Subgroups — C₂×C₃×#397 #84254

Order 384.

820 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#397	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#95	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#397	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁵×C₃	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄	C₂²	1	2	yes	—
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊C₄	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
15	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
16	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
17	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	—
18	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₃×#95	C₂²	1	8	yes	complemented
20	C₃×#95	C₂²	1	4	yes	complemented
21	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²×C₂²⋊C₄	C₂×C₃	1	2	yes	—
23	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
24	#397	C₂×C₃	1	8	yes	direct product factor
25	C₂×#95	C₂×C₃	1	2	yes	complemented
26	C₂×#95	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂⁴×C₃	—	2	2	no	—
28	C₂⁴×C₃	—	2	2	no	—
29	C₂⁴×C₃	—	2	4	no	—
30	C₂⁴×C₃	—	2	4	no	—
31	C₂⁴×C₃	D₄	1	4	yes	—
32	C₂⁴×C₃	C₂³	1	2	yes	—
33	C₂⁴×C₃	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄	—	2	4	no	—
35	C₂²×C₃×C₄	—	2	4	no	—
36	C₂²×C₃×C₄	—	2	4	no	—
37	C₂²×C₃×C₄	—	2	2	no	—
38	C₂²×C₃×C₄	—	2	4	no	—
39	C₂²×C₃×C₄	—	2	4	no	—
40	C₂²×C₃×C₄	—	2	4	no	—
41	C₂²×C₃×C₄	—	2	4	no	—
42	C₂²×C₃×C₄	—	2	4	no	—
43	C₂²×C₃×C₄	D₄	1	4	yes	—
44	C₂²×C₃×C₄	C₂³	1	4	yes	—
45	C₂²×C₃×C₄	C₂³	1	2	yes	—
46	C₂²×C₃×C₄	C₂³	1	4	yes	—
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
48	C₃×C₄⋊C₄	—	2	8	no	—
49	C₃×C₂²⋊C₄	—	2	16	no	—
50	C₃×C₂²⋊C₄	—	2	8	no	—
51	C₂⁵	C₂²×C₃	1	1	yes	characteristic
52	C₂³×C₄	C₂²×C₃	1	2	yes	—
53	C₂³×C₄	C₂²×C₃	1	2	yes	—
54	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄⋊C₄	—	2	4	no	—
57	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
58	C₂×C₂²⋊C₄	—	2	8	no	—
59	C₂×C₂²⋊C₄	—	2	4	no	—
60	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
61	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
62	#95	C₂²×C₃	1	8	yes	complemented
63	#95	C₂²×C₃	1	4	yes	complemented
64	C₂³×C₃	—	2	8	no	—
65	C₂³×C₃	—	2	16	no	—
66	C₂³×C₃	—	2	2	no	—
67	C₂³×C₃	—	2	2	no	—
68	C₂³×C₃	—	2	4	no	—
69	C₂³×C₃	—	2	4	no	—
70	C₂³×C₃	—	4	4	no	—
71	C₂³×C₃	—	4	4	no	—
72	C₂³×C₃	—	4	8	no	—
73	C₂³×C₃	C₂×D₄	1	4	yes	—
74	C₂³×C₃	C₂×D₄	1	2	yes	—
75	C₂³×C₃	C₂×D₄	1	2	yes	—
76	C₂³×C₃	#41	1	2	yes	—
77	C₂³×C₃	#41	1	4	yes	—
78	C₂³×C₃	#41	1	4	yes	—
79	C₂³×C₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₃×C₄	—	2	8	no	—
81	C₂×C₃×C₄	—	2	4	no	—
82	C₂×C₃×C₄	—	2	4	no	—
83	C₂×C₃×C₄	—	2	4	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	2	4	no	—
86	C₂×C₃×C₄	—	2	4	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	—	2	4	no	—
89	C₂×C₃×C₄	—	2	4	no	—
90	C₂×C₃×C₄	—	4	8	no	—
91	C₂×C₃×C₄	—	4	8	no	—
92	C₂×C₃×C₄	—	4	4	no	—
93	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
94	C₂⁴	—	2	2	no	—
95	C₂⁴	—	2	2	no	—
96	C₂⁴	—	2	4	no	—
97	C₂⁴	—	2	4	no	—
98	C₂⁴	C₃×D₄	1	4	yes	—
99	C₂⁴	C₂³×C₃	1	2	yes	—
100	C₂⁴	C₂³×C₃	1	1	yes	characteristic
