Subgroups — C₃×#2317 #84181

Order 384.

600 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#389	C₂	1	4	yes	maximal, complemented
5	C₃×#385	C₂	1	4	yes	maximal
6	C₃×#384	C₂	1	2	yes	maximal, complemented
7	C₃×#382	C₂	1	2	yes	maximal, complemented
8	#2317	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂³×C₃×C₄	C₂²	1	2	yes	—
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₃×#117	—	2	4	no	—
12	C₃×#117	C₄	1	8	yes	complemented
13	C₂×C₃×C₄⋊C₄	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	2	yes	—
24	C₃×#95	C₂²	1	8	yes	—
25	C₂×#117	C₂×C₃	1	1	yes	characteristic
26	C₂×#117	C₂×C₃	1	1	yes	characteristic
27	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
28	#389	C₂×C₃	1	4	yes	complemented
29	#385	C₂×C₃	1	4	yes	—
30	#384	C₂×C₃	1	2	yes	complemented
31	#382	C₂×C₃	1	2	yes	complemented
32	C₂⁴×C₃	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄	—	2	4	no	—
34	C₂²×C₃×C₄	—	4	4	no	—
35	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
36	C₂²×C₃×C₄	D₄	1	4	yes	—
37	C₂²×C₃×C₄	Q₈	1	4	yes	—
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	4	yes	—
40	C₂²×C₃×C₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	C₂³	1	4	yes	—
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₃×C₄⋊C₄	—	2	4	no	—
45	C₃×C₄⋊C₄	—	2	4	no	—
46	C₃×C₄⋊C₄	—	2	2	no	—
47	C₃×C₄⋊C₄	—	2	2	no	—
48	C₃×C₄⋊C₄	—	4	8	no	—
49	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
50	C₃×C₂²⋊C₄	—	2	4	no	—
51	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
52	C₃×C₄²	—	2	4	no	—
53	C₃×C₄²	C₂×C₄	1	8	yes	complemented
54	C₂³×C₄	C₂²×C₃	1	2	yes	—
55	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
56	#117	—	2	4	no	—
57	#117	C₃×C₄	1	8	yes	complemented
58	C₂×C₄⋊C₄	—	2	4	no	—
59	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
60	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	—
64	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂×C₄²	C₂²×C₃	1	2	yes	—
68	C₂×C₄²	C₂²×C₃	1	2	yes	—
69	#95	C₂²×C₃	1	8	yes	—
70	C₂³×C₃	—	2	2	no	—
71	C₂³×C₃	—	2	2	no	—
72	C₂³×C₃	C₂²×C₄	1	2	yes	—
73	C₂³×C₃	C₂×D₄	1	2	yes	—
74	C₂³×C₃	C₂×Q₈	1	2	yes	—
75	C₂³×C₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₃×C₄	—	2	4	no	—
77	C₂×C₃×C₄	—	2	4	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	4	no	—
81	C₂×C₃×C₄	—	2	4	no	—
82	C₂×C₃×C₄	—	2	8	no	—
83	C₂×C₃×C₄	—	2	4	no	—
84	C₂×C₃×C₄	—	4	8	no	—
85	C₂×C₃×C₄	C₄⋊C₄	1	16	yes	complemented
86	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
87	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
92	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
93	C₂⁴	C₂³×C₃	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	4	4	no	—
96	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
97	C₂²×C₄	C₃×D₄	1	4	yes	—
98	C₂²×C₄	C₃×Q₈	1	4	yes	—
99	C₂²×C₄	C₂³×C₃	1	2	yes	—
100	C₂²×C₄	C₂³×C₃	1	4	yes	—
101	C₂²×C₄	C₂³×C₃	1	2	yes	—
102	C₂²×C₄	C₂³×C₃	1	4	yes	—
103	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
104	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	4	8	no	—
110	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
111	C₂²⋊C₄	—	2	4	no	—
112	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
113	C₄²	—	2	4	no	—
114	C₄²	C₂×C₃×C₄	1	8	yes	complemented
115	C₂²×C₃	—	2	4	no	—
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	—	2	4	no	—
119	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
121	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
124	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
125	C₃×C₄	—	2	8	no	—
126	C₃×C₄	—	4	2	no	—
127	C₃×C₄	—	4	4	no	—
128	C₃×C₄	—	4	2	no	—
129	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
130	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	C₂²×C₃×C₄	1	2	yes	—
134	C₂³	C₂×C₃×D₄	1	2	yes	—
135	C₂³	C₂×C₃×Q₈	1	2	yes	—
136	C₂³	C₂⁴×C₃	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	C₃×C₄⋊C₄	1	16	yes	complemented
147	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
148	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
149	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
150	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
151	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
152	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
153	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
154	C₂×C₃	—	2	4	no	—
155	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₃	#520	1	2	yes	—
157	C₂×C₃	#535	1	1	yes	characteristic
158	C₂×C₃	#536	1	1	yes	characteristic
159	C₂×C₃	#544	1	2	yes	—
160	C₂²	—	2	4	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	4	no	—
164	C₂²	C₂×C₃×C₄⋊C₄	1	4	yes	—
165	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
166	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
167	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
168	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
169	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
170	C₄	—	2	8	no	—
171	C₄	—	4	4	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	2	no	—
174	C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
175	C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
176	C₃	#2317	1	1	yes	characteristic, direct product factor, Sylow
177	C₂	—	2	4	no	—
178	C₂	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
179	C₂	C₃×#520	1	2	yes	—
180	C₂	C₃×#535	1	1	yes	characteristic
181	C₂	C₃×#536	1	1	yes	characteristic
182	C₂	C₃×#544	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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
4. C₃×#389	C₂	semidirect
6. C₃×#384	C₂	semidirect
7. C₃×#382	C₂	semidirect
8. #2317	C₃	direct (N × H)
12. C₃×#117	C₄	semidirect
28. #389	C₂×C₃	semidirect
30. #384	C₂×C₃	semidirect
31. #382	C₂×C₃	semidirect
49. C₃×C₄⋊C₄	C₂×C₄	semidirect
53. C₃×C₄²	C₂×C₄	semidirect
57. #117	C₃×C₄	semidirect
85. C₂×C₃×C₄	C₄⋊C₄	semidirect
110. C₄⋊C₄	C₂×C₃×C₄	semidirect
114. C₄²	C₂×C₃×C₄	semidirect
146. C₂×C₄	C₃×C₄⋊C₄	semidirect
176. C₃	#2317	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×#117	2
C₂²×C₃×C₄⋊C₄	2
C₃×#389	2
C₃×#385	2
C₃×#384	2
C₃×#382	2
#2317	3

Subgroups — C3×#2317 #84181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2317 #84181