Subgroups — C₃×#2357 #84221

Order 384.

560 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#390	C₂	1	2	yes	maximal
3	C₃×#388	C₂	1	4	yes	maximal, complemented
4	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#381	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
9	#2357	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₃×C₄×D₄	C₄	1	8	yes	complemented
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
17	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
18	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄²	C₂²	1	2	yes	—
20	C₂×C₃×C₄²	C₂²	1	4	yes	—
21	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
22	C₃×#95	C₂²	1	1	yes	characteristic
23	C₃×#95	C₂²	1	2	yes	—
24	C₃×#95	C₂²	1	1	yes	characteristic
25	C₃×#95	C₂²	1	4	yes	—
26	C₃×#95	C₂²	1	2	yes	—
27	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
28	#390	C₂×C₃	1	2	yes	—
29	#388	C₂×C₃	1	4	yes	complemented
30	#382	C₂×C₃	1	1	yes	characteristic, complemented
31	#381	C₂×C₃	1	2	yes	complemented
32	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
33	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
34	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
35	C₂⁴×C₃	C₂³	1	2	yes	—
36	C₂×C₃×D₄	—	2	2	no	—
37	C₂×C₃×D₄	—	2	2	no	—
38	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
39	C₂²×C₃×C₄	—	4	4	no	—
40	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
41	C₂²×C₃×C₄	C₂³	1	2	yes	—
42	C₂²×C₃×C₄	C₂³	1	4	yes	—
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₃×C₄⋊C₄	—	2	4	no	—
49	C₃×C₄⋊C₄	—	2	4	no	—
50	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
51	C₃×C₂²⋊C₄	—	2	8	no	—
52	C₃×C₂²⋊C₄	—	4	4	no	—
53	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
54	C₃×C₄²	—	2	4	no	—
55	C₃×C₄²	—	4	4	no	—
56	C₃×C₄²	C₂×C₄	1	4	yes	—
57	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
58	C₂³×C₄	C₂²×C₃	1	2	yes	—
59	C₄×D₄	C₃×C₄	1	8	yes	complemented
60	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
64	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
65	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₄²	C₂²×C₃	1	2	yes	—
67	C₂×C₄²	C₂²×C₃	1	4	yes	—
68	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
69	#95	C₂²×C₃	1	1	yes	characteristic
70	#95	C₂²×C₃	1	2	yes	—
71	#95	C₂²×C₃	1	1	yes	characteristic
72	#95	C₂²×C₃	1	4	yes	—
73	#95	C₂²×C₃	1	2	yes	—
74	C₂³×C₃	—	2	4	no	—
75	C₂³×C₃	—	2	2	no	—
76	C₂³×C₃	—	2	2	no	—
77	C₂³×C₃	—	2	4	no	—
78	C₂³×C₃	C₂²×C₄	1	4	yes	—
79	C₂³×C₃	C₂⁴	1	1	yes	characteristic
80	C₃×D₄	—	2	8	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	—	2	2	no	—
89	C₂×C₃×C₄	—	2	4	no	—
90	C₂×C₃×C₄	—	2	4	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	4	8	no	—
93	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
94	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
95	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
96	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₄	#41	1	4	yes	—
98	C₂×C₃×C₄	#41	1	2	yes	—
99	C₂×C₃×C₄	#41	1	2	yes	—
100	C₂⁴	C₂³×C₃	1	2	yes	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
104	C₂²×C₄	—	4	4	no	—
105	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
106	C₂²×C₄	C₂³×C₃	1	2	yes	—
107	C₂²×C₄	C₂³×C₃	1	4	yes	—
108	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₂²×C₄	C₂³×C₃	1	2	yes	—
110	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₂²×C₄	C₂³×C₃	1	2	yes	—
112	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
116	C₂²⋊C₄	—	2	8	no	—
117	C₂²⋊C₄	—	4	4	no	—
118	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
119	C₄²	—	2	4	no	—
120	C₄²	—	4	4	no	—
121	C₄²	C₂×C₃×C₄	1	4	yes	—
122	C₂²×C₃	—	2	4	no	—
123	C₂²×C₃	—	2	4	no	—
124	C₂²×C₃	—	2	4	no	—
125	C₂²×C₃	—	4	4	no	—
126	C₂²×C₃	—	4	4	no	—
127	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
128	C₂²×C₃	C₂×#41	1	1	yes	characteristic
129	C₂²×C₃	C₂×#41	1	1	yes	characteristic
130	C₂²×C₃	C₂×#41	1	2	yes	—
131	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	—	2	2	no	—
134	C₃×C₄	—	2	4	no	—
135	C₃×C₄	—	2	4	no	—
136	C₃×C₄	—	4	2	no	—
137	C₃×C₄	—	4	4	no	—
138	C₃×C₄	#117	1	4	yes	—
139	C₂³	—	2	4	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	4	no	—
143	C₂³	C₂²×C₃×C₄	1	4	yes	—
144	C₂³	C₂⁴×C₃	1	1	yes	characteristic
145	D₄	—	2	8	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
159	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
160	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
161	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
162	C₂×C₄	C₃×#41	1	4	yes	—
163	C₂×C₄	C₃×#41	1	2	yes	—
164	C₂×C₄	C₃×#41	1	2	yes	—
165	C₂×C₃	—	4	4	no	—
166	C₂×C₃	C₂×#117	1	1	yes	characteristic
167	C₂×C₃	C₄×#41	1	1	yes	characteristic
168	C₂×C₃	#518	1	1	yes	characteristic
169	C₂×C₃	#553	1	2	yes	—
170	C₂×C₃	#559	1	2	yes	—
171	C₂²	—	2	4	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	4	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
177	C₂²	C₂×C₃×#41	1	1	yes	characteristic
178	C₂²	C₂×C₃×#41	1	2	yes	—
179	C₂²	C₂×C₃×#41	1	1	yes	characteristic
180	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₄	—	4	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₃×#117	1	4	yes	—
188	C₃	#2357	1	1	yes	characteristic, direct product factor, Sylow
189	C₂	—	4	4	no	—
190	C₂	C₂×C₃×#117	1	1	yes	characteristic
191	C₂	C₃×C₄×#41	1	1	yes	characteristic
192	C₂	C₃×#518	1	1	yes	characteristic
193	C₂	C₃×#553	1	2	yes	—
194	C₂	C₃×#559	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×#388	C₂	semidirect
4. C₃×#382	C₂	semidirect
5. C₃×#381	C₂	semidirect
6. C₃×C₄×C₄⋊C₄	C₂	semidirect
7. C₃×C₄×C₄⋊C₄	C₂	semidirect
8. C₃×C₄×C₂²⋊C₄	C₂	semidirect
9. #2357	C₃	direct (N × H)
12. C₃×C₄×D₄	C₄	semidirect
29. #388	C₂×C₃	semidirect
30. #382	C₂×C₃	semidirect
31. #381	C₂×C₃	semidirect
32. C₄×C₄⋊C₄	C₂×C₃	semidirect
33. C₄×C₄⋊C₄	C₂×C₃	semidirect
34. C₄×C₂²⋊C₄	C₂×C₃	semidirect
59. C₄×D₄	C₃×C₄	semidirect
188. C₃	#2357	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×D₄	2
C₃×#390	2
C₃×#388	2
C₃×#382	2
C₃×#381	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2357	3

Subgroups — C3×#2357 #84221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2357 #84221