C₄	C₃⋊#95	1	4	yes	—
155	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
156	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
157	C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
158	C₄	C₂×#217	1	1	yes	characteristic
159	C₃	#1729	1	1	yes	characteristic, complemented, Sylow
160	C₂	—	2	4	no	—
161	C₂	#5119	1	2	yes	—
162	C₂	#5122	1	2	yes	—
163	C₂	#5129	1	2	yes	—
164	C₂	C₂×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#1729
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #4550	C₂	semidirect
9. C₂×C₃×M₁₆	C₄	semidirect
18. C₂²×C₃⋊C₈	C₄	semidirect
49. C₂×M₁₆	Dic₃	semidirect
159. C₃	#1729	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×M₁₆	2
C₂×#926	2
C₂³×C₃⋊C₈	2
#4550	2
#1729	3