Subgroups — C₃×#2365 #84229

Order 384.

552 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⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	2	yes	maximal
3	C₃×#391	C₂	1	2	yes	maximal
4	C₃×#389	C₂	1	2	yes	maximal
5	C₃×#387	C₂	1	4	yes	maximal
6	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal
9	#2365	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₈	C₂²	1	2	yes	—
11	C₃×C₄⋊Q₈	C₄	1	8	yes	complemented
12	C₃×C₄×Q₈	—	2	8	no	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄²	C₂²	1	2	yes	—
21	C₂×C₃×C₄²	C₂²	1	4	yes	—
22	C₂×C₃×C₄²	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
24	C₃×#95	C₂²	1	2	yes	—
25	C₃×#95	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	4	yes	—
27	C₃×#95	C₂²	1	4	yes	—
28	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
29	C₂×C₄×Q₈	C₂×C₃	1	2	yes	—
30	#391	C₂×C₃	1	2	yes	—
31	#389	C₂×C₃	1	2	yes	—
32	#387	C₂×C₃	1	4	yes	—
33	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
34	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
35	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	—
36	C₂×C₃×Q₈	—	2	4	no	—
37	C₂×C₃×Q₈	—	2	4	no	—
38	C₂×C₃×Q₈	C₂×C₄	1	8	yes	—
39	C₂²×C₃×C₄	C₂³	1	2	yes	—
40	C₂²×C₃×C₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	4	yes	—
43	C₂²×C₃×C₄	C₂³	1	2	yes	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
47	C₃×C₄⋊C₄	—	2	8	no	—
48	C₃×C₄⋊C₄	—	2	2	no	—
49	C₃×C₄⋊C₄	—	2	2	no	—
50	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
51	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
52	C₃×C₄⋊C₄	D₄	1	4	yes	—
53	C₃×C₄⋊C₄	Q₈	1	4	yes	complemented
54	C₃×C₄²	—	2	8	no	—
55	C₃×C₄²	—	2	4	no	—
56	C₃×C₄²	—	2	4	no	—
57	C₃×C₄²	C₂×C₄	1	4	yes	—
58	C₂²×Q₈	C₂²×C₃	1	2	yes	—
59	C₄⋊Q₈	C₃×C₄	1	8	yes	complemented
60	C₄×Q₈	—	2	8	no	—
61	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
62	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄²	C₂²×C₃	1	4	yes	—
69	C₂×C₄²	C₂²×C₃	1	2	yes	—
70	C₂×C₄²	C₂²×C₃	1	2	yes	—
71	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
72	#95	C₂²×C₃	1	2	yes	—
73	#95	C₂²×C₃	1	2	yes	—
74	#95	C₂²×C₃	1	4	yes	—
75	#95	C₂²×C₃	1	4	yes	—
76	C₂³×C₃	C₂⁴	1	1	yes	characteristic
77	C₃×Q₈	—	2	16	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	4	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	2	4	no	—
82	C₂×C₃×C₄	—	2	1	no	—
83	C₂×C₃×C₄	—	2	1	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	2	2	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	C₂²×C₄	1	2	yes	—
93	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
94	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
95	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
97	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
98	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
99	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
100	C₂×C₃×C₄	#41	1	2	yes	—
101	C₂×C₃×C₄	#41	1	2	yes	—
102	C₂×Q₈	—	2	4	no	—
103	C₂×Q₈	—	2	4	no	—
104	C₂×Q₈	C₂×C₃×C₄	1	8	yes	—
105	C₂²×C₄	C₂³×C₃	1	2	yes	—
106	C₂²×C₄	C₂³×C₃	1	2	yes	—
107	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₂²×C₄	C₂³×C₃	1	4	yes	—
109	C₂²×C₄	C₂³×C₃	1	2	yes	—
110	C₂²×C₄	C₂³×C₃	1	2	yes	—
111	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₄⋊C₄	—	2	8	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
117	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
118	C₄⋊C₄	C₃×D₄	1	4	yes	—
119	C₄⋊C₄	C₃×Q₈	1	4	yes	complemented
120	C₄²	—	2	8	no	—
121	C₄²	—	2	4	no	—
122	C₄²	—	2	4	no	—
123	C₄²	C₂×C₃×C₄	1	4	yes	—
124	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
125	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂²×C₃	C₂×#41	1	1	yes	characteristic
128	C₂²×C₃	C₂×#41	1	1	yes	characteristic
129	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
131	C₃×C₄	—	2	8	no	—
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	—	2	4	no	—
134	C₃×C₄	—	2	4	no	—
135	C₃×C₄	—	4	2	no	—
136	C₃×C₄	—	4	2	no	—
137	C₃×C₄	C₄×D₄	1	4	yes	—
138	C₃×C₄	C₄×Q₈	1	4	yes	complemented
139	C₂³	C₂⁴×C₃	1	1	yes	characteristic
140	Q₈	—	2	16	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
156	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
157	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
158	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
159	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
160	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
161	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
162	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
163	C₂×C₄	C₃×#41	1	2	yes	—
164	C₂×C₄	C₃×#41	1	2	yes	—
165	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₂×C₃	#519	1	1	yes	characteristic
168	C₂×C₃	#547	1	1	yes	characteristic
169	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
170	C₂×C₃	#556	1	1	yes	characteristic
171	C₂×C₃	#557	1	1	yes	characteristic
172	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
173	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
174	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
175	C₂²	C₂×C₃×#41	1	1	yes	characteristic
176	C₂²	C₂×C₃×#41	1	1	yes	characteristic
177	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
178	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
179	C₄	—	2	8	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	4	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	C₃×C₄×D₄	1	4	yes	—
186	C₄	C₃×C₄×Q₈	1	4	yes	complemented
187	C₃	#2365	1	1	yes	characteristic, direct product factor, Sylow
188	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
189	C₂	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
190	C₂	C₃×#519	1	1	yes	characteristic
191	C₂	C₃×#547	1	1	yes	characteristic
192	C₂	C₃×D₄×Q₈	1	1	yes	characteristic
193	C₂	C₃×#556	1	1	yes	characteristic
194	C₂	C₃×#557	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
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
9. #2365	C₃	direct (N × H)
11. C₃×C₄⋊Q₈	C₄	semidirect
53. C₃×C₄⋊C₄	Q₈	semidirect
59. C₄⋊Q₈	C₃×C₄	semidirect
119. C₄⋊C₄	C₃×Q₈	semidirect
138. C₃×C₄	C₄×Q₈	semidirect
186. C₄	C₃×C₄×Q₈	semidirect
187. C₃	#2365	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⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₃×#391	2
C₃×#389	2
C₃×#387	2
C₃×C₄×C₄⋊C₄	2
#2365	3

Subgroups — C3×#2365 #84229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2365 #84229