Subgroups — C₃×#2561 #84425

Order 384.

672 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#397	C₂	1	2	yes	maximal, complemented
4	C₃×#393	C₂	1	2	yes	maximal, complemented
5	C₃×#388	C₂	1	4	yes	maximal, complemented
6	C₃×#381	C₂	1	2	yes	maximal, complemented
7	C₃×#379	C₂	1	2	yes	maximal
8	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#2561	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂⁵×C₃	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₃×C₂²⋊Q₈	—	2	4	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
18	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
25	C₃×#95	C₂²	1	2	yes	complemented
26	C₃×#95	C₂²	1	4	yes	complemented
27	C₃×#95	C₂²	1	4	yes	complemented
28	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
30	#397	C₂×C₃	1	2	yes	complemented
31	#393	C₂×C₃	1	2	yes	complemented
32	#388	C₂×C₃	1	4	yes	complemented
33	#381	C₂×C₃	1	2	yes	complemented
34	#379	C₂×C₃	1	2	yes	—
35	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂⁴×C₃	—	2	2	no	—
37	C₂⁴×C₃	—	2	4	no	—
38	C₂⁴×C₃	—	2	4	no	—
39	C₂⁴×C₃	—	4	2	no	—
40	C₂⁴×C₃	C₂³	1	2	yes	—
41	C₂⁴×C₃	C₂³	1	1	yes	characteristic
42	C₂×C₃×Q₈	—	2	2	no	—
43	C₂×C₃×Q₈	—	4	2	no	—
44	C₂²×C₃×C₄	—	2	2	no	—
45	C₂²×C₃×C₄	—	2	4	no	—
46	C₂²×C₃×C₄	—	4	2	no	—
47	C₂²×C₃×C₄	D₄	1	4	yes	—
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	C₂³	1	4	yes	—
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	2	yes	—
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₃×C₄⋊C₄	—	2	2	no	—
55	C₃×C₄⋊C₄	—	2	4	no	—
56	C₃×C₂²⋊C₄	—	2	4	no	—
57	C₃×C₂²⋊C₄	—	2	8	no	—
58	C₃×C₂²⋊C₄	—	2	4	no	—
59	C₃×C₂²⋊C₄	—	2	4	no	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	—	4	4	no	—
62	C₃×C₄²	—	4	2	no	—
63	C₂⁵	C₂²×C₃	1	1	yes	characteristic
64	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
65	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂²⋊Q₈	—	2	4	no	—
68	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
69	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
70	C₂×C₂²⋊C₄	—	2	4	no	—
71	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
76	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
77	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
78	#95	C₂²×C₃	1	2	yes	complemented
79	#95	C₂²×C₃	1	4	yes	complemented
80	#95	C₂²×C₃	1	4	yes	complemented
81	C₂³×C₃	—	2	2	no	—
82	C₂³×C₃	—	2	2	no	—
83	C₂³×C₃	—	2	8	no	—
84	C₂³×C₃	—	2	2	no	—
85	C₂³×C₃	—	2	2	no	—
86	C₂³×C₃	—	2	4	no	—
87	C₂³×C₃	—	2	4	no	—
88	C₂³×C₃	—	4	16	no	—
89	C₂³×C₃	—	4	4	no	—
90	C₂³×C₃	—	4	4	no	—
91	C₂³×C₃	C₂×D₄	1	2	yes	—
92	C₂³×C₃	#41	1	4	yes	—
93	C₂³×C₃	#41	1	4	yes	—
94	C₂³×C₃	C₂⁴	1	1	yes	characteristic
95	C₃×Q₈	—	4	4	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	4	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	4	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	4	4	no	—
108	C₂×C₃×C₄	—	4	4	no	—
109	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
110	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
111	C₂⁴	—	2	2	no	—
112	C₂⁴	—	2	4	no	—
113	C₂⁴	—	2	4	no	—
114	C₂⁴	—	4	2	no	—
115	C₂⁴	C₂³×C₃	1	2	yes	—
116	C₂⁴	C₂³×C₃	1	1	yes	characteristic
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	C₃×D₄	1	4	yes	—
