Subgroups — C₂²×#67⋊C₄ #85682

Order 384.

2528 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#67	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄×D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#67⋊C₄	C₂	1	24	yes	maximal, direct product factor
9	C₂²×C₄×D₄	—	3	1	no	maximal, Sylow
10	C₂⁵×C₃	C₂²	1	1	yes	characteristic
11	C₂⁴×D₃	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂²×#67	C₄	1	16	yes	complemented
15	C₂²×#67	C₂²	1	12	yes	—
16	C₂³×Dic₃	—	2	8	no	—
17	C₂³×Dic₃	C₂²	1	1	yes	characteristic
18	C₂³×Dic₃	C₂²	1	1	yes	characteristic
19	C₂³×Dic₃	C₂²	1	1	yes	characteristic
20	C₂³×Dic₃	C₂²	1	12	yes	complemented
21	C₂²×C₄×D₃	—	2	8	no	—
22	C₂²×C₄×D₃	C₂²	1	12	yes	complemented
23	C₂×C₃×C₂²⋊C₄	C₂²	1	12	yes	complemented
24	C₂×(C₂×D₃)⋊C₄	C₂²	1	12	yes	complemented
25	C₂×Dic₃⋊C₄	C₂²	1	12	yes	complemented
26	C₂×C₄×Dic₃	C₂²	1	12	yes	complemented
27	#67⋊C₄	C₂²	1	64	yes	direct product factor
28	C₂³×D₄	—	3	1	no	—
29	C₂⁴×C₄	—	3	1	no	—
30	C₂⁴×C₄	—	3	1	no	—
31	C₂×C₄×D₄	—	3	24	no	—
32	C₂²×C₄⋊C₄	—	3	1	no	—
33	C₂²×C₂²⋊C₄	—	3	1	no	—
34	C₂²×C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
35	C₂²×C₄²	—	3	1	no	—
36	C₂⁴×C₃	—	2	4	no	—
37	C₂⁴×C₃	—	2	4	no	—
38	C₂⁴×C₃	C₂×C₄	1	8	yes	complemented
39	C₂⁴×C₃	C₂³	1	6	yes	—
40	C₂⁴×C₃	C₂³	1	1	yes	characteristic
41	C₂³×D₃	—	2	4	no	—
42	C₂³×D₃	—	2	4	no	—
43	C₂³×D₃	C₂×C₄	1	8	yes	complemented
44	C₂³×D₃	C₂³	1	6	yes	—
45	C₂²×C₃×C₄	—	2	4	no	—
46	C₂²×C₃×C₄	—	2	4	no	—
47	C₂²×C₃×C₄	C₂³	1	6	yes	—
48	C₂²×C₃×C₄	C₂³	1	6	yes	—
49	C₂×#67	C₂×C₄	1	96	yes	complemented
50	C₂×#67	C₂³	1	16	yes	—
51	C₂²×Dic₃	—	2	4	no	—
52	C₂²×Dic₃	—	2	48	no	—
53	C₂²×Dic₃	C₂×C₄	1	8	yes	complemented
54	C₂²×Dic₃	D₄	1	8	yes	complemented
55	C₂²×Dic₃	C₂³	1	6	yes	—
56	C₂²×Dic₃	C₂³	1	6	yes	—
57	C₂²×Dic₃	C₂³	1	6	yes	—
58	C₂²×Dic₃	C₂³	1	16	yes	complemented
59	C₂×C₄×D₃	—	2	48	no	—
60	C₂×C₄×D₃	C₂³	1	16	yes	complemented
61	C₃×C₂²⋊C₄	C₂³	1	16	yes	complemented
62	(C₂×D₃)⋊C₄	C₂³	1	16	yes	complemented
63	Dic₃⋊C₄	C₂³	1	16	yes	complemented
