Subgroups — C₂×C₃×#552 #87354

Order 384.

1124 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#552	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#126	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#125	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#123	C₂	1	2	yes	maximal
6	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#117	C₂	1	2	yes	maximal, complemented
12	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#552	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
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×#126	C₂²	1	16	yes	complemented
20	C₃×#125	C₂²	1	16	yes	complemented
21	C₃×#125	C₂²	1	8	yes	complemented
22	C₃×#123	C₂²	1	16	yes	—
23	C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
24	C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
25	C₃×C₄×Q₈	C₂²	1	8	yes	complemented
26	C₃×C₄×D₄	C₂²	1	8	yes	—
27	C₃×#117	C₂²	1	8	yes	complemented
28	C₃×#117	C₂²	1	16	yes	complemented
29	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
32	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
33	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
35	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
38	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
39	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	complemented
40	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
41	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
43	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
44	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
45	C₂×C₃×C₄²	C₂²	1	2	yes	—
46	C₂×C₃×C₄²	C₂²	1	2	yes	—
47	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
48	#552	C₂×C₃	1	16	yes	direct product factor
49	C₂×#126	C₂×C₃	1	2	yes	complemented
50	C₂×#125	C₂×C₃	1	2	yes	complemented
51	C₂×#125	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×#123	C₂×C₃	1	2	yes	—
53	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
56	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
57	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
58	C₂×#117	C₂×C₃	1	2	yes	complemented
59	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂⁴×C₃	C₂³	1	1	yes	characteristic
61	C₂⁴×C₃	C₂³	1	1	yes	characteristic
62	C₂×C₃×Q₈	—	2	4	no	—
63	C₂×C₃×Q₈	C₂³	1	4	yes	—
64	C₂×C₃×D₄	—	2	4	no	—
65	C₂×C₃×D₄	C₂³	1	4	yes	—
66	C₂²×C₃×C₄	—	2	8	no	—
67	C₂²×C₃×C₄	—	2	4	no	—
68	C₂²×C₃×C₄	—	2	4	no	—
69	C₂²×C₃×C₄	C₂³	1	8	yes	—
70	C₂²×C₃×C₄	C₂³	1	4	yes	—
71	C₂²×C₃×C₄	C₂³	1	4	yes	—
72	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂³	1	2	yes	—
75	C₂²×C₃×C₄	C₂³	1	2	yes	—
76	C₂²×C₃×C₄	C₂³	1	2	yes	—
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂³	1	2	yes	—
80	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
81	C₃×C₄⋊C₄	C₂³	1	4	yes	—
82	C₃×C₄⋊C₄	C₂³	1	4	yes	—
83	C₃×C₄⋊C₄	C₂³	1	8	yes	—
84	C₃×C₄⋊C₄	C₂³	1	8	yes	—
85	C₃×C₄⋊C₄	C₂³	1	4	yes	—
86	C₃×C₄⋊C₄	C₂³	1	8	yes	—
87	C₃×C₄⋊C₄	C₂³	1	4	yes	—
88	C₃×C₄⋊C₄	C₂³	1	8	yes	—
89	C₃×C₄⋊C₄	C₂³	1	8	yes	—
90	C₃×C₄⋊C₄	C₂³	1	8	yes	—
91	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
92	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
93	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
