Subgroups — C₂×#14521604 #16859000

Order 3968.

1168 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521604	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11758446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11758467	C₂	1	2	yes	maximal, complemented
4	C₂×#11758474	C₂	1	2	yes	maximal
5	C₂×#11758523	C₂	1	2	yes	maximal
6	C₂×Q₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
7	C₂×#11758464	C₂	1	2	yes	maximal, complemented
8	C₂×#11758469	C₂	1	2	yes	maximal
9	C₂×C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₆₂	C₂²	1	2	yes	—
16	C₃₁×C₄⋊Q₈	C₂²	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	#11758523	C₂²	1	16	yes	—
21	Q₈×Dic₃₁	C₂²	1	16	yes	complemented
22	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
23	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
26	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
31	#11758474	C₂²	1	16	yes	—
32	#11758469	C₂²	1	16	yes	—
33	#11758467	C₂²	1	16	yes	complemented
34	#11758464	C₂²	1	16	yes	complemented
35	C₄²×D₃₁	C₂²	1	8	yes	—
36	#11758446	C₂²	1	8	yes	complemented
37	C₂³×D₃₁	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₃₁	—	2	8	no	—
39	C₂×Q₈×C₃₁	C₂³	1	8	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	4	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	1	yes	characteristic
44	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
46	C₂²×Dic₃₁	C₂³	1	4	yes	—
47	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₃₁	—	2	4	no	—
49	C₂×C₄×D₃₁	—	2	4	no	—
50	C₂×C₄×D₃₁	Q₈	1	8	yes	complemented
51	C₂×C₄×D₃₁	C₂³	1	4	yes	—
52	C₂×C₄×D₃₁	C₂³	1	4	yes	—
53	C₂×C₄×D₃₁	C₂³	1	4	yes	—
54	C₂×Dic₆₂	—	2	8	no	—
55	C₂×Dic₆₂	C₂³	1	8	yes	—
56	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
57	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
58	C₄²×C₃₁	C₂³	1	4	yes	—
59	(C₂×D₃₁)⋊C₄	C₂³	1	16	yes	—
60	C₄⋊Dic₃₁	C₂³	1	8	yes	—
61	C₄⋊Dic₃₁	C₂³	1	8	yes	—
62	C₄⋊Dic₃₁	C₂³	1	16	yes	—
63	Dic₃₁⋊C₄	C₂³	1	16	yes	—
64	C₄×Dic₃₁	C₂³	1	4	yes	—
65	C₄×Dic₃₁	C₂³	1	4	yes	—
66	C₄×Dic₃₁	C₂³	1	4	yes	—
67	C₄×Dic₃₁	C₂³	1	16	yes	—
68	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
69	C₂²×D₃₁	—	2	2	no	—
70	C₂²×D₃₁	—	2	2	no	—
71	C₂²×D₃₁	C₂×Q₈	1	4	yes	—
72	C₂²×D₃₁	C₂⁴	1	2	yes	—
73	Q₈×C₃₁	—	2	16	no	—
74	C₂×C₄×C₃₁	—	2	8	no	—
75	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₃₁	#41	1	4	yes	—
77	C₂×C₄×C₃₁	#41	1	4	yes	—
78	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
79	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
80	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
81	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
