Subgroups — C₂²×#16840813 #30625502

Order 15616.

2222 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840813	C₂	1	24	yes	maximal, direct product factor
2	C₂²×#14519844	C₂	1	2	yes	maximal
3	C₂²×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic
4	C₂²×#14519832	C₂	1	2	yes	maximal
5	C₂²×C₄×#92420	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×#92420	C₂²	1	1	yes	characteristic
8	C₂³×#92420	C₂²	1	2	yes	—
9	C₂³×#92420	C₂²	1	1	yes	characteristic
10	#16840813	C₂²	1	64	yes	direct product factor
11	C₂×#14519844	C₂²	1	24	yes	—
12	C₂×C₄⋊#92420	C₂²	1	12	yes	—
13	C₂×#14519832	C₂²	1	24	yes	—
14	C₂×C₄×#92420	C₂²	1	12	yes	—
15	C₂×Q₈×Dic₆₁	C₄	1	12	yes	—
16	C₂×Q₈×Dic₆₁	C₂²	1	12	yes	—
17	C₂²×C₄⋊Dic₆₁	C₄	1	2	yes	—
18	C₂²×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
19	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄×Dic₆₁	C₂²	1	2	yes	—
21	C₂³×Q₈×C₆₁	C₄	1	1	yes	characteristic
22	C₂²×#92420	—	2	4	no	—
23	C₂²×#92420	—	2	8	no	—
24	C₂²×#92420	Q₈	1	8	yes	complemented
25	C₂²×#92420	C₂³	1	6	yes	—
26	C₂²×#92420	C₂³	1	12	yes	—
27	C₂²×#92420	C₂³	1	6	yes	—
28	#14519844	C₂³	1	32	yes	—
29	C₄⋊#92420	C₂³	1	16	yes	—
30	#14519832	C₂³	1	32	yes	—
31	C₄×#92420	C₂³	1	16	yes	—
32	C₂³×Dic₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×Dic₆₁	C₂×C₄	1	2	yes	—
34	C₂³×Dic₆₁	C₂³	1	1	yes	characteristic
35	Q₈×Dic₆₁	C₂×C₄	1	48	yes	—
36	Q₈×Dic₆₁	C₂³	1	16	yes	—
37	C₂×C₄⋊Dic₆₁	C₂×C₄	1	24	yes	—
38	C₂×C₄⋊Dic₆₁	C₂×C₄	1	12	yes	—
39	C₂×C₄×Dic₆₁	C₂×C₄	1	12	yes	—
40	C₂×C₄×Dic₆₁	C₂×C₄	1	6	yes	—
41	C₂×C₄×Dic₆₁	C₂³	1	6	yes	—
42	C₂×C₄×Dic₆₁	C₂³	1	12	yes	—
43	C₂²×Q₈×C₆₁	—	2	8	no	—
44	C₂²×Q₈×C₆₁	C₂×C₄	1	12	yes	—
45	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×Q₈×C₆₁	—	2	48	no	—
49	C₂×Q₈×C₆₁	C₂²×C₄	1	16	yes	—
50	C₂²×C₄×C₆₁	—	2	8	no	—
51	C₂²×C₄×C₆₁	M₁₆	1	8	yes	—
52	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	6	yes	—
54	C₂²×Dic₆₁	—	2	4	no	—
55	C₂²×Dic₆₁	—	2	8	no	—
56	C₂²×Dic₆₁	C₂²×C₄	1	6	yes	—
57	C₂²×Dic₆₁	C₂²×C₄	1	12	yes	—
58	C₂²×Dic₆₁	C₂²×C₄	1	3	yes	—
59	C₂²×Dic₆₁	C₂×Q₈	1	4	yes	—
60	C₂²×Dic₆₁	#41	1	4	yes	—
61	C₂²×Dic₆₁	C₂⁴	1	3	yes	—
62	C₂×#92420	—	2	12	no	—
63	C₂×#92420	—	2	24	no	—
64	C₂×#92420	C₂×Q₈	1	24	yes	complemented
65	C₂×#92420	C₂⁴	1	4	yes	—
66	C₂×#92420	C₂⁴	1	8	yes	—
67	C₂×#92420	C₂⁴	1	4	yes	—
68	C₄⋊Dic₆₁	C₂²×C₄	1	32	yes	—
69	C₄⋊Dic₆₁	C₂²×C₄	1	16	yes	—
70	C₄×Dic₆₁	C₂²×C₄	1	24	yes	—
71	C₄×Dic₆₁	C₂²×C₄	1	12	yes	—
72	C₄×Dic₆₁	C₂⁴	1	4	yes	—
73	C₄×Dic₆₁	C₂⁴	1	8	yes	—
74	C₂³×C₆₁	C₄×Q₈	1	4	yes	—
75	C₂³×C₆₁	C₂×M₁₆	1	4	yes	—
