Subgroups — C₃×#2677 #84541

Order 384.

544 conjugacy classes of subgroups.

The subgroup lattice has 234 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	2	yes	maximal, complemented
2	C₂×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal
3	C₃×#399	C₂	1	1	yes	maximal, characteristic
4	C₃×#398	C₂	1	2	yes	maximal, complemented
5	C₃×#395	C₂	1	2	yes	maximal, complemented
6	C₃×#393	C₂	1	1	yes	maximal, characteristic
7	C₃×#388	C₂	1	2	yes	maximal
8	C₃×#387	C₂	1	2	yes	maximal, complemented
9	C₃×#385	C₂	1	1	yes	maximal, characteristic
10	#2677	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Q₈	C₂²	1	2	yes	—
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Q₈	—	2	8	no	—
15	C₃×C₂²⋊Q₈	—	2	8	no	—
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	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄²	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	1	yes	characteristic
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	2	yes	—
32	C₃×#95	C₂²	1	2	yes	—
33	C₃×#95	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Q₈	C₂×C₃	1	2	yes	complemented
35	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
36	#399	C₂×C₃	1	1	yes	characteristic
37	#398	C₂×C₃	1	2	yes	complemented
38	#395	C₂×C₃	1	2	yes	complemented
39	#393	C₂×C₃	1	1	yes	characteristic
40	#388	C₂×C₃	1	2	yes	—
41	#387	C₂×C₃	1	2	yes	complemented
42	#385	C₂×C₃	1	1	yes	characteristic
43	C₂⁴×C₃	C₂³	1	1	yes	characteristic
44	C₂×C₃×Q₈	—	2	4	no	—
45	C₂×C₃×Q₈	—	2	4	no	—
46	C₂×C₃×Q₈	—	2	4	no	—
47	C₂×C₃×Q₈	—	4	2	no	—
48	C₂×C₃×Q₈	—	4	1	no	—
49	C₂²×C₃×C₄	—	2	4	no	—
50	C₂²×C₃×C₄	—	4	1	no	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂³	1	2	yes	—
56	C₂²×C₃×C₄	C₂³	1	2	yes	—
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₃×C₄⋊C₄	—	2	4	no	—
60	C₃×C₄⋊C₄	—	2	4	no	—
61	C₃×C₄⋊C₄	—	2	4	no	—
62	C₃×C₄⋊C₄	—	2	4	no	—
63	C₃×C₄⋊C₄	—	2	4	no	—
64	C₃×C₄⋊C₄	—	4	1	no	—
65	C₃×C₄⋊C₄	D₄	1	8	yes	—
66	C₃×C₂²⋊C₄	—	2	4	no	—
67	C₃×C₂²⋊C₄	—	2	4	no	—
68	C₃×C₂²⋊C₄	—	4	1	no	—
69	C₃×C₂²⋊C₄	Q₈	1	4	yes	complemented
70	C₃×C₄²	—	2	4	no	—
71	C₂²×Q₈	C₂²×C₃	1	2	yes	—
72	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
73	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
74	C₄⋊Q₈	—	2	8	no	—
75	C₂²⋊Q₈	—	2	8	no	—
76	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₄²	C₂²×C₃	1	2	yes	—
88	#95	C₂²×C₃	1	2	yes	—
89	#95	C₂²×C₃	1	1	yes	characteristic
90	#95	C₂²×C₃	1	2	yes	—
91	#95	C₂²×C₃	1	2	yes	—
92	#95	C₂²×C₃	1	2	yes	—
93	#95	C₂²×C₃	1	1	yes	characteristic
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	—	2	1	no	—
96	C₂³×C₃	—	2	1	no	—
97	C₂³×C₃	—	2	2	no	—
98	C₂³×C₃	C₂×Q₈	1	2	yes	—
99	C₂³×C₃	C₂⁴	1	1	yes	characteristic
100	C₃×Q₈	—	4	8	no	—
101	C₃×Q₈	—	4	4	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	4	4	no	—
121	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
122	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
123	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
124	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
125	C₂⁴	C₂³×C₃	1	1	yes	characteristic
126	C₂×Q₈	—	2	4	no	—
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	4	2	no	—
130	C₂×Q₈	—	4	1	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	4	1	no	—
133	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₄	C₂³×C₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₃	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	C₃×D₄	1	8	yes	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	—	4	1	no	—
151	C₂²⋊C₄	C₃×Q₈	1	4	yes	complemented
152	C₄²	—	2	4	no	—
153	C₂²×C₃	—	2	2	no	—
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	—	4	2	no	—
156	C₂²×C₃	—	4	1	no	—
157	C₂²×C₃	—	4	1	no	—
158	C₂²×C₃	C₂²×D₄	1	2	yes	—
159	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
160	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	—	2	4	no	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	2	4	no	—
167	C₃×C₄	—	4	2	no	—
168	C₃×C₄	—	4	2	no	—
169	C₃×C₄	—	4	1	no	—
170	C₃×C₄	—	4	1	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	1	no	—
173	C₂³	—	2	1	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₂×C₃×Q₈	1	2	yes	—
176	C₂³	C₂⁴×C₃	1	1	yes	characteristic
177	Q₈	—	4	8	no	—
178	Q₈	—	4	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
199	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
200	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
201	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
202	C₂×C₃	—	4	2	no	—
203	C₂×C₃	#536	1	2	yes	—
204	C₂×C₃	#544	1	1	yes	characteristic
205	C₂×C₃	D₄²	1	1	yes	characteristic
206	C₂×C₃	D₄×Q₈	1	2	yes	—
207	C₂×C₃	#563	1	1	yes	characteristic
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	1	no	—
213	C₂²	C₂²×C₃×D₄	1	2	yes	—
214	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
215	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
216	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
217	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	4	no	—
220	C₄	—	2	4	no	—
221	C₄	—	2	4	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	1	no	—
226	C₃	#2677	1	1	yes	characteristic, direct product factor, Sylow
227	C₂	—	4	2	no	—
228	C₂	C₃×#536	1	2	yes	—
229	C₂	C₃×#544	1	1	yes	characteristic
230	C₂	C₃×D₄²	1	1	yes	characteristic
231	C₂	C₃×D₄×Q₈	1	2	yes	—
232	C₂	C₃×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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
4. C₃×#398	C₂	semidirect
5. C₃×#395	C₂	semidirect
8. C₃×#387	C₂	semidirect
10. #2677	C₃	direct (N × H)
34. C₂×C₄⋊Q₈	C₂×C₃	semidirect
37. #398	C₂×C₃	semidirect
38. #395	C₂×C₃	semidirect
41. #387	C₂×C₃	semidirect
69. C₃×C₂²⋊C₄	Q₈	semidirect
151. C₂²⋊C₄	C₃×Q₈	semidirect
226. C₃	#2677	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
C₃×#399	2
C₃×#398	2
C₃×#395	2
C₃×#393	2
C₃×#388	2
C₃×#387	2
C₃×#385	2
#2677	3

Subgroups — C3×#2677 #84541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2677 #84541