Subgroups — C₄×#11758496 #16854483

Order 3968.

500 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758496	C₂	1	1	yes	maximal, characteristic
2	C₄×#92499	C₂	1	2	yes	maximal, complemented
3	Q₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#92481	C₂	1	2	yes	maximal, complemented
5	#14521134	C₂	1	2	yes	maximal, complemented
6	C₄×D₂₄₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521131	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521086	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521044	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#92499	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	Q₈×Dic₃₁	C₂²	1	2	yes	—
16	C₂×#92481	C₂²	1	2	yes	—
17	#11758496	C₄	1	8	yes	direct product factor
18	C₂×D₂₄₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
21	#11758470	—	2	2	no	—
22	#11758470	C₂²	1	2	yes	—
23	#11758469	—	2	2	no	—
24	#11758469	C₂²	1	2	yes	—
25	C₄×D₁₂₄	—	2	2	no	—
26	C₄×D₁₂₄	C₂²	1	2	yes	—
27	C₄²×D₃₁	—	2	2	no	—
28	C₄²×D₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#107	C₂²	1	1	yes	characteristic
30	C₃₁×#103	C₂²	1	2	yes	—
31	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	#11758423	C₂²	1	2	yes	—
33	#11758409	C₂²	1	2	yes	—
34	#11758406	C₂²	1	1	yes	characteristic
35	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
36	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
37	#11758397	C₂²	1	2	yes	—
38	#11758396	C₂²	1	2	yes	—
39	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
41	#92499	—	2	4	no	—
42	#92499	C₂×C₄	1	8	yes	complemented
43	C₂×D₁₂₄	—	2	2	no	—
44	C₂×D₁₂₄	C₂³	1	2	yes	—
45	C₂×C₄×D₃₁	—	2	2	no	—
46	C₂×C₄×D₃₁	—	2	2	no	—
47	C₂×C₄×D₃₁	—	2	2	no	—
48	C₂×C₄×D₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄×D₃₁	D₄	1	1	yes	characteristic
50	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
51	Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₄²×C₃₁	—	2	2	no	—
57	C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	#92481	C₂×C₄	1	8	yes	complemented
59	(C₂×D₃₁)⋊C₄	—	2	2	no	—
60	(C₂×D₃₁)⋊C₄	—	2	2	no	—
61	(C₂×D₃₁)⋊C₄	—	2	2	no	—
62	C₄⋊Dic₃₁	—	2	2	no	—
63	C₄⋊Dic₃₁	C₂³	1	2	yes	—
64	C₄×Dic₃₁	—	2	2	no	—
65	C₄×Dic₃₁	—	2	2	no	—
66	C₄×Dic₃₁	D₄	1	1	yes	characteristic
67	C₄×Dic₃₁	D₄	1	1	yes	characteristic
68	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	D₂₄₈	C₂×C₄	1	4	yes	complemented
72	C₈×D₃₁	—	2	2	no	—
73	C₈×D₃₁	C₂×C₄	1	4	yes	complemented
74	C₂²×D₃₁	—	2	2	no	—
75	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₃₁	—	2	2	no	—
77	Q₈×C₃₁	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₃₁	—	2	2	no	—
84	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
87	D₁₂₄	—	2	8	no	—
88	D₁₂₄	—	2	2	no	—
