Subgroups — C₃×#2445 #84309

Order 384.

696 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#123	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#394	C₂	1	2	yes	maximal, complemented
5	C₃×#394	C₂	1	2	yes	maximal, complemented
6	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#386	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#2445	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×D₄	C₂²	1	2	yes	—
16	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₄	C₂²	1	2	yes	—
18	C₃×#127	—	2	4	no	—
19	C₃×#123	—	2	8	no	—
20	C₃×C₄×D₄	—	2	4	no	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
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×C₄²	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
35	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
36	C₃×#95	C₂²	1	1	yes	characteristic
37	C₃×#95	C₂²	1	1	yes	characteristic
38	C₃×#95	C₂²	1	2	yes	—
39	C₂×#127	C₂×C₃	1	1	yes	characteristic
40	C₂×#123	C₂×C₃	1	2	yes	complemented
41	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
42	#394	C₂×C₃	1	2	yes	complemented
43	#394	C₂×C₃	1	2	yes	complemented
44	#390	C₂×C₃	1	1	yes	characteristic, complemented
45	#390	C₂×C₃	1	1	yes	characteristic, complemented
46	#390	C₂×C₃	1	1	yes	characteristic, complemented
47	#389	C₂×C₃	1	1	yes	characteristic, complemented
48	#386	C₂×C₃	1	2	yes	complemented
49	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂⁴×C₃	C₂³	1	2	yes	—
51	C₂⁴×C₃	C₂³	1	2	yes	—
52	C₂×C₃×D₄	—	2	4	no	—
53	C₂×C₃×D₄	—	2	4	no	—
54	C₂×C₃×D₄	—	2	4	no	—
55	C₂×C₃×D₄	—	2	2	no	—
56	C₂×C₃×D₄	—	2	2	no	—
57	C₂×C₃×D₄	—	2	2	no	—
58	C₂×C₃×D₄	—	2	4	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	4	2	no	—
62	C₂×C₃×D₄	—	4	2	no	—
63	C₂×C₃×D₄	D₄	1	4	yes	—
64	C₂²×C₃×C₄	—	2	4	no	—
65	C₂²×C₃×C₄	—	2	4	no	—
66	C₂²×C₃×C₄	—	4	2	no	—
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	1	yes	characteristic
72	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
76	C₃×C₄⋊C₄	—	2	2	no	—
77	C₃×C₄⋊C₄	—	2	4	no	—
78	C₃×C₄⋊C₄	—	2	2	no	—
79	C₃×C₄⋊C₄	—	2	2	no	—
80	C₃×C₄⋊C₄	D₄	1	4	yes	complemented
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	4	no	—
85	C₃×C₂²⋊C₄	—	4	2	no	—
86	C₃×C₂²⋊C₄	—	4	2	no	—
87	C₃×C₂²⋊C₄	—	4	2	no	—
88	C₃×C₄²	—	2	2	no	—
89	C₃×C₄²	—	2	2	no	—
90	C₃×C₄²	—	2	2	no	—
91	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
92	C₂²×D₄	C₂²×C₃	1	2	yes	—
93	C₂²×D₄	C₂²×C₃	1	2	yes	—
94	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
95	C₂³×C₄	C₂²×C₃	1	2	yes	—
96	#127	—	2	4	no	—
97	#123	—	2	8	no	—
98	C₄×D₄	—	2	4	no	—
99	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
102	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
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	2	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
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	1	yes	characteristic
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	2	no	—
118	C₂³×C₃	—	2	2	no	—
119	C₂³×C₃	—	2	2	no	—
120	C₂³×C₃	—	2	2	no	—
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	—	2	2	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	C₂×D₄	1	4	yes	—
131	C₂³×C₃	C₂⁴	1	1	yes	characteristic
132	C₃×D₄	—	2	8	no	—
133	C₃×D₄	—	4	8	no	—
134	C₃×D₄	—	4	8	no	—
135	C₃×D₄	—	4	4	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	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	2	no	—
147	C₂×C₃×C₄	—	2	2	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	1	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	4	8	no	—
157	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
158	C₂×C₃×C₄	C₂×D₄	1	2	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	C₂×C₃×C₄	#41	1	2	yes	—
164	C₂⁴	C₂³×C₃	1	2	yes	—
165	C₂⁴	C₂³×C₃	1	2	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	C₃×D₄	1	4	yes	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₃	1	2	yes	—
184	C₂²×C₄	C₂³×C₃	1	2	yes	—
