Subgroups — Dic₃×#117 #82222

Order 384.

752 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#926	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
4	Dic₃×C₄⋊C₄	C₂	1	4	yes	maximal, complemented
5	Dic₃×C₂²⋊C₄	C₂	1	4	yes	maximal
6	#4932	C₂	1	2	yes	maximal, complemented
7	C₄²×Dic₃	C₂	1	2	yes	maximal, complemented
8	C₄×#117	—	3	1	no	maximal, Sylow
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂³×Dic₃	C₂²	1	2	yes	—
11	C₃×#117	C₄	1	8	yes	complemented
12	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	2	yes	—
15	C₂×#217	C₂²	1	2	yes	—
16	#926	C₄	1	8	yes	complemented
17	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
18	C₂×C₄×Dic₃	C₄	1	8	yes	complemented
19	C₂×C₄×Dic₃	C₂²	1	8	yes	—
20	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₃	C₂²	1	2	yes	—
23	C₂×C₄×Dic₃	C₂²	1	2	yes	—
24	C₃⋊#95	C₂²	1	8	yes	—
25	C₂×#117	—	3	1	no	—
26	C₂×#117	D₃	1	1	yes	characteristic
27	C₂²×C₄²	—	3	1	no	—
28	C₄×C₄⋊C₄	—	3	4	no	—
29	C₄×C₂²⋊C₄	—	3	4	no	—
30	#376	—	3	2	no	—
31	C₄³	—	3	2	no	—
32	C₂⁴×C₃	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
34	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
35	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
36	C₂²×C₃×C₄	C₂³	1	4	yes	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×Dic₃	—	2	8	no	—
40	C₂²×Dic₃	C₂×C₄	1	8	yes	—
41	C₂²×Dic₃	C₂³	1	2	yes	—
42	C₂²×Dic₃	C₂³	1	4	yes	—
43	C₂²×Dic₃	C₂³	1	2	yes	—
44	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
45	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
46	C₃×C₄²	C₂×C₄	1	8	yes	complemented
47	#217	C₂×C₄	1	8	yes	—
48	C₄⋊Dic₃	C₂×C₄	1	8	yes	complemented
49	C₄×Dic₃	—	2	16	no	—
50	C₄×Dic₃	C₂×C₄	1	4	yes	—
51	C₄×Dic₃	C₂×C₄	1	8	yes	—
52	C₄×Dic₃	C₂×C₄	1	4	yes	—
53	C₄×Dic₃	C₂×C₄	1	8	yes	complemented
54	C₂³×C₄	—	3	2	no	—
55	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
56	#117	—	3	8	no	—
57	#117	Dic₃	1	8	yes	direct product factor
58	C₂×C₄⋊C₄	—	3	2	no	—
59	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
60	C₂×C₂²⋊C₄	—	3	2	no	—
61	C₂×C₂²⋊C₄	C₂×D₃	1	2	yes	—
62	C₂×C₄²	—	3	8	no	—
63	C₂×C₄²	—	3	2	no	—
64	C₂×C₄²	—	3	2	no	—
65	C₂×C₄²	—	3	1	no	—
66	C₂×C₄²	—	3	1	no	—
67	C₂×C₄²	—	3	8	no	—
68	C₂×C₄²	C₂×D₃	1	2	yes	—
69	#95	—	3	8	no	—
70	C₂³×C₃	—	2	4	no	—
71	C₂³×C₃	C₂²×C₄	1	2	yes	—
72	C₂³×C₃	C₂²×C₄	1	2	yes	—
73	C₂³×C₃	C₂²×C₄	1	2	yes	—
74	C₂³×C₃	C₂⁴	1	1	yes	characteristic
75	C₂×C₃×C₄	—	2	8	no	—
76	C₂×C₃×C₄	C₄²	1	16	yes	complemented
77	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
82	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
84	C₂×Dic₃	—	2	8	no	—
85	C₂×Dic₃	—	2	4	no	—
86	C₂×Dic₃	—	2	16	no	—
87	C₂×Dic₃	C₂²×C₄	1	4	yes	—
88	C₂×Dic₃	C₂²×C₄	1	8	yes	—
89	C₂×Dic₃	C₂²×C₄	1	4	yes	—
90	C₂×Dic₃	#41	1	8	yes	—
91	C₂⁴	C₂²×D₃	1	1	yes	characteristic
92	C₂²×C₄	—	3	2	no	—
93	C₂²×C₄	—	3	8	no	—
94	C₂²×C₄	—	3	2	no	—
95	C₂²×C₄	—	3	4	no	—
96	C₂²×C₄	—	6	8	no	—
97	C₂²×C₄	C₄×D₃	1	4	yes	—
98	C₂²×C₄	C₄×D₃	1	4	yes	—
99	C₂²×C₄	C₂×Dic₃	1	4	yes	—
100	C₂²×C₄	C₂²×D₃	1	4	yes	—
101	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₄⋊C₄	—	3	8	no	—