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	4	no	—
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	C₃×D₄	1	4	yes	—
111	C₂²×C₄	C₂³×C₃	1	4	yes	—
112	C₂²×C₄	C₂³×C₃	1	2	yes	—
113	C₂²×C₄	C₂³×C₃	1	4	yes	—
114	C₂²×C₄	C₂³×C₃	1	2	yes	—
115	C₄⋊C₄	—	2	8	no	—
116	C₂²⋊C₄	—	2	16	no	—
117	C₂²⋊C₄	—	2	8	no	—
118	C₂²×C₃	—	2	4	no	—
119	C₂²×C₃	—	2	8	no	—
120	C₂²×C₃	—	4	2	no	—
121	C₂²×C₃	—	4	2	no	—
122	C₂²×C₃	—	4	4	no	—
123	C₂²×C₃	—	4	16	no	—
124	C₂²×C₃	#121	1	4	yes	—
125	C₂²×C₃	#123	1	4	yes	—
126	C₂²×C₃	#123	1	8	yes	—
127	C₂²×C₃	#124	1	4	yes	—
128	C₂²×C₃	#126	1	8	yes	—
129	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₃	C₂×#41	1	1	yes	characteristic
132	C₂²×C₃	C₂×#41	1	2	yes	—
133	C₂²×C₃	C₂×#41	1	2	yes	—
134	C₃×C₄	—	2	4	no	—
135	C₃×C₄	—	4	4	no	—
136	C₃×C₄	—	4	4	no	—
137	C₃×C₄	—	4	2	no	—
138	C₂³	—	2	8	no	—
139	C₂³	—	2	16	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	4	no	—
143	C₂³	—	2	4	no	—
144	C₂³	—	4	4	no	—
145	C₂³	—	4	4	no	—
146	C₂³	—	4	8	no	—
147	C₂³	C₂×C₃×D₄	1	4	yes	—
148	C₂³	C₂×C₃×D₄	1	2	yes	—
149	C₂³	C₂×C₃×D₄	1	2	yes	—
150	C₂³	C₃×#41	1	2	yes	—
151	C₂³	C₃×#41	1	4	yes	—
152	C₂³	C₃×#41	1	4	yes	—
153	C₂³	C₂⁴×C₃	1	1	yes	characteristic
154	C₂×C₄	—	2	8	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
168	C₂×C₃	—	4	4	no	—
169	C₂×C₃	#397	1	8	yes	direct product factor
170	C₂×C₃	C₂×#121	1	1	yes	characteristic
171	C₂×C₃	C₂×#123	1	1	yes	characteristic
172	C₂×C₃	C₂×#123	1	2	yes	—
173	C₂×C₃	C₂×#124	1	1	yes	characteristic
174	C₂×C₃	C₂×#126	1	2	yes	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	8	no	—
177	C₂²	—	4	16	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₃×#121	1	4	yes	—
182	C₂²	C₃×#123	1	4	yes	—
183	C₂²	C₃×#123	1	8	yes	—
184	C₂²	C₃×#124	1	4	yes	—
185	C₂²	C₃×#126	1	8	yes	—
186	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
187	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
188	C₂²	C₂×C₃×#41	1	1	yes	characteristic
189	C₂²	C₂×C₃×#41	1	2	yes	—
190	C₂²	C₂×C₃×#41	1	2	yes	—
191	C₄	—	2	4	no	—
192	C₄	—	4	4	no	—
193	C₄	—	4	4	no	—
194	C₄	—	4	2	no	—
195	C₃	C₂×#397	1	1	yes	characteristic, direct product factor, Sylow
196	C₂	—	4	4	no	—
197	C₂	C₃×#397	1	8	yes	direct product factor
198	C₂	C₂×C₃×#121	1	1	yes	characteristic
199	C₂	C₂×C₃×#123	1	1	yes	characteristic
200	C₂	C₂×C₃×#123	1	2	yes	—
201	C₂	C₂×C₃×#124	1	1	yes	characteristic
202	C₂	C₂×C₃×#126	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₄⋊C₄	C₂	semidirect
4. C₃×#397	C₂	direct (N × H)
5. C₂×C₃×#95	C₂	semidirect
6. C₂×C₃×#95	C₂	semidirect
7. C₂×#397	C₃	direct (N × H)
14. C₂×C₃×C₄⋊C₄	C₂²	semidirect
19. C₃×#95	C₂²	semidirect
20. C₃×#95	C₂²	semidirect
21. C₂²×C₄⋊C₄	C₂×C₃	semidirect
24. #397	C₂×C₃	direct (N × H)
25. C₂×#95	C₂×C₃	semidirect
26. C₂×#95	C₂×C₃	semidirect
57. C₂×C₄⋊C₄	C₂²×C₃	semidirect
62. #95	C₂²×C₃	semidirect
63. #95	C₂²×C₃	semidirect
169. C₂×C₃	#397	direct (N × H)
195. C₃	C₂×#397	direct (N × H)
197. C₂	C₃×#397	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×C₄⋊C₄	2
C₂²×C₃×C₂²⋊C₄	2
C₃×#397	2
C₂×C₃×#95	2
C₂×#397	3

Subgroups — C2×C3×#397 #84254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#397 #84254