Subgroups — C₃×#3274 #85138

Order 384.

556 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#469	C₂	1	2	yes	maximal, complemented
6	C₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#449	C₂	1	2	yes	maximal, complemented
9	C₃×#447	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	#3274	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	complemented
14	C₂²×C₃×D₄	—	2	1	no	—
15	C₂²×C₃×D₄	C₂²	1	2	yes	—
16	C₃×AGL(1,ℤ/8)	—	2	8	no	—
17	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
19	C₂×C₃×D₈	C₂²	1	2	yes	—
20	C₂×C₃×D₈	C₂²	1	2	yes	complemented
21	C₂×C₃×D₈	C₂²	1	2	yes	complemented
22	C₂×C₃×M₁₆	C₂²	1	2	yes	complemented
23	C₃×#127	—	2	1	no	—
24	C₃×#127	C₂²	1	1	yes	characteristic
25	C₃×#127	C₂²	1	1	yes	characteristic, complemented
26	C₃×#121	—	2	1	no	—
27	C₃×#121	—	2	2	no	—
28	C₃×#121	C₂²	1	2	yes	—
29	C₃×#121	C₂²	1	2	yes	complemented
30	C₃×#120	—	2	2	no	—
31	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
32	C₃×C₄×D₄	—	2	1	no	—
33	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
34	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
35	C₃×#106	C₂²	1	1	yes	characteristic, complemented
36	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₃×#103	C₂²	1	1	yes	characteristic, complemented
38	C₃×#102	C₂²	1	1	yes	characteristic
39	C₃×#102	C₂²	1	2	yes	complemented
40	C₃×#102	C₂²	1	2	yes	complemented
41	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
42	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
43	C₂×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
44	#550	C₂×C₃	1	1	yes	characteristic, complemented
45	D₄²	C₂×C₃	1	1	yes	characteristic, complemented
46	#493	C₂×C₃	1	1	yes	characteristic, complemented
47	#469	C₂×C₃	1	2	yes	complemented
48	#460	C₂×C₃	1	1	yes	characteristic, complemented
49	#459	C₂×C₃	1	1	yes	characteristic, complemented
50	#449	C₂×C₃	1	2	yes	complemented
51	#447	C₂×C₃	1	2	yes	complemented
52	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
53	#436	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂⁴×C₃	—	2	2	no	—
55	C₃×#41	—	2	4	no	—
56	C₃×#41	—	4	2	no	—
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×D₄	—	2	4	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	2	2	no	—
62	C₂×C₃×D₄	—	2	1	no	—
63	C₂×C₃×D₄	—	2	2	no	—
64	C₂×C₃×D₄	—	2	1	no	—
65	C₂×C₃×D₄	—	2	1	no	—
66	C₂×C₃×D₄	—	2	1	no	—
67	C₂×C₃×D₄	—	4	2	no	—
68	C₂×C₃×D₄	—	4	4	no	—
69	C₂×C₃×D₄	D₄	1	1	yes	characteristic
70	C₂×C₃×D₄	C₂³	1	2	yes	—
71	C₂×C₃×D₄	C₂³	1	2	yes	—
72	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
73	C₂²×C₃×C₄	—	2	2	no	—
74	C₂²×C₃×C₄	—	2	1	no	—
75	C₂²×C₃×C₄	C₂³	1	2	yes	—
76	C₃×SD₁₆	—	2	4	no	—
77	C₃×SD₁₆	D₄	1	4	yes	complemented
78	C₃×D₈	—	2	4	no	—
79	C₃×D₈	—	2	4	no	—
80	C₃×D₈	—	4	2	no	—
81	C₃×M₁₆	—	2	4	no	—
82	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂³	1	2	yes	—
84	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
85	C₃×C₄⋊C₄	—	2	1	no	—
86	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₃×C₂²⋊C₄	—	2	2	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₂²⋊C₄	—	2	2	no	—
92	C₃×C₂²⋊C₄	D₄	1	2	yes	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	C₂³	1	1	yes	characteristic
95	C₂×#41	C₂²×C₃	1	2	yes	complemented
96	C₂²×D₄	—	2	1	no	—
97	C₂²×D₄	C₂²×C₃	1	2	yes	—
98	AGL(1,ℤ/8)	—	2	8	no	—
