Subgroups — #83750

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 180 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	C₂²×#208	C₂	1	1	yes	maximal, characteristic
4	#5119	C₂	1	4	yes	maximal, complemented
5	#5118	C₂	1	4	yes	maximal, complemented
6	#5102	C₂	1	4	yes	maximal, complemented
7	#2905	—	3	1	no	maximal, Sylow
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×M₁₆	—	2	4	no	—
10	C₂×C₃×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₃×C₈	C₂²	1	2	yes	—
13	C₂×#217	C₄	1	2	yes	—
14	#926	C₄	1	2	yes	—
15	#926	C₄	1	2	yes	—
16	#926	C₂²	1	2	yes	—
17	#926	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₃	C₄	1	2	yes	—
19	C₂×C₄×Dic₃	C₂²	1	2	yes	—
20	C₂×#208	—	2	4	no	—
21	C₂×#208	C₂²	1	2	yes	—
22	C₂×#208	C₂²	1	2	yes	—
23	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
24	Dic₃⋊C₈	C₂²	1	16	yes	complemented
25	C₂²×M₁₆	—	3	1	no	—
26	C₂²×M₁₆	D₃	1	1	yes	characteristic
27	C₂×#117	—	3	1	no	—
28	#424	—	3	4	no	—
29	#424	—	3	4	no	—
30	#423	—	3	4	no	—
31	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	D₄	1	4	yes	—
36	C₂²×C₃×C₄	Q₈	1	4	yes	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×Dic₃	C₂×C₄	1	4	yes	—
41	C₃×M₁₆	—	2	4	no	—
42	C₃×M₁₆	—	2	4	no	—
43	C₂×C₃×C₈	—	2	4	no	—
44	C₂×C₃×C₈	C₂³	1	4	yes	—
45	#217	C₂×C₄	1	4	yes	—
46	#217	C₂×C₄	1	4	yes	—
47	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
48	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
49	C₄×Dic₃	C₂×C₄	1	4	yes	—
50	C₄×Dic₃	C₂³	1	4	yes	—
51	#208	—	2	4	no	—
52	#208	—	2	4	no	—
53	C₂×C₃⋊C₈	—	2	4	no	—
54	C₂×C₃⋊C₈	C₂³	1	4	yes	—
55	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
56	C₂×M₁₆	—	2	4	no	—
57	C₂×M₁₆	—	3	2	no	—
58	C₂×M₁₆	—	3	2	no	—
59	C₂×M₁₆	—	6	4	no	—
60	C₂×M₁₆	C₂×D₃	1	2	yes	—
61	C₂×M₁₆	C₂×D₃	1	2	yes	—
62	C₂²×C₈	—	3	2	no	—
63	C₂²×C₈	C₂×D₃	1	2	yes	—
64	#117	—	3	2	no	—
65	#117	—	3	2	no	—
66	#117	—	3	2	no	—
67	#117	—	3	2	no	—
68	C₂×C₄⋊C₄	—	3	2	no	—
69	C₂×C₂²⋊C₄	—	3	2	no	—
70	C₂×C₄²	—	3	2	no	—
71	C₄⋊C₈	—	3	16	no	—
72	C₂³×C₃	—	4	2	no	—
73	C₂³×C₃	C₄⋊C₄	1	4	yes	—
74	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
78	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
79	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
80	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
87	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
88	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
89	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₃	—	4	4	no	—
91	C₂×Dic₃	C₂²×C₄	1	8	yes	—
92	C₃×C₈	—	2	4	no	—
93	C₃⋊C₈	—	2	4	no	—
94	C₂⁴	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₄	—	3	4	no	—
96	C₂²×C₄	Dic₆	1	4	yes	—
97	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
98	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
100	C₂²×C₄	#67	1	4	yes	—
101	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
104	M₁₆	—	2	4	no	—
105	M₁₆	—	2	4	no	—
106	M₁₆	—	6	4	no	—
107	M₁₆	—	6	4	no	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	—	3	4	no	—
110	C₂×C₈	—	6	4	no	—
111	C₂×C₈	C₂²×D₃	1	4	yes	—
112	C₄⋊C₄	—	3	4	no	—
113	C₄⋊C₄	—	3	4	no	—
114	C₂²⋊C₄	—	3	4	no	—
115	C₂²⋊C₄	—	3	4	no	—
116	C₄²	—	3	4	no	—
117	C₄²	—	3	4	no	—
118	C₂²×C₃	—	2	2	no	—
119	C₂²×C₃	—	2	2	no	—
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	4	4	no	—
122	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
126	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
127	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
128	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
129	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
130	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
131	Dic₃	—	2	8	no	—
132	C₂³	—	4	2	no	—
133	C₂³	Dic₃⋊C₄	1	4	yes	—
134	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂×C₄	—	3	8	no	—
138	C₂×C₄	—	12	4	no	—
139	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
140	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
141	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
142	C₂×C₄	C₂×Dic₆	1	2	yes	—
143	C₂×C₄	C₂×Dic₆	1	2	yes	—
144	C₂×C₄	C₂×Dic₆	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₄	C₂×#67	1	2	yes	—
149	C₂×C₄	C₂×#67	1	2	yes	—
150	C₂×C₄	C₂×#67	1	2	yes	—
151	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	6	4	no	—
154	C₂×C₃	—	2	2	no	—
155	C₂×C₃	—	2	2	no	—
156	C₂×C₃	—	2	2	no	—
157	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₃	#568	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	4	4	no	—
163	C₂²	C₂×Dic₃⋊C₄	1	2	yes	—
164	C₂²	C₂×Dic₃⋊C₄	1	2	yes	—
165	C₂²	C₂×Dic₃⋊C₄	1	2	yes	—
166	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
167	C₄	—	6	8	no	—
168	C₄	C₂×Dic₃⋊C₄	1	2	yes	—
169	C₄	C₂×Dic₃⋊C₄	1	2	yes	—
170	C₄	C₂×Dic₃⋊C₄	1	2	yes	—
171	C₄	C₂²×Dic₆	1	1	yes	characteristic
172	C₄	C₂²×#67	1	1	yes	characteristic
173	C₃	#2905	1	1	yes	characteristic, complemented, Sylow
174	C₂	—	2	2	no	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	#5745	1	2	yes	—
178	C₂	C₂²×Dic₃⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂²×C₃×M₁₆	C₂
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#2905
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #5119	C₂	semidirect
5. #5118	C₂	semidirect
6. #5102	C₂	semidirect
24. Dic₃⋊C₈	C₂²	semidirect
173. C₃	#2905	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
C₂²×#208	2
#5119	2
#5118	2
#5102	2
#2905	3