Subgroups — C₄×#11758464 #16848060

Order 3968.

684 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758464	C₂	1	2	yes	maximal
3	C₄⋊C₄×Dic₃₁	C₂	1	2	yes	maximal
4	C₄×Dic₃₁⋊C₄	C₂	1	2	yes	maximal
5	#14520900	C₂	1	4	yes	maximal
6	C₄²×Dic₃₁	C₂	1	1	yes	maximal, characteristic
7	C₄²×Dic₃₁	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₁⋊C₄	C₂²	1	2	yes	—
15	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
16	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
17	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
18	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
20	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
23	#11758464	C₄	1	16	yes	direct product factor
24	C₄×Dic₆₂	C₄	1	8	yes	complemented
25	C₃₁×#95	C₂²	1	2	yes	—
26	C₃₁⋊#95	C₂²	1	4	yes	—
27	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
28	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃₁	C₂³	1	2	yes	—
32	C₂²×Dic₃₁	C₂³	1	2	yes	—
33	C₂²×Dic₃₁	C₂³	1	2	yes	—
34	C₂²×Dic₃₁	C₂³	1	2	yes	—
35	C₂×Dic₆₂	C₂×C₄	1	4	yes	—
36	C₂×Dic₆₂	C₂×C₄	1	8	yes	—
37	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
38	C₄²×C₃₁	—	2	4	no	—
39	C₄²×C₃₁	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
41	Dic₃₁⋊C₄	C₂×C₄	1	8	yes	—
42	Dic₃₁⋊C₄	C₂×C₄	1	16	yes	—
43	C₄×Dic₃₁	—	2	2	no	—
44	C₄×Dic₃₁	—	2	4	no	—
45	C₄×Dic₃₁	—	2	8	no	—
46	C₄×Dic₃₁	C₂×C₄	1	8	yes	—
47	C₄×Dic₃₁	C₂×C₄	1	8	yes	—
48	C₄×Dic₃₁	C₂×C₄	1	16	yes	—
49	C₄×Dic₃₁	Q₈	1	4	yes	complemented
50	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₃₁	—	2	4	no	—
52	C₂×C₄×C₃₁	—	2	2	no	—
53	C₂×C₄×C₃₁	—	2	4	no	—
54	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₃₁	#41	1	2	yes	—
61	C₂×Dic₃₁	—	2	2	no	—
62	C₂×Dic₃₁	—	2	2	no	—
63	C₂×Dic₃₁	—	2	4	no	—
64	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
65	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
66	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
67	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
68	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
69	C₂×Dic₃₁	C₂×Q₈	1	4	yes	—
70	C₂×Dic₃₁	#41	1	4	yes	—
71	Dic₆₂	C₄²	1	16	yes	complemented
72	C₄²×Q₈	—	31	1	no	maximal, Sylow
73	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
74	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
77	C₂²×C₃₁	C₂×#41	1	2	yes	—
78	C₄×C₃₁	—	2	8	no	—
79	C₄×C₃₁	—	2	2	no	—
80	C₄×C₃₁	C₂×C₄²	1	4	yes	—
81	C₄×C₃₁	C₄×Q₈	1	4	yes	complemented
82	Dic₃₁	—	2	4	no	—
83	Dic₃₁	—	2	4	no	—
84	Dic₃₁	C₂×C₄²	1	8	yes	—
85	Dic₃₁	C₄×Q₈	1	8	yes	complemented
86	C₂×C₄×Q₈	—	31	2	no	—
87	C₂×C₄×Q₈	—	31	1	no	—
88	C₄×C₄⋊C₄	—	31	2	no	—
89	C₄×C₄⋊C₄	—	31	2	no	—
90	C₄×C₄⋊C₄	—	31	4	no	—
91	C₄×C₄⋊C₄	D₃₁	1	1	yes	characteristic
92	C₄³	—	31	2	no	—
93	C₄³	—	31	1	no	—
94	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
95	C₂×C₃₁	C₂×C₄×Q₈	1	2	yes	—
96	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₃₁	C₄×#41	1	2	yes	—
98	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
99	C₂²×Q₈	—	31	1	no	—
100	C₄×Q₈	—	31	16	no	—