82	C₂×Dic₃₁	—	2	8	no	—
83	C₂×Dic₃₁	—	2	2	no	—
84	C₂×Dic₃₁	—	2	2	no	—
85	C₂×Dic₃₁	C₂×Q₈	1	4	yes	—
86	C₂×Dic₃₁	C₂⁴	1	2	yes	—
87	C₂×Dic₃₁	C₂⁴	1	2	yes	—
88	C₂×Dic₃₁	C₂⁴	1	8	yes	—
89	C₂×Dic₃₁	C₂⁴	1	2	yes	—
90	C₄×D₃₁	—	2	8	no	—
91	C₄×D₃₁	—	2	8	no	—
92	C₄×D₃₁	C₂×Q₈	1	16	yes	complemented
93	Dic₆₂	—	2	16	no	—
94	C₂×#533	—	31	1	no	maximal, Sylow
95	C₂²×C₃₁	C₂²×Q₈	1	2	yes	—
96	C₂²×C₃₁	C₂×#41	1	2	yes	—
97	C₂²×C₃₁	C₂×#41	1	2	yes	—
98	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
99	C₂×D₃₁	—	2	8	no	—
100	C₂×D₃₁	—	2	2	no	—
101	C₂×D₃₁	—	2	2	no	—
102	C₂×D₃₁	C₂²×Q₈	1	4	yes	—
103	C₄×C₃₁	—	2	8	no	—
104	C₄×C₃₁	C₂²×Q₈	1	4	yes	—
105	C₄×C₃₁	C₂×#41	1	4	yes	—
106	C₄×C₃₁	C₂×#41	1	4	yes	—
107	Dic₃₁	—	2	8	no	—
108	Dic₃₁	—	2	2	no	—
109	Dic₃₁	—	2	2	no	—
110	Dic₃₁	C₂²×Q₈	1	4	yes	—
111	#533	—	31	16	no	—
112	C₂×C₄⋊Q₈	—	31	1	no	—
113	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
114	C₂×#125	—	31	2	no	—
115	C₂×C₂²⋊Q₈	—	31	2	no	—
116	C₂×C₂²⋊Q₈	—	31	2	no	—
117	C₂×C₄×Q₈	—	31	2	no	—
118	C₂×C₄×Q₈	—	31	2	no	—
119	C₂×#117	—	31	2	no	—
120	C₂²×C₄²	—	31	1	no	—
121	C₂×C₃₁	#533	1	4	yes	complemented
122	C₂×C₃₁	C₂³×Q₈	1	1	yes	characteristic
123	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
124	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
125	D₃₁	—	2	4	no	—
126	C₂²×Q₈	—	31	2	no	—
127	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
128	C₂³×C₄	—	31	1	no	—
129	C₂³×C₄	—	31	1	no	—
130	C₂³×C₄	—	31	1	no	—
131	C₄⋊Q₈	—	31	8	no	—
132	C₄⋊Q₈	C₂×D₃₁	1	8	yes	complemented
133	#125	—	31	16	no	—
134	C₂²⋊Q₈	—	31	16	no	—
135	C₂²⋊Q₈	—	31	16	no	—
136	C₄×Q₈	—	31	16	no	—
137	C₄×Q₈	—	31	16	no	—
138	#117	—	31	16	no	—
139	C₂×C₄⋊C₄	—	31	2	no	—
140	C₂×C₄⋊C₄	—	31	4	no	—
141	C₂×C₄⋊C₄	—	31	4	no	—
142	C₂×C₄⋊C₄	—	31	2	no	—
143	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
145	C₂×C₂²⋊C₄	—	31	4	no	—
146	C₂×C₄²	—	31	4	no	—
147	C₂×C₄²	—	31	1	no	—
148	C₂×C₄²	—	31	1	no	—
149	C₂×C₄²	—	31	1	no	—
150	C₂×C₄²	—	31	8	no	—
151	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
152	C₃₁	C₂×#533	1	1	yes	characteristic, complemented, Sylow
153	C₂⁴	—	31	1	no	—
154	C₂×Q₈	—	2	8	no	—
155	C₂×Q₈	—	31	8	no	—
156	C₂×Q₈	—	62	8	no	—
157	C₂×Q₈	C₂²×D₃₁	1	8	yes	—
158	C₂²×C₄	—	31	1	no	—
159	C₂²×C₄	—	31	8	no	—
160	C₂²×C₄	—	31	4	no	—
161	C₂²×C₄	—	31	1	no	—
162	C₂²×C₄	—	31	1	no	—
163	C₂²×C₄	—	31	4	no	—
164	C₂²×C₄	—	31	4	no	—
165	C₂²×C₄	—	31	4	no	—
166	C₂²×C₄	—	62	4	no	—
167	C₂²×C₄	—	62	4	no	—
168	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
169	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
172	C₄⋊C₄	—	31	16	no	—
173	C₄⋊C₄	—	31	8	no	—
174	C₄⋊C₄	—	31	16	no	—