64	C₄×Dic₃	C₂³	1	16	yes	complemented
65	C₂⁵	—	3	1	no	—
66	C₂⁵	C₂×D₃	1	1	yes	characteristic
67	C₂²×D₄	—	3	16	no	—
68	C₂²×D₄	—	3	12	no	—
69	C₂³×C₄	—	3	1	no	—
70	C₂³×C₄	—	3	1	no	—
71	C₂³×C₄	—	3	12	no	—
72	C₂³×C₄	—	3	1	no	—
73	C₂³×C₄	—	3	12	no	—
74	C₂³×C₄	—	6	8	no	—
75	C₂³×C₄	—	6	8	no	—
76	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
77	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
78	C₄×D₄	—	3	64	no	—
79	C₂×C₄⋊C₄	—	3	12	no	—
80	C₂×C₂²⋊C₄	—	3	12	no	—
81	C₂×C₂²⋊C₄	C₂×D₃	1	12	yes	complemented
82	C₂×C₄²	—	3	12	no	—
83	C₂³×C₃	—	2	32	no	—
84	C₂³×C₃	—	2	12	no	—
85	C₂³×C₃	—	2	12	no	—
86	C₂³×C₃	C₂²×C₄	1	4	yes	—
87	C₂³×C₃	C₂²×C₄	1	24	yes	complemented
88	C₂³×C₃	C₂×D₄	1	4	yes	—
89	C₂³×C₃	#41	1	4	yes	—
90	C₂³×C₃	C₂⁴	1	4	yes	—
91	C₂³×C₃	C₂⁴	1	3	yes	—
92	C₂²×D₃	—	2	32	no	—
93	C₂²×D₃	—	2	12	no	—
94	C₂²×D₃	—	2	12	no	—
95	C₂²×D₃	C₂²×C₄	1	24	yes	complemented
96	C₂²×D₃	C₂⁴	1	4	yes	—
97	C₂×C₃×C₄	—	2	12	no	—
98	C₂×C₃×C₄	—	2	12	no	—
99	C₂×C₃×C₄	C₂⁴	1	4	yes	—
100	C₂×C₃×C₄	C₂⁴	1	4	yes	—
101	#67	C₂²×C₄	1	64	yes	complemented
102	C₂×Dic₃	—	2	12	no	—
103	C₂×Dic₃	—	2	32	no	—
104	C₂×Dic₃	C₂²×C₄	1	24	yes	complemented
105	C₂×Dic₃	C₂×D₄	1	24	yes	complemented
106	C₂×Dic₃	C₂⁴	1	4	yes	—
107	C₂×Dic₃	C₂⁴	1	4	yes	—
108	C₂×Dic₃	C₂⁴	1	4	yes	—
109	C₄×D₃	—	2	32	no	—
110	C₂⁴	—	2	4	no	—
111	C₂⁴	—	2	4	no	—
112	C₂⁴	—	3	8	no	—
113	C₂⁴	—	3	6	no	—
114	C₂⁴	—	6	4	no	—
115	C₂⁴	—	6	4	no	—
116	C₂⁴	C₄×D₃	1	8	yes	complemented
117	C₂⁴	C₂²×D₃	1	6	yes	—
118	C₂⁴	C₂²×D₃	1	1	yes	characteristic
119	C₂×D₄	—	3	96	no	—
120	C₂×D₄	—	3	16	no	—
121	C₂²×C₄	—	2	4	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	3	8	no	—
124	C₂²×C₄	—	3	8	no	—
125	C₂²×C₄	—	3	6	no	—
126	C₂²×C₄	—	3	6	no	—
127	C₂²×C₄	—	3	6	no	—
128	C₂²×C₄	—	3	16	no	—
129	C₂²×C₄	—	3	16	no	—
130	C₂²×C₄	—	6	48	no	—
131	C₂²×C₄	—	6	48	no	—
132	C₂²×C₄	—	6	4	no	—
133	C₂²×C₄	C₂²×D₃	1	6	yes	—
134	C₂²×C₄	C₂²×D₃	1	6	yes	—
135	C₄⋊C₄	—	3	16	no	—
136	C₂²⋊C₄	—	3	16	no	—
137	C₂²⋊C₄	C₂²×D₃	1	16	yes	complemented
138	C₄²	—	3	16	no	—