101	C₄×Q₈	—	31	8	no	—
102	C₂×C₄⋊C₄	—	31	4	no	—
103	C₂×C₄⋊C₄	—	31	2	no	—
104	C₂×C₄⋊C₄	—	31	1	no	—
105	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
106	C₂×C₄²	—	31	4	no	—
107	C₂×C₄²	—	31	2	no	—
108	C₂×C₄²	—	31	2	no	—
109	C₂×C₄²	—	31	1	no	—
110	C₂×C₄²	—	31	4	no	—
111	C₂×C₄²	—	31	2	no	—
112	C₂×C₄²	—	31	1	no	—
113	C₂×C₄²	C₂×D₃₁	1	2	yes	—
114	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
115	#95	—	31	4	no	—
116	#95	C₂×D₃₁	1	2	yes	—
117	C₃₁	C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
118	C₂×Q₈	—	31	8	no	—
119	C₂×Q₈	—	31	4	no	—
120	C₂²×C₄	—	31	2	no	—
121	C₂²×C₄	—	31	2	no	—
122	C₂²×C₄	—	31	2	no	—
123	C₂²×C₄	—	31	2	no	—
124	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
125	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
128	C₄⋊C₄	—	31	16	no	—
129	C₄⋊C₄	—	31	8	no	—
130	C₄⋊C₄	—	31	4	no	—
131	C₄⋊C₄	C₄×D₃₁	1	8	yes	—
132	C₄²	—	2	4	no	—
133	C₄²	—	31	16	no	—
134	C₄²	—	31	8	no	—
135	C₄²	—	31	8	no	—
136	C₄²	—	31	4	no	—
137	C₄²	—	62	8	no	—
138	C₄²	—	62	4	no	—
139	C₄²	—	62	2	no	—
140	C₄²	C₄×D₃₁	1	4	yes	—
141	C₂³	C₂³×D₃₁	1	1	yes	characteristic
142	Q₈	—	31	16	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	31	4	no	—
147	C₂×C₄	—	31	8	no	—
148	C₂×C₄	—	31	4	no	—
149	C₂×C₄	—	31	4	no	—
150	C₂×C₄	—	31	4	no	—
151	C₂×C₄	—	31	4	no	—
152	C₂×C₄	—	31	4	no	—
153	C₂×C₄	—	62	2	no	—
154	C₂×C₄	—	62	2	no	—
155	C₂×C₄	—	62	4	no	—
156	C₂×C₄	C₂×C₄×D₃₁	1	8	yes	—
157	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
159	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
160	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
161	C₂×C₄	#92497	1	2	yes	—
162	C₂×C₄	Q₈×D₃₁	1	2	yes	—
163	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
164	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
165	C₂²	C₂×#92497	1	1	yes	characteristic
166	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
167	C₂²	#41×D₃₁	1	2	yes	—
168	C₄	—	2	8	no	—
169	C₄	—	2	2	no	—
170	C₄	—	31	8	no	—
171	C₄	—	31	8	no	—
172	C₄	—	62	4	no	—
173	C₄	—	62	4	no	—
174	C₄	C₄²×D₃₁	1	4	yes	—
175	C₄	#11758464	1	4	yes	direct product factor
176	C₂	C₂×C₄²×D₃₁	1	1	yes	characteristic
177	C₂	C₄×Q₈×D₃₁	1	2	yes	—
178	C₂	C₂×#11758464	1	1	yes	characteristic
179	C₂	C₄×#92497	1	2	yes	—
180	C₂	#14521517	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Frattini	C₂³	C₂³×D₃₁
Fitting	C₄×C₄⋊C₄×C₃₁	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₃₁
hypocenter	C₃₁	C₄²×Q₈
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
23. #11758464	C₄	direct (N × H)
24. C₄×Dic₆₂	C₄	semidirect
49. C₄×Dic₃₁	Q₈	semidirect
71. Dic₆₂	C₄²	semidirect
81. C₄×C₃₁	C₄×Q₈	semidirect
85. Dic₃₁	C₄×Q₈	semidirect
117. C₃₁	C₄²×Q₈	semidirect
175. C₄	#11758464	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×C₄×Dic₆₂	2
C₂×#11758464	2
C₄⋊C₄×Dic₃₁	2
C₄×Dic₃₁⋊C₄	2
#14520900	2
C₄²×Dic₃₁	2
C₄×C₄⋊C₄×C₃₁	2
C₄²×Q₈	31

Subgroups — C4×#11758464 #16848060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11758464 #16848060