99	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
100	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
101	C₂×D₈	C₂²×C₃	1	2	yes	—
102	C₂×D₈	C₂²×C₃	1	2	yes	complemented
103	C₂×D₈	C₂²×C₃	1	2	yes	complemented
104	C₂×M₁₆	C₂²×C₃	1	2	yes	complemented
105	#127	—	2	1	no	—
106	#127	C₂²×C₃	1	1	yes	characteristic
107	#127	C₂²×C₃	1	1	yes	characteristic, complemented
108	#121	—	2	1	no	—
109	#121	—	2	2	no	—
110	#121	C₂²×C₃	1	2	yes	—
111	#121	C₂²×C₃	1	2	yes	complemented
112	#120	—	2	2	no	—
113	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
114	C₄×D₄	—	2	1	no	—
115	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
116	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
117	#106	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
119	#103	C₂²×C₃	1	1	yes	characteristic, complemented
120	#102	C₂²×C₃	1	1	yes	characteristic
121	#102	C₂²×C₃	1	2	yes	complemented
122	#102	C₂²×C₃	1	2	yes	complemented
123	C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
124	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	—	2	2	no	—
127	C₂³×C₃	—	2	2	no	—
128	C₂³×C₃	—	4	4	no	—
129	C₂³×C₃	—	4	4	no	—
130	C₂³×C₃	—	8	2	no	—
131	C₂³×C₃	C₂×D₄	1	2	yes	—
132	C₃×Q₈	—	2	1	no	—
133	C₃×Q₈	C₂×D₄	1	2	yes	complemented
134	C₃×D₄	—	2	2	no	—
135	C₃×D₄	—	2	2	no	—
136	C₃×D₄	—	2	2	no	—
137	C₃×D₄	—	2	1	no	—
138	C₃×D₄	—	2	2	no	—
139	C₃×D₄	—	2	2	no	—
140	C₃×D₄	—	4	4	no	—
141	C₃×D₄	—	4	4	no	—
142	C₃×D₄	—	4	4	no	—
143	C₃×D₄	—	4	2	no	—
144	C₃×D₄	—	4	2	no	—
145	C₃×D₄	—	8	1	no	—
146	C₃×D₄	—	8	1	no	—
147	C₃×D₄	—	8	2	no	—
148	C₃×D₄	C₂×D₄	1	2	yes	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	4	4	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
158	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
160	C₃×C₈	—	2	2	no	—
161	C₃×C₈	—	2	1	no	—
162	C₃×C₈	C₂×D₄	1	2	yes	complemented
163	C₂⁴	—	2	2	no	—
164	#41	—	2	4	no	—
165	#41	—	4	2	no	—
166	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	4	no	—
178	C₂×D₄	C₃×D₄	1	1	yes	characteristic
179	C₂×D₄	C₂³×C₃	1	2	yes	—
180	C₂×D₄	C₂³×C₃	1	2	yes	—
181	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	C₂³×C₃	1	2	yes	—
185	SD₁₆	—	2	4	no	—
186	SD₁₆	C₃×D₄	1	4	yes	complemented
187	D₈	—	2	4	no	—
188	D₈	—	2	4	no	—
189	D₈	—	4	2	no	—
190	M₁₆	—	2	4	no	—
191	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
192	C₂×C₈	C₂³×C₃	1	2	yes	—
193	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
196	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
197	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	C₃×D₄	1	2	yes	—
202	C₄²	—	2	1	no	—
203	C₄²	C₂³×C₃	1	1	yes	characteristic
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	2	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	4	1	no	—
209	C₂²×C₃	—	4	1	no	—
210	C₂²×C₃	—	4	2	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	—	4	2	no	—
214	C₂²×C₃	—	4	2	no	—
215	C₂²×C₃	—	4	2	no	—
216	C₂²×C₃	—	8	8	no	—
217	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	4	1	no	—
222	C₃×C₄	—	4	1	no	—
223	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	complemented
224	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
225	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
226	C₂³	—	2	1	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	8	2	no	—
232	C₂³	C₂×C₃×D₄	1	2	yes	—
233	Q₈	—	2	1	no	—