185	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	C₃×D₄	1	4	yes	complemented
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	2	4	no	—
197	C₂²⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	4	2	no	—
200	C₂²⋊C₄	—	4	2	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	4	2	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	—	4	2	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	—	4	2	no	—
219	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
220	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
221	C₂²×C₃	C₂×#41	1	1	yes	characteristic
222	C₂²×C₃	C₂×#41	1	1	yes	characteristic
223	C₂²×C₃	C₂×#41	1	1	yes	characteristic
224	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	4	1	no	—
231	C₃×C₄	—	4	2	no	—
232	C₃×C₄	—	4	1	no	—
233	C₃×C₄	—	4	2	no	—
234	C₃×C₄	#123	1	4	yes	complemented
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	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	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	C₂×C₃×D₄	1	4	yes	—
249	C₂³	C₂⁴×C₃	1	1	yes	characteristic
250	D₄	—	2	8	no	—
251	D₄	—	4	8	no	—
252	D₄	—	4	8	no	—
253	D₄	—	4	4	no	—
254	C₂×C₄	—	2	1	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	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	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	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
276	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
277	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
278	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
279	C₂×C₄	C₃×#41	1	2	yes	—
280	C₂×C₄	C₃×#41	1	2	yes	—
281	C₂×C₄	C₃×#41	1	2	yes	—
282	C₂×C₃	—	4	4	no	—
283	C₂×C₃	—	8	2	no	—
284	C₂×C₃	C₂×#123	1	1	yes	characteristic
285	C₂×C₃	#532	1	1	yes	characteristic
286	C₂×C₃	#540	1	1	yes	characteristic
287	C₂×C₃	D₄²	1	1	yes	characteristic
288	C₂×C₃	#546	1	1	yes	characteristic
289	C₂×C₃	#550	1	1	yes	characteristic
290	C₂×C₃	#559	1	1	yes	characteristic
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	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	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
306	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
307	C₂²	C₂×C₃×#41	1	1	yes	characteristic
308	C₂²	C₂×C₃×#41	1	1	yes	characteristic
309	C₂²	C₂×C₃×#41	1	1	yes	characteristic
310	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
311	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	2	no	—
320	C₄	C₃×#123	1	4	yes	complemented
321	C₃	#2445	1	1	yes	characteristic, direct product factor, Sylow
322	C₂	—	4	4	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×C₃×#123	1	1	yes	characteristic
325	C₂	C₃×#532	1	1	yes	characteristic
326	C₂	C₃×#540	1	1	yes	characteristic
327	C₂	C₃×D₄²	1	1	yes	characteristic
328	C₂	C₃×#546	1	1	yes	characteristic
329	C₂	C₃×#550	1	1	yes	characteristic
330	C₂	C₃×#559	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂×C₃×#127	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
2. C₂×C₃×#123	C₂	semidirect
3. C₂×C₃×C₄×D₄	C₂	semidirect
4. C₃×#394	C₂	semidirect
5. C₃×#394	C₂	semidirect
6. C₃×#390	C₂	semidirect
7. C₃×#390	C₂	semidirect
8. C₃×#390	C₂	semidirect
9. C₃×#389	C₂	semidirect
10. C₃×#386	C₂	semidirect
11. C₃×C₄×C₄⋊C₄	C₂	semidirect
12. #2445	C₃	direct (N × H)
40. C₂×#123	C₂×C₃	semidirect
41. C₂×C₄×D₄	C₂×C₃	semidirect
42. #394	C₂×C₃	semidirect
43. #394	C₂×C₃	semidirect
44. #390	C₂×C₃	semidirect
45. #390	C₂×C₃	semidirect
46. #390	C₂×C₃	semidirect
47. #389	C₂×C₃	semidirect
48. #386	C₂×C₃	semidirect
49. C₄×C₄⋊C₄	C₂×C₃	semidirect
80. C₃×C₄⋊C₄	D₄	semidirect
194. C₄⋊C₄	C₃×D₄	semidirect
234. C₃×C₄	#123	semidirect
320. C₄	C₃×#123	semidirect
321. C₃	#2445	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×#127	2
C₂×C₃×#123	2
C₂×C₃×C₄×D₄	2
C₃×#394	2
C₃×#390	2
C₃×#389	2
C₃×#386	2
C₃×C₄×C₄⋊C₄	2
#2445	3

Subgroups — C3×#2445 #84309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2445 #84309