76	C₂³×C₆₁	#131	1	4	yes	—
77	C₂³×C₆₁	C₂³×C₄	1	3	yes	—
78	Q₈×C₆₁	—	2	32	no	—
79	C₂×C₄×C₆₁	—	2	24	no	—
80	C₂×C₄×C₆₁	C₂×M₁₆	1	24	yes	—
81	C₂×C₄×C₆₁	C₂³×C₄	1	8	yes	—
82	C₂×C₄×C₆₁	C₂³×C₄	1	4	yes	—
83	C₂×Dic₆₁	—	2	12	no	—
84	C₂×Dic₆₁	—	2	24	no	—
85	C₂×Dic₆₁	C₄×Q₈	1	12	yes	—
86	C₂×Dic₆₁	C₂³×C₄	1	4	yes	—
87	C₂×Dic₆₁	C₂³×C₄	1	8	yes	—
88	C₂×Dic₆₁	C₂³×C₄	1	3	yes	—
89	C₂×Dic₆₁	C₂²×Q₈	1	6	yes	—
90	C₂×Dic₆₁	C₂×#41	1	6	yes	—
91	C₂×Dic₆₁	C₂⁵	1	1	yes	characteristic
92	#92420	—	2	4	no	—
93	#92420	—	2	8	no	—
94	#92420	C₂²×Q₈	1	8	yes	complemented
95	C₂²×#446	—	61	1	no	maximal, Sylow
96	C₂²×C₆₁	#446	1	16	yes	complemented
97	C₂²×C₆₁	C₂×C₄×Q₈	1	6	yes	—
98	C₂²×C₆₁	C₂²×M₁₆	1	6	yes	—
99	C₂²×C₆₁	C₂×#131	1	6	yes	—
100	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₆₁	—	2	8	no	—
102	C₄×C₆₁	C₂²×M₁₆	1	8	yes	—
103	Dic₆₁	—	2	4	no	—
104	Dic₆₁	—	2	8	no	—
105	Dic₆₁	C₂×C₄×Q₈	1	6	yes	—
106	Dic₆₁	C₂³×Q₈	1	1	yes	characteristic
107	Dic₆₁	C₂²×#41	1	1	yes	characteristic
108	C₂²×C₄×Q₈	—	61	1	no	—
109	C₂×#446	—	61	24	no	—
110	C₂²×C₄⋊C₈	—	61	1	no	—
111	C₂²×C₄⋊C₈	—	61	2	no	—
112	C₂²×#97	—	61	2	no	—
113	C₂²×C₄×C₈	—	61	1	no	—
114	C₂×C₆₁	C₂×#446	1	12	yes	complemented
115	C₂×C₆₁	C₂²×C₄×Q₈	1	1	yes	characteristic
116	C₂×C₆₁	C₂³×M₁₆	1	1	yes	characteristic
117	C₂×C₆₁	C₂²×#131	1	1	yes	characteristic
118	C₂³×Q₈	#6923	1	1	yes	characteristic
119	C₂³×C₈	—	61	1	no	—
120	C₂³×C₈	—	61	1	no	—
121	C₂³×C₈	—	61	2	no	—
122	C₂×C₄×Q₈	—	61	12	no	—
123	C₂×C₄×Q₈	—	61	12	no	—
124	C₂²×C₄⋊C₄	—	61	2	no	—
125	C₂²×C₄⋊C₄	—	61	1	no	—
126	C₂²×C₄²	—	61	1	no	—
127	C₂²×C₄²	—	61	2	no	—
128	#446	—	61	64	no	—
129	C₂×C₄⋊C₈	—	61	12	no	—
130	C₂×C₄⋊C₈	—	61	24	no	—
131	C₂×#97	—	61	24	no	—
132	C₂×C₄×C₈	—	61	12	no	—
133	C₆₁	C₂²×#446	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₈	—	2	8	no	—
135	C₂²×Q₈	C₂×#6923	1	12	yes	—
136	C₂³×C₄	—	61	1	no	—
137	C₂³×C₄	—	61	1	no	—
138	C₂³×C₄	—	61	2	no	—
139	C₂³×C₄	C₂×#6923	1	2	yes	—
140	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
141	C₂²×C₈	—	61	8	no	—
142	C₂²×C₈	—	61	6	no	—
143	C₂²×C₈	—	61	6	no	—
144	C₂²×C₈	—	61	12	no	—
145	C₂²×C₈	—	122	4	no	—
146	C₂²×C₈	—	122	8	no	—
147	C₄×Q₈	—	61	16	no	—
148	C₄×Q₈	—	61	48	no	—
149	C₂×C₄⋊C₄	—	61	24	no	—
150	C₂×C₄⋊C₄	—	61	12	no	—
151	C₂×C₄²	—	61	6	no	—
152	C₂×C₄²	—	61	12	no	—
153	C₂×C₄²	—	61	12	no	—
154	C₂×C₄²	—	61	6	no	—
155	C₄⋊C₈	—	61	16	no	—
156	C₄⋊C₈	—	61	32	no	—
157	#97	—	61	32	no	—
158	C₄×C₈	—	61	16	no	—
159	C₂⁴	C₂²×#6923	1	1	yes	characteristic
160	C₂×Q₈	—	2	48	no	—
