Subgroups — C₃×#2728 #84592

Order 384.

512 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₃×#400	C₂	1	2	yes	maximal, complemented
3	C₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#397	C₂	1	2	yes	maximal, complemented
5	C₃×#394	C₂	1	1	yes	maximal, characteristic
6	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#388	C₂	1	2	yes	maximal, complemented
8	C₃×#386	C₂	1	2	yes	maximal
9	C₃×#385	C₂	1	2	yes	maximal, complemented
10	#2728	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₃×C₂²⋊Q₈	—	2	8	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	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×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊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×C₄²	C₂²	1	1	yes	characteristic
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	1	yes	characteristic
32	C₃×#95	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
34	#400	C₂×C₃	1	2	yes	complemented
35	#398	C₂×C₃	1	1	yes	characteristic, complemented
36	#397	C₂×C₃	1	2	yes	complemented
37	#394	C₂×C₃	1	1	yes	characteristic
38	#391	C₂×C₃	1	1	yes	characteristic, complemented
39	#388	C₂×C₃	1	2	yes	complemented
40	#386	C₂×C₃	1	2	yes	—
41	#385	C₂×C₃	1	2	yes	complemented
42	C₂⁴×C₃	C₂³	1	2	yes	—
43	C₂×C₃×Q₈	—	4	1	no	—
44	C₂×C₃×Q₈	—	4	1	no	—
45	C₂×C₃×Q₈	D₄	1	4	yes	complemented
46	C₂×C₃×D₄	—	2	4	no	—
47	C₂×C₃×D₄	—	4	1	no	—
48	C₂²×C₃×C₄	—	2	4	no	—
49	C₂²×C₃×C₄	—	4	2	no	—
50	C₂²×C₃×C₄	—	4	2	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	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	2	yes	—
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
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	—	4	1	no	—
63	C₃×C₄⋊C₄	—	4	1	no	—
64	C₃×C₂²⋊C₄	—	2	4	no	—
65	C₃×C₂²⋊C₄	—	2	4	no	—
66	C₃×C₂²⋊C₄	—	2	4	no	—
67	C₃×C₂²⋊C₄	—	2	4	no	—
68	C₃×C₂²⋊C₄	—	4	2	no	—
69	C₃×C₄²	—	4	1	no	—
70	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
72	C₂³×C₄	C₂²×C₃	1	2	yes	—
73	C₂²⋊Q₈	—	2	8	no	—
74	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
85	#95	C₂²×C₃	1	2	yes	—
86	#95	C₂²×C₃	1	2	yes	—
87	#95	C₂²×C₃	1	2	yes	—
88	#95	C₂²×C₃	1	2	yes	—
89	#95	C₂²×C₃	1	2	yes	—
90	#95	C₂²×C₃	1	1	yes	characteristic
91	#95	C₂²×C₃	1	1	yes	characteristic
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	—	2	2	no	—
98	C₂³×C₃	#41	1	4	yes	—
99	C₂³×C₃	C₂⁴	1	1	yes	characteristic
100	C₃×Q₈	—	4	4	no	—
101	C₃×D₄	—	4	4	no	—
102	C₂×C₃×C₄	—	2	1	no	—
103	C₂×C₃×C₄	—	2	1	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	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	4	8	no	—
123	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
124	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
125	C₂⁴	C₂³×C₃	1	2	yes	—
126	C₂×Q₈	—	4	1	no	—
127	C₂×Q₈	—	4	1	no	—
128	C₂×Q₈	C₃×D₄	1	4	yes	complemented
129	C₂×D₄	—	2	4	no	—
130	C₂×D₄	—	4	1	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₃	1	2	yes	—
137	C₂²×C₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	4	2	no	—
152	C₄²	—	4	1	no	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	—	2	4	no	—
156	C₂²×C₃	—	4	2	no	—
157	C₂²×C₃	—	4	2	no	—
158	C₂²×C₃	—	4	2	no	—
159	C₂²×C₃	—	4	2	no	—
160	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
161	C₂²×C₃	C₂×#41	1	2	yes	—
162	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	—	4	2	no	—
168	C₃×C₄	—	4	2	no	—
169	C₃×C₄	—	4	2	no	—
170	C₃×C₄	—	4	2	no	—
171	C₃×C₄	—	4	1	no	—
172	C₃×C₄	—	4	1	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	C₃×#41	1	4	yes	—
180	C₂³	C₂⁴×C₃	1	1	yes	characteristic
181	Q₈	—	4	4	no	—
182	D₄	—	4	4	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	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	2	no	—
192	C₂×C₄	—	2	2	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	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
205	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
206	C₂×C₃	—	4	4	no	—
207	C₂×C₃	#534	1	1	yes	characteristic
208	C₂×C₃	#539	1	2	yes	—
209	C₂×C₃	#548	1	2	yes	—
210	C₂×C₃	#551	1	1	yes	characteristic
211	C₂×C₃	#563	1	1	yes	characteristic
212	C₂²	—	2	4	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
220	C₂²	C₂×C₃×#41	1	2	yes	—
221	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
222	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
223	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
224	C₄	—	2	4	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₃	#2728	1	1	yes	characteristic, direct product factor, Sylow
233	C₂	—	4	4	no	—
234	C₂	C₃×#534	1	1	yes	characteristic
235	C₂	C₃×#539	1	2	yes	—
236	C₂	C₃×#548	1	2	yes	—
237	C₂	C₃×#551	1	1	yes	characteristic
238	C₂	C₃×#563	1	1	yes	characteristic

	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×C₂²⋊Q₈	C₂	semidirect
2. C₃×#400	C₂	semidirect
3. C₃×#398	C₂	semidirect
4. C₃×#397	C₂	semidirect
6. C₃×#391	C₂	semidirect
7. C₃×#388	C₂	semidirect
9. C₃×#385	C₂	semidirect
10. #2728	C₃	direct (N × H)
33. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
34. #400	C₂×C₃	semidirect
35. #398	C₂×C₃	semidirect
36. #397	C₂×C₃	semidirect
38. #391	C₂×C₃	semidirect
39. #388	C₂×C₃	semidirect
41. #385	C₂×C₃	semidirect
45. C₂×C₃×Q₈	D₄	semidirect
128. C₂×Q₈	C₃×D₄	semidirect
232. C₃	#2728	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×C₂²⋊Q₈	2
C₃×#400	2
C₃×#398	2
C₃×#397	2
C₃×#394	2
C₃×#391	2
C₃×#388	2
C₃×#386	2
C₃×#385	2
#2728	3

Subgroups — C3×#2728 #84592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2728 #84592