Subgroups — C₃×#2545 #84409

Order 384.

576 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#396	C₂	1	4	yes	maximal, complemented
6	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#390	C₂	1	1	yes	maximal, characteristic
8	C₃×#388	C₂	1	4	yes	maximal, complemented
9	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#2545	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Q₈	—	2	4	no	—
15	C₃×#124	—	2	4	no	—
16	C₃×#124	—	2	4	no	—
17	C₃×C₄×Q₈	—	2	4	no	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂²	1	2	yes	—
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	4	yes	—
32	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×#124	C₂×C₃	1	1	yes	characteristic
34	C₂×#124	C₂×C₃	1	1	yes	characteristic
35	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
36	#396	C₂×C₃	1	4	yes	complemented
37	#391	C₂×C₃	1	1	yes	characteristic, complemented
38	#390	C₂×C₃	1	1	yes	characteristic
39	#388	C₂×C₃	1	4	yes	complemented
40	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂⁴×C₃	C₂³	1	2	yes	—
42	C₂×C₃×Q₈	—	2	2	no	—
43	C₂×C₃×Q₈	—	2	2	no	—
44	C₂×C₃×Q₈	—	2	2	no	—
45	C₂×C₃×Q₈	—	2	2	no	—
46	C₂×C₃×Q₈	—	2	2	no	—
47	C₂×C₃×Q₈	D₄	1	4	yes	—
48	C₂×C₃×D₄	—	2	2	no	—
49	C₂×C₃×D₄	—	2	2	no	—
50	C₂×C₃×D₄	—	2	2	no	—
51	C₂²×C₃×C₄	C₂³	1	2	yes	—
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	2	yes	—
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₃×C₄⋊C₄	—	2	4	no	—
60	C₃×C₄⋊C₄	—	2	2	no	—
61	C₃×C₄⋊C₄	—	2	8	no	—
62	C₃×C₄⋊C₄	—	2	4	no	—
63	C₃×C₂²⋊C₄	—	2	8	no	—
64	C₃×C₂²⋊C₄	—	2	4	no	—
65	C₃×C₂²⋊C₄	—	2	4	no	—
66	C₃×C₂²⋊C₄	—	2	8	no	—
67	C₃×C₄²	—	2	4	no	—
68	C₃×C₄²	—	2	2	no	—
69	C₃×C₄²	—	2	2	no	—
70	C₃×C₄²	D₄	1	4	yes	—
71	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
72	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
73	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
74	C₄⋊Q₈	—	2	4	no	—
75	#124	—	2	4	no	—
76	#124	—	2	4	no	—
77	C₄×Q₈	—	2	4	no	—
78	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
84	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
86	C₂×C₄²	C₂²×C₃	1	2	yes	—
87	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
90	#95	C₂²×C₃	1	2	yes	—
91	#95	C₂²×C₃	1	4	yes	—
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	4	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	4	no	—
97	C₂³×C₃	C₂⁴	1	1	yes	characteristic
98	C₃×Q₈	—	2	8	no	—
99	C₃×Q₈	—	4	4	no	—
100	C₃×D₄	—	4	4	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	4	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
116	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
117	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
118	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
119	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
120	C₂×C₃×C₄	#41	1	4	yes	—
121	C₂×C₃×C₄	#41	1	2	yes	—
122	C₂⁴	C₂³×C₃	1	2	yes	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	2	2	no	—
125	C₂×Q₈	—	2	2	no	—
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	C₃×D₄	1	4	yes	—
129	C₂×D₄	—	2	2	no	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂²×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₄	C₂³×C₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	8	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	2	8	no	—
145	C₂²⋊C₄	—	2	4	no	—
146	C₂²⋊C₄	—	2	4	no	—
147	C₂²⋊C₄	—	2	8	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	C₃×D₄	1	4	yes	—
152	C₂²×C₃	—	4	2	no	—
153	C₂²×C₃	—	4	2	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	—	4	4	no	—
156	C₂²×C₃	—	4	4	no	—
157	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
158	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
159	C₂²×C₃	C₂×#41	1	1	yes	characteristic
160	C₂²×C₃	C₂×#41	1	2	yes	—
161	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
162	C₃×C₄	—	2	4	no	—
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	4	2	no	—
167	C₃×C₄	—	4	2	no	—
168	C₃×C₄	—	4	2	no	—
169	C₃×C₄	#121	1	4	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	4	no	—
175	C₂³	C₂⁴×C₃	1	1	yes	characteristic
176	Q₈	—	2	8	no	—
177	Q₈	—	4	4	no	—
178	D₄	—	4	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
194	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
195	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
196	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
197	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
198	C₂×C₄	C₃×#41	1	4	yes	—
199	C₂×C₄	C₃×#41	1	2	yes	—
200	C₂×C₃	—	8	2	no	—
201	C₂×C₃	C₂×#121	1	1	yes	characteristic
202	C₂×C₃	#532	1	1	yes	characteristic
203	C₂×C₃	#536	1	1	yes	characteristic
204	C₂×C₃	#548	1	2	yes	—
205	C₂×C₃	#556	1	2	yes	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	4	no	—
210	C₂²	—	4	4	no	—
211	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
212	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₂²	C₂×C₃×#41	1	1	yes	characteristic
214	C₂²	C₂×C₃×#41	1	2	yes	—
215	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	4	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	C₃×#121	1	4	yes	—
224	C₃	#2545	1	1	yes	characteristic, direct product factor, Sylow
225	C₂	—	8	2	no	—
226	C₂	C₂×C₃×#121	1	1	yes	characteristic
227	C₂	C₃×#532	1	1	yes	characteristic
228	C₂	C₃×#536	1	1	yes	characteristic
229	C₂	C₃×#548	1	2	yes	—
230	C₂	C₃×#556	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. C₂×C₃×C₄⋊Q₈	C₂	semidirect
4. C₂×C₃×C₄×Q₈	C₂	semidirect
5. C₃×#396	C₂	semidirect
6. C₃×#391	C₂	semidirect
8. C₃×#388	C₂	semidirect
9. C₃×C₄×C₄⋊C₄	C₂	semidirect
10. #2545	C₃	direct (N × H)
32. C₂×C₄⋊Q₈	C₂×C₃	semidirect
35. C₂×C₄×Q₈	C₂×C₃	semidirect
36. #396	C₂×C₃	semidirect
37. #391	C₂×C₃	semidirect
39. #388	C₂×C₃	semidirect
40. C₄×C₄⋊C₄	C₂×C₃	semidirect
224. C₃	#2545	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×C₄⋊Q₈	2
C₂×C₃×#124	2
C₂×C₃×C₄×Q₈	2
C₃×#396	2
C₃×#391	2
C₃×#390	2
C₃×#388	2
C₃×C₄×C₄⋊C₄	2
#2545	3

Subgroups — C3×#2545 #84409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2545 #84409