Subgroups — C₃×D₄×C₂²⋊C₄ #84202

Order 384.

1252 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₃×#390	C₂	1	2	yes	maximal, complemented
5	C₃×#386	C₂	1	4	yes	maximal, complemented
6	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#379	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂⁵×C₃	C₂²	1	2	yes	—
11	C₂²×C₃×D₄	—	2	8	no	—
12	C₂²×C₃×D₄	C₄	1	8	yes	complemented
13	C₂²×C₃×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₄	C₂²	1	4	yes	complemented
18	C₃×C₄×D₄	—	2	8	no	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
25	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
26	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
27	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
28	C₃×#95	C₂²	1	2	yes	complemented
29	C₃×#95	C₂²	1	2	yes	complemented
30	C₂³×D₄	C₂×C₃	1	1	yes	characteristic
31	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
32	C₂²×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
33	#390	C₂×C₃	1	2	yes	complemented
34	#386	C₂×C₃	1	4	yes	complemented
35	#380	C₂×C₃	1	1	yes	characteristic, complemented
36	#379	C₂×C₃	1	2	yes	complemented
37	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂⁴×C₃	—	2	4	no	—
39	C₂⁴×C₃	—	2	8	no	—
40	C₂⁴×C₃	—	2	4	no	—
41	C₂⁴×C₃	—	4	4	no	—
42	C₂⁴×C₃	C₂×C₄	1	8	yes	complemented
43	C₂⁴×C₃	C₂³	1	2	yes	—
44	C₂⁴×C₃	C₂³	1	2	yes	—
45	C₂⁴×C₃	C₂³	1	1	yes	characteristic
46	C₂×C₃×D₄	—	2	8	no	—
47	C₂×C₃×D₄	—	2	32	no	—
48	C₂×C₃×D₄	—	2	4	no	—
49	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
50	C₂×C₃×D₄	C₂×C₄	1	8	yes	complemented
51	C₂×C₃×D₄	C₂×C₄	1	4	yes	complemented
52	C₂×C₃×D₄	D₄	1	8	yes	—
53	C₂²×C₃×C₄	—	2	4	no	—
54	C₂²×C₃×C₄	—	2	8	no	—
55	C₂²×C₃×C₄	—	2	8	no	—
56	C₂²×C₃×C₄	—	4	4	no	—
57	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄	C₂³	1	4	yes	—
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	2	yes	—
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₃×C₄⋊C₄	—	2	4	no	—
64	C₃×C₂²⋊C₄	—	2	4	no	—
65	C₃×C₂²⋊C₄	—	2	8	no	—
66	C₃×C₂²⋊C₄	—	2	2	no	—
67	C₃×C₂²⋊C₄	—	2	8	no	—
68	C₃×C₂²⋊C₄	—	4	4	no	—
69	C₃×C₂²⋊C₄	D₄	1	4	yes	direct product factor
70	C₃×C₄²	—	2	4	no	—
71	C₂⁵	C₂²×C₃	1	2	yes	—
72	C₂²×D₄	—	2	8	no	—
73	C₂²×D₄	C₃×C₄	1	8	yes	complemented
74	C₂²×D₄	C₂²×C₃	1	2	yes	—
75	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
76	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
77	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂³×C₄	C₂²×C₃	1	4	yes	complemented
79	C₄×D₄	—	2	8	no	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
81	C₂×C₂²⋊C₄	—	2	8	no	—
82	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
86	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
87	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
88	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
89	#95	C₂²×C₃	1	2	yes	complemented
90	#95	C₂²×C₃	1	2	yes	complemented
91	C₂³×C₃	—	2	2	no	—
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	4	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	16	no	—
97	C₂³×C₃	—	2	16	no	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	2	4	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	2	4	no	—
102	C₂³×C₃	—	4	8	no	—
103	C₂³×C₃	—	4	16	no	—
104	C₂³×C₃	—	4	16	no	—
