Subgroups — C₃×#3105 #84969

Order 384.

454 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#481	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#478	C₂	1	2	yes	maximal, complemented
6	C₃×#463	C₂	1	2	yes	maximal, complemented
7	C₃×#450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#447	C₂	1	1	yes	maximal, characteristic
9	C₃×C₄×SD₁₆	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×D₈	C₂	1	2	yes	maximal, complemented
11	C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	#3105	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	—	2	1	no	—
15	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₃×D₈	C₂²	1	2	yes	—
17	C₃×#126	—	2	1	no	—
18	C₃×#125	C₂²	1	1	yes	characteristic, complemented
19	C₃×#124	C₂²	1	1	yes	characteristic, complemented
20	C₃×#123	—	2	1	no	—
21	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₃×#121	C₂²	1	1	yes	characteristic
23	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₃×C₄×D₄	—	2	1	no	—
25	C₃×C₄×D₄	—	2	1	no	—
26	C₃×C₄×D₄	—	2	1	no	—
27	C₃×C₄×D₄	C₂²	1	2	yes	—
28	C₃×C₄×D₄	C₂²	1	2	yes	—
29	C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
30	C₃×#117	—	2	1	no	—
31	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
33	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
34	C₃×#107	C₂²	1	2	yes	complemented
35	C₃×#106	C₂²	1	2	yes	complemented
36	C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₃×#103	C₂²	1	2	yes	complemented
38	C₃×#102	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#102	C₂²	1	2	yes	complemented
41	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₄×C₈	C₂²	1	2	yes	complemented
44	#529	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
46	#484	C₂×C₃	1	1	yes	characteristic, complemented
47	#481	C₂×C₃	1	1	yes	characteristic, complemented
48	#478	C₂×C₃	1	2	yes	complemented
49	#463	C₂×C₃	1	2	yes	complemented
50	#450	C₂×C₃	1	1	yes	characteristic, complemented
51	#447	C₂×C₃	1	1	yes	characteristic
52	C₄×SD₁₆	C₂×C₃	1	2	yes	complemented
53	C₄×D₈	C₂×C₃	1	2	yes	complemented
54	#431	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂⁴×C₃	—	2	1	no	—
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×D₄	—	2	4	no	—
58	C₂×C₃×D₄	—	2	1	no	—
59	C₂×C₃×D₄	—	2	1	no	—
60	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
61	C₂×C₃×D₄	C₂³	1	2	yes	—
62	C₂²×C₃×C₄	—	2	1	no	—
63	C₂²×C₃×C₄	—	2	2	no	—
64	C₂²×C₃×C₄	—	2	1	no	—
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	—	4	4	no	—
67	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₃×SD₁₆	—	2	4	no	—
71	C₃×D₈	—	2	4	no	—
72	C₂×C₃×C₈	C₂³	1	2	yes	—
73	C₂×C₃×C₈	C₂³	1	2	yes	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	—	2	1	no	—
76	C₃×C₄⋊C₄	—	2	1	no	—
77	C₃×C₄⋊C₄	—	2	1	no	—
78	C₃×C₄⋊C₄	—	2	1	no	—
79	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₃×C₄⋊C₄	C₂³	1	2	yes	—
82	C₃×C₂²⋊C₄	—	2	1	no	—
83	C₃×C₂²⋊C₄	—	2	2	no	—
84	C₃×C₂²⋊C₄	—	2	2	no	—
85	C₃×C₂²⋊C₄	—	2	1	no	—
86	C₃×C₂²⋊C₄	—	2	1	no	—
87	C₃×C₂²⋊C₄	—	2	1	no	—
88	C₃×C₂²⋊C₄	—	2	1	no	—
89	C₃×C₄²	—	2	1	no	—
90	C₃×C₄²	D₄	1	1	yes	characteristic
91	C₃×C₄²	D₄	1	1	yes	characteristic
92	C₃×C₄²	C₂³	1	1	yes	characteristic
93	C₃×C₄²	C₂³	1	1	yes	characteristic
94	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
95	C₂³×C₄	—	2	1	no	—
96	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
97	C₂×D₈	C₂²×C₃	1	2	yes	—
98	#126	—	2	1	no	—
99	#125	C₂²×C₃	1	1	yes	characteristic, complemented
100	#124	C₂²×C₃	1	1	yes	characteristic, complemented
101	#123	—	2	1	no	—
102	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
103	#121	C₂²×C₃	1	1	yes	characteristic
104	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
105	C₄×D₄	—	2	1	no	—
106	C₄×D₄	—	2	1	no	—
107	C₄×D₄	—	2	1	no	—
108	C₄×D₄	C₂²×C₃	1	2	yes	—
109	C₄×D₄	C₂²×C₃	1	2	yes	—
110	C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
111	#117	—	2	1	no	—
112	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
113	C₂×C₂²⋊C₄	—	2	1	no	—
114	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
115	#107	C₂²×C₃	1	2	yes	complemented
116	#106	C₂²×C₃	1	2	yes	complemented
117	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