94	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
95	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
96	C₃×C₄²	C₂³	1	8	yes	—
97	C₃×C₄²	C₂³	1	8	yes	—
98	C₃×C₄²	C₂³	1	4	yes	—
99	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
100	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
101	C₂³×C₄	C₂²×C₃	1	2	yes	—
102	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
103	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
104	#126	C₂²×C₃	1	16	yes	complemented
105	#125	C₂²×C₃	1	16	yes	complemented
106	#125	C₂²×C₃	1	8	yes	complemented
107	#123	C₂²×C₃	1	16	yes	—
108	C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
109	C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
110	C₄×Q₈	C₂²×C₃	1	8	yes	complemented
111	C₄×D₄	C₂²×C₃	1	8	yes	—
112	#117	C₂²×C₃	1	8	yes	complemented
113	#117	C₂²×C₃	1	16	yes	complemented
114	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
125	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
130	C₂×C₄²	C₂²×C₃	1	2	yes	—
131	C₂×C₄²	C₂²×C₃	1	2	yes	—
132	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
133	C₂³×C₃	—	2	2	no	—
134	C₂³×C₃	—	2	2	no	—
135	C₂³×C₃	—	2	2	no	—
136	C₂³×C₃	—	2	2	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	#41	1	4	yes	—
139	C₂³×C₃	C₂⁴	1	2	yes	—
140	C₂³×C₃	C₂⁴	1	2	yes	—
141	C₂³×C₃	C₂⁴	1	1	yes	characteristic
142	C₃×Q₈	—	2	8	no	—
143	C₃×D₄	—	2	8	no	—
144	C₂×C₃×C₄	—	2	16	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	8	no	—
154	C₂×C₃×C₄	—	2	8	no	—
155	C₂×C₃×C₄	#41	1	4	yes	—
156	C₂×C₃×C₄	C₂⁴	1	2	yes	—
157	C₂×C₃×C₄	C₂⁴	1	2	yes	—
158	C₂×C₃×C₄	C₂⁴	1	4	yes	—
159	C₂×C₃×C₄	C₂⁴	1	4	yes	—
160	C₂×C₃×C₄	C₂⁴	1	4	yes	—
161	C₂×C₃×C₄	C₂⁴	1	2	yes	—
162	C₂×C₃×C₄	C₂⁴	1	2	yes	—
163	C₂×C₃×C₄	C₂⁴	1	4	yes	—
164	C₂×C₃×C₄	C₂⁴	1	2	yes	—
165	C₂⁴	C₂³×C₃	1	1	yes	characteristic
166	C₂⁴	C₂³×C₃	1	1	yes	characteristic
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	C₂³×C₃	1	4	yes	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	C₂³×C₃	1	4	yes	—
171	C₂²×C₄	—	2	8	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₃	1	2	yes	—
178	C₂²×C₄	C₂³×C₃	1	2	yes	—
179	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₃	1	2	yes	—
182	C₂²×C₄	C₂³×C₃	1	8	yes	—
183	C₂²×C₄	C₂³×C₃	1	4	yes	—
184	C₂²×C₄	C₂³×C₃	1	4	yes	—
185	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₃	1	4	yes	—
187	C₄⋊C₄	C₂³×C₃	1	4	yes	—
188	C₄⋊C₄	C₂³×C₃	1	8	yes	—
189	C₄⋊C₄	C₂³×C₃	1	8	yes	—
190	C₄⋊C₄	C₂³×C₃	1	4	yes	—
191	C₄⋊C₄	C₂³×C₃	1	8	yes	—
192	C₄⋊C₄	C₂³×C₃	1	4	yes	—
193	C₄⋊C₄	C₂³×C₃	1	8	yes	—
194	C₄⋊C₄	C₂³×C₃	1	8	yes	—
195	C₄⋊C₄	C₂³×C₃	1	8	yes	—
196	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
197	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
198	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
199	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
200	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
201	C₄²	C₂³×C₃	1	8	yes	—
202	C₄²	C₂³×C₃	1	8	yes	—
203	C₄²	C₂³×C₃	1	4	yes	—
204	C₂²×C₃	—	2	8	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	2	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	4	4	no	—
211	C₂²×C₃	C₂×#41	1	4	yes	—
212	C₂²×C₃	C₂×#41	1	2	yes	—
213	C₂²×C₃	C₂×#41	1	2	yes	—
214	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
