Subgroups — #79314

Order 384.

356 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#432	C₂	1	1	yes	maximal, characteristic
3	#4926	C₂	1	1	yes	maximal, characteristic
4	#4828	C₂	1	4	yes	maximal, complemented
5	#4719	C₂	1	2	yes	maximal
6	#4718	C₂	1	2	yes	maximal
7	#4701	C₂	1	4	yes	maximal, complemented
8	#2983	—	3	1	no	maximal, Sylow
9	C₂²×Dic₆	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₄	1	1	yes	characteristic
12	C₂×Dic₃⋊C₄	C₄	1	1	yes	characteristic
13	C₂×Dic₃⋊C₄	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
16	C₄×Dic₆	C₄	1	2	yes	—
17	C₄×Dic₆	C₄	1	2	yes	—
18	C₄×Dic₆	C₂²	1	2	yes	—
19	C₄×Dic₆	C₂²	1	2	yes	—
20	C₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₃×#97	C₂²	1	2	yes	—
23	#830	C₂²	1	2	yes	—
24	#815	C₂²	1	4	yes	—
25	Dic₃⋊C₈	C₂²	1	4	yes	—
26	Dic₃⋊C₈	C₂²	1	4	yes	—
27	C₈×Dic₃	C₂²	1	4	yes	—
28	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
29	#803	C₂²	1	2	yes	—
30	C₂×C₄×Q₈	—	3	1	no	—
31	#446	—	3	4	no	—
32	#446	—	3	4	no	—
33	#432	—	3	1	no	—
34	#432	—	3	2	no	—
35	#432	D₃	1	1	yes	characteristic
36	#431	—	3	2	no	—
37	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₆	—	2	4	no	—
45	C₂×Dic₆	C₂×C₄	1	2	yes	—
46	C₂×Dic₆	C₂×C₄	1	2	yes	—
47	C₂×C₃×C₈	C₂³	1	4	yes	—
48	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
50	C₃×C₄²	C₂³	1	1	yes	characteristic
51	C₃×C₄²	C₂³	1	1	yes	characteristic
52	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
54	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
58	C₄×Dic₃	C₂×C₄	1	2	yes	—
59	C₄×Dic₃	C₂×C₄	1	2	yes	—
60	C₄×Dic₃	C₂³	1	2	yes	—
61	C₄×Dic₃	C₂³	1	2	yes	—
62	C₂×C₃⋊C₈	C₂³	1	4	yes	—
63	C₂²×Q₈	—	3	1	no	—
64	C₄×Q₈	—	3	2	no	—
65	C₄×Q₈	—	3	2	no	—
66	C₄×Q₈	—	3	2	no	—
67	C₄×Q₈	—	3	2	no	—
68	C₂×C₄⋊C₄	—	3	1	no	—
69	C₂×C₄⋊C₄	—	3	1	no	—
70	C₂×C₄⋊C₄	—	3	1	no	—
71	C₂×C₄²	—	3	1	no	—
72	C₂×C₄²	—	3	1	no	—
73	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
74	C₄⋊C₈	—	3	4	no	—
75	C₄⋊C₈	—	3	4	no	—
76	C₄⋊C₈	—	3	2	no	—
77	C₄⋊C₈	C₂×D₃	1	2	yes	—
78	C₂²⋊C₈	—	3	2	no	—
79	C₂²⋊C₈	C₂×D₃	1	2	yes	—
80	#97	—	3	4	no	—
81	#97	—	3	2	no	—
82	#97	C₂×D₃	1	2	yes	—
83	C₄×C₈	—	3	4	no	—
84	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	4	1	no	—
88	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₃	—	2	2	no	—
95	C₂×Dic₃	—	4	1	no	—
96	C₂×Dic₃	—	4	1	no	—
97	C₂×Dic₃	M₁₆	1	4	yes	—
98	C₂×Dic₃	C₂²×C₄	1	2	yes	—
99	C₂×Dic₃	C₂²×C₄	1	2	yes	—
100	C₂×Dic₃	C₂²×C₄	1	2	yes	—
101	C₂×Dic₃	C₂²×C₄	1	2	yes	—
102	Dic₆	—	2	4	no	—
103	Dic₆	—	2	4	no	—
104	C₃×C₈	—	2	4	no	—
