Subgroups — C₃×#2392 #84256

Order 384.

696 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#396	C₂	1	2	yes	maximal, complemented
7	C₃×#394	C₂	1	2	yes	maximal, complemented
8	C₃×#394	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#392	C₂	1	1	yes	maximal, characteristic
10	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	#2392	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	C₂³×C₃×C₄	C₂²	1	2	yes	—
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₃×C₄×D₄	—	2	8	no	—
18	C₃×C₄×D₄	—	2	4	no	—
19	C₃×#117	—	2	4	no	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
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	2	yes	—
24	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
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×C₄²	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂²	1	2	yes	—
34	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
35	C₃×#95	C₂²	1	2	yes	—
36	C₃×#95	C₂²	1	2	yes	—
37	C₃×#95	C₂²	1	1	yes	characteristic
38	C₃×#95	C₂²	1	1	yes	characteristic
39	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
40	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
42	#398	C₂×C₃	1	1	yes	characteristic, complemented
43	#397	C₂×C₃	1	1	yes	characteristic, complemented
44	#396	C₂×C₃	1	2	yes	complemented
45	#394	C₂×C₃	1	2	yes	complemented
46	#394	C₂×C₃	1	1	yes	characteristic, complemented
47	#392	C₂×C₃	1	1	yes	characteristic
48	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
49	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
50	C₂⁴×C₃	C₂³	1	1	yes	characteristic
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	4	no	—
57	C₂²×C₃×C₄	—	2	2	no	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	—	2	4	no	—
60	C₂²×C₃×C₄	—	2	4	no	—
61	C₂²×C₃×C₄	—	2	4	no	—
62	C₂²×C₃×C₄	D₄	1	4	yes	complemented
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂³	1	2	yes	—
71	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	4	no	—
75	C₃×C₄⋊C₄	—	2	2	no	—
76	C₃×C₄⋊C₄	—	2	4	no	—
77	C₃×C₂²⋊C₄	—	2	4	no	—
78	C₃×C₂²⋊C₄	—	2	4	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	2	2	no	—
81	C₃×C₂²⋊C₄	—	2	4	no	—
82	C₃×C₂²⋊C₄	—	2	4	no	—
83	C₃×C₂²⋊C₄	—	2	4	no	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	—	2	4	no	—
86	C₃×C₄²	D₄	1	4	yes	complemented
87	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
88	C₂²×D₄	C₂²×C₃	1	2	yes	—
89	C₂³×C₄	C₂²×C₃	1	2	yes	—
90	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
91	C₄×D₄	—	2	8	no	—
92	C₄×D₄	—	2	4	no	—
93	#117	—	2	4	no	—
94	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
100	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
101	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
102	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
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×C₄²	C₂²×C₃	1	1	yes	characteristic
107	C₂×C₄²	C₂²×C₃	1	2	yes	—
108	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
109	#95	C₂²×C₃	1	2	yes	—
110	#95	C₂²×C₃	1	2	yes	—
111	#95	C₂²×C₃	1	1	yes	characteristic
112	#95	C₂²×C₃	1	1	yes	characteristic
113	C₂³×C₃	—	2	1	no	—
114	C₂³×C₃	—	2	2	no	—
115	C₂³×C₃	—	2	2	no	—
116	C₂³×C₃	—	2	2	no	—
117	C₂³×C₃	—	2	1	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	C₂×D₄	1	2	yes	—
124	C₂³×C₃	#41	1	4	yes	—
125	C₂³×C₃	C₂⁴	1	1	yes	characteristic
126	C₃×D₄	—	2	8	no	—
127	C₃×D₄	—	2	16	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	8	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	4	8	no	—
146	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
147	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
148	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
