Subgroups — C₃×#3384 #85248

Order 384.

374 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#501	C₂	1	2	yes	maximal, complemented
4	C₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#478	C₂	1	2	yes	maximal, complemented
6	C₃×#476	C₂	1	2	yes	maximal, complemented
7	C₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#442	C₂	1	2	yes	maximal
11	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic
12	#3384	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	—	2	1	no	—
14	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×#125	C₂²	1	2	yes	—
19	C₃×C₂²⋊Q₈	—	2	1	no	—
20	C₃×C₂²⋊Q₈	—	2	1	no	—
21	C₃×C₂²⋊Q₈	—	2	2	no	—
22	C₃×C₂²⋊Q₈	—	2	2	no	—
23	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
25	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
26	C₃×C₄×D₄	C₂²	1	2	yes	—
27	C₃×C₄×D₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄⋊C₄	—	2	1	no	—
29	C₃×#107	C₂²	1	1	yes	characteristic
30	C₃×#107	C₂²	1	2	yes	—
31	C₃×#106	C₂²	1	2	yes	—
32	C₃×#106	C₂²	1	2	yes	—
33	C₃×#106	C₂²	1	2	yes	—
34	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₃×#102	C₂²	1	1	yes	characteristic
37	C₃×#102	C₂²	1	1	yes	characteristic
38	C₃×#102	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#97	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#554	C₂×C₃	1	1	yes	characteristic, complemented
43	D₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
44	#501	C₂×C₃	1	2	yes	complemented
45	#498	C₂×C₃	1	1	yes	characteristic, complemented
46	#478	C₂×C₃	1	2	yes	complemented
47	#476	C₂×C₃	1	2	yes	complemented
48	#476	C₂×C₃	1	1	yes	characteristic, complemented
49	#474	C₂×C₃	1	1	yes	characteristic, complemented
50	#446	C₂×C₃	1	1	yes	characteristic, complemented
51	#442	C₂×C₃	1	2	yes	—
52	C₄×D₈	C₂×C₃	1	1	yes	characteristic
53	C₂×C₃×Q₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	1	no	—
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	—	4	4	no	—
57	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
58	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
59	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	—	2	1	no	—
61	C₂²×C₃×C₄	—	2	2	no	—
62	C₂²×C₃×C₄	—	2	1	no	—
63	C₂²×C₃×C₄	—	2	2	no	—
64	C₃×D₈	Q₈	1	4	yes	complemented
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂³	1	2	yes	—
67	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	1	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	1	no	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₃×C₄⋊C₄	D₄	1	2	yes	—
75	C₃×C₄⋊C₄	C₂³	1	2	yes	—
76	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₃×C₄⋊C₄	C₂³	1	2	yes	—
78	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₃×C₂²⋊C₄	—	2	1	no	—
80	C₃×C₂²⋊C₄	—	2	2	no	—
81	C₃×C₂²⋊C₄	—	2	1	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₄²	C₂³	1	1	yes	characteristic
84	C₃×C₄²	C₂³	1	2	yes	—
85	C₂²×Q₈	—	2	1	no	—
86	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
87	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
88	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
89	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
90	#125	C₂²×C₃	1	2	yes	—
91	C₂²⋊Q₈	—	2	1	no	—
92	C₂²⋊Q₈	—	2	1	no	—
93	C₂²⋊Q₈	—	2	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
96	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
97	C₄×D₄	C₂²×C₃	1	2	yes	—
98	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
99	C₄×D₄	C₂²×C₃	1	2	yes	—
100	C₂×C₄⋊C₄	—	2	1	no	—
101	#107	C₂²×C₃	1	1	yes	characteristic
102	#107	C₂²×C₃	1	2	yes	—
103	#106	C₂²×C₃	1	2	yes	—
104	#106	C₂²×C₃	1	2	yes	—
105	#106	C₂²×C₃	1	2	yes	—
106	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
107	C₄⋊C₈	C₂²×C₃	1	2	yes	—
108	#102	C₂²×C₃	1	1	yes	characteristic
109	#102	C₂²×C₃	1	1	yes	characteristic
110	#102	C₂²×C₃	1	2	yes	—
111	#102	C₂²×C₃	1	2	yes	—
112	#97	C₂²×C₃	1	2	yes	—