215	C₂²×C₃	C₂⁵	1	1	yes	characteristic
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	4	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	C₂×#41	1	4	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₃×#41	1	4	yes	—
231	C₂³	C₂⁴×C₃	1	2	yes	—
232	C₂³	C₂⁴×C₃	1	2	yes	—
233	C₂³	C₂⁴×C₃	1	1	yes	characteristic
234	Q₈	—	2	8	no	—
235	D₄	—	2	8	no	—
236	C₂×C₄	—	2	16	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	C₃×#41	1	4	yes	—
248	C₂×C₄	C₂⁴×C₃	1	2	yes	—
249	C₂×C₄	C₂⁴×C₃	1	2	yes	—
250	C₂×C₄	C₂⁴×C₃	1	4	yes	—
251	C₂×C₄	C₂⁴×C₃	1	4	yes	—
252	C₂×C₄	C₂⁴×C₃	1	4	yes	—
253	C₂×C₄	C₂⁴×C₃	1	2	yes	—
254	C₂×C₄	C₂⁴×C₃	1	2	yes	—
255	C₂×C₄	C₂⁴×C₃	1	4	yes	—
256	C₂×C₄	C₂⁴×C₃	1	2	yes	—
257	C₂×C₃	—	2	4	no	—
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	#552	1	4	yes	direct product factor
260	C₂×C₃	C₂²×#41	1	1	yes	characteristic
261	C₂×C₃	C₂²×#41	1	1	yes	characteristic
262	C₂×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	4	no	—
270	C₂²	C₂×C₃×#41	1	4	yes	—
271	C₂²	C₂×C₃×#41	1	2	yes	—
272	C₂²	C₂×C₃×#41	1	2	yes	—
273	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
274	C₂²	C₂⁵×C₃	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	4	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	4	no	—
283	C₄	C₂×C₃×#41	1	4	yes	—
284	C₃	C₂×#552	1	1	yes	characteristic, direct product factor, Sylow
285	C₂	—	2	4	no	—
286	C₂	—	4	2	no	—
287	C₂	C₃×#552	1	4	yes	direct product factor
288	C₂	C₂²×C₃×#41	1	1	yes	characteristic
289	C₂	C₂²×C₃×#41	1	1	yes	characteristic
290	C₂	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#552	C₂	direct (N × H)
2. C₂×C₃×#126	C₂	semidirect
3. C₂×C₃×#125	C₂	semidirect
4. C₂×C₃×#125	C₂	semidirect
6. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
7. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
8. C₂×C₃×C₄×Q₈	C₂	semidirect
10. C₂×C₃×#117	C₂	semidirect
11. C₂×C₃×#117	C₂	semidirect
12. C₂²×C₃×C₄⋊C₄	C₂	semidirect
13. C₂×#552	C₃	direct (N × H)
19. C₃×#126	C₂²	semidirect
20. C₃×#125	C₂²	semidirect
21. C₃×#125	C₂²	semidirect
23. C₃×C₂²⋊Q₈	C₂²	semidirect
24. C₃×C₂²⋊Q₈	C₂²	semidirect
25. C₃×C₄×Q₈	C₂²	semidirect
27. C₃×#117	C₂²	semidirect
28. C₃×#117	C₂²	semidirect
39. C₂×C₃×C₄⋊C₄	C₂²	semidirect
48. #552	C₂×C₃	direct (N × H)
49. C₂×#126	C₂×C₃	semidirect
50. C₂×#125	C₂×C₃	semidirect
51. C₂×#125	C₂×C₃	semidirect
53. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
54. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
55. C₂×C₄×Q₈	C₂×C₃	semidirect
57. C₂×#117	C₂×C₃	semidirect
58. C₂×#117	C₂×C₃	semidirect
59. C₂²×C₄⋊C₄	C₂×C₃	semidirect
104. #126	C₂²×C₃	semidirect
105. #125	C₂²×C₃	semidirect
106. #125	C₂²×C₃	semidirect
108. C₂²⋊Q₈	C₂²×C₃	semidirect
109. C₂²⋊Q₈	C₂²×C₃	semidirect
110. C₄×Q₈	C₂²×C₃	semidirect
112. #117	C₂²×C₃	semidirect
113. #117	C₂²×C₃	semidirect
124. C₂×C₄⋊C₄	C₂²×C₃	semidirect
259. C₂×C₃	#552	direct (N × H)
284. C₃	C₂×#552	direct (N × H)
287. C₂	C₃×#552	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#552	2
C₂×C₃×#126	2
C₂×C₃×#125	2
C₂×C₃×#123	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×C₄×D₄	2
C₂×C₃×#117	2
C₂²×C₃×C₄⋊C₄	2
C₂×#552	3

Subgroups — C2×C3×#552 #87354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#552 #87354