Subgroups — C₃×#3367 #85231

Order 384.

372 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#489	C₂	1	2	yes	maximal, complemented
4	C₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#477	C₂	1	2	yes	maximal, complemented
6	C₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#463	C₂	1	2	yes	maximal
8	C₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#440	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic
12	#3367	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	—	2	1	no	—
14	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₃×#126	—	2	2	no	—
20	C₃×#124	C₂²	1	2	yes	—
21	C₃×C₂²⋊Q₈	—	2	1	no	—
22	C₃×C₂²⋊Q₈	—	2	1	no	—
23	C₃×C₂²⋊Q₈	—	2	2	no	—
24	C₃×C₄×Q₈	C₂²	1	2	yes	—
25	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
26	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
27	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
28	C₃×C₄×D₄	C₂²	1	2	yes	—
29	C₃×#117	—	2	1	no	—
30	C₃×#107	C₂²	1	1	yes	characteristic
31	C₃×#107	C₂²	1	2	yes	—
32	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃×C₄⋊C₈	C₂²	1	2	yes	—
34	C₃×#103	C₂²	1	2	yes	—
35	C₃×#103	C₂²	1	2	yes	—
36	C₃×#103	C₂²	1	2	yes	—
37	C₃×#102	C₂²	1	1	yes	characteristic
38	C₃×#102	C₂²	1	1	yes	characteristic
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#97	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#556	C₂×C₃	1	1	yes	characteristic, complemented
43	D₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
44	#489	C₂×C₃	1	2	yes	complemented
45	#487	C₂×C₃	1	1	yes	characteristic, complemented
46	#477	C₂×C₃	1	2	yes	complemented
47	#474	C₂×C₃	1	1	yes	characteristic, complemented
48	#463	C₂×C₃	1	2	yes	—
49	#461	C₂×C₃	1	1	yes	characteristic, complemented
50	#446	C₂×C₃	1	1	yes	characteristic, complemented
51	#440	C₂×C₃	1	2	yes	complemented
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	—	4	4	no	—
56	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³	1	2	yes	—
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	C₃×SD₁₆	—	2	4	no	—
64	C₃×D₈	—	2	2	no	—
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂³	1	2	yes	—
68	C₃×C₄⋊C₄	—	2	1	no	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	1	no	—
72	C₃×C₄⋊C₄	—	2	1	no	—
73	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₃×C₄⋊C₄	D₄	1	2	yes	—
75	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₃×C₄⋊C₄	C₂³	1	2	yes	—
77	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₃×C₂²⋊C₄	—	2	2	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	2	1	no	—
81	C₃×C₂²⋊C₄	—	2	1	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₄²	—	2	2	no	—
84	C₃×C₄²	C₂³	1	1	yes	characteristic
85	C₃×C₄²	C₂³	1	2	yes	—
86	C₂²×Q₈	—	2	1	no	—
87	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
88	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
89	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
91	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
92	#126	—	2	2	no	—
93	#124	C₂²×C₃	1	2	yes	—
94	C₂²⋊Q₈	—	2	1	no	—
95	C₂²⋊Q₈	—	2	1	no	—
96	C₂²⋊Q₈	—	2	2	no	—
97	C₄×Q₈	C₂²×C₃	1	2	yes	—
98	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
99	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
100	C₄×D₄	C₂²×C₃	1	2	yes	—
101	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
102	#117	—	2	1	no	—
103	#107	C₂²×C₃	1	1	yes	characteristic
104	#107	C₂²×C₃	1	2	yes	—
105	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
106	C₄⋊C₈	C₂²×C₃	1	2	yes	—
107	#103	C₂²×C₃	1	2	yes	—
108	#103	C₂²×C₃	1	2	yes	—
109	#103	C₂²×C₃	1	2	yes	—
110	#102	C₂²×C₃	1	1	yes	characteristic
111	#102	C₂²×C₃	1	1	yes	characteristic
112	#102	C₂²×C₃	1	2	yes	—
