Subgroups — C₃×#2542 #84406

Order 384.

676 conjugacy classes of subgroups.

The subgroup lattice has 186 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, complemented
2	C₂×C₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#399	C₂	1	4	yes	maximal, complemented
5	C₃×#390	C₂	1	2	yes	maximal
6	C₃×#384	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#2542	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₈	C₂²	1	2	yes	—
10	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₃×C₄⋊Q₈	—	2	4	no	—
13	C₃×C₂²⋊Q₈	—	2	16	no	—
14	C₃×C₄×D₄	—	2	4	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	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	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
27	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
28	#399	C₂×C₃	1	4	yes	complemented
29	#390	C₂×C₃	1	2	yes	—
30	#384	C₂×C₃	1	2	yes	complemented
31	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂⁴×C₃	C₂³	1	2	yes	—
33	C₂×C₃×Q₈	—	2	8	no	—
34	C₂×C₃×Q₈	—	2	4	no	—
35	C₂×C₃×D₄	—	2	2	no	—
36	C₂×C₃×D₄	—	2	2	no	—
37	C₂×C₃×D₄	Q₈	1	4	yes	complemented
38	C₂²×C₃×C₄	—	2	4	no	—
39	C₂²×C₃×C₄	—	2	8	no	—
40	C₂²×C₃×C₄	C₂³	1	4	yes	—
41	C₂²×C₃×C₄	C₂³	1	2	yes	—
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂³	1	4	yes	—
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₃×C₄⋊C₄	—	2	2	no	—
47	C₃×C₄⋊C₄	—	2	16	no	—
48	C₃×C₄⋊C₄	—	2	4	no	—
49	C₃×C₄⋊C₄	—	2	4	no	—
50	C₃×C₄⋊C₄	D₄	1	8	yes	complemented
51	C₃×C₂²⋊C₄	—	2	4	no	—
52	C₃×C₂²⋊C₄	—	2	16	no	—
53	C₃×C₄²	—	2	4	no	—
54	C₃×C₄²	D₄	1	4	yes	—
55	C₂²×Q₈	C₂²×C₃	1	2	yes	—
56	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₂³×C₄	C₂²×C₃	1	2	yes	—
58	C₄⋊Q₈	—	2	4	no	—
59	C₂²⋊Q₈	—	2	16	no	—
60	C₄×D₄	—	2	4	no	—
61	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	—
63	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
68	C₂×C₄²	C₂²×C₃	1	2	yes	—
69	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
70	#95	C₂²×C₃	1	2	yes	—
71	C₂³×C₃	—	2	4	no	—
72	C₂³×C₃	—	2	2	no	—
73	C₂³×C₃	—	2	2	no	—
74	C₂³×C₃	—	2	4	no	—
75	C₂³×C₃	C₂×Q₈	1	4	yes	—
76	C₂³×C₃	C₂⁴	1	1	yes	characteristic
77	C₃×Q₈	—	4	8	no	—
78	C₃×D₄	—	2	8	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	2	no	—
83	C₂×C₃×C₄	—	2	8	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	—	2	4	no	—
86	C₂×C₃×C₄	—	4	8	no	—
87	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄	C₂×D₄	1	8	yes	—
89	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
92	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
93	C₂×C₃×C₄	#41	1	2	yes	—
94	C₂⁴	C₂³×C₃	1	2	yes	—
95	C₂×Q₈	—	2	8	no	—
96	C₂×Q₈	—	2	4	no	—
97	C₂×D₄	—	2	2	no	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	C₃×Q₈	1	4	yes	complemented
100	C₂²×C₄	—	2	4	no	—
101	C₂²×C₄	—	2	8	no	—
102	C₂²×C₄	C₂³×C₃	1	4	yes	—
103	C₂²×C₄	C₂³×C₃	1	2	yes	—
104	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₂²×C₄	C₂³×C₃	1	4	yes	—
106	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	16	no	—
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	C₃×D₄	1	8	yes	complemented
113	C₂²⋊C₄	—	2	4	no	—
114	C₂²⋊C₄	—	2	16	no	—
115	C₄²	—	2	4	no	—
116	C₄²	C₃×D₄	1	4	yes	—
117	C₂²×C₃	—	2	4	no	—
118	C₂²×C₃	—	2	8	no	—
119	C₂²×C₃	—	4	2	no	—
120	C₂²×C₃	—	4	2	no	—
121	C₂²×C₃	—	4	4	no	—
122	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₃	C₂²×D₄	1	2	yes	—
124	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₃	C₂×#41	1	1	yes	characteristic
126	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
127	C₃×C₄	—	2	2	no	—
128	C₃×C₄	—	2	8	no	—
129	C₃×C₄	—	4	2	no	—
130	C₃×C₄	—	4	4	no	—
131	C₃×C₄	C₂²⋊Q₈	1	4	yes	complemented
132	C₃×C₄	#127	1	4	yes	—
133	C₂³	—	2	4	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	2	4	no	—
137	C₂³	C₂×C₃×Q₈	1	4	yes	—
138	C₂³	C₂⁴×C₃	1	1	yes	characteristic
139	Q₈	—	4	8	no	—
140	D₄	—	2	8	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
150	C₂×C₄	C₂×C₃×D₄	1	8	yes	—
151	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
152	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
153	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
154	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
155	C₂×C₄	C₃×#41	1	2	yes	—
156	C₂×C₃	—	4	4	no	—
157	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₃	C₂×#127	1	1	yes	characteristic
159	C₂×C₃	#536	1	1	yes	characteristic
160	C₂×C₃	#547	1	2	yes	—
161	C₂×C₃	D₄×Q₈	1	2	yes	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	8	no	—
164	C₂²	—	4	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	4	no	—
167	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
168	C₂²	C₂²×C₃×D₄	1	2	yes	—
169	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
170	C₂²	C₂×C₃×#41	1	1	yes	characteristic
171	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	8	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	4	no	—
176	C₄	C₃×C₂²⋊Q₈	1	4	yes	complemented
177	C₄	C₃×#127	1	4	yes	—
178	C₃	#2542	1	1	yes	characteristic, direct product factor, Sylow
179	C₂	—	4	4	no	—
180	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
181	C₂	C₂×C₃×#127	1	1	yes	characteristic
182	C₂	C₃×#536	1	1	yes	characteristic
183	C₂	C₃×#547	1	2	yes	—
184	C₂	C₃×D₄×Q₈	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	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×C₄⋊Q₈	C₂	semidirect
2. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
4. C₃×#399	C₂	semidirect
6. C₃×#384	C₂	semidirect
7. C₃×C₄×C₄⋊C₄	C₂	semidirect
8. #2542	C₃	direct (N × H)
25. C₂×C₄⋊Q₈	C₂×C₃	semidirect
26. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
28. #399	C₂×C₃	semidirect
30. #384	C₂×C₃	semidirect
31. C₄×C₄⋊C₄	C₂×C₃	semidirect
37. C₂×C₃×D₄	Q₈	semidirect
50. C₃×C₄⋊C₄	D₄	semidirect
99. C₂×D₄	C₃×Q₈	semidirect
112. C₄⋊C₄	C₃×D₄	semidirect
131. C₃×C₄	C₂²⋊Q₈	semidirect
176. C₄	C₃×C₂²⋊Q₈	semidirect
178. C₃	#2542	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×C₂²⋊Q₈	2
C₂×C₃×C₄×D₄	2
C₃×#399	2
C₃×#390	2
C₃×#384	2
C₃×C₄×C₄⋊C₄	2
#2542	3

Subgroups — C3×#2542 #84406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2542 #84406