Subgroups — C₃×#2426 #84290

Order 384.

632 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#396	C₂	1	2	yes	maximal, complemented
6	C₃×#394	C₂	1	2	yes	maximal, complemented
7	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#390	C₂	1	2	yes	maximal, complemented
9	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#388	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#2426	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×D₄	C₂²	1	2	yes	—
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₃×#124	—	2	4	no	—
18	C₃×C₂²⋊Q₈	—	2	4	no	—
19	C₃×#121	—	2	4	no	—
20	C₃×C₄×Q₈	—	2	4	no	—
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×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
34	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
35	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
36	C₂×C₃×C₄²	C₂²	1	2	yes	—
37	C₃×#95	C₂²	1	1	yes	characteristic
38	C₃×#95	C₂²	1	1	yes	characteristic
39	C₃×#95	C₂²	1	2	yes	—
40	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×#121	C₂×C₃	1	1	yes	characteristic
43	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
44	#396	C₂×C₃	1	2	yes	complemented
45	#394	C₂×C₃	1	2	yes	complemented
46	#390	C₂×C₃	1	1	yes	characteristic, complemented
47	#390	C₂×C₃	1	2	yes	complemented
48	#389	C₂×C₃	1	1	yes	characteristic, complemented
49	#388	C₂×C₃	1	2	yes	complemented
50	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂⁴×C₃	C₂³	1	2	yes	—
52	C₂⁴×C₃	C₂³	1	1	yes	characteristic
53	C₂×C₃×Q₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	2	no	—
55	C₂×C₃×Q₈	D₄	1	4	yes	complemented
56	C₂×C₃×D₄	—	2	4	no	—
57	C₂×C₃×D₄	—	2	4	no	—
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	4	2	no	—
62	C₂²×C₃×C₄	—	2	2	no	—
63	C₂²×C₃×C₄	—	2	2	no	—
64	C₂²×C₃×C₄	—	2	2	no	—
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	2	yes	—
70	C₂²×C₃×C₄	C₂³	1	2	yes	—
71	C₂²×C₃×C₄	C₂³	1	2	yes	—
72	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
74	C₃×C₄⋊C₄	—	2	4	no	—
75	C₃×C₄⋊C₄	—	2	2	no	—
76	C₃×C₄⋊C₄	—	2	4	no	—
77	C₃×C₄⋊C₄	—	4	2	no	—
78	C₃×C₄⋊C₄	D₄	1	4	yes	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	2	4	no	—
81	C₃×C₂²⋊C₄	—	2	4	no	—
82	C₃×C₂²⋊C₄	—	2	4	no	—
83	C₃×C₂²⋊C₄	—	2	4	no	—
84	C₃×C₂²⋊C₄	—	2	2	no	—
85	C₃×C₂²⋊C₄	—	2	4	no	—
86	C₃×C₂²⋊C₄	—	4	2	no	—
87	C₃×C₄²	—	2	4	no	—
88	C₃×C₄²	—	2	2	no	—
89	C₃×C₄²	—	2	2	no	—
90	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
91	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
92	C₂²×D₄	C₂²×C₃	1	2	yes	—
93	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
94	#124	—	2	4	no	—
95	C₂²⋊Q₈	—	2	4	no	—
96	#121	—	2	4	no	—
97	C₄×Q₈	—	2	4	no	—
98	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
104	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
105	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
106	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
107	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
109	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
110	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
111	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₃	1	2	yes	—
113	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
114	#95	C₂²×C₃	1	1	yes	characteristic
115	#95	C₂²×C₃	1	1	yes	characteristic
116	#95	C₂²×C₃	1	2	yes	—
117	C₂³×C₃	—	2	1	no	—
118	C₂³×C₃	—	2	1	no	—
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	—	2	1	no	—
121	C₂³×C₃	—	2	1	no	—
122	C₂³×C₃	—	2	1	no	—
123	C₂³×C₃	—	2	2	no	—
124	C₂³×C₃	—	2	2	no	—
125	C₂³×C₃	—	2	2	no	—
126	C₂³×C₃	—	2	2	no	—
127	C₂³×C₃	—	2	2	no	—
128	C₂³×C₃	—	2	2	no	—
129	C₂³×C₃	—	2	2	no	—
130	C₂³×C₃	#41	1	2	yes	—
131	C₂³×C₃	C₂⁴	1	1	yes	characteristic
132	C₃×Q₈	—	2	8	no	—
133	C₃×D₄	—	4	8	no	—
134	C₃×D₄	—	4	4	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	4	4	no	—
157	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
158	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