139	C₂²×C₃	—	2	48	no	—
140	C₂²×C₃	—	2	4	no	—
141	C₂²×C₃	—	2	4	no	—
142	C₂²×C₃	C₄×D₄	1	16	yes	complemented
143	C₂²×C₃	C₂³×C₄	1	6	yes	—
144	C₂²×C₃	C₂³×C₄	1	8	yes	complemented
145	C₂²×C₃	C₂²×D₄	1	6	yes	—
146	C₂²×C₃	C₂×#41	1	6	yes	—
147	C₂²×C₃	C₂⁵	1	1	yes	characteristic
148	C₂×D₃	—	2	48	no	—
149	C₂×D₃	—	2	4	no	—
150	C₂×D₃	—	2	4	no	—
151	C₂×D₃	C₂³×C₄	1	8	yes	complemented
152	C₃×C₄	—	2	4	no	—
153	C₃×C₄	—	2	4	no	—
154	Dic₃	—	2	4	no	—
155	Dic₃	C₂³×C₄	1	8	yes	complemented
156	Dic₃	C₂²×D₄	1	8	yes	complemented
157	C₂³	—	2	32	no	—
158	C₂³	—	2	12	no	—
159	C₂³	—	2	12	no	—
160	C₂³	—	3	24	no	—
161	C₂³	—	3	4	no	—
162	C₂³	—	6	32	no	—
163	C₂³	—	6	12	no	—
164	C₂³	—	6	12	no	—
165	C₂³	C₂×C₄×D₃	1	4	yes	—
166	C₂³	C₂×C₄×D₃	1	24	yes	complemented
167	C₂³	D₃×D₄	1	4	yes	—
168	C₂³	#237	1	4	yes	—
169	C₂³	C₂³×D₃	1	4	yes	—
170	C₂³	C₂³×D₃	1	3	yes	—
171	D₄	—	3	64	no	—
172	C₂×C₄	—	2	12	no	—
173	C₂×C₄	—	2	12	no	—
174	C₂×C₄	—	3	24	no	—
175	C₂×C₄	—	3	24	no	—
176	C₂×C₄	—	3	4	no	—
177	C₂×C₄	—	3	4	no	—
178	C₂×C₄	—	3	4	no	—
179	C₂×C₄	—	6	32	no	—
180	C₂×C₄	—	6	32	no	—
181	C₂×C₄	—	6	12	no	—
182	C₂×C₄	C₂³×D₃	1	4	yes	—
183	C₂×C₄	C₂³×D₃	1	4	yes	—
184	C₂×C₃	—	2	8	no	—
185	C₂×C₃	C₂×C₄×D₄	1	12	yes	complemented
186	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
187	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
188	C₂×C₃	C₂²×#41	1	1	yes	characteristic
189	D₃	—	2	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	48	no	—
193	C₂²	—	3	8	no	—
194	C₂²	—	6	4	no	—
195	C₂²	—	6	4	no	—
196	C₂²	—	6	48	no	—
197	C₂²	#67⋊C₄	1	16	yes	direct product factor
198	C₂²	C₂²×C₄×D₃	1	6	yes	—
199	C₂²	C₂²×C₄×D₃	1	8	yes	complemented
200	C₂²	C₂×D₃×D₄	1	6	yes	—
201	C₂²	C₂×#237	1	6	yes	—
202	C₂²	C₂⁴×D₃	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	4	no	—
205	C₄	—	3	8	no	—
206	C₄	—	3	8	no	—
207	C₄	—	6	4	no	—
208	C₃	C₂²×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
209	C₂	—	2	8	no	—
210	C₂	—	6	8	no	—
211	C₂	C₂×#67⋊C₄	1	12	yes	direct product factor
212	C₂	C₂³×C₄×D₃	1	1	yes	characteristic