113	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
114	C₂³×C₃	—	2	1	no	—
115	C₂³×C₃	—	2	1	no	—
116	C₃×Q₈	—	2	2	no	—
117	C₃×Q₈	—	4	4	no	—
118	C₃×Q₈	—	4	2	no	—
119	C₃×Q₈	—	8	1	no	—
120	C₃×D₄	—	2	1	no	—
121	C₃×D₄	—	2	1	no	—
122	C₃×D₄	C₂×Q₈	1	2	yes	—
123	C₃×D₄	C₂×Q₈	1	2	yes	complemented
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	4	4	no	—
129	C₂×C₃×C₄	—	4	2	no	—
130	C₂×C₃×C₄	—	4	2	no	—
131	C₂×C₃×C₄	—	8	2	no	—
132	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
134	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
137	C₃×C₈	—	2	1	no	—
138	C₃×C₈	—	2	2	no	—
139	C₃×C₈	C₂×Q₈	1	2	yes	complemented
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
145	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
146	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	2	no	—
151	D₈	C₃×Q₈	1	4	yes	complemented
152	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
153	C₂×C₈	C₂³×C₃	1	2	yes	—
154	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
161	C₄⋊C₄	C₃×D₄	1	2	yes	—
162	C₄⋊C₄	C₂³×C₃	1	2	yes	—
163	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
164	C₄⋊C₄	C₂³×C₃	1	2	yes	—
165	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
166	C₂²⋊C₄	—	2	1	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	1	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₄²	C₂³×C₃	1	1	yes	characteristic
171	C₄²	C₂³×C₃	1	2	yes	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	4	1	no	—
175	C₂²×C₃	—	4	1	no	—
176	C₂²×C₃	—	4	1	no	—
177	C₂²×C₃	—	4	1	no	—
178	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	4	1	no	—
183	C₃×C₄	—	4	2	no	—
184	C₃×C₄	—	4	1	no	—
185	C₃×C₄	—	4	2	no	—
186	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
187	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
188	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂³	—	2	1	no	—
190	C₂³	—	2	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	4	4	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	8	1	no	—
195	D₄	—	2	1	no	—
196	D₄	—	2	1	no	—
197	D₄	C₂×C₃×Q₈	1	2	yes	—
198	D₄	C₂×C₃×Q₈	1	2	yes	complemented
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
209	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
210	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	2	no	—
214	C₈	C₂×C₃×Q₈	1	2	yes	complemented
215	C₂×C₃	—	4	2	no	—
216	C₂×C₃	—	4	2	no	—
217	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
218	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₂×C₃	#577	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
235	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
236	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
237	C₃	#3384	1	1	yes	characteristic, direct product factor, Sylow
238	C₂	—	4	2	no	—
239	C₂	—	4	2	no	—
240	C₂	C₃×D₄×Q₈	1	1	yes	characteristic
241	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×#554	C₂	semidirect
2. C₃×D₄×Q₈	C₂	semidirect
3. C₃×#501	C₂	semidirect
4. C₃×#498	C₂	semidirect
5. C₃×#478	C₂	semidirect
6. C₃×#476	C₂	semidirect
7. C₃×#476	C₂	semidirect
8. C₃×#474	C₂	semidirect
9. C₃×#446	C₂	semidirect
12. #3384	C₃	direct (N × H)
42. #554	C₂×C₃	semidirect
43. D₄×Q₈	C₂×C₃	semidirect
44. #501	C₂×C₃	semidirect
45. #498	C₂×C₃	semidirect
46. #478	C₂×C₃	semidirect
47. #476	C₂×C₃	semidirect
48. #476	C₂×C₃	semidirect
49. #474	C₂×C₃	semidirect
50. #446	C₂×C₃	semidirect
64. C₃×D₈	Q₈	semidirect
123. C₃×D₄	C₂×Q₈	semidirect
139. C₃×C₈	C₂×Q₈	semidirect
151. D₈	C₃×Q₈	semidirect
198. D₄	C₂×C₃×Q₈	semidirect
214. C₈	C₂×C₃×Q₈	semidirect
237. C₃	#3384	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#554	2
C₃×D₄×Q₈	2
C₃×#501	2
C₃×#498	2
C₃×#478	2
C₃×#476	2
C₃×#474	2
C₃×#446	2
C₃×#442	2
C₃×C₄×D₈	2
#3384	3

Subgroups — C3×#3384 #85248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3384 #85248