Subgroups — C₂²×#16839512 #29648978

Order 15616.

942 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#92418	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839512	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×#92418	—	2	1	no	—
6	C₂³×#92418	C₄	1	1	yes	characteristic
7	C₂³×#92418	C₄	1	1	yes	characteristic
8	C₂×C₄⋊#92418	—	2	3	no	—
9	C₂×C₄⋊#92418	C₂²	1	6	yes	—
10	C₂×C₄×#92418	C₄	1	6	yes	—
11	C₂×C₄×#92418	C₂²	1	6	yes	—
12	#16839512	C₂²	1	16	yes	direct product factor
13	C₂³×C₈×C₆₁	—	2	1	no	—
14	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₁	—	2	3	no	—
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	6	yes	—
17	C₂²×#92418	—	2	4	no	—
18	C₂²×#92418	—	2	6	no	—
19	C₂²×#92418	—	4	4	no	—
20	C₂²×#92418	C₈	1	8	yes	complemented
21	C₂²×#92418	C₂×C₄	1	6	yes	—
22	C₂²×#92418	C₂×C₄	1	6	yes	—
23	C₄⋊#92418	—	2	6	no	—
24	C₄⋊#92418	C₂³	1	4	yes	—
25	C₄×#92418	C₂×C₄	1	12	yes	—
26	C₄×#92418	C₂³	1	4	yes	—
27	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	—	2	6	no	—
31	C₂²×C₈×C₆₁	—	4	4	no	—
32	C₂×C₄²×C₆₁	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₆₁	D₄	1	3	yes	—
34	C₂×C₄²×C₆₁	Q₈	1	3	yes	—
35	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
36	C₄⋊C₈×C₆₁	—	2	6	no	—
37	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
38	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	—	2	4	no	—
40	C₂²×C₄×C₆₁	C₄⋊C₄	1	3	yes	—
41	C₂²×C₄×C₆₁	C₄⋊C₄	1	3	yes	—
42	C₂²×C₄×C₆₁	C₄⋊C₄	1	3	yes	—
43	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
45	C₂²×C₄×C₆₁	SD₁₆	1	4	yes	—
46	C₂²×C₄×C₆₁	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₆₁	C₂×Q₈	1	3	yes	—
50	C₂×C₈×C₆₁	—	2	4	no	—
51	C₂×C₈×C₆₁	—	4	12	no	—
52	C₄²×C₆₁	C₄⋊C₄	1	6	yes	—
53	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
54	C₄²×C₆₁	C₂×D₄	1	3	yes	—
55	C₄²×C₆₁	C₂×Q₈	1	3	yes	—
56	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
57	C₂×#92418	—	2	12	no	—
58	C₂×#92418	—	2	4	no	—
59	C₂×#92418	—	4	12	no	—
60	C₂×#92418	C₂×C₈	1	24	yes	complemented
61	C₂×#92418	C₂²×C₄	1	4	yes	—
62	C₂×#92418	C₂²×C₄	1	4	yes	—
63	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
64	C₂³×C₆₁	#106	1	4	yes	—
65	C₂³×C₆₁	#108	1	4	yes	—
66	C₂³×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
67	C₂×C₄×C₆₁	—	2	6	no	—
68	C₂×C₄×C₆₁	—	2	6	no	—
69	C₂×C₄×C₆₁	C₄⋊C₈	1	12	yes	—
70	C₂×C₄×C₆₁	#106	1	12	yes	—
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₆₁	C₂²×C₈	1	6	yes	—
75	C₂×C₄×C₆₁	C₂×M₁₆	1	6	yes	—
76	C₂×C₄×C₆₁	C₂×SD₁₆	1	6	yes	—
77	C₂×C₄×C₆₁	C₂×SD₁₆	1	6	yes	—
78	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
81	C₈×C₆₁	—	4	4	no	—
82	#92418	—	2	4	no	—
83	#92418	—	4	4	no	—
84	#92418	C₂²×C₈	1	8	yes	complemented
85	C₂²×#334	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	#334	1	16	yes	complemented
87	C₂²×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
88	C₂²×C₆₁	C₂×#106	1	6	yes	—
89	C₂²×C₆₁	C₂×#108	1	6	yes	—
90	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₆₁	—	2	3	no	—
92	C₄×C₆₁	—	2	1	no	—
93	C₄×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
94	C₄×C₆₁	C₂×#106	1	6	yes	—
95	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
97	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
98	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
99	C₂²×C₄⋊C₈	—	61	1	no	—
100	C₂²×C₄⋊C₈	D₆₁	1	1	yes	characteristic
101	C₂²×C₄×C₈	—	61	1	no	—
102	C₂×#334	—	61	12	no	—
103	C₂×C₆₁	C₂×#334	1	12	yes	complemented
104	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₆₁	C₂²×#106	1	1	yes	characteristic
106	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
107	C₂³×C₈	—	2	1	no	—
108	C₂³×C₈	—	61	1	no	—