175	C₄⋊C₄	—	31	8	no	—
176	C₄⋊C₄	C₂²×D₃₁	1	8	yes	—
177	C₄⋊C₄	C₂²×D₃₁	1	8	yes	—
178	C₂²⋊C₄	—	31	16	no	—
179	C₄²	—	31	4	no	—
180	C₄²	—	31	4	no	—
181	C₄²	—	31	4	no	—
182	C₄²	—	31	16	no	—
183	C₄²	C₂²×D₃₁	1	4	yes	—
184	C₂³	—	31	4	no	—
185	C₂³	—	31	2	no	—
186	C₂³	—	62	2	no	—
187	C₂³	—	62	2	no	—
188	C₂³	C₂³×D₃₁	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	62	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	31	4	no	—
193	C₂×C₄	—	31	2	no	—
194	C₂×C₄	—	31	16	no	—
195	C₂×C₄	—	31	2	no	—
196	C₂×C₄	—	31	2	no	—
197	C₂×C₄	—	31	8	no	—
198	C₂×C₄	—	62	2	no	—
199	C₂×C₄	—	62	2	no	—
200	C₂×C₄	—	62	8	no	—
201	C₂×C₄	—	62	8	no	—
202	C₂×C₄	—	62	8	no	—
203	C₂×C₄	#92497	1	4	yes	—
204	C₂×C₄	Q₈×D₃₁	1	4	yes	—
205	C₂×C₄	#92499	1	4	yes	—
206	C₂×C₄	C₂³×D₃₁	1	8	yes	—
207	C₂×C₄	C₂³×D₃₁	1	2	yes	—
208	C₂×C₄	C₂³×D₃₁	1	2	yes	—
209	C₂×C₄	C₂³×D₃₁	1	2	yes	—
210	C₂²	—	31	4	no	—
211	C₂²	—	62	8	no	—
212	C₂²	—	62	2	no	—
213	C₂²	—	62	2	no	—
214	C₂²	C₂×#92497	1	2	yes	—
215	C₂²	C₂×Q₈×D₃₁	1	2	yes	—
216	C₂²	C₂×#92499	1	2	yes	—
217	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
218	C₄	—	2	8	no	—
219	C₄	—	31	4	no	—
220	C₄	—	62	2	no	—
221	C₄	—	62	2	no	—
222	C₄	—	62	8	no	—
223	C₄	C₂×#92497	1	4	yes	—
224	C₄	C₂×Q₈×D₃₁	1	4	yes	—
225	C₄	C₂×#92499	1	4	yes	—
226	C₂	—	62	4	no	—
227	C₂	#14521604	1	4	yes	direct product factor
228	C₂	C₂²×Q₈×D₃₁	1	1	yes	characteristic
229	C₂	C₂²×#92497	1	1	yes	characteristic
230	C₂	C₂²×#92499	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₁
commutator, cosocle	C₂²×C₃₁	C₂⁵
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂³×D₃₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	C₂×#533
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14521604	C₂	direct (N × H)
2. C₂×#11758446	C₂	semidirect
3. C₂×#11758467	C₂	semidirect
6. C₂×Q₈×Dic₃₁	C₂	semidirect
7. C₂×#11758464	C₂	semidirect
10. C₂×C₃₁×C₄⋊Q₈	C₂	semidirect
16. C₃₁×C₄⋊Q₈	C₂²	semidirect
21. Q₈×Dic₃₁	C₂²	semidirect
33. #11758467	C₂²	semidirect
34. #11758464	C₂²	semidirect
36. #11758446	C₂²	semidirect
50. C₂×C₄×D₃₁	Q₈	semidirect
92. C₄×D₃₁	C₂×Q₈	semidirect
113. C₂×C₄⋊Q₈	D₃₁	semidirect
121. C₂×C₃₁	#533	semidirect
132. C₄⋊Q₈	C₂×D₃₁	semidirect
152. C₃₁	C₂×#533	semidirect
227. C₂	#14521604	direct (N × H)

Maximal subgroups

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

subgroup	index
#14521604	2
C₂×#11758446	2
C₂×#11758467	2
C₂×#11758474	2
C₂×#11758523	2
C₂×Q₈×Dic₃₁	2
C₂×#11758464	2
C₂×#11758469	2
C₂×C₄²×D₃₁	2
C₂×C₃₁×C₄⋊Q₈	2
C₂×#533	31

Subgroups — C2×#14521604 #16859000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521604 #16859000