Subgroups — #79265

Order 384.

364 conjugacy classes of subgroups.

The subgroup lattice has 198 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×#431	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	#4679	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₆	C₂	1	2	yes	maximal, complemented
9	#2984	—	3	1	no	maximal, Sylow
10	C₂²×Dic₆	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃	C₄	1	1	yes	characteristic
13	C₂×Dic₃⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₃	C₂²	1	2	yes	—
15	C₄×Dic₆	C₄	1	1	yes	characteristic
16	C₄×Dic₆	C₄	1	2	yes	—
17	C₄×Dic₆	C₄	1	1	yes	characteristic
18	C₄×Dic₆	C₂²	1	1	yes	characteristic
19	C₄×Dic₆	C₂²	1	1	yes	characteristic
20	C₄×Dic₆	C₂²	1	2	yes	—
21	C₃×C₄⋊C₈	C₂²	1	2	yes	—
22	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₈	C₂²	1	2	yes	—
25	#830	C₂²	1	2	yes	—
26	#815	C₂²	1	4	yes	—
27	Dic₃⋊C₈	C₂²	1	4	yes	—
28	Dic₃⋊C₈	C₂²	1	4	yes	—
29	C₈×Dic₃	C₂²	1	4	yes	—
30	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
31	#803	C₂²	1	2	yes	—
32	C₂×C₄×Q₈	—	3	1	no	—
33	#446	—	3	4	no	—
34	#446	—	3	2	no	—
35	C₈×Q₈	—	3	2	no	—
36	#432	—	3	1	no	—
37	#432	—	3	2	no	—
38	#431	—	3	2	no	—
39	#431	D₃	1	1	yes	characteristic
40	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂²×Dic₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₃	C₂×C₄	1	2	yes	—
45	C₂×Dic₆	—	2	4	no	—
46	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₆	C₂×C₄	1	2	yes	—
48	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃×C₈	C₂³	1	2	yes	—
50	C₂×C₃×C₈	C₂³	1	2	yes	—
51	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄²	C₂³	1	1	yes	characteristic
54	C₃×C₄²	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
58	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
62	C₄×Dic₃	C₂×C₄	1	2	yes	—
63	C₄×Dic₃	C₂×C₄	1	2	yes	—
64	C₄×Dic₃	C₂³	1	2	yes	—
65	C₄×Dic₃	C₂³	1	2	yes	—
66	C₂×C₃⋊C₈	C₂³	1	4	yes	—
67	C₂²×Q₈	—	3	1	no	—
68	C₄×Q₈	—	3	1	no	—
69	C₄×Q₈	—	3	2	no	—
70	C₄×Q₈	—	3	1	no	—
71	C₄×Q₈	—	3	1	no	—
72	C₄×Q₈	—	3	2	no	—
73	C₄×Q₈	—	3	1	no	—
74	C₂×C₄⋊C₄	—	3	2	no	—
75	C₂×C₄⋊C₄	—	3	1	no	—
76	C₂×C₄²	—	3	2	no	—
77	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
78	C₄⋊C₈	—	3	4	no	—
79	C₄⋊C₈	—	3	4	no	—
80	C₄⋊C₈	—	3	2	no	—
81	C₄⋊C₈	C₂×D₃	1	2	yes	—
82	C₂²⋊C₈	—	3	2	no	—
83	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
84	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
85	#97	—	3	4	no	—
86	#97	—	3	2	no	—
87	C₄×C₈	—	3	4	no	—
88	C₄×C₈	C₂×D₃	1	2	yes	—
89	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₃	—	2	4	no	—
100	C₂×Dic₃	—	2	4	no	—
