Subgroups — C₂×C₃×#388 #84153

Order 384.

928 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#388	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#388	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂⁵×C₃	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	—	2	4	no	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
19	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
20	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
21	C₂×C₃×C₂²⋊C₄	C₄	1	8	yes	complemented
22	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂×C₃×C₄²	—	2	4	no	—
26	C₂×C₃×C₄²	C₂²	1	4	yes	complemented
27	C₃×#95	C₂²	1	4	yes	complemented
28	C₃×#95	C₂²	1	4	yes	complemented
29	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
33	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic, complemented
34	#388	C₂×C₃	1	8	yes	direct product factor
35	C₂×#95	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×#95	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂⁴×C₃	—	2	2	no	—
38	C₂⁴×C₃	—	2	2	no	—
39	C₂⁴×C₃	—	2	2	no	—
40	C₂⁴×C₃	—	2	2	no	—
41	C₂⁴×C₃	—	2	2	no	—
42	C₂⁴×C₃	—	2	2	no	—
43	C₂⁴×C₃	C₂×C₄	1	4	yes	—
44	C₂⁴×C₃	C₂³	1	2	yes	—
45	C₂⁴×C₃	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₄	—	2	2	no	—
47	C₂²×C₃×C₄	—	2	2	no	—
48	C₂²×C₃×C₄	—	2	2	no	—
49	C₂²×C₃×C₄	—	2	2	no	—
50	C₂²×C₃×C₄	—	2	2	no	—
51	C₂²×C₃×C₄	—	2	2	no	—
52	C₂²×C₃×C₄	—	2	2	no	—
53	C₂²×C₃×C₄	—	2	2	no	—
54	C₂²×C₃×C₄	—	2	2	no	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	—	2	2	no	—
57	C₂²×C₃×C₄	—	2	2	no	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
61	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
62	C₂²×C₃×C₄	D₄	1	4	yes	—
63	C₂²×C₃×C₄	D₄	1	4	yes	—
64	C₂²×C₃×C₄	C₂³	1	2	yes	—
65	C₂²×C₃×C₄	C₂³	1	2	yes	—
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₂²×C₃×C₄	C₂³	1	2	yes	—
68	C₂²×C₃×C₄	C₂³	1	2	yes	—
69	C₂²×C₃×C₄	C₂³	1	2	yes	—
70	C₃×C₄⋊C₄	—	2	8	no	—
71	C₃×C₂²⋊C₄	—	2	8	no	—
72	C₃×C₂²⋊C₄	—	2	8	no	—
73	C₃×C₂²⋊C₄	C₂×C₄	1	16	yes	complemented
74	C₃×C₄²	—	2	8	no	—
75	C₂⁵	C₂²×C₃	1	1	yes	characteristic
76	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
80	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
81	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	2	4	no	—
83	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
84	C₂×C₂²⋊C₄	—	2	4	no	—
85	C₂×C₂²⋊C₄	—	2	4	no	—
86	C₂×C₂²⋊C₄	C₃×C₄	1	8	yes	complemented
87	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
88	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
90	C₂×C₄²	—	2	4	no	—
91	C₂×C₄²	C₂²×C₃	1	4	yes	complemented
92	#95	C₂²×C₃	1	4	yes	complemented
93	#95	C₂²×C₃	1	4	yes	complemented
94	C₂³×C₃	—	2	8	no	—
95	C₂³×C₃	—	2	8	no	—
96	C₂³×C₃	—	2	8	no	—
97	C₂³×C₃	—	2	2	no	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	2	2	no	—
102	C₂³×C₃	—	2	2	no	—
103	C₂³×C₃	—	4	4	no	—
104	C₂³×C₃	—	4	4	no	—
105	C₂³×C₃	—	4	4	no	—
106	C₂³×C₃	—	4	4	no	—
107	C₂³×C₃	C₂²×C₄	1	4	yes	—
108	C₂³×C₃	C₂²×C₄	1	2	yes	—
109	C₂³×C₃	C₂×D₄	1	2	yes	—
110	C₂³×C₃	C₂×D₄	1	2	yes	—
111	C₂³×C₃	#41	1	2	yes	—
112	C₂³×C₃	#41	1	2	yes	—
113	C₂³×C₃	#41	1	2	yes	—
114	C₂³×C₃	#41	1	2	yes	—
115	C₂³×C₃	C₂⁴	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	8	no	—
117	C₂×C₃×C₄	—	2	8	no	—
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	8	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	4	4	no	—
135	C₂×C₃×C₄	—	4	4	no	—
136	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
137	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
138	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
139	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
140	C₂⁴	—	2	2	no	—
141	C₂⁴	—	2	2	no	—
142	C₂⁴	—	2	2	no	—
143	C₂⁴	—	2	2	no	—
144	C₂⁴	—	2	2	no	—
145	C₂⁴	—	2	2	no	—
146	C₂⁴	C₂×C₃×C₄	1	4	yes	—
147	C₂⁴	C₂³×C₃	1	2	yes	—
148	C₂⁴	C₂³×C₃	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
164	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
165	C₂²×C₄	C₃×D₄	1	4	yes	—
166	C₂²×C₄	C₃×D₄	1	4	yes	—
167	C₂²×C₄	C₂³×C₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₃	1	2	yes	—
171	C₂²×C₄	C₂³×C₃	1	2	yes	—
172	C₂²×C₄	C₂³×C₃	1	2	yes	—
173	C₄⋊C₄	—	2	8	no	—
174	C₂²⋊C₄	—	2	8	no	—
175	C₂²⋊C₄	—	2	8	no	—
176	C₂²⋊C₄	C₂×C₃×C₄	1	16	yes	complemented