161	C₂×Q₈	C₂²×#6923	1	16	yes	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	61	4	no	—
164	C₂²×C₄	—	61	4	no	—
165	C₂²×C₄	—	61	3	no	—
166	C₂²×C₄	—	61	3	no	—
167	C₂²×C₄	—	61	6	no	—
168	C₂²×C₄	—	61	12	no	—
169	C₂²×C₄	—	122	4	no	—
170	C₂²×C₄	—	122	8	no	—
171	C₂²×C₄	#11758217	1	8	yes	—
172	C₂²×C₄	C₂²×#6923	1	12	yes	—
173	C₂²×C₄	C₂²×#6923	1	6	yes	—
174	C₂×C₈	—	61	4	no	—
175	C₂×C₈	—	61	24	no	—
176	C₂×C₈	—	61	4	no	—
177	C₂×C₈	—	61	8	no	—
178	C₂×C₈	—	122	12	no	—
179	C₂×C₈	—	122	24	no	—
180	C₄⋊C₄	—	61	32	no	—
181	C₄⋊C₄	—	61	16	no	—
182	C₄²	—	61	4	no	—
183	C₄²	—	61	8	no	—
184	C₄²	—	61	24	no	—
185	C₄²	—	61	12	no	—
186	C₂³	Q₈×#6923	1	4	yes	—
187	C₂³	C₂×#11758217	1	4	yes	—
188	C₂³	#14519865	1	4	yes	—
189	C₂³	C₂³×#6923	1	3	yes	—
190	Q₈	—	2	32	no	—
191	C₂×C₄	—	2	24	no	—
192	C₂×C₄	—	61	12	no	—
193	C₂×C₄	—	61	6	no	—
194	C₂×C₄	—	61	6	no	—
195	C₂×C₄	—	61	3	no	—
196	C₂×C₄	—	61	1	no	—
197	C₂×C₄	—	61	4	no	—
198	C₂×C₄	—	61	8	no	—
199	C₂×C₄	—	122	12	no	—
200	C₂×C₄	—	122	24	no	—
201	C₂×C₄	C₂×#11758217	1	24	yes	—
202	C₂×C₄	C₂³×#6923	1	8	yes	—
203	C₂×C₄	C₂³×#6923	1	4	yes	—
204	C₈	—	61	8	no	—
205	C₈	—	122	4	no	—
206	C₈	—	122	8	no	—
207	C₂²	#16840813	1	16	yes	direct product factor
208	C₂²	C₂×Q₈×#6923	1	6	yes	—
209	C₂²	C₂²×#11758217	1	6	yes	—
210	C₂²	C₂×#14519865	1	6	yes	—
211	C₂²	C₂⁴×#6923	1	1	yes	characteristic
212	C₄	—	2	8	no	—
213	C₄	—	61	6	no	—
214	C₄	—	61	1	no	—
215	C₄	—	61	1	no	—
216	C₄	—	122	4	no	—
217	C₄	—	122	8	no	—
218	C₄	C₂²×#11758217	1	8	yes	—
219	C₂	C₂×#16840813	1	12	yes	direct product factor
220	C₂	C₂²×Q₈×#6923	1	1	yes	characteristic
221	C₂	C₂³×#11758217	1	1	yes	characteristic
222	C₂	C₂²×#14519865	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×#6923
commutator	C₂²×C₆₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×#6923
Fitting	C₂³×Q₈×C₆₁	C₄
socle, prime radical	C₂⁴×C₆₁	C₂²×C₄
cosocle	C₂×Dic₆₁	C₂⁵
hypercenter	C₂³×Q₈	#6923
hypocenter	C₆₁	C₂²×#446
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16840813	C₂	direct (N × H)
10. #16840813	C₂²	direct (N × H)
24. C₂²×#92420	Q₈	semidirect
64. C₂×#92420	C₂×Q₈	semidirect
94. #92420	C₂²×Q₈	semidirect
96. C₂²×C₆₁	#446	semidirect
114. C₂×C₆₁	C₂×#446	semidirect
133. C₆₁	C₂²×#446	semidirect
207. C₂²	#16840813	direct (N × H)
219. C₂	C₂×#16840813	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×#16840813	2
C₂²×#14519844	2
C₂²×C₄⋊#92420	2
C₂²×#14519832	2
C₂²×C₄×#92420	2
C₂²×Q₈×Dic₆₁	2
C₂²×#446	61

Subgroups — C22×#16840813 #30625502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840813 #30625502