109	C₂³×C₈	—	61	1	no	—
110	C₂³×C₈	—	122	1	no	—
111	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
112	C₂×C₄⋊C₈	—	2	3	no	—
113	C₂×C₄⋊C₈	—	61	6	no	—
114	C₂×C₄⋊C₈	—	122	3	no	—
115	C₂×C₄⋊C₈	C₂×D₆₁	1	6	yes	—
116	C₂×C₄×C₈	—	61	6	no	—
117	C₂×C₄×C₈	—	61	6	no	—
118	#334	—	61	16	no	—
119	C₆₁	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	Dic₁₂₂	1	1	yes	characteristic
121	C₂³×C₄	C₄×D₆₁	1	1	yes	characteristic
122	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₂²×C₈	—	2	6	no	—
124	C₂²×C₈	—	4	4	no	—
125	C₂²×C₈	—	61	8	no	—
126	C₂²×C₈	—	61	6	no	—
127	C₂²×C₈	—	61	6	no	—
128	C₂²×C₈	—	122	4	no	—
129	C₂²×C₈	—	122	6	no	—
130	C₂²×C₈	—	244	4	no	—
131	C₂×C₄²	Dic₁₂₂	1	3	yes	—
132	C₂×C₄²	C₄×D₆₁	1	3	yes	—
133	C₂×C₄²	C₂²⋊D₆₁	1	3	yes	—
134	C₂×C₄²	C₂²×D₆₁	1	3	yes	—
135	C₄⋊C₈	—	2	6	no	—
136	C₄⋊C₈	—	61	4	no	—
137	C₄⋊C₈	—	122	6	no	—
138	C₄⋊C₈	C₂²×D₆₁	1	4	yes	—
139	C₄×C₈	—	61	12	no	—
140	C₄×C₈	—	61	4	no	—
141	C₂⁴	#11758196	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	C₈×D₆₁	1	4	yes	—
144	C₂²×C₄	#11758189	1	4	yes	—
145	C₂²×C₄	#11758196	1	3	yes	—
146	C₂²×C₄	#11758196	1	3	yes	—
147	C₂²×C₄	#11758196	1	3	yes	—
148	C₂²×C₄	#11758200	1	4	yes	—
149	C₂²×C₄	#11758201	1	4	yes	—
150	C₂²×C₄	C₂×Dic₁₂₂	1	3	yes	—
151	C₂²×C₄	C₂×C₄×D₆₁	1	3	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	12	no	—
155	C₂×C₈	—	61	4	no	—
156	C₂×C₈	—	61	24	no	—
157	C₂×C₈	—	61	4	no	—
158	C₂×C₈	—	122	12	no	—
159	C₂×C₈	—	122	4	no	—
160	C₂×C₈	—	244	12	no	—
161	C₄²	#11758196	1	6	yes	—
162	C₄²	C₂×Dic₁₂₂	1	3	yes	—
163	C₄²	C₂×C₄×D₆₁	1	3	yes	—
164	C₄²	C₂×C₂²⋊D₆₁	1	3	yes	—
165	C₄²	C₂³×D₆₁	1	1	yes	characteristic
166	C₂³	#14519706	1	4	yes	—
167	C₂³	#14519708	1	4	yes	—
168	C₂³	#14519712	1	4	yes	—
169	C₂³	C₂×#11758196	1	3	yes	—
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	—	2	6	no	—
172	C₂×C₄	#14519706	1	12	yes	—
173	C₂×C₄	#14519708	1	12	yes	—
174	C₂×C₄	C₂×#11758196	1	3	yes	—
175	C₂×C₄	C₂×#11758196	1	3	yes	—
176	C₂×C₄	C₂×#11758196	1	3	yes	—
177	C₂×C₄	C₂×C₈×D₆₁	1	6	yes	—
178	C₂×C₄	C₂×#11758189	1	6	yes	—
179	C₂×C₄	C₂×#11758201	1	6	yes	—
180	C₂×C₄	C₂×#11758200	1	6	yes	—
181	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
184	C₈	—	4	4	no	—
185	C₈	—	61	8	no	—
186	C₈	—	122	4	no	—
187	C₈	—	244	4	no	—
188	C₂²	#16839512	1	16	yes	direct product factor
189	C₂²	C₂×#14519706	1	6	yes	—
190	C₂²	C₂×#14519708	1	6	yes	—
191	C₂²	C₂×#14519712	1	6	yes	—
192	C₂²	C₂²×#11758196	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×#14519706	1	6	yes	—
196	C₄	C₂×#14519708	1	6	yes	—
197	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
198	C₄	C₂²×#11758189	1	1	yes	characteristic
199	C₄	C₂²×#11758201	1	1	yes	characteristic
200	C₄	C₂²×#11758200	1	1	yes	characteristic
201	C₂	C₂×#16839512	1	12	yes	direct product factor
202	C₂	C₂²×#14519706	1	1	yes	characteristic
203	C₂	C₂²×#14519708	1	1	yes	characteristic
204	C₂	C₂²×#14519712	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂²×C₄⋊C₈×C₆₁	C₂
socle, prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₆₁
hypocenter	C₆₁	C₂²×#334
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#16839512	C₂	direct (N × H)
12. #16839512	C₂²	direct (N × H)
20. C₂²×#92418	C₈	semidirect
60. C₂×#92418	C₂×C₈	semidirect
84. #92418	C₂²×C₈	semidirect
86. C₂²×C₆₁	#334	semidirect
103. C₂×C₆₁	C₂×#334	semidirect
119. C₆₁	C₂²×#334	semidirect
188. C₂²	#16839512	direct (N × H)
201. C₂	C₂×#16839512	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×C₄⋊#92418	2
C₂²×C₄×#92418	2
C₂×#16839512	2
C₂²×C₄⋊C₈×C₆₁	2
C₂²×#334	61

Subgroups — C22×#16839512 #29648978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16839512 #29648978