Subgroups — C₃×#2293 #84157

Order 384.

652 conjugacy classes of subgroups.

The subgroup lattice has 206 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	C₂×C₃×#117	C₂	1	2	yes	maximal, complemented
3	C₃×#390	C₂	1	2	yes	maximal
4	C₃×#389	C₂	1	2	yes	maximal, complemented
5	C₃×#388	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₃×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
9	#2293	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	C₃×#117	—	2	8	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
18	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄²	C₂²	1	2	yes	—
21	C₂×C₃×C₄²	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄²	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
26	C₃×#95	C₂²	1	4	yes	—
27	C₃×#95	C₂²	1	2	yes	—
28	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
29	C₂×#117	C₂×C₃	1	2	yes	complemented
30	#390	C₂×C₃	1	2	yes	—
31	#389	C₂×C₃	1	2	yes	complemented
32	#388	C₂×C₃	1	4	yes	complemented
33	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
34	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
35	C₄³	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂⁴×C₃	C₂³	1	2	yes	—
37	C₂×C₃×D₄	—	2	4	no	—
38	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
39	C₂²×C₃×C₄	—	2	4	no	—
40	C₂²×C₃×C₄	—	2	4	no	—
41	C₂²×C₃×C₄	C₂×C₄	1	8	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	2	yes	—
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₃×C₄⋊C₄	—	2	8	no	—
50	C₃×C₄⋊C₄	—	2	4	no	—
51	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
52	C₃×C₂²⋊C₄	—	2	8	no	—
53	C₃×C₂²⋊C₄	—	2	8	no	—
54	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
55	C₃×C₄²	—	2	4	no	—
56	C₃×C₄²	—	2	4	no	—
57	C₃×C₄²	—	2	4	no	—
58	C₃×C₄²	—	2	2	no	—
59	C₃×C₄²	C₂×C₄	1	4	yes	—
60	C₃×C₄²	D₄	1	4	yes	complemented
61	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
62	C₂³×C₄	C₂²×C₃	1	2	yes	—
63	C₄×D₄	C₃×C₄	1	8	yes	complemented
64	#117	—	2	8	no	—
65	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
66	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
69	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
70	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₄²	C₂²×C₃	1	2	yes	—
72	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄²	C₂²×C₃	1	2	yes	—
74	C₂×C₄²	C₂²×C₃	1	2	yes	—
75	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
77	#95	C₂²×C₃	1	4	yes	—
78	#95	C₂²×C₃	1	2	yes	—
79	C₂³×C₃	—	2	2	no	—
80	C₂³×C₃	—	2	4	no	—
81	C₂³×C₃	—	2	2	no	—
82	C₂³×C₃	—	2	4	no	—
83	C₂³×C₃	C₂²×C₄	1	4	yes	—
84	C₂³×C₃	C₂⁴	1	1	yes	characteristic
85	C₃×D₄	—	2	8	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	—	2	2	no	—
89	C₂×C₃×C₄	—	2	2	no	—
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	16	no	—
94	C₂×C₃×C₄	—	2	4	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
98	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
99	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
100	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
101	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
102	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
103	C₂×C₃×C₄	#41	1	4	yes	—
104	C₂×C₃×C₄	#41	1	4	yes	—
105	C₂×C₃×C₄	#41	1	2	yes	—
106	C₂⁴	C₂³×C₃	1	2	yes	—
107	C₂×D₄	—	2	4	no	—
108	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
112	C₂²×C₄	C₂³×C₃	1	4	yes	—
113	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₂²×C₄	C₂³×C₃	1	2	yes	—
