Subgroups — C₃×#2453 #84317

Order 384.

452 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#400	C₂	1	2	yes	maximal
4	C₃×#398	C₂	1	2	yes	maximal
5	C₃×#391	C₂	1	1	yes	maximal, characteristic
6	C₃×#391	C₂	1	1	yes	maximal, characteristic
7	C₃×#389	C₂	1	1	yes	maximal, characteristic
8	C₃×#389	C₂	1	2	yes	maximal
9	C₃×#389	C₂	1	1	yes	maximal, characteristic
10	C₃×#387	C₂	1	2	yes	maximal
11	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	#2453	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₃×#125	—	2	4	no	—
15	C₃×C₄×Q₈	—	2	4	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	1	yes	characteristic
31	C₃×#95	C₂²	1	1	yes	characteristic
32	C₃×#95	C₂²	1	2	yes	—
33	C₃×#95	C₂²	1	2	yes	—
34	C₃×#95	C₂²	1	2	yes	—
35	C₃×#95	C₂²	1	2	yes	—
36	C₃×#95	C₂²	1	2	yes	—
37	C₃×#95	C₂²	1	2	yes	—
38	C₂×#125	C₂×C₃	1	1	yes	characteristic
39	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
40	#400	C₂×C₃	1	2	yes	—
41	#398	C₂×C₃	1	2	yes	—
42	#391	C₂×C₃	1	1	yes	characteristic
43	#391	C₂×C₃	1	1	yes	characteristic
44	#389	C₂×C₃	1	1	yes	characteristic
45	#389	C₂×C₃	1	2	yes	—
46	#389	C₂×C₃	1	1	yes	characteristic
47	#387	C₂×C₃	1	2	yes	—
48	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
49	C₂×C₃×Q₈	—	2	2	no	—
50	C₂×C₃×Q₈	—	2	2	no	—
51	C₂×C₃×Q₈	Q₈	1	4	yes	—
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₃×C₄⋊C₄	—	2	4	no	—
64	C₃×C₄⋊C₄	—	2	4	no	—
65	C₃×C₄⋊C₄	—	2	4	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	4	no	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	4	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	D₄	1	4	yes	—
73	C₃×C₄²	—	2	4	no	—
74	C₃×C₄²	—	2	2	no	—
75	C₃×C₄²	—	2	2	no	—
76	C₃×C₄²	—	2	2	no	—
77	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
78	#125	—	2	4	no	—
79	C₄×Q₈	—	2	4	no	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₄²	C₂²×C₃	1	2	yes	—
92	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
93	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
94	#95	C₂²×C₃	1	1	yes	characteristic
95	#95	C₂²×C₃	1	1	yes	characteristic
96	#95	C₂²×C₃	1	2	yes	—
97	#95	C₂²×C₃	1	2	yes	—
98	#95	C₂²×C₃	1	2	yes	—
99	#95	C₂²×C₃	1	2	yes	—
100	#95	C₂²×C₃	1	2	yes	—
101	#95	C₂²×C₃	1	2	yes	—
102	C₂³×C₃	C₂⁴	1	1	yes	characteristic
103	C₃×Q₈	—	2	8	no	—
104	C₂×C₃×C₄	—	2	1	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	1	no	—
107	C₂×C₃×C₄	—	2	1	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	1	no	—
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
131	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
133	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
134	C₂×C₃×C₄	#41	1	2	yes	—
135	C₂×C₃×C₄	#41	1	2	yes	—
136	C₂×C₃×C₄	#41	1	2	yes	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	C₃×Q₈	1	4	yes	—
140	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	C₃×D₄	1	4	yes	—
161	C₄²	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
166	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
167	C₂²×C₃	C₂×#41	1	1	yes	characteristic
168	C₂²×C₃	C₂×#41	1	1	yes	characteristic
169	C₂²×C₃	C₂×#41	1	1	yes	characteristic
170	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
171	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	4	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	4	2	no	—
177	C₃×C₄	—	4	1	no	—
178	C₃×C₄	—	4	1	no	—
179	C₃×C₄	—	4	1	no	—
180	C₃×C₄	—	4	1	no	—
181	C₃×C₄	—	4	2	no	—
182	C₃×C₄	C₂²⋊Q₈	1	4	yes	—
183	C₂³	C₂⁴×C₃	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
212	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
214	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
215	C₂×C₄	C₃×#41	1	2	yes	—
216	C₂×C₄	C₃×#41	1	2	yes	—
217	C₂×C₄	C₃×#41	1	2	yes	—
218	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
219	C₂×C₃	#529	1	1	yes	characteristic
220	C₂×C₃	#541	1	1	yes	characteristic
221	C₂×C₃	#547	1	1	yes	characteristic
222	C₂×C₃	#548	1	1	yes	characteristic
223	C₂×C₃	#557	1	1	yes	characteristic
224	C₂×C₃	#557	1	1	yes	characteristic
225	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
226	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
227	C₂²	C₂×C₃×#41	1	1	yes	characteristic
228	C₂²	C₂×C₃×#41	1	1	yes	characteristic
229	C₂²	C₂×C₃×#41	1	1	yes	characteristic
230	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
231	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
232	C₄	—	2	4	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	2	no	—
242	C₄	C₃×C₂²⋊Q₈	1	4	yes	—
243	C₃	#2453	1	1	yes	characteristic, direct product factor, Sylow
244	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂	C₃×#529	1	1	yes	characteristic
246	C₂	C₃×#541	1	1	yes	characteristic
247	C₂	C₃×#547	1	1	yes	characteristic
248	C₂	C₃×#548	1	1	yes	characteristic
249	C₂	C₃×#557	1	1	yes	characteristic
250	C₂	C₃×#557	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
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
12. #2453	C₃	direct (N × H)
243. C₃	#2453	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×#125	2
C₂×C₃×C₄×Q₈	2
C₃×#400	2
C₃×#398	2
C₃×#391	2
C₃×#389	2
C₃×#387	2
C₃×C₄×C₄⋊C₄	2
#2453	3

Subgroups — C3×#2453 #84317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2453 #84317