101	C₂×Dic₃	C₂²×C₄	1	2	yes	—
102	C₂×Dic₃	C₂²×C₄	1	2	yes	—
103	C₂×Dic₃	C₂²×C₄	1	2	yes	—
104	C₂×Dic₃	C₂²×C₄	1	2	yes	—
105	Dic₆	—	2	2	no	—
106	Dic₆	—	2	4	no	—
107	Dic₆	M₁₆	1	4	yes	—
108	C₃×C₈	—	2	2	no	—
109	C₃×C₈	—	2	2	no	—
110	C₃⋊C₈	—	2	4	no	—
111	C₂×Q₈	—	3	1	no	—
112	C₂×Q₈	—	3	2	no	—
113	C₂×Q₈	—	3	1	no	—
114	C₂×Q₈	—	6	4	no	—
115	C₂²×C₄	—	3	2	no	—
116	C₂²×C₄	—	3	2	no	—
117	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₈	—	3	4	no	—
121	C₂×C₈	C₂²×D₃	1	2	yes	—
122	C₂×C₈	C₂²×D₃	1	2	yes	—
123	C₄⋊C₄	—	3	2	no	—
124	C₄⋊C₄	—	3	2	no	—
125	C₄⋊C₄	—	3	2	no	—
126	C₄⋊C₄	—	3	2	no	—
127	C₄⋊C₄	—	3	2	no	—
128	C₄⋊C₄	—	3	1	no	—
129	C₄⋊C₄	—	3	1	no	—
130	C₄²	—	3	2	no	—
131	C₄²	—	3	2	no	—
132	C₄²	—	3	2	no	—
133	C₄²	—	3	2	no	—
134	C₄²	C₄×D₃	1	1	yes	characteristic
135	C₄²	C₄×D₃	1	1	yes	characteristic
136	C₄²	C₂²×D₃	1	1	yes	characteristic
137	C₄²	C₂²×D₃	1	1	yes	characteristic
138	C₂²×C₃	—	2	1	no	—
139	C₂²×C₃	—	2	1	no	—
140	C₂²×C₃	#131	1	2	yes	—
141	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
142	C₃×C₄	—	2	1	no	—
143	C₃×C₄	—	2	1	no	—
144	C₃×C₄	—	2	1	no	—
145	C₃×C₄	—	2	1	no	—
146	C₃×C₄	C₂×M₁₆	1	2	yes	—
147	C₃×C₄	2₋¹⁺⁴	1	2	yes	—
148	Dic₃	—	2	2	no	—
149	Dic₃	—	2	2	no	—
150	Dic₃	—	2	2	no	—
151	Dic₃	C₂×M₁₆	1	4	yes	—
152	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
153	Q₈	—	3	4	no	—
154	Q₈	—	6	4	no	—
155	Q₈	—	6	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	3	2	no	—
160	C₂×C₄	—	3	2	no	—
161	C₂×C₄	—	3	2	no	—
162	C₂×C₄	—	3	2	no	—
163	C₂×C₄	—	6	4	no	—
164	C₂×C₄	—	6	4	no	—
165	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	—	6	4	no	—
174	C₂×C₃	—	2	2	no	—
175	C₂×C₃	#519	1	1	yes	characteristic
176	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
177	C₂×C₃	C₂×#131	1	1	yes	characteristic
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	#901	1	2	yes	—
181	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	3	4	no	—
187	C₄	—	6	2	no	—
188	C₄	—	6	2	no	—
189	C₄	—	6	2	no	—
190	C₄	D₃×M₁₆	1	2	yes	—
191	C₄	#1010	1	2	yes	—
192	C₃	#2984	1	1	yes	characteristic, complemented, Sylow
193	C₂	—	2	2	no	—
194	C₂	#5516	1	1	yes	characteristic
195	C₂	C₂×#901	1	1	yes	characteristic
196	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#431	C₂
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#431	D₃
hypocenter	C₃	#2984
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
4. #4828	C₂	semidirect
7. #4679	C₂	semidirect
8. C₈×Dic₆	C₂	semidirect
192. C₃	#2984	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×#431	2
#4926	2
#4828	2
#4719	2
#4718	2
#4679	2
C₈×Dic₆	2
#2984	3