Subgroups — C₂×C₄×#92495 #16858891

Order 3968.

1728 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92495	C₂	1	1	yes	maximal, characteristic
2	C₄×#92495	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₆₂	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₁₂₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₂²⋊D₃₁	C₂	1	4	yes	maximal, complemented
6	C₂×#11758503	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11758449	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₃₁	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
11	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₃₁	C₂²	1	2	yes	—
13	C₂×#92495	C₄	1	16	yes	direct product factor
14	C₂×#92495	C₂²	1	8	yes	—
15	C₂²×D₁₂₄	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₃₁	C₂²	1	4	yes	—
17	C₂²×C₄×D₃₁	C₂²	1	2	yes	—
18	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₃₁	C₂²	1	8	yes	complemented
23	C₂×C₂²⋊Dic₃₁	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₃₁	C₂²	1	32	yes	complemented
25	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
26	#11758503	C₂²	1	8	yes	complemented
27	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
28	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
29	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
30	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
31	C₄×D₁₂₄	C₂²	1	16	yes	complemented
32	#11758449	C₂²	1	16	yes	complemented
33	C₄²×D₃₁	C₂²	1	16	yes	complemented
34	C₄×Dic₆₂	C₂²	1	16	yes	complemented
35	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
36	C₂³×D₃₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₃₁	—	2	8	no	—
38	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
39	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	8	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
44	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₃₁	C₂×C₄	1	16	yes	complemented
46	C₂×C₂²⋊D₃₁	C₂³	1	8	yes	—
47	C₂²×Dic₃₁	C₂³	1	4	yes	—
48	C₂²×Dic₃₁	C₂³	1	2	yes	—
49	#92495	C₂×C₄	1	64	yes	direct product factor
50	C₂×D₁₂₄	C₂×C₄	1	8	yes	complemented
51	C₂×D₁₂₄	C₂³	1	4	yes	—
52	C₂×C₄×D₃₁	—	2	16	no	—
53	C₂×C₄×D₃₁	C₂×C₄	1	16	yes	complemented
54	C₂×C₄×D₃₁	C₂³	1	16	yes	—
55	C₂×C₄×D₃₁	C₂³	1	8	yes	—
56	C₂×Dic₆₂	C₂×C₄	1	8	yes	complemented
57	C₂×Dic₆₂	C₂³	1	4	yes	—
58	C₄²×C₃₁	C₂³	1	8	yes	—
59	C₄²×C₃₁	C₂³	1	4	yes	—
60	C₄²×C₃₁	C₂³	1	4	yes	—
61	C₂²⋊Dic₃₁	C₂³	1	8	yes	—
62	(C₂×D₃₁)⋊C₄	C₂³	1	16	yes	—
63	C₄⋊Dic₃₁	C₂³	1	8	yes	—
64	Dic₃₁⋊C₄	C₂³	1	16	yes	—
65	C₄×Dic₃₁	C₂³	1	16	yes	—
66	C₄×Dic₃₁	C₂³	1	8	yes	—
67	C₂³×C₃₁	—	2	4	no	—
68	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
69	C₂³×C₃₁	C₂⁴	1	2	yes	—
70	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
71	C₂²×D₃₁	—	2	8	no	—
72	C₂²×D₃₁	C₂²×C₄	1	8	yes	—
73	C₂²×D₃₁	C₂⁴	1	4	yes	—
74	C₂×C₄×C₃₁	—	2	4	no	—
75	C₂×C₄×C₃₁	—	2	16	no	—
76	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₃₁	C₂²×C₄	1	16	yes	complemented
79	C₂×C₄×C₃₁	#41	1	8	yes	complemented
80	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
