Subgroups — C₃×#2437 #84301

Order 384.

780 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	C₂×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#396	C₂	1	2	yes	maximal, complemented
5	C₃×#393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#385	C₂	1	2	yes	maximal, complemented
7	C₃×#385	C₂	1	2	yes	maximal, complemented
8	C₃×#380	C₂	1	2	yes	maximal, complemented
9	C₃×#379	C₂	1	1	yes	maximal, characteristic
10	#2437	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂⁵×C₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₂³×C₃×C₄	C₂²	1	2	yes	—
15	C₃×C₂²⋊Q₈	—	2	8	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
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×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	1	yes	characteristic, complemented
31	C₃×#95	C₂²	1	2	yes	complemented
32	C₃×#95	C₂²	1	2	yes	complemented
33	C₃×#95	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
35	C₂²×C₂²⋊C₄	C₂×C₃	1	2	yes	—
36	#399	C₂×C₃	1	1	yes	characteristic, complemented
37	#396	C₂×C₃	1	2	yes	complemented
38	#393	C₂×C₃	1	1	yes	characteristic, complemented
39	#385	C₂×C₃	1	2	yes	complemented
40	#385	C₂×C₃	1	2	yes	complemented
41	#380	C₂×C₃	1	2	yes	complemented
42	#379	C₂×C₃	1	1	yes	characteristic
43	C₂⁴×C₃	—	2	4	no	—
44	C₂⁴×C₃	—	2	4	no	—
45	C₂⁴×C₃	—	4	1	no	—
46	C₂⁴×C₃	—	4	1	no	—
47	C₂⁴×C₃	Q₈	1	4	yes	complemented
48	C₂⁴×C₃	C₂³	1	1	yes	characteristic
49	C₂⁴×C₃	C₂³	1	2	yes	—
50	C₂×C₃×Q₈	—	2	4	no	—
51	C₂×C₃×Q₈	—	4	1	no	—
52	C₂²×C₃×C₄	—	2	4	no	—
53	C₂²×C₃×C₄	—	2	4	no	—
54	C₂²×C₃×C₄	—	2	4	no	—
55	C₂²×C₃×C₄	—	2	4	no	—
56	C₂²×C₃×C₄	—	4	2	no	—
57	C₂²×C₃×C₄	—	4	2	no	—
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄	C₂³	1	2	yes	—
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂³	1	2	yes	—
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	2	yes	—
65	C₃×C₄⋊C₄	—	2	4	no	—
66	C₃×C₄⋊C₄	—	2	4	no	—
67	C₃×C₄⋊C₄	—	2	4	no	—
68	C₃×C₄⋊C₄	—	4	2	no	—
69	C₃×C₄⋊C₄	—	4	1	no	—
70	C₃×C₂²⋊C₄	—	2	4	no	—
71	C₃×C₂²⋊C₄	—	2	4	no	—
72	C₃×C₂²⋊C₄	—	2	4	no	—
73	C₃×C₂²⋊C₄	—	2	4	no	—
74	C₃×C₂²⋊C₄	—	2	4	no	—
75	C₃×C₂²⋊C₄	—	4	1	no	—
76	C₃×C₂²⋊C₄	—	4	1	no	—
77	C₃×C₂²⋊C₄	D₄	1	8	yes	—
78	C₂⁵	C₂²×C₃	1	1	yes	characteristic
79	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
80	C₂³×C₄	C₂²×C₃	1	2	yes	—
81	C₂³×C₄	C₂²×C₃	1	2	yes	—
82	C₂²⋊Q₈	—	2	8	no	—
83	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
84	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
85	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
86	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
87	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
88	C₂×C₂²⋊C₄	—	2	8	no	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
90	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
91	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
92	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
93	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
94	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
97	#95	C₂²×C₃	1	1	yes	characteristic, complemented
98	#95	C₂²×C₃	1	2	yes	complemented
99	#95	C₂²×C₃	1	2	yes	complemented
100	#95	C₂²×C₃	1	1	yes	characteristic, complemented
101	C₂³×C₃	—	2	2	no	—
102	C₂³×C₃	—	2	2	no	—
103	C₂³×C₃	—	2	2	no	—
104	C₂³×C₃	—	2	2	no	—
105	C₂³×C₃	—	2	16	no	—
106	C₂³×C₃	—	2	2	no	—
107	C₂³×C₃	—	2	2	no	—
108	C₂³×C₃	—	2	1	no	—
109	C₂³×C₃	—	2	1	no	—
110	C₂³×C₃	—	4	4	no	—
111	C₂³×C₃	—	4	4	no	—
112	C₂³×C₃	—	4	8	no	—
113	C₂³×C₃	—	4	4	no	—
114	C₂³×C₃	C₂×D₄	1	4	yes	—
115	C₂³×C₃	C₂×Q₈	1	2	yes	—
116	C₂³×C₃	C₂×Q₈	1	4	yes	—
117	C₂³×C₃	#41	1	4	yes	—
118	C₂³×C₃	C₂⁴	1	1	yes	characteristic
119	C₃×Q₈	—	4	4	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	4	8	no	—
138	C₂×C₃×C₄	—	4	8	no	—
139	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
140	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