149	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
150	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
151	C₂×C₃×C₄	#41	1	4	yes	—
152	C₂×C₃×C₄	#41	1	2	yes	—
153	C₂⁴	C₂³×C₃	1	1	yes	characteristic
154	C₂⁴	C₂³×C₃	1	2	yes	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	C₃×D₄	1	4	yes	complemented
166	C₂²×C₄	C₂³×C₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₃	1	2	yes	—
174	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	4	no	—
189	C₄²	C₃×D₄	1	4	yes	complemented
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	—	2	4	no	—
193	C₂²×C₃	—	2	4	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	—	4	2	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	—	4	2	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	—	4	1	no	—
201	C₂²×C₃	—	4	2	no	—
202	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
203	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
204	C₂²×C₃	C₂×#41	1	1	yes	characteristic
205	C₂²×C₃	C₂×#41	1	2	yes	—
206	C₂²×C₃	C₂×#41	1	2	yes	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	4	2	no	—
212	C₃×C₄	—	4	2	no	—
213	C₃×C₄	—	4	1	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	#121	1	4	yes	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₃×D₄	1	2	yes	—
227	C₂³	C₃×#41	1	4	yes	—
228	C₂³	C₂⁴×C₃	1	1	yes	characteristic
229	D₄	—	2	8	no	—
230	D₄	—	2	16	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
250	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
251	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
252	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
253	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
254	C₂×C₄	C₃×#41	1	4	yes	—
255	C₂×C₄	C₃×#41	1	2	yes	—
256	C₂×C₃	—	4	2	no	—
257	C₂×C₃	—	4	4	no	—
258	C₂×C₃	C₂×#121	1	1	yes	characteristic
259	C₂×C₃	#525	1	1	yes	characteristic
260	C₂×C₃	#525	1	2	yes	—
261	C₂×C₃	#529	1	2	yes	—
262	C₂×C₃	#532	1	1	yes	characteristic
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
276	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
277	C₂²	C₂×C₃×#41	1	1	yes	characteristic
278	C₂²	C₂×C₃×#41	1	2	yes	—
279	C₂²	C₂×C₃×#41	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	4	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	1	no	—
288	C₄	C₃×#121	1	4	yes	—
289	C₃	#2392	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	—	4	2	no	—
291	C₂	—	4	4	no	—
292	C₂	C₂×C₃×#121	1	1	yes	characteristic
293	C₂	C₃×#525	1	1	yes	characteristic
294	C₂	C₃×#525	1	2	yes	—
295	C₂	C₃×#529	1	2	yes	—
296	C₂	C₃×#532	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₃×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×D₄	C₂	semidirect
2. C₂×C₃×C₄×D₄	C₂	semidirect
3. C₂×C₃×#117	C₂	semidirect
4. C₃×#398	C₂	semidirect
5. C₃×#397	C₂	semidirect
6. C₃×#396	C₂	semidirect
7. C₃×#394	C₂	semidirect
8. C₃×#394	C₂	semidirect
10. C₃×C₄×C₄⋊C₄	C₂	semidirect
11. C₃×C₄×C₂²⋊C₄	C₂	semidirect
12. #2392	C₃	direct (N × H)
39. C₂×C₄×D₄	C₂×C₃	semidirect
40. C₂×C₄×D₄	C₂×C₃	semidirect
41. C₂×#117	C₂×C₃	semidirect
42. #398	C₂×C₃	semidirect
43. #397	C₂×C₃	semidirect
44. #396	C₂×C₃	semidirect
45. #394	C₂×C₃	semidirect
46. #394	C₂×C₃	semidirect
48. C₄×C₄⋊C₄	C₂×C₃	semidirect
49. C₄×C₂²⋊C₄	C₂×C₃	semidirect
62. C₂²×C₃×C₄	D₄	semidirect
86. C₃×C₄²	D₄	semidirect
165. C₂²×C₄	C₃×D₄	semidirect
189. C₄²	C₃×D₄	semidirect
289. C₃	#2392	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×D₄	2
C₂×C₃×#117	2
C₃×#398	2
C₃×#397	2
C₃×#396	2
C₃×#394	2
C₃×#392	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2392	3

Subgroups — C3×#2392 #84256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2392 #84256