115	C₂²×C₄	C₂³×C₃	1	2	yes	—
116	C₂²×C₄	C₂³×C₃	1	2	yes	—
117	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	8	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
122	C₂²⋊C₄	—	2	8	no	—
123	C₂²⋊C₄	—	2	8	no	—
124	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
125	C₄²	—	2	4	no	—
126	C₄²	—	2	4	no	—
127	C₄²	—	2	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂×C₃×C₄	1	4	yes	—
130	C₄²	C₃×D₄	1	4	yes	complemented
131	C₂²×C₃	—	2	4	no	—
132	C₂²×C₃	—	2	4	no	—
133	C₂²×C₃	—	2	4	no	—
134	C₂²×C₃	—	4	4	no	—
135	C₂²×C₃	—	4	4	no	—
136	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
137	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₃	C₂×#41	1	2	yes	—
139	C₂²×C₃	C₂×#41	1	1	yes	characteristic
140	C₂²×C₃	C₂×#41	1	2	yes	—
141	C₃×C₄	—	2	4	no	—
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	2	4	no	—
144	C₃×C₄	—	2	4	no	—
145	C₃×C₄	—	4	4	no	—
146	C₃×C₄	#117	1	4	yes	complemented
147	C₃×C₄	C₄×D₄	1	4	yes	—
148	C₂³	—	2	2	no	—
149	C₂³	—	2	4	no	—
150	C₂³	—	2	2	no	—
151	C₂³	—	2	4	no	—
152	C₂³	C₂²×C₃×C₄	1	4	yes	—
153	C₂³	C₂⁴×C₃	1	1	yes	characteristic
154	D₄	—	2	8	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	16	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
167	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
168	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
169	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
170	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
171	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
172	C₂×C₄	C₃×#41	1	4	yes	—
173	C₂×C₄	C₃×#41	1	4	yes	—
174	C₂×C₄	C₃×#41	1	2	yes	—
175	C₂×C₃	—	4	4	no	—
176	C₂×C₃	C₂×#117	1	1	yes	characteristic
177	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×C₃	C₄×#41	1	1	yes	characteristic
179	C₂×C₃	#529	1	2	yes	—
180	C₂×C₃	#532	1	2	yes	—
181	C₂²	—	2	4	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	4	4	no	—
185	C₂²	—	4	4	no	—
186	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
187	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
188	C₂²	C₂×C₃×#41	1	2	yes	—
189	C₂²	C₂×C₃×#41	1	1	yes	characteristic
190	C₂²	C₂×C₃×#41	1	2	yes	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	4	no	—
195	C₄	—	4	4	no	—
196	C₄	C₃×#117	1	4	yes	complemented
197	C₄	C₃×C₄×D₄	1	4	yes	—
198	C₃	#2293	1	1	yes	characteristic, direct product factor, Sylow
199	C₂	—	4	4	no	—
200	C₂	C₂×C₃×#117	1	1	yes	characteristic
201	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
202	C₂	C₃×C₄×#41	1	1	yes	characteristic
203	C₂	C₃×#529	1	2	yes	—
204	C₂	C₃×#532	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#117	C₂	semidirect
4. C₃×#389	C₂	semidirect
5. C₃×#388	C₂	semidirect
6. C₃×C₄×C₄⋊C₄	C₂	semidirect
7. C₃×C₄×C₂²⋊C₄	C₂	semidirect
8. C₃×C₄³	C₂	semidirect
9. #2293	C₃	direct (N × H)
12. C₃×C₄×D₄	C₄	semidirect
29. C₂×#117	C₂×C₃	semidirect
31. #389	C₂×C₃	semidirect
32. #388	C₂×C₃	semidirect
33. C₄×C₄⋊C₄	C₂×C₃	semidirect
34. C₄×C₂²⋊C₄	C₂×C₃	semidirect
35. C₄³	C₂×C₃	semidirect
60. C₃×C₄²	D₄	semidirect
63. C₄×D₄	C₃×C₄	semidirect
130. C₄²	C₃×D₄	semidirect
146. C₃×C₄	#117	semidirect
196. C₄	C₃×#117	semidirect
198. C₃	#2293	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
C₂×C₃×#117	2
C₃×#390	2
C₃×#389	2
C₃×#388	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
C₃×C₄³	2
#2293	3

Subgroups — C3×#2293 #84157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2293 #84157