123	C₂²×C₄	C₂³×C₃	1	2	yes	—
124	C₂²×C₄	C₂³×C₃	1	4	yes	—
125	C₂²×C₄	C₂³×C₃	1	2	yes	—
126	C₂²×C₄	C₂³×C₃	1	2	yes	—
127	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
128	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	8	no	—
133	C₂²⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₂²⋊C₄	—	4	4	no	—
137	C₄²	—	4	2	no	—
138	C₂²×C₃	—	2	2	no	—
139	C₂²×C₃	—	2	2	no	—
140	C₂²×C₃	—	2	8	no	—
141	C₂²×C₃	—	2	4	no	—
142	C₂²×C₃	—	2	8	no	—
143	C₂²×C₃	—	4	16	no	—
144	C₂²×C₃	—	4	2	no	—
145	C₂²×C₃	—	4	2	no	—
146	C₂²×C₃	—	4	4	no	—
147	C₂²×C₃	#124	1	4	yes	—
148	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
149	C₂²×C₃	C₂×#41	1	2	yes	—
150	C₂²×C₃	C₂×#41	1	2	yes	—
151	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
152	C₃×C₄	—	2	2	no	—
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	4	2	no	—
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	—	4	2	no	—
157	C₃×C₄	—	4	4	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	8	no	—
161	C₂³	—	2	2	no	—
162	C₂³	—	2	2	no	—
163	C₂³	—	2	4	no	—
164	C₂³	—	2	4	no	—
165	C₂³	—	4	16	no	—
166	C₂³	—	4	4	no	—
167	C₂³	—	4	4	no	—
168	C₂³	C₂×C₃×D₄	1	2	yes	—
169	C₂³	C₃×#41	1	4	yes	—
170	C₂³	C₃×#41	1	4	yes	—
171	C₂³	C₂⁴×C₃	1	1	yes	characteristic
172	Q₈	—	4	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
187	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
188	C₂×C₃	—	2	4	no	—
189	C₂×C₃	—	4	4	no	—
190	C₂×C₃	#525	1	1	yes	characteristic
191	C₂×C₃	C₂×#124	1	1	yes	characteristic
192	C₂×C₃	#534	1	1	yes	characteristic
193	C₂×C₃	#551	1	4	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	8	no	—
199	C₂²	—	4	16	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	4	no	—
203	C₂²	C₃×#124	1	4	yes	—
204	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
205	C₂²	C₂×C₃×#41	1	2	yes	—
206	C₂²	C₂×C₃×#41	1	2	yes	—
207	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	4	no	—
214	C₃	#2561	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	—	2	4	no	—
216	C₂	—	4	4	no	—
217	C₂	C₃×#525	1	1	yes	characteristic
218	C₂	C₂×C₃×#124	1	1	yes	characteristic
219	C₂	C₃×#534	1	1	yes	characteristic
220	C₂	C₃×#551	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
3. C₃×#397	C₂	semidirect
4. C₃×#393	C₂	semidirect
5. C₃×#388	C₂	semidirect
6. C₃×#381	C₂	semidirect
8. C₃×C₄×C₂²⋊C₄	C₂	semidirect
9. #2561	C₃	direct (N × H)
11. C₂²×C₃×Q₈	C₂²	semidirect
15. C₂×C₃×C₄⋊C₄	C₂²	semidirect
16. C₂×C₃×C₄⋊C₄	C₂²	semidirect
24. C₂×C₃×C₄²	C₂²	semidirect
25. C₃×#95	C₂²	semidirect
26. C₃×#95	C₂²	semidirect
27. C₃×#95	C₂²	semidirect
28. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
30. #397	C₂×C₃	semidirect
31. #393	C₂×C₃	semidirect
32. #388	C₂×C₃	semidirect
33. #381	C₂×C₃	semidirect
35. C₄×C₂²⋊C₄	C₂×C₃	semidirect
64. C₂²×Q₈	C₂²×C₃	semidirect
68. C₂×C₄⋊C₄	C₂²×C₃	semidirect
69. C₂×C₄⋊C₄	C₂²×C₃	semidirect
77. C₂×C₄²	C₂²×C₃	semidirect
78. #95	C₂²×C₃	semidirect
79. #95	C₂²×C₃	semidirect
80. #95	C₂²×C₃	semidirect
214. C₃	#2561	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×C₂²⋊Q₈	2
C₂²×C₃×C₂²⋊C₄	2
C₃×#397	2
C₃×#393	2
C₃×#388	2
C₃×#381	2
C₃×#379	2
C₃×C₄×C₂²⋊C₄	2
#2561	3

Subgroups — C3×#2561 #84425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2561 #84425