105	C₂³×C₃	C₂²×C₄	1	4	yes	—
106	C₂³×C₃	C₂²×C₄	1	4	yes	—
107	C₂³×C₃	C₂²×C₄	1	2	yes	—
108	C₂³×C₃	C₂×D₄	1	8	yes	—
109	C₂³×C₃	C₂×D₄	1	2	yes	—
110	C₂³×C₃	#41	1	2	yes	—
111	C₂³×C₃	C₂⁴	1	1	yes	characteristic
112	C₃×D₄	—	2	16	no	—
113	C₃×D₄	—	2	8	no	—
114	C₃×D₄	C₂²⋊C₄	1	16	yes	direct product factor
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	4	16	no	—
126	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
127	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
128	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
129	C₂×C₃×C₄	C₂×D₄	1	4	yes	complemented
130	C₂⁴	—	2	4	no	—
131	C₂⁴	—	2	8	no	—
132	C₂⁴	—	2	4	no	—
133	C₂⁴	—	4	4	no	—
134	C₂⁴	C₂×C₃×C₄	1	8	yes	complemented
135	C₂⁴	C₂³×C₃	1	2	yes	—
136	C₂⁴	C₂³×C₃	1	2	yes	—
137	C₂⁴	C₂³×C₃	1	1	yes	characteristic
138	C₂×D₄	—	2	8	no	—
139	C₂×D₄	—	2	32	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
142	C₂×D₄	C₂×C₃×C₄	1	8	yes	complemented
143	C₂×D₄	C₂×C₃×C₄	1	4	yes	complemented
144	C₂×D₄	C₃×D₄	1	8	yes	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	2	8	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
150	C₂²×C₄	C₂³×C₃	1	2	yes	—
151	C₂²×C₄	C₂³×C₃	1	4	yes	—
152	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₃	1	2	yes	—
154	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₂²⋊C₄	—	2	4	no	—
157	C₂²⋊C₄	—	2	8	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	2	8	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₂²⋊C₄	C₃×D₄	1	4	yes	direct product factor
162	C₄²	—	2	4	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	—	2	4	no	—
165	C₂²×C₃	—	2	2	no	—
166	C₂²×C₃	—	2	4	no	—
167	C₂²×C₃	—	2	8	no	—
168	C₂²×C₃	—	2	4	no	—
169	C₂²×C₃	—	2	4	no	—
170	C₂²×C₃	—	2	8	no	—
171	C₂²×C₃	—	2	4	no	—
172	C₂²×C₃	—	2	4	no	—
173	C₂²×C₃	—	2	4	no	—
174	C₂²×C₃	—	4	32	no	—
175	C₂²×C₃	—	4	4	no	—
176	C₂²×C₃	—	4	4	no	—
177	C₂²×C₃	C₂×C₂²⋊C₄	1	8	yes	complemented
178	C₂²×C₃	C₄×D₄	1	4	yes	complemented
179	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
180	C₂²×C₃	C₂²×D₄	1	2	yes	—
181	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
182	C₂²×C₃	C₂×#41	1	1	yes	characteristic
183	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	4	4	no	—
187	C₃×C₄	—	4	2	no	—
188	C₃×C₄	C₂×C₂²⋊C₄	1	4	yes	complemented
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	4	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	16	no	—
195	C₂³	—	2	16	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	4	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	4	no	—
200	C₂³	—	4	8	no	—
201	C₂³	—	4	16	no	—
202	C₂³	—	4	16	no	—
203	C₂³	C₂²×C₃×C₄	1	4	yes	—
204	C₂³	C₂²×C₃×C₄	1	4	yes	—
205	C₂³	C₂²×C₃×C₄	1	2	yes	—
206	C₂³	C₂×C₃×D₄	1	8	yes	—
207	C₂³	C₂×C₃×D₄	1	2	yes	—
208	C₂³	C₃×#41	1	2	yes	—
209	C₂³	C₂⁴×C₃	1	1	yes	characteristic
210	D₄	—	2	16	no	—
211	D₄	—	2	8	no	—
212	D₄	C₃×C₂²⋊C₄	1	16	yes	direct product factor
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	16	no	—
224	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
225	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
226	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
227	C₂×C₄	C₂×C₃×D₄	1	4	yes	complemented
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	—	2	8	no	—