141	C₂⁴	—	2	4	no	—
142	C₂⁴	—	2	4	no	—
143	C₂⁴	—	4	1	no	—
144	C₂⁴	—	4	1	no	—
145	C₂⁴	C₃×Q₈	1	4	yes	complemented
146	C₂⁴	C₂³×C₃	1	1	yes	characteristic
147	C₂⁴	C₂³×C₃	1	2	yes	—
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₃	1	2	yes	—
158	C₂²×C₄	C₂³×C₃	1	2	yes	—
159	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₃	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₂²⋊C₄	—	4	1	no	—
175	C₂²⋊C₄	C₃×D₄	1	8	yes	—
176	C₂²×C₃	—	2	4	no	—
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	2	4	no	—
181	C₂²×C₃	—	2	4	no	—
182	C₂²×C₃	—	2	4	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	4	16	no	—
185	C₂²×C₃	—	4	1	no	—
186	C₂²×C₃	—	4	2	no	—
187	C₂²×C₃	—	4	1	no	—
188	C₂²×C₃	C₂²⋊Q₈	1	8	yes	complemented
189	C₂²×C₃	C₂²×D₄	1	2	yes	—
190	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
191	C₂²×C₃	C₂×#41	1	2	yes	—
192	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
194	C₃×C₄	—	2	4	no	—
195	C₃×C₄	—	2	4	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	—	4	1	no	—
198	C₃×C₄	—	4	2	no	—
199	C₃×C₄	—	4	2	no	—
200	C₃×C₄	—	4	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	16	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	4	4	no	—
212	C₂³	—	4	8	no	—
213	C₂³	—	4	4	no	—
214	C₂³	C₂×C₃×D₄	1	4	yes	—
215	C₂³	C₂×C₃×Q₈	1	2	yes	—
216	C₂³	C₂×C₃×Q₈	1	4	yes	—
217	C₂³	C₃×#41	1	4	yes	—
218	C₂³	C₂⁴×C₃	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
240	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
241	C₂×C₃	—	2	8	no	—
242	C₂×C₃	—	4	2	no	—
243	C₂×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
244	C₂×C₃	#543	1	1	yes	characteristic
245	C₂×C₃	D₄²	1	1	yes	characteristic
246	C₂×C₃	#546	1	2	yes	—
247	C₂×C₃	#551	1	1	yes	characteristic
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	16	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	C₃×C₂²⋊Q₈	1	8	yes	complemented
261	C₂²	C₂²×C₃×D₄	1	2	yes	—
262	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
263	C₂²	C₂×C₃×#41	1	2	yes	—
264	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
266	C₄	—	2	4	no	—
267	C₄	—	2	4	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₃	#2437	1	1	yes	characteristic, direct product factor, Sylow
274	C₂	—	2	8	no	—
275	C₂	—	4	2	no	—
276	C₂	C₂×C₃×C₂²⋊Q₈	1	2	yes	—
277	C₂	C₃×#543	1	1	yes	characteristic
278	C₂	C₃×D₄²	1	1	yes	characteristic
279	C₂	C₃×#546	1	2	yes	—
280	C₂	C₃×#551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	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
3. C₃×#399	C₂	semidirect
4. C₃×#396	C₂	semidirect
5. C₃×#393	C₂	semidirect
6. C₃×#385	C₂	semidirect
7. C₃×#385	C₂	semidirect
8. C₃×#380	C₂	semidirect
10. #2437	C₃	direct (N × H)
12. C₂²×C₃×Q₈	C₂²	semidirect
16. C₂×C₃×C₄⋊C₄	C₂²	semidirect
17. C₂×C₃×C₄⋊C₄	C₂²	semidirect
18. C₂×C₃×C₄⋊C₄	C₂²	semidirect
19. C₂×C₃×C₄⋊C₄	C₂²	semidirect
20. C₂×C₃×C₄⋊C₄	C₂²	semidirect
30. C₃×#95	C₂²	semidirect
31. C₃×#95	C₂²	semidirect
32. C₃×#95	C₂²	semidirect
33. C₃×#95	C₂²	semidirect
34. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
36. #399	C₂×C₃	semidirect
37. #396	C₂×C₃	semidirect
38. #393	C₂×C₃	semidirect
39. #385	C₂×C₃	semidirect
40. #385	C₂×C₃	semidirect
41. #380	C₂×C₃	semidirect
47. C₂⁴×C₃	Q₈	semidirect
79. C₂²×Q₈	C₂²×C₃	semidirect
83. C₂×C₄⋊C₄	C₂²×C₃	semidirect
84. C₂×C₄⋊C₄	C₂²×C₃	semidirect
85. C₂×C₄⋊C₄	C₂²×C₃	semidirect
86. C₂×C₄⋊C₄	C₂²×C₃	semidirect
87. C₂×C₄⋊C₄	C₂²×C₃	semidirect
97. #95	C₂²×C₃	semidirect
98. #95	C₂²×C₃	semidirect
99. #95	C₂²×C₃	semidirect
100. #95	C₂²×C₃	semidirect
145. C₂⁴	C₃×Q₈	semidirect
188. C₂²×C₃	C₂²⋊Q₈	semidirect
260. C₂²	C₃×C₂²⋊Q₈	semidirect
273. C₃	#2437	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
C₂²×C₃×C₂²⋊C₄	2
C₃×#399	2
C₃×#396	2
C₃×#393	2
C₃×#385	2
C₃×#380	2
C₃×#379	2
#2437	3

Subgroups — C3×#2437 #84301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2437 #84301