Subgroups — C₂²×C₃×#104 #84763

Order 384.

856 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₃×C₄	D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂²⋊C₄
prime radical	C₂²×C₃×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×M₁₆	C₂	semidirect
3. C₂²×C₃×C₄²	C₂	semidirect
4. C₂×C₃×#104	C₂	direct (N × H)
5. C₂²×#104	C₃	direct (N × H)
8. C₂×C₃×#41	C₄	semidirect
10. C₂²×C₃×Q₈	C₄	semidirect
12. C₂²×C₃×D₄	C₄	semidirect
17. C₂×C₃×M₁₆	C₂²	semidirect
20. C₂×C₃×C₄²	C₂²	semidirect
21. C₃×#104	C₂²	direct (N × H)
23. C₂²×M₁₆	C₂×C₃	semidirect
24. C₂²×C₄²	C₂×C₃	semidirect
25. C₂×#104	C₂×C₃	direct (N × H)
31. C₃×#41	C₂×C₄	semidirect
34. C₂×C₃×Q₈	C₂×C₄	semidirect
38. C₂×C₃×D₄	C₂×C₄	semidirect
51. C₃×M₁₆	C₂³	semidirect
54. C₃×C₄²	C₂³	semidirect
57. C₂×#41	C₃×C₄	semidirect
59. C₂²×Q₈	C₃×C₄	semidirect
61. C₂²×D₄	C₃×C₄	semidirect
66. C₂×M₁₆	C₂²×C₃	semidirect
69. C₂×C₄²	C₂²×C₃	semidirect
70. #104	C₂²×C₃	direct (N × H)
78. C₃×Q₈	C₂²×C₄	semidirect
82. C₃×D₄	C₂²×C₄	semidirect
103. #41	C₂×C₃×C₄	semidirect
106. C₂×Q₈	C₂×C₃×C₄	semidirect
110. C₂×D₄	C₂×C₃×C₄	semidirect
123. M₁₆	C₂³×C₃	semidirect
126. C₄²	C₂³×C₃	semidirect
131. C₂²×C₃	#104	direct (N × H)
148. Q₈	C₂²×C₃×C₄	semidirect
152. D₄	C₂²×C₃×C₄	semidirect
171. C₂×C₃	C₂×#104	direct (N × H)
177. C₂²	C₃×#104	direct (N × H)
187. C₃	C₂²×#104	direct (N × H)
191. C₂	C₂×C₃×#104	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#41	2
C₂²×C₃×M₁₆	2
C₂²×C₃×C₄²	2
C₂×C₃×#104	2
C₂²×#104	3

Subgroups — C22×C3×#104 #84763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#104 #84763