89	D₁₂₄	C₂²×C₄	1	4	yes	—
90	C₄×D₃₁	—	2	2	no	—
91	C₄×D₃₁	—	2	8	no	—
92	C₄×D₃₁	—	2	2	no	—
93	C₄×D₃₁	—	4	4	no	—
94	C₄×D₃₁	—	4	4	no	—
95	C₄×D₃₁	C₂²×C₄	1	2	yes	—
96	C₄×D₃₁	#41	1	2	yes	—
97	C₈×C₃₁	—	2	1	no	—
98	C₈×C₃₁	C₂²×C₄	1	2	yes	—
99	C₃₁⋊C₈	—	2	1	no	—
100	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
101	C₄×#135	—	31	1	no	maximal, Sylow
102	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
103	C₂×D₃₁	—	2	1	no	—
104	C₂×D₃₁	—	2	1	no	—
105	C₂×D₃₁	—	2	4	no	—
106	C₂×D₃₁	—	4	2	no	—
107	C₂×D₃₁	—	4	2	no	—
108	C₂×D₃₁	C₄×D₄	1	2	yes	—
109	C₄×C₃₁	—	2	4	no	—
110	C₄×C₃₁	—	2	1	no	—
111	C₄×C₃₁	—	4	2	no	—
112	C₄×C₃₁	#135	1	2	yes	complemented
113	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
114	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
115	Dic₃₁	—	2	1	no	—
116	Dic₃₁	—	4	2	no	—
117	Dic₃₁	C₄×D₄	1	2	yes	—
118	Dic₃₁	#135	1	2	yes	complemented
119	C₂×#135	—	31	1	no	—
120	C₄×#41	—	31	2	no	—
121	C₄×Q₁₆	—	31	1	no	—
122	C₄×Q₁₆	D₃₁	1	1	yes	characteristic, complemented
123	C₄×SD₁₆	—	31	2	no	—
124	C₄×SD₁₆	—	31	2	no	—
125	C₄×D₈	—	31	1	no	—
126	C₄×D₈	—	31	1	no	—
127	#427	—	31	1	no	—
128	#416	—	31	2	no	—
129	C₂×C₄×C₈	—	31	1	no	—
130	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
132	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
133	D₃₁	—	2	2	no	—
134	D₃₁	—	4	4	no	—
135	C₂×#41	—	31	2	no	—
136	#135	—	31	8	no	—
137	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
138	C₂×SD₁₆	—	31	2	no	—
139	C₂×D₈	—	31	1	no	—
140	C₂²×C₈	—	31	1	no	—
141	C₂²×C₈	—	31	1	no	—
142	C₄×Q₈	—	31	2	no	—
143	C₄×Q₈	C₂×D₃₁	1	2	yes	—
144	C₄×D₄	—	31	2	no	—
145	C₄×D₄	—	31	2	no	—
146	C₄×D₄	—	62	2	no	—
147	C₄×D₄	—	62	2	no	—
148	#117	—	31	2	no	—
149	#117	—	62	2	no	—
150	C₂×C₄²	—	31	1	no	—
151	C₂×C₄²	—	62	2	no	—
152	#107	—	31	1	no	—
153	#107	C₂×D₃₁	1	1	yes	characteristic
154	#106	—	31	2	no	—
155	#103	—	31	2	no	—
156	#103	C₂×D₃₁	1	2	yes	—
157	#102	—	31	2	no	—
158	#102	—	31	2	no	—
159	C₄×C₈	—	31	1	no	—
160	C₄×C₈	—	31	1	no	—
161	C₄×C₈	—	31	1	no	—
162	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
163	C₃₁	C₄×#135	1	1	yes	characteristic, complemented, Sylow
164	#41	—	31	8	no	—
165	#41	—	62	4	no	—
166	C₂×Q₈	C₂²×D₃₁	1	2	yes	—
167	C₂×D₄	—	31	2	no	—
168	C₂×D₄	—	62	2	no	—
169	C₂²×C₄	—	31	1	no	—
170	C₂²×C₄	—	31	1	no	—
171	C₂²×C₄	—	31	1	no	—
172	C₂²×C₄	—	62	2	no	—
173	C₂²×C₄	—	62	2	no	—
174	C₂²×C₄	—	62	2	no	—
175	Q₁₆	C₄×D₃₁	1	4	yes	complemented
176	SD₁₆	—	31	8	no	—
177	D₈	—	31	4	no	—
178	C₂×C₈	—	31	1	no	—
179	C₂×C₈	—	31	4	no	—
180	C₂×C₈	—	31	1	no	—
181	C₂×C₈	—	62	2	no	—
182	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	31	2	no	—
186	C₄⋊C₄	—	62	2	no	—
187	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