177	C₄²	—	2	8	no	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	2	4	no	—
181	C₂²×C₃	—	4	2	no	—
182	C₂²×C₃	—	4	2	no	—
183	C₂²×C₃	—	4	2	no	—
184	C₂²×C₃	—	4	2	no	—
185	C₂²×C₃	—	4	16	no	—
186	C₂²×C₃	#117	1	4	yes	—
187	C₂²×C₃	C₄×D₄	1	4	yes	—
188	C₂²×C₃	C₄×D₄	1	4	yes	—
189	C₂²×C₃	#121	1	4	yes	—
190	C₂²×C₃	#123	1	4	yes	—
191	C₂²×C₃	#124	1	4	yes	—
192	C₂²×C₃	#126	1	4	yes	—
193	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
194	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
195	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
196	C₂²×C₃	C₂×#41	1	1	yes	characteristic
197	C₂²×C₃	C₂×#41	1	1	yes	characteristic
198	C₂²×C₃	C₂×#41	1	1	yes	characteristic
199	C₂²×C₃	C₂×#41	1	1	yes	characteristic
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	—	2	4	no	—
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	4	2	no	—
205	C₃×C₄	—	4	2	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	8	no	—
208	C₂³	—	2	8	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	4	no	—
216	C₂³	—	4	4	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	4	no	—
219	C₂³	C₂²×C₃×C₄	1	4	yes	—
220	C₂³	C₂²×C₃×C₄	1	2	yes	—
221	C₂³	C₂×C₃×D₄	1	2	yes	—
222	C₂³	C₂×C₃×D₄	1	2	yes	—
223	C₂³	C₃×#41	1	2	yes	—
224	C₂³	C₃×#41	1	2	yes	—
225	C₂³	C₃×#41	1	2	yes	—
226	C₂³	C₃×#41	1	2	yes	—
227	C₂³	C₂⁴×C₃	1	1	yes	characteristic
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	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	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
249	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
250	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
251	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
252	C₂×C₃	—	4	4	no	—
253	C₂×C₃	#388	1	8	yes	direct product factor
254	C₂×C₃	C₂×#117	1	1	yes	characteristic
255	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
257	C₂×C₃	C₂×#121	1	1	yes	characteristic
258	C₂×C₃	C₂×#123	1	1	yes	characteristic
259	C₂×C₃	C₂×#124	1	1	yes	characteristic
260	C₂×C₃	C₂×#126	1	1	yes	characteristic
261	C₂²	—	2	4	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	16	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	C₃×#117	1	4	yes	—
270	C₂²	C₃×C₄×D₄	1	4	yes	—
271	C₂²	C₃×C₄×D₄	1	4	yes	—
272	C₂²	C₃×#121	1	4	yes	—
273	C₂²	C₃×#123	1	4	yes	—
274	C₂²	C₃×#124	1	4	yes	—
275	C₂²	C₃×#126	1	4	yes	—
276	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
277	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
278	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
279	C₂²	C₂×C₃×#41	1	1	yes	characteristic
280	C₂²	C₂×C₃×#41	1	1	yes	characteristic
281	C₂²	C₂×C₃×#41	1	1	yes	characteristic
282	C₂²	C₂×C₃×#41	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	4	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₃	C₂×#388	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	—	4	4	no	—
291	C₂	C₃×#388	1	8	yes	direct product factor
292	C₂	C₂×C₃×#117	1	1	yes	characteristic
293	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
294	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
295	C₂	C₂×C₃×#121	1	1	yes	characteristic
296	C₂	C₂×C₃×#123	1	1	yes	characteristic
297	C₂	C₂×C₃×#124	1	1	yes	characteristic
298	C₂	C₂×C₃×#126	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₄⋊C₄	C₂	semidirect
5. C₂²×C₃×C₄²	C₂	semidirect
6. C₃×#388	C₂	direct (N × H)
7. C₂×C₃×#95	C₂	semidirect
8. C₂×C₃×#95	C₂	semidirect
9. C₂×#388	C₃	direct (N × H)
18. C₂×C₃×C₄⋊C₄	C₂²	semidirect
21. C₂×C₃×C₂²⋊C₄	C₄	semidirect
26. C₂×C₃×C₄²	C₂²	semidirect
27. C₃×#95	C₂²	semidirect
28. C₃×#95	C₂²	semidirect
29. C₂²×C₄⋊C₄	C₂×C₃	semidirect
33. C₂²×C₄²	C₂×C₃	semidirect
34. #388	C₂×C₃	direct (N × H)
35. C₂×#95	C₂×C₃	semidirect
36. C₂×#95	C₂×C₃	semidirect
73. C₃×C₂²⋊C₄	C₂×C₄	semidirect
83. C₂×C₄⋊C₄	C₂²×C₃	semidirect
86. C₂×C₂²⋊C₄	C₃×C₄	semidirect
91. C₂×C₄²	C₂²×C₃	semidirect
92. #95	C₂²×C₃	semidirect
93. #95	C₂²×C₃	semidirect
176. C₂²⋊C₄	C₂×C₃×C₄	semidirect
253. C₂×C₃	#388	direct (N × H)
289. C₃	C₂×#388	direct (N × H)
291. C₂	C₃×#388	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₂²⋊C₄	2
C₂²×C₃×C₄²	2
C₃×#388	2
C₂×C₃×#95	2
C₂×#388	3

Subgroups — C2×C3×#388 #84153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#388 #84153