Subgroups — C₃×#2409 #84273

Order 384.

840 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#395	C₂	1	2	yes	maximal, complemented
5	C₃×#393	C₂	1	2	yes	maximal, complemented
6	C₃×#391	C₂	1	2	yes	maximal, complemented
7	C₃×#386	C₂	1	2	yes	maximal
8	C₃×#385	C₂	1	2	yes	maximal, complemented
9	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	#2409	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	—	2	8	no	—
12	C₂²×C₃×Q₈	—	2	4	no	—
13	C₂²×C₃×Q₈	C₂²	1	2	yes	—
14	C₂²×C₃×Q₈	C₂²	1	2	yes	—
15	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	2	yes	—
17	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
18	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
19	C₃×C₂²⋊Q₈	—	2	8	no	—
20	C₃×C₄×D₄	—	2	4	no	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₃×#95	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	4	yes	—
31	C₃×#95	C₂²	1	4	yes	—
32	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
34	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
35	#395	C₂×C₃	1	2	yes	complemented
36	#393	C₂×C₃	1	2	yes	complemented
37	#391	C₂×C₃	1	2	yes	complemented
38	#386	C₂×C₃	1	2	yes	—
39	#385	C₂×C₃	1	2	yes	complemented
40	#380	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂⁴×C₃	C₂³	1	1	yes	characteristic
42	C₂⁴×C₃	C₂³	1	1	yes	characteristic
43	C₂×C₃×Q₈	—	2	32	no	—
44	C₂×C₃×Q₈	—	2	8	no	—
45	C₂×C₃×Q₈	—	2	4	no	—
46	C₂×C₃×Q₈	—	2	4	no	—
47	C₂×C₃×Q₈	—	2	4	no	—
48	C₂×C₃×Q₈	D₄	1	8	yes	complemented
49	C₂×C₃×D₄	—	2	2	no	—
50	C₂×C₃×D₄	—	2	2	no	—
51	C₂×C₃×D₄	Q₈	1	4	yes	complemented
52	C₂²×C₃×C₄	—	2	4	no	—
53	C₂²×C₃×C₄	—	2	8	no	—
54	C₂²×C₃×C₄	—	2	4	no	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	—	2	4	no	—
57	C₂²×C₃×C₄	D₄	1	4	yes	—
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	4	yes	—
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	4	yes	—
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄⋊C₄	—	2	4	no	—
67	C₃×C₄⋊C₄	—	2	8	no	—
68	C₃×C₂²⋊C₄	—	2	2	no	—
69	C₃×C₂²⋊C₄	—	2	4	no	—
70	C₃×C₂²⋊C₄	—	2	2	no	—
71	C₃×C₂²⋊C₄	—	2	8	no	—
72	C₃×C₄²	—	2	2	no	—
73	C₂²×Q₈	—	2	8	no	—
74	C₂²×Q₈	—	2	4	no	—
75	C₂²×Q₈	C₂²×C₃	1	2	yes	—
76	C₂²×Q₈	C₂²×C₃	1	2	yes	—
77	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
78	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂³×C₄	C₂²×C₃	1	2	yes	—
80	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
81	C₂²⋊Q₈	—	2	8	no	—
82	C₄×D₄	—	2	4	no	—
83	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
86	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
90	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
91	#95	C₂²×C₃	1	2	yes	—
92	#95	C₂²×C₃	1	4	yes	—
93	#95	C₂²×C₃	1	4	yes	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	—	2	1	no	—
98	C₂³×C₃	—	2	1	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	C₂×D₄	1	2	yes	—
101	C₂³×C₃	C₂×Q₈	1	2	yes	—
102	C₂³×C₃	C₂×Q₈	1	2	yes	complemented
103	C₂³×C₃	#41	1	2	yes	—
104	C₂³×C₃	C₂⁴	1	1	yes	characteristic
105	C₃×Q₈	—	2	16	no	—
106	C₃×Q₈	—	2	16	no	—
107	C₃×D₄	—	2	8	no	—
108	C₂×C₃×C₄	—	2	16	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	2	4	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	8	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	4	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	4	4	no	—
123	C₂×C₃×C₄	C₂×D₄	1	8	yes	—
124	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
126	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
127	C₂×C₃×C₄	C₂×Q₈	1	2	yes	complemented
128	C₂⁴	C₂³×C₃	1	1	yes	characteristic
129	C₂⁴	C₂³×C₃	1	1	yes	characteristic
130	C₂×Q₈	—	2	32	no	—
131	C₂×Q₈	—	2	8	no	—
132	C₂×Q₈	—	2	4	no	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₃×D₄	1	8	yes	complemented
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	C₃×Q₈	1	4	yes	complemented