81	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
82	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
83	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
84	C₂²⋊D₃₁	C₂²×C₄	1	32	yes	complemented
85	C₂×Dic₃₁	—	2	8	no	—
86	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
87	C₂×Dic₃₁	C₂⁴	1	8	yes	—
88	C₂×Dic₃₁	C₂⁴	1	4	yes	—
89	D₁₂₄	C₂²×C₄	1	16	yes	complemented
90	C₄×D₃₁	—	2	32	no	—
91	C₄×D₃₁	C₂²×C₄	1	32	yes	complemented
92	Dic₆₂	C₂²×C₄	1	16	yes	complemented
93	C₂×C₄×#41	—	31	1	no	maximal, Sylow
94	C₂²×C₃₁	—	2	8	no	—
95	C₂²×C₃₁	—	2	4	no	—
96	C₂²×C₃₁	C₂³×C₄	1	4	yes	—
97	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
98	C₂²×C₃₁	C₂×#41	1	4	yes	—
99	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
100	C₂×D₃₁	—	2	16	no	—
101	C₂×D₃₁	—	2	8	no	—
102	C₂×D₃₁	C₂³×C₄	1	8	yes	—
103	C₄×C₃₁	—	2	4	no	—
104	C₄×C₃₁	C₂³×C₄	1	4	yes	—
105	C₄×C₃₁	C₂³×C₄	1	4	yes	—
106	C₄×C₃₁	C₂×#41	1	8	yes	complemented
107	Dic₃₁	—	2	8	no	—
108	Dic₃₁	C₂³×C₄	1	8	yes	—
109	C₂²×#41	—	31	1	no	—
110	C₄×#41	—	31	16	no	—
111	C₂×C₄×Q₈	—	31	2	no	—
112	C₂×C₄×D₄	—	31	4	no	—
113	C₂×C₄×D₄	—	31	2	no	—
114	C₂×#117	—	31	1	no	—
115	C₂×#117	—	31	2	no	—
116	C₂²×C₄²	—	31	2	no	—
117	C₂²×C₄²	D₃₁	1	1	yes	characteristic, complemented
118	C₂×C₃₁	—	2	4	no	—
119	C₂×C₃₁	C₄×#41	1	4	yes	complemented
120	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
121	C₂×C₃₁	C₂²×#41	1	2	yes	—
122	D₃₁	—	2	8	no	—
123	C₂×#41	—	31	16	no	—
124	C₂×#41	—	31	8	no	—
125	C₂²×Q₈	—	31	1	no	—
126	C₂²×D₄	—	31	2	no	—
127	C₂²×D₄	—	31	1	no	—
128	C₂³×C₄	—	31	2	no	—
129	C₂³×C₄	—	31	4	no	—
130	C₂³×C₄	C₂×D₃₁	1	2	yes	—
131	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
132	C₄×Q₈	—	31	16	no	—
133	C₄×D₄	—	31	32	no	—
134	C₄×D₄	—	31	16	no	—
135	#117	—	31	8	no	—
136	#117	—	31	16	no	—
137	C₂×C₄⋊C₄	—	31	4	no	—
138	C₂×C₄⋊C₄	—	31	2	no	—
139	C₂×C₂²⋊C₄	—	31	2	no	—
140	C₂×C₂²⋊C₄	—	31	4	no	—
141	C₂×C₄²	—	31	2	no	—
142	C₂×C₄²	—	31	4	no	—
143	C₂×C₄²	—	31	16	no	—
144	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
145	C₂×C₄²	C₂×D₃₁	1	2	yes	—
146	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
147	C₂×C₄²	C₂×D₃₁	1	8	yes	complemented
148	C₃₁	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
149	C₂⁴	—	31	2	no	—
150	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
151	#41	—	31	64	no	—
152	C₂×Q₈	—	31	8	no	—
153	C₂×Q₈	—	31	4	no	—
154	C₂×D₄	—	31	16	no	—
155	C₂×D₄	—	31	8	no	—
156	C₂×D₄	—	31	8	no	—
157	C₂×D₄	—	31	4	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	31	2	no	—
160	C₂²×C₄	—	31	16	no	—
161	C₂²×C₄	—	31	8	no	—
162	C₂²×C₄	—	31	4	no	—
163	C₂²×C₄	—	31	16	no	—
164	C₂²×C₄	—	62	16	no	—
165	C₂²×C₄	C₄×D₃₁	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
167	C₂²×C₄	C₂²×D₃₁	1	8	yes	—
168	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
171	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
172	C₄⋊C₄	—	31	16	no	—
173	C₄⋊C₄	—	31	8	no	—
174	C₂²⋊C₄	—	31	8	no	—
175	C₂²⋊C₄	—	31	16	no	—
176	C₄²	—	31	8	no	—
177	C₄²	—	31	16	no	—
178	C₄²	C₂²×D₃₁	1	4	yes	—