105	C₃⋊C₈	—	2	4	no	—
106	C₂×Q₈	—	3	2	no	—
107	C₂×Q₈	—	3	2	no	—
108	C₂×Q₈	—	6	4	no	—
109	C₂²×C₄	—	3	1	no	—
110	C₂²×C₄	—	3	1	no	—
111	C₂²×C₄	—	3	1	no	—
112	C₂²×C₄	—	3	1	no	—
113	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
114	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₈	—	3	4	no	—
117	C₂×C₈	C₂²×D₃	1	4	yes	—
118	C₄⋊C₄	—	3	2	no	—
119	C₄⋊C₄	—	3	2	no	—
120	C₄⋊C₄	—	3	2	no	—
121	C₄⋊C₄	—	3	2	no	—
122	C₄⋊C₄	—	3	2	no	—
123	C₄⋊C₄	—	3	2	no	—
124	C₄²	—	3	2	no	—
125	C₄²	—	3	2	no	—
126	C₄²	—	3	2	no	—
127	C₄²	—	3	2	no	—
128	C₄²	C₄×D₃	1	1	yes	characteristic
129	C₄²	C₄×D₃	1	1	yes	characteristic
130	C₄²	C₂²×D₃	1	1	yes	characteristic
131	C₄²	C₂²×D₃	1	1	yes	characteristic
132	C₂²×C₃	—	2	1	no	—
133	C₂²×C₃	—	2	1	no	—
134	C₂²×C₃	C₂×M₁₆	1	2	yes	—
135	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
136	C₃×C₄	—	2	1	no	—
137	C₃×C₄	—	2	1	no	—
138	C₃×C₄	—	2	1	no	—
139	C₃×C₄	—	2	1	no	—
140	C₃×C₄	—	2	1	no	—
141	C₃×C₄	2₋¹⁺⁴	1	2	yes	—
142	Dic₃	—	2	2	no	—
143	Dic₃	—	2	2	no	—
144	Dic₃	—	2	2	no	—
145	Dic₃	C₂×M₁₆	1	4	yes	—
146	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
147	Q₈	—	6	4	no	—
148	Q₈	—	6	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	3	4	no	—
152	C₂×C₄	—	3	2	no	—
153	C₂×C₄	—	3	2	no	—
154	C₂×C₄	—	3	2	no	—
155	C₂×C₄	—	3	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	6	2	no	—
158	C₂×C₄	—	12	1	no	—
159	C₂×C₄	—	12	1	no	—
160	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
165	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
166	C₈	—	2	4	no	—
167	C₈	—	6	4	no	—
168	C₂×C₃	—	2	2	no	—
169	C₂×C₃	#519	1	1	yes	characteristic
170	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
171	C₂×C₃	#568	1	1	yes	characteristic
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	D₃×M₁₆	1	2	yes	—
175	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	3	4	no	—
182	C₄	—	6	2	no	—
183	C₄	—	6	2	no	—
184	C₄	—	6	2	no	—
185	C₄	#1010	1	2	yes	—
186	C₃	#2983	1	1	yes	characteristic, complemented, Sylow
187	C₂	—	2	2	no	—
188	C₂	#5516	1	1	yes	characteristic
189	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic
190	C₂	#5745	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#432	C₂
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#432	D₃
hypocenter	C₃	#2983
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
4. #4828	C₂	semidirect
7. #4701	C₂	semidirect
186. C₃	#2983	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₆	2
C₃×#432	2
#4926	2
#4828	2
#4719	2
#4718	2
#4701	2
#2983	3