159	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
160	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
161	C₂×C₃×C₄	#41	1	2	yes	—
162	C₂×C₃×C₄	#41	1	2	yes	—
163	C₂⁴	C₂³×C₃	1	2	yes	—
164	C₂⁴	C₂³×C₃	1	1	yes	characteristic
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	C₃×D₄	1	4	yes	complemented
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
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	2	yes	—
183	C₂²×C₄	C₂³×C₃	1	2	yes	—
184	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	C₃×D₄	1	4	yes	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	—	4	1	no	—
207	C₂²×C₃	—	4	1	no	—
208	C₂²×C₃	—	4	1	no	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	—	4	2	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	—	4	2	no	—
214	C₂²×C₃	—	4	2	no	—
215	C₂²×C₃	—	4	2	no	—
216	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
217	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
218	C₂²×C₃	C₂×#41	1	1	yes	characteristic
219	C₂²×C₃	C₂×#41	1	1	yes	characteristic
220	C₂²×C₃	C₂×#41	1	1	yes	characteristic
221	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	4	1	no	—
227	C₃×C₄	—	4	1	no	—
228	C₃×C₄	—	4	1	no	—
229	C₃×C₄	—	4	1	no	—
230	C₃×C₄	—	4	2	no	—
231	C₃×C₄	#121	1	4	yes	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	C₃×#41	1	2	yes	—
246	C₂³	C₂⁴×C₃	1	1	yes	characteristic
247	Q₈	—	2	8	no	—
248	D₄	—	4	8	no	—
249	D₄	—	4	4	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
273	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
274	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
275	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
276	C₂×C₄	C₃×#41	1	2	yes	—
277	C₂×C₄	C₃×#41	1	2	yes	—
278	C₂×C₃	—	4	2	no	—
279	C₂×C₃	—	8	2	no	—
280	C₂×C₃	C₂×#121	1	1	yes	characteristic
281	C₂×C₃	#529	1	1	yes	characteristic
282	C₂×C₃	#542	1	1	yes	characteristic
283	C₂×C₃	#546	1	1	yes	characteristic
284	C₂×C₃	#548	1	1	yes	characteristic
285	C₂×C₃	#548	1	1	yes	characteristic
286	C₂×C₃	#550	1	1	yes	characteristic
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
302	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
303	C₂²	C₂×C₃×#41	1	1	yes	characteristic
304	C₂²	C₂×C₃×#41	1	1	yes	characteristic
305	C₂²	C₂×C₃×#41	1	1	yes	characteristic
306	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
307	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	2	4	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	2	no	—
316	C₄	C₃×#121	1	4	yes	—
317	C₃	#2426	1	1	yes	characteristic, direct product factor, Sylow
318	C₂	—	4	2	no	—
319	C₂	—	8	2	no	—
320	C₂	C₂×C₃×#121	1	1	yes	characteristic
321	C₂	C₃×#529	1	1	yes	characteristic
322	C₂	C₃×#542	1	1	yes	characteristic
323	C₂	C₃×#546	1	1	yes	characteristic
324	C₂	C₃×#548	1	1	yes	characteristic
325	C₂	C₃×#548	1	1	yes	characteristic
326	C₂	C₃×#550	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂×C₃×#121	C₂
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×#124	C₂	semidirect
2. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
4. C₂×C₃×C₄×Q₈	C₂	semidirect
5. C₃×#396	C₂	semidirect
6. C₃×#394	C₂	semidirect
7. C₃×#390	C₂	semidirect
8. C₃×#390	C₂	semidirect
9. C₃×#389	C₂	semidirect
10. C₃×#388	C₂	semidirect
11. C₃×C₄×C₂²⋊C₄	C₂	semidirect
12. #2426	C₃	direct (N × H)
40. C₂×#124	C₂×C₃	semidirect
41. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
43. C₂×C₄×Q₈	C₂×C₃	semidirect
44. #396	C₂×C₃	semidirect
45. #394	C₂×C₃	semidirect
46. #390	C₂×C₃	semidirect
47. #390	C₂×C₃	semidirect
48. #389	C₂×C₃	semidirect
49. #388	C₂×C₃	semidirect
50. C₄×C₂²⋊C₄	C₂×C₃	semidirect
55. C₂×C₃×Q₈	D₄	semidirect
167. C₂×Q₈	C₃×D₄	semidirect
317. C₃	#2426	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×#124	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×#121	2
C₂×C₃×C₄×Q₈	2
C₃×#396	2
C₃×#394	2
C₃×#390	2
C₃×#389	2
C₃×#388	2
C₃×C₄×C₂²⋊C₄	2
#2426	3

Subgroups — C3×#2426 #84290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2426 #84290