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₃×D₄	1	4	yes	—
145	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₃	1	4	yes	—
147	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₃	1	4	yes	—
149	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	8	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	—	2	4	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	8	no	—
159	C₄²	—	2	2	no	—
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	2	4	no	—
163	C₂²×C₃	—	2	4	no	—
164	C₂²×C₃	—	2	2	no	—
165	C₂²×C₃	—	2	4	no	—
166	C₂²×C₃	—	4	1	no	—
167	C₂²×C₃	—	4	1	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	C₂²⋊Q₈	1	4	yes	complemented
170	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
171	C₂²×C₃	C₂²×D₄	1	2	yes	—
172	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
173	C₂²×C₃	C₂×#41	1	1	yes	characteristic
174	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
175	C₃×C₄	—	2	8	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	4	1	no	—
179	C₃×C₄	—	4	1	no	—
180	C₃×C₄	—	4	4	no	—
181	C₃×C₄	#120	1	4	yes	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	1	no	—
186	C₂³	—	2	1	no	—
187	C₂³	—	2	2	no	—
188	C₂³	C₂×C₃×D₄	1	2	yes	—
189	C₂³	C₂×C₃×Q₈	1	2	yes	—
190	C₂³	C₂×C₃×Q₈	1	2	yes	complemented
191	C₂³	C₃×#41	1	2	yes	—
192	C₂³	C₂⁴×C₃	1	1	yes	characteristic
193	Q₈	—	2	16	no	—
194	Q₈	—	2	16	no	—
195	D₄	—	2	8	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	C₂×C₃×D₄	1	8	yes	—
212	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
215	C₂×C₄	C₂×C₃×Q₈	1	2	yes	complemented
216	C₂×C₃	—	2	4	no	—
217	C₂×C₃	—	4	2	no	—
218	C₂×C₃	C₂×#120	1	1	yes	characteristic
219	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
220	C₂×C₃	#536	1	1	yes	characteristic
221	C₂×C₃	#548	1	2	yes	—
222	C₂×C₃	D₄×Q₈	1	2	yes	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	C₃×C₂²⋊Q₈	1	4	yes	complemented
233	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
234	C₂²	C₂²×C₃×D₄	1	2	yes	—
235	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
236	C₂²	C₂×C₃×#41	1	1	yes	characteristic
237	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
238	C₄	—	2	8	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	4	no	—
244	C₄	C₃×#120	1	4	yes	—
245	C₃	#2409	1	1	yes	characteristic, direct product factor, Sylow
246	C₂	—	2	4	no	—
247	C₂	—	4	2	no	—
248	C₂	C₂×C₃×#120	1	1	yes	characteristic
249	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂	C₃×#536	1	1	yes	characteristic
251	C₂	C₃×#548	1	2	yes	—
252	C₂	C₃×D₄×Q₈	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	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. C₂³×C₃×Q₈	C₂	semidirect
2. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
4. C₃×#395	C₂	semidirect
5. C₃×#393	C₂	semidirect
6. C₃×#391	C₂	semidirect
8. C₃×#385	C₂	semidirect
9. C₃×#380	C₂	semidirect
10. #2409	C₃	direct (N × H)
32. C₂³×Q₈	C₂×C₃	semidirect
33. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
35. #395	C₂×C₃	semidirect
36. #393	C₂×C₃	semidirect
37. #391	C₂×C₃	semidirect
39. #385	C₂×C₃	semidirect
40. #380	C₂×C₃	semidirect
48. C₂×C₃×Q₈	D₄	semidirect
51. C₂×C₃×D₄	Q₈	semidirect
102. C₂³×C₃	C₂×Q₈	semidirect
127. C₂×C₃×C₄	C₂×Q₈	semidirect
135. C₂×Q₈	C₃×D₄	semidirect
138. C₂×D₄	C₃×Q₈	semidirect
169. C₂²×C₃	C₂²⋊Q₈	semidirect
190. C₂³	C₂×C₃×Q₈	semidirect
215. C₂×C₄	C₂×C₃×Q₈	semidirect
232. C₂²	C₃×C₂²⋊Q₈	semidirect
245. C₃	#2409	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×Q₈	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×C₄×D₄	2
C₃×#395	2
C₃×#393	2
C₃×#391	2
C₃×#386	2
C₃×#385	2
C₃×#380	2
#2409	3

Subgroups — C3×#2409 #84273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2409 #84273