113	#97	C₂²×C₃	1	2	yes	—
114	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
115	C₂³×C₃	—	2	1	no	—
116	C₂³×C₃	—	2	1	no	—
117	C₃×Q₈	—	2	2	no	—
118	C₃×Q₈	—	2	2	no	—
119	C₃×Q₈	—	2	2	no	—
120	C₃×Q₈	—	4	4	no	—
121	C₃×Q₈	—	4	2	no	—
122	C₃×D₄	—	2	1	no	—
123	C₃×D₄	—	2	1	no	—
124	C₃×D₄	—	2	1	no	—
125	C₃×D₄	#41	1	2	yes	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	4	4	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	—	4	2	no	—
133	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
134	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
136	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
138	C₃×C₈	—	2	1	no	—
139	C₃×C₈	—	2	2	no	—
140	C₃×C₈	—	2	1	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
145	C₂×Q₈	C₂³×C₃	1	2	yes	—
146	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
147	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	1	no	—
151	SD₁₆	—	2	4	no	—
152	D₈	—	2	2	no	—
153	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
154	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
155	C₂×C₈	C₂³×C₃	1	2	yes	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
162	C₄⋊C₄	C₃×D₄	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	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	1	no	—
169	C₂²⋊C₄	—	2	1	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	C₂³×C₃	1	1	yes	characteristic
173	C₄²	C₂³×C₃	1	2	yes	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	4	1	no	—
176	C₂²×C₃	—	4	1	no	—
177	C₂²×C₃	—	4	1	no	—
178	C₂²×C₃	—	4	1	no	—
179	C₂²×C₃	—	4	1	no	—
180	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	4	2	no	—
185	C₃×C₄	—	4	1	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	—	4	1	no	—
188	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
189	C₃×C₄	C₂×#41	1	1	yes	characteristic
190	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	4	4	no	—
197	Q₈	—	4	2	no	—
198	D₄	—	2	1	no	—
199	D₄	—	2	1	no	—
200	D₄	—	2	1	no	—
201	D₄	C₃×#41	1	2	yes	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
210	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
212	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₂×C₃	—	4	2	no	—
218	C₂×C₃	—	8	1	no	—
219	C₂×C₃	#548	1	1	yes	characteristic
220	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂×C₃	#578	1	1	yes	characteristic
222	C₂²	—	2	2	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	2	no	—
236	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
237	C₄	C₂×C₃×#41	1	1	yes	characteristic
238	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
239	C₃	#3367	1	1	yes	characteristic, direct product factor, Sylow
240	C₂	—	4	2	no	—
241	C₂	—	8	1	no	—
242	C₂	C₃×#548	1	1	yes	characteristic
243	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂	C₃×#578	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×#556	C₂	semidirect
2. C₃×D₄×Q₈	C₂	semidirect
3. C₃×#489	C₂	semidirect
4. C₃×#487	C₂	semidirect
5. C₃×#477	C₂	semidirect
6. C₃×#474	C₂	semidirect
8. C₃×#461	C₂	semidirect
9. C₃×#446	C₂	semidirect
10. C₃×#440	C₂	semidirect
12. #3367	C₃	direct (N × H)
42. #556	C₂×C₃	semidirect
43. D₄×Q₈	C₂×C₃	semidirect
44. #489	C₂×C₃	semidirect
45. #487	C₂×C₃	semidirect
46. #477	C₂×C₃	semidirect
47. #474	C₂×C₃	semidirect
49. #461	C₂×C₃	semidirect
50. #446	C₂×C₃	semidirect
51. #440	C₂×C₃	semidirect
239. C₃	#3367	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×#556	2
C₃×D₄×Q₈	2
C₃×#489	2
C₃×#487	2
C₃×#477	2
C₃×#474	2
C₃×#463	2
C₃×#461	2
C₃×#446	2
C₃×#440	2
C₃×C₄×D₈	2
#3367	3

Subgroups — C3×#3367 #85231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3367 #85231