213	C₂	C₂²×D₃×D₄	1	1	yes	characteristic
214	C₂	C₂²×#237	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator	C₂×C₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₂²⋊C₄	C₂
socle	C₂⁴×C₃	C₂³
prime radical	C₂³×#67	C₂
cosocle	C₂²×C₃	C₂⁵
hypercenter	C₂²×C₂²⋊C₄	D₃
hypocenter	C₃	C₂²×C₄×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Dic₃	C₂	semidirect
3. C₂³×C₄×D₃	C₂	semidirect
4. C₂²×C₃×C₂²⋊C₄	C₂	semidirect
5. C₂²×(C₂×D₃)⋊C₄	C₂	semidirect
6. C₂²×Dic₃⋊C₄	C₂	semidirect
7. C₂²×C₄×Dic₃	C₂	semidirect
8. C₂×#67⋊C₄	C₂	direct (N × H)
14. C₂²×#67	C₄	semidirect
20. C₂³×Dic₃	C₂²	semidirect
22. C₂²×C₄×D₃	C₂²	semidirect
23. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
24. C₂×(C₂×D₃)⋊C₄	C₂²	semidirect
25. C₂×Dic₃⋊C₄	C₂²	semidirect
26. C₂×C₄×Dic₃	C₂²	semidirect
27. #67⋊C₄	C₂²	direct (N × H)
34. C₂²×C₂²⋊C₄	D₃	semidirect
38. C₂⁴×C₃	C₂×C₄	semidirect
43. C₂³×D₃	C₂×C₄	semidirect
49. C₂×#67	C₂×C₄	semidirect
53. C₂²×Dic₃	C₂×C₄	semidirect
54. C₂²×Dic₃	D₄	semidirect
58. C₂²×Dic₃	C₂³	semidirect
60. C₂×C₄×D₃	C₂³	semidirect
61. C₃×C₂²⋊C₄	C₂³	semidirect
62. (C₂×D₃)⋊C₄	C₂³	semidirect
63. Dic₃⋊C₄	C₂³	semidirect
64. C₄×Dic₃	C₂³	semidirect
81. C₂×C₂²⋊C₄	C₂×D₃	semidirect
87. C₂³×C₃	C₂²×C₄	semidirect
95. C₂²×D₃	C₂²×C₄	semidirect
101. #67	C₂²×C₄	semidirect
104. C₂×Dic₃	C₂²×C₄	semidirect
105. C₂×Dic₃	C₂×D₄	semidirect
116. C₂⁴	C₄×D₃	semidirect
137. C₂²⋊C₄	C₂²×D₃	semidirect
142. C₂²×C₃	C₄×D₄	semidirect
144. C₂²×C₃	C₂³×C₄	semidirect
151. C₂×D₃	C₂³×C₄	semidirect
155. Dic₃	C₂³×C₄	semidirect
156. Dic₃	C₂²×D₄	semidirect
166. C₂³	C₂×C₄×D₃	semidirect
185. C₂×C₃	C₂×C₄×D₄	semidirect
197. C₂²	#67⋊C₄	direct (N × H)
199. C₂²	C₂²×C₄×D₃	semidirect
208. C₃	C₂²×C₄×D₄	semidirect
211. C₂	C₂×#67⋊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×#67	2
C₂⁴×Dic₃	2
C₂³×C₄×D₃	2
C₂²×C₃×C₂²⋊C₄	2
C₂²×(C₂×D₃)⋊C₄	2
C₂²×Dic₃⋊C₄	2
C₂²×C₄×Dic₃	2
C₂×#67⋊C₄	2
C₂²×C₄×D₄	3

Subgroups — C22×#67⋊C4 #85682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#67⋊C₄ #85682