188	C₂²⋊C₄	—	62	2	no	—
189	C₂²⋊C₄	—	62	2	no	—
190	C₂²⋊C₄	—	62	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	31	1	no	—
193	C₄²	—	31	1	no	—
194	C₄²	—	31	1	no	—
195	C₄²	—	62	2	no	—
196	C₄²	—	62	2	no	—
197	C₄²	C₂²×D₃₁	1	1	yes	characteristic
198	C₂³	—	31	1	no	—
199	C₂³	—	62	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	C₂×C₄×D₃₁	1	4	yes	—
202	D₄	—	31	4	no	—
203	D₄	—	62	8	no	—
204	D₄	—	62	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	31	1	no	—
208	C₂×C₄	—	31	1	no	—
209	C₂×C₄	—	31	2	no	—
210	C₂×C₄	—	31	2	no	—
211	C₂×C₄	—	31	1	no	—
212	C₂×C₄	—	62	2	no	—
213	C₂×C₄	—	62	8	no	—
214	C₂×C₄	—	62	2	no	—
215	C₂×C₄	—	62	2	no	—
216	C₂×C₄	—	124	4	no	—
217	C₂×C₄	—	124	4	no	—
218	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
219	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	31	2	no	—
223	C₈	—	62	1	no	—
224	C₈	C₂×C₄×D₃₁	1	2	yes	—
225	C₂²	—	31	2	no	—
226	C₂²	—	62	1	no	—
227	C₂²	—	62	1	no	—
228	C₂²	—	62	4	no	—
229	C₂²	—	124	2	no	—
230	C₂²	—	124	2	no	—
231	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
232	C₄	—	2	4	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	31	2	no	—
236	C₄	—	31	2	no	—
237	C₄	—	62	1	no	—
238	C₄	—	124	2	no	—
239	C₄	#11758496	1	2	yes	direct product factor
240	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
241	C₄	#41×D₃₁	1	1	yes	characteristic
242	C₂	—	62	2	no	—
243	C₂	—	124	4	no	—
244	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic
245	C₂	#135×D₃₁	1	1	yes	characteristic
246	C₂	C₂×#11758496	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×Q₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#11758496	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×Q₁₆	D₃₁
hypocenter	C₃₁	C₄×#135
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#92499	C₂	semidirect
3. Q₁₆×Dic₃₁	C₂	semidirect
4. C₄×#92481	C₂	semidirect
5. #14521134	C₂	semidirect
6. C₄×D₂₄₈	C₂	semidirect
7. #14521131	C₂	semidirect
8. #14521086	C₂	semidirect
9. #14521044	C₂	semidirect
10. C₄×C₈×D₃₁	C₂	semidirect
11. C₄×Q₁₆×C₃₁	C₂	semidirect
17. #11758496	C₄	direct (N × H)
42. #92499	C₂×C₄	semidirect
51. Q₁₆×C₃₁	C₂×C₄	semidirect
58. #92481	C₂×C₄	semidirect
71. D₂₄₈	C₂×C₄	semidirect
73. C₈×D₃₁	C₂×C₄	semidirect
112. C₄×C₃₁	#135	semidirect
118. Dic₃₁	#135	semidirect
122. C₄×Q₁₆	D₃₁	semidirect
163. C₃₁	C₄×#135	semidirect
175. Q₁₆	C₄×D₃₁	semidirect
239. C₄	#11758496	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×#11758496	2
C₄×#92499	2
Q₁₆×Dic₃₁	2
C₄×#92481	2
#14521134	2
C₄×D₂₄₈	2
#14521131	2
#14521086	2
#14521044	2
C₄×C₈×D₃₁	2
C₄×Q₁₆×C₃₁	2
C₄×#135	31

Subgroups — C4×#11758496 #16854483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11758496 #16854483