179	C₄²	C₂²×D₃₁	1	4	yes	—
180	C₄²	C₂²×D₃₁	1	8	yes	—
181	C₂³	—	2	4	no	—
182	C₂³	—	31	8	no	—
183	C₂³	—	31	4	no	—
184	C₂³	—	62	8	no	—
185	C₂³	C₂×C₄×D₃₁	1	4	yes	—
186	C₂³	C₂³×D₃₁	1	2	yes	—
187	C₂³	C₂³×D₃₁	1	1	yes	characteristic
188	Q₈	—	31	16	no	—
189	D₄	—	31	32	no	—
190	D₄	—	31	16	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	31	8	no	—
194	C₂×C₄	—	31	4	no	—
195	C₂×C₄	—	31	32	no	—
196	C₂×C₄	—	31	8	no	—
197	C₂×C₄	—	62	8	no	—
198	C₂×C₄	—	62	32	no	—
199	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
201	C₂×C₄	C₂×C₄×D₃₁	1	16	yes	complemented
202	C₂×C₄	#92495	1	8	yes	direct product factor
203	C₂×C₄	C₂³×D₃₁	1	4	yes	—
204	C₂×C₄	C₂³×D₃₁	1	2	yes	—
205	C₂×C₄	C₂³×D₃₁	1	2	yes	—
206	C₂×C₄	C₂³×D₃₁	1	4	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	—	31	8	no	—
210	C₂²	—	62	8	no	—
211	C₂²	—	62	16	no	—
212	C₂²	C₂²×C₄×D₃₁	1	4	yes	—
213	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
214	C₂²	C₂×#92495	1	4	yes	—
215	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	31	8	no	—
218	C₄	—	62	8	no	—
219	C₄	C₂²×C₄×D₃₁	1	4	yes	—
220	C₄	C₂²×C₄×D₃₁	1	4	yes	—
221	C₄	C₂×#92495	1	8	yes	direct product factor
222	C₂	—	2	4	no	—
223	C₂	—	62	8	no	—
224	C₂	C₄×#92495	1	4	yes	direct product factor
225	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
226	C₂	C₂²×#92495	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂²×C₄²×C₃₁	C₂
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂²×#92495	C₂
cosocle	C₂²×C₃₁	C₂⁵
hypercenter	C₂²×C₄²	D₃₁
hypocenter	C₃₁	C₂×C₄×#41
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#92495	C₂	direct (N × H)
3. C₂×C₄×Dic₆₂	C₂	semidirect
4. C₂×C₄×D₁₂₄	C₂	semidirect
5. C₂×C₄×C₂²⋊D₃₁	C₂	semidirect
6. C₂×#11758503	C₂	semidirect
7. C₂×#11758449	C₂	semidirect
8. C₂×C₄²×D₃₁	C₂	semidirect
9. C₂²×C₄²×C₃₁	C₂	semidirect
13. C₂×#92495	C₄	direct (N × H)
22. C₂×C₄²×C₃₁	C₂²	semidirect
24. C₄×C₂²⋊D₃₁	C₂²	semidirect
26. #11758503	C₂²	semidirect
31. C₄×D₁₂₄	C₂²	semidirect
32. #11758449	C₂²	semidirect
33. C₄²×D₃₁	C₂²	semidirect
34. C₄×Dic₆₂	C₂²	semidirect
38. C₂²×C₄×C₃₁	C₂×C₄	semidirect
45. C₂×C₂²⋊D₃₁	C₂×C₄	semidirect
49. #92495	C₂×C₄	direct (N × H)
50. C₂×D₁₂₄	C₂×C₄	semidirect
53. C₂×C₄×D₃₁	C₂×C₄	semidirect
56. C₂×Dic₆₂	C₂×C₄	semidirect
78. C₂×C₄×C₃₁	C₂²×C₄	semidirect
79. C₂×C₄×C₃₁	#41	semidirect
84. C₂²⋊D₃₁	C₂²×C₄	semidirect
89. D₁₂₄	C₂²×C₄	semidirect
91. C₄×D₃₁	C₂²×C₄	semidirect
92. Dic₆₂	C₂²×C₄	semidirect
106. C₄×C₃₁	C₂×#41	semidirect
117. C₂²×C₄²	D₃₁	semidirect
119. C₂×C₃₁	C₄×#41	semidirect
147. C₂×C₄²	C₂×D₃₁	semidirect
148. C₃₁	C₂×C₄×#41	semidirect
165. C₂²×C₄	C₄×D₃₁	semidirect
201. C₂×C₄	C₂×C₄×D₃₁	semidirect
202. C₂×C₄	#92495	direct (N × H)
221. C₄	C₂×#92495	direct (N × H)
224. C₂	C₄×#92495	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×#92495	2
C₄×#92495	2
C₂×C₄×Dic₆₂	2
C₂×C₄×D₁₂₄	2
C₂×C₄×C₂²⋊D₃₁	2
C₂×#11758503	2
C₂×#11758449	2
C₂×C₄²×D₃₁	2
C₂²×C₄²×C₃₁	2
C₂×C₄×#41	31

Subgroups — C2×C4×#92495 #16858891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#92495 #16858891