230	C₂×C₃	—	4	4	no	—
231	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
233	C₂×C₃	#518	1	1	yes	characteristic
234	C₂×C₃	D₄²	1	2	yes	—
235	C₂×C₃	#546	1	2	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	32	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	4	4	no	—
250	C₂²	C₂×C₃×C₂²⋊C₄	1	8	yes	complemented
251	C₂²	C₃×C₄×D₄	1	4	yes	complemented
252	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
253	C₂²	C₂²×C₃×D₄	1	2	yes	—
254	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
255	C₂²	C₂×C₃×#41	1	1	yes	characteristic
256	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	4	4	no	—
260	C₄	—	4	2	no	—
261	C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	complemented
262	C₃	D₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	—	2	4	no	—
264	C₂	—	2	8	no	—
265	C₂	—	4	4	no	—
266	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
267	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
268	C₂	C₃×#518	1	1	yes	characteristic
269	C₂	C₃×D₄²	1	2	yes	—
270	C₂	C₃×#546	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂³×C₃×D₄	C₂
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
2. C₂×C₃×C₄×D₄	C₂	semidirect
3. C₂²×C₃×C₂²⋊C₄	C₂	semidirect
4. C₃×#390	C₂	semidirect
5. C₃×#386	C₂	semidirect
6. C₃×#380	C₂	semidirect
7. C₃×#379	C₂	semidirect
8. C₃×C₄×C₂²⋊C₄	C₂	semidirect
9. D₄×C₂²⋊C₄	C₃	direct (N × H)
12. C₂²×C₃×D₄	C₄	semidirect
17. C₂³×C₃×C₄	C₂²	semidirect
19. C₂×C₃×C₄⋊C₄	C₂²	semidirect
24. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
25. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
26. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
27. C₂×C₃×C₄²	C₂²	semidirect
28. C₃×#95	C₂²	semidirect
29. C₃×#95	C₂²	semidirect
31. C₂×C₄×D₄	C₂×C₃	semidirect
32. C₂²×C₂²⋊C₄	C₂×C₃	semidirect
33. #390	C₂×C₃	semidirect
34. #386	C₂×C₃	semidirect
35. #380	C₂×C₃	semidirect
36. #379	C₂×C₃	semidirect
37. C₄×C₂²⋊C₄	C₂×C₃	semidirect
42. C₂⁴×C₃	C₂×C₄	semidirect
50. C₂×C₃×D₄	C₂×C₄	semidirect
51. C₂×C₃×D₄	C₂×C₄	semidirect
57. C₂²×C₃×C₄	C₂×C₄	semidirect
69. C₃×C₂²⋊C₄	D₄	direct (N × H)
73. C₂²×D₄	C₃×C₄	semidirect
78. C₂³×C₄	C₂²×C₃	semidirect
80. C₂×C₄⋊C₄	C₂²×C₃	semidirect
85. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
86. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
87. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
88. C₂×C₄²	C₂²×C₃	semidirect
89. #95	C₂²×C₃	semidirect
90. #95	C₂²×C₃	semidirect
114. C₃×D₄	C₂²⋊C₄	direct (N × H)
129. C₂×C₃×C₄	C₂×D₄	semidirect
134. C₂⁴	C₂×C₃×C₄	semidirect
142. C₂×D₄	C₂×C₃×C₄	semidirect
143. C₂×D₄	C₂×C₃×C₄	semidirect
149. C₂²×C₄	C₂×C₃×C₄	semidirect
161. C₂²⋊C₄	C₃×D₄	direct (N × H)
177. C₂²×C₃	C₂×C₂²⋊C₄	semidirect
178. C₂²×C₃	C₄×D₄	semidirect
188. C₃×C₄	C₂×C₂²⋊C₄	semidirect
212. D₄	C₃×C₂²⋊C₄	direct (N × H)
227. C₂×C₄	C₂×C₃×D₄	semidirect
250. C₂²	C₂×C₃×C₂²⋊C₄	semidirect
251. C₂²	C₃×C₄×D₄	semidirect
261. C₄	C₂×C₃×C₂²⋊C₄	semidirect
262. C₃	D₄×C₂²⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×D₄	2
C₂×C₃×C₄×D₄	2
C₂²×C₃×C₂²⋊C₄	2
C₃×#390	2
C₃×#386	2
C₃×#380	2
C₃×#379	2
C₃×C₄×C₂²⋊C₄	2
D₄×C₂²⋊C₄	3

Subgroups — C3×D4×C22⋊C4 #84202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×C₂²⋊C₄ #84202