234	Q₈	C₂×C₃×D₄	1	2	yes	complemented
235	D₄	—	2	2	no	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	2	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	4	4	no	—
242	D₄	—	4	4	no	—
243	D₄	—	4	4	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	8	1	no	—
247	D₄	—	8	1	no	—
248	D₄	—	8	2	no	—
249	D₄	C₂×C₃×D₄	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
259	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
260	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	2	1	no	—
263	C₈	C₂×C₃×D₄	1	2	yes	complemented
264	C₂×C₃	—	4	2	no	—
265	C₂×C₃	—	4	2	no	—
266	C₂×C₃	—	8	2	no	—
267	C₂×C₃	—	8	2	no	—
268	C₂×C₃	D₄²	1	1	yes	characteristic
269	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₂×C₃	#576	1	1	yes	characteristic
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	8	8	no	—
284	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	complemented
291	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
292	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
293	C₃	#3274	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	—	4	2	no	—
295	C₂	—	4	2	no	—
296	C₂	—	8	2	no	—
297	C₂	—	8	2	no	—
298	C₂	C₃×D₄²	1	1	yes	characteristic
299	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₂	C₃×#576	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₃×#550	C₂	semidirect
3. C₃×D₄²	C₂	semidirect
4. C₃×#493	C₂	semidirect
5. C₃×#469	C₂	semidirect
6. C₃×#460	C₂	semidirect
7. C₃×#459	C₂	semidirect
8. C₃×#449	C₂	semidirect
9. C₃×#447	C₂	semidirect
10. C₃×C₄×SD₁₆	C₂	semidirect
11. C₃×#436	C₂	semidirect
12. #3274	C₃	direct (N × H)
13. C₂×C₃×#41	C₂²	semidirect
18. C₂×C₃×SD₁₆	C₂²	semidirect
20. C₂×C₃×D₈	C₂²	semidirect
21. C₂×C₃×D₈	C₂²	semidirect
22. C₂×C₃×M₁₆	C₂²	semidirect
25. C₃×#127	C₂²	semidirect
29. C₃×#121	C₂²	semidirect
31. C₃×C₄×Q₈	C₂²	semidirect
35. C₃×#106	C₂²	semidirect
36. C₃×C₄⋊C₈	C₂²	semidirect
37. C₃×#103	C₂²	semidirect
39. C₃×#102	C₂²	semidirect
40. C₃×#102	C₂²	semidirect
41. C₃×C₂²⋊C₈	C₂²	semidirect
42. C₃×C₄×C₈	C₂²	semidirect
43. C₂×AGL(1,ℤ/8)	C₂×C₃	semidirect
44. #550	C₂×C₃	semidirect
45. D₄²	C₂×C₃	semidirect
46. #493	C₂×C₃	semidirect
47. #469	C₂×C₃	semidirect
48. #460	C₂×C₃	semidirect
49. #459	C₂×C₃	semidirect
50. #449	C₂×C₃	semidirect
51. #447	C₂×C₃	semidirect
52. C₄×SD₁₆	C₂×C₃	semidirect
53. #436	C₂×C₃	semidirect
77. C₃×SD₁₆	D₄	semidirect
95. C₂×#41	C₂²×C₃	semidirect
100. C₂×SD₁₆	C₂²×C₃	semidirect
102. C₂×D₈	C₂²×C₃	semidirect
103. C₂×D₈	C₂²×C₃	semidirect
104. C₂×M₁₆	C₂²×C₃	semidirect
107. #127	C₂²×C₃	semidirect
111. #121	C₂²×C₃	semidirect
113. C₄×Q₈	C₂²×C₃	semidirect
117. #106	C₂²×C₃	semidirect
118. C₄⋊C₈	C₂²×C₃	semidirect
119. #103	C₂²×C₃	semidirect
121. #102	C₂²×C₃	semidirect
122. #102	C₂²×C₃	semidirect
123. C₂²⋊C₈	C₂²×C₃	semidirect
124. C₄×C₈	C₂²×C₃	semidirect
133. C₃×Q₈	C₂×D₄	semidirect
162. C₃×C₈	C₂×D₄	semidirect
186. SD₁₆	C₃×D₄	semidirect
223. C₃×C₄	AGL(1,ℤ/8)	semidirect
234. Q₈	C₂×C₃×D₄	semidirect
263. C₈	C₂×C₃×D₄	semidirect
290. C₄	C₃×AGL(1,ℤ/8)	semidirect
293. C₃	#3274	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×AGL(1,ℤ/8)	2
C₃×#550	2
C₃×D₄²	2
C₃×#493	2
C₃×#469	2
C₃×#460	2
C₃×#459	2
C₃×#449	2
C₃×#447	2
C₃×C₄×SD₁₆	2
C₃×#436	2
#3274	3

Subgroups — C3×#3274 #85138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3274 #85138