104	C₄⋊C₄	C₂×Dic₃	1	8	yes	complemented
105	C₂²⋊C₄	—	3	8	no	—
106	C₂²⋊C₄	C₂×Dic₃	1	8	yes	—
107	C₄²	—	3	4	no	—
108	C₄²	—	3	8	no	—
109	C₄²	—	3	4	no	—
110	C₄²	—	3	8	no	—
111	C₄²	—	6	16	no	—
112	C₄²	C₂×Dic₃	1	8	yes	complemented
113	C₂²×C₃	—	2	4	no	—
114	C₂²×C₃	—	2	4	no	—
115	C₂²×C₃	—	2	4	no	—
116	C₂²×C₃	C₂×C₄²	1	4	yes	—
117	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
118	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
119	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
120	C₂²×C₃	C₂×#41	1	4	yes	—
121	C₃×C₄	—	2	8	no	—
122	C₃×C₄	C₂×C₄²	1	4	yes	—
123	C₃×C₄	C₂×C₄²	1	4	yes	—
124	Dic₃	—	2	4	no	—
125	Dic₃	—	2	8	no	—
126	Dic₃	#117	1	8	yes	direct product factor
127	C₂³	—	2	4	no	—
128	C₂³	C₂×C₄×D₃	1	2	yes	—
129	C₂³	C₂×C₄×D₃	1	2	yes	—
130	C₂³	C₂²×Dic₃	1	2	yes	—
131	C₂³	C₂³×D₃	1	1	yes	characteristic
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	3	4	no	—
134	C₂×C₄	—	3	8	no	—
135	C₂×C₄	—	3	4	no	—
136	C₂×C₄	—	3	8	no	—
137	C₂×C₄	—	6	16	no	—
138	C₂×C₄	—	6	8	no	—
139	C₂×C₄	—	6	4	no	—
140	C₂×C₄	C₄×Dic₃	1	16	yes	complemented
141	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
142	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
143	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
144	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
145	C₂×C₄	C₂²×Dic₃	1	8	yes	—
146	C₂×C₄	C₂²×Dic₃	1	2	yes	—
147	C₂×C₄	C₂²×Dic₃	1	2	yes	—
148	C₂×C₃	—	2	4	no	—
149	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
150	C₂×C₃	C₂×#117	1	2	yes	—
151	C₂×C₃	C₄×#41	1	2	yes	—
152	C₂×C₃	C₄×#41	1	2	yes	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	C₂×C₄×Dic₃	1	4	yes	—
157	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
158	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
159	C₂²	D₃×#41	1	4	yes	—
160	C₂²	C₂³×Dic₃	1	1	yes	characteristic
161	C₄	—	2	8	no	—
162	C₄	—	3	8	no	—
163	C₄	—	6	4	no	—
164	C₄	—	6	8	no	—
165	C₄	C₂×C₄×Dic₃	1	4	yes	—
166	C₄	C₂×C₄×Dic₃	1	4	yes	—
167	C₃	C₄×#117	1	1	yes	characteristic, complemented, Sylow
168	C₂	—	2	4	no	—
169	C₂	D₃×#117	1	2	yes	—
170	C₂	#5522	1	2	yes	—
171	C₂	C₂²×C₄×Dic₃	1	1	yes	characteristic
172	C₂	Dic₃×#41	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₃
Fitting	C₂×C₃×#117	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
hypercenter	C₂×#117	D₃
hypocenter	C₃	C₄×#117
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
4. Dic₃×C₄⋊C₄	C₂	semidirect
6. #4932	C₂	semidirect
7. C₄²×Dic₃	C₂	semidirect
11. C₃×#117	C₄	semidirect
16. #926	C₄	semidirect
18. C₂×C₄×Dic₃	C₄	semidirect
44. C₃×C₄⋊C₄	C₂×C₄	semidirect
46. C₃×C₄²	C₂×C₄	semidirect
48. C₄⋊Dic₃	C₂×C₄	semidirect
53. C₄×Dic₃	C₂×C₄	semidirect
57. #117	Dic₃	direct (N × H)
76. C₂×C₃×C₄	C₄²	semidirect
104. C₄⋊C₄	C₂×Dic₃	semidirect
112. C₄²	C₂×Dic₃	semidirect
126. Dic₃	#117	direct (N × H)
140. C₂×C₄	C₄×Dic₃	semidirect
167. C₃	C₄×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#117	2
C₂×#926	2
C₂²×C₄×Dic₃	2
Dic₃×C₄⋊C₄	2
Dic₃×C₂²⋊C₄	2
#4932	2
C₄²×Dic₃	2
C₄×#117	3

Subgroups — Dic3×#117 #82222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×#117 #82222