118	#103	C₂²×C₃	1	2	yes	complemented
119	#102	C₂²×C₃	1	2	yes	—
120	#102	C₂²×C₃	1	2	yes	—
121	#102	C₂²×C₃	1	2	yes	complemented
122	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
123	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
124	C₄×C₈	C₂²×C₃	1	2	yes	complemented
125	C₂³×C₃	—	2	4	no	—
126	C₂³×C₃	—	2	2	no	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	—	4	2	no	—
129	C₂³×C₃	—	4	2	no	—
130	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
131	C₃×Q₈	—	2	2	no	—
132	C₃×D₄	—	2	4	no	—
133	C₃×D₄	—	2	2	no	—
134	C₃×D₄	—	2	2	no	—
135	C₃×D₄	—	4	2	no	—
136	C₃×D₄	#41	1	4	yes	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	4	4	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	—	4	4	no	—
148	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
154	C₃×C₈	—	2	2	no	—
155	C₃×C₈	—	2	2	no	—
156	C₂⁴	—	2	1	no	—
157	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
162	C₂×D₄	C₂³×C₃	1	2	yes	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
169	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	SD₁₆	—	2	4	no	—
172	D₈	—	2	4	no	—
173	C₂×C₈	C₂³×C₃	1	2	yes	—
174	C₂×C₈	C₂³×C₃	1	2	yes	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₃	1	2	yes	—
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	1	no	—
187	C₂²⋊C₄	—	2	1	no	—
188	C₂²⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	C₃×D₄	1	1	yes	characteristic
192	C₄²	C₃×D₄	1	1	yes	characteristic
193	C₄²	C₂³×C₃	1	1	yes	characteristic
194	C₄²	C₂³×C₃	1	1	yes	characteristic
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	4	8	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	—	4	1	no	—
203	C₂²×C₃	—	4	1	no	—
204	C₂²×C₃	#135	1	2	yes	—
205	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	4	1	no	—
211	C₃×C₄	—	4	1	no	—
212	C₃×C₄	—	4	1	no	—
213	C₃×C₄	—	4	2	no	—
214	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
215	C₃×C₄	C₂×#41	1	2	yes	—
216	C₂³	—	2	4	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	2	no	—
227	D₄	C₃×#41	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	4	4	no	—
249	C₂×C₃	—	8	1	no	—
250	C₂×C₃	#525	1	1	yes	characteristic
251	C₂×C₃	C₂×#135	1	1	yes	characteristic
252	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	8	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	C₃×#135	1	2	yes	—
263	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
273	C₄	C₂×C₃×#41	1	2	yes	—
274	C₃	#3105	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	4	4	no	—
277	C₂	—	8	1	no	—
278	C₂	C₃×#525	1	1	yes	characteristic
279	C₂	C₂×C₃×#135	1	1	yes	characteristic
280	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₄
prime radical	C₃×#447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#529	C₂	semidirect
2. C₂×C₃×C₄×D₄	C₂	semidirect
3. C₃×#484	C₂	semidirect
4. C₃×#481	C₂	semidirect
5. C₃×#478	C₂	semidirect
6. C₃×#463	C₂	semidirect
7. C₃×#450	C₂	semidirect
9. C₃×C₄×SD₁₆	C₂	semidirect
10. C₃×C₄×D₈	C₂	semidirect
11. C₃×#431	C₂	semidirect
12. #3105	C₃	direct (N × H)
18. C₃×#125	C₂²	semidirect
19. C₃×#124	C₂²	semidirect
23. C₃×C₄×Q₈	C₂²	semidirect
29. C₃×C₄×D₄	C₂²	semidirect
34. C₃×#107	C₂²	semidirect
35. C₃×#106	C₂²	semidirect
36. C₃×C₄⋊C₈	C₂²	semidirect
37. C₃×#103	C₂²	semidirect
40. C₃×#102	C₂²	semidirect
43. C₃×C₄×C₈	C₂²	semidirect
44. #529	C₂×C₃	semidirect
45. C₂×C₄×D₄	C₂×C₃	semidirect
46. #484	C₂×C₃	semidirect
47. #481	C₂×C₃	semidirect
48. #478	C₂×C₃	semidirect
49. #463	C₂×C₃	semidirect
50. #450	C₂×C₃	semidirect
52. C₄×SD₁₆	C₂×C₃	semidirect
53. C₄×D₈	C₂×C₃	semidirect
54. #431	C₂×C₃	semidirect
99. #125	C₂²×C₃	semidirect
100. #124	C₂²×C₃	semidirect
104. C₄×Q₈	C₂²×C₃	semidirect
110. C₄×D₄	C₂²×C₃	semidirect
115. #107	C₂²×C₃	semidirect
116. #106	C₂²×C₃	semidirect
117. C₄⋊C₈	C₂²×C₃	semidirect
118. #103	C₂²×C₃	semidirect
121. #102	C₂²×C₃	semidirect
124. C₄×C₈	C₂²×C₃	semidirect
274. C₃	#3105	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×#529	2
C₂×C₃×C₄×D₄	2
C₃×#484	2
C₃×#481	2
C₃×#478	2
C₃×#463	2
C₃×#450	2
C₃×#447	2
C₃×C₄×SD₁₆	2
C₃×C₄×D₈	2
C₃×#431	2
#3105	3

Subgroups — C3×#3105 #84969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3105 #84969