Subgroups — #30607893

Order 15616.

350 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758187	C₂	1	2	yes	maximal
2	C₂×C₄⋊#11758187	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11758187	C₂	1	1	yes	maximal, characteristic
4	#21460857	C₂	1	2	yes	maximal
5	C₄⋊C₄×#92418	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758187	C₂²	1	2	yes	—
7	C₂²×#11758187	C₂²	1	1	yes	characteristic
8	C₂²×#11758187	C₂²	1	1	yes	characteristic
9	C₄⋊#11758187	—	2	2	no	—
10	C₄⋊#11758187	—	2	4	no	—
11	C₄×#11758187	—	2	2	no	—
12	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
13	C₂×C₄×#92418	C₂²	1	2	yes	—
14	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
15	#16839517	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
17	C₂×#11758187	—	2	1	no	—
18	C₂×#11758187	—	2	1	no	—
19	C₂×#11758187	—	2	1	no	—
20	C₂×#11758187	—	2	1	no	—
21	C₂×#11758187	—	2	2	no	—
22	C₂×#11758187	—	2	2	no	—
23	C₂×#11758187	—	2	2	no	—
24	C₂×#11758187	D₄	1	2	yes	—
25	C₂×#11758187	Q₈	1	2	yes	—
26	C₂²×#92418	C₂×C₄	1	2	yes	—
27	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
28	C₂²×#92418	C₂³	1	1	yes	characteristic
29	C₄⋊#92418	—	2	2	no	—
30	C₄×#92418	—	2	2	no	—
31	C₄×#92418	—	2	2	no	—
32	C₄×#92418	D₄	1	2	yes	—
33	C₄×#92418	Q₈	1	2	yes	—
34	C₂×C₄⋊C₄×C₆₁	C₈	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₁	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	#95×C₆₁	C₈	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₄⋊C₄×C₆₁	—	2	2	no	—
45	C₄⋊C₄×C₆₁	C₁₆	1	4	yes	complemented
46	C₄²×C₆₁	—	2	2	no	—
47	C₄²×C₆₁	—	2	2	no	—
48	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
49	C₄²×C₆₁	M₁₆	1	2	yes	—
50	C₂×#92418	—	2	1	no	—
51	C₂×#92418	—	2	1	no	—
52	C₂×#92418	—	2	2	no	—
53	C₂×#92418	—	2	2	no	—
54	C₂×#92418	—	2	2	no	—
55	C₂×#92418	C₄⋊C₄	1	2	yes	—
56	C₂×#92418	C₂×D₄	1	1	yes	characteristic
57	C₂×#92418	C₂×D₄	1	1	yes	characteristic
58	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
59	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
60	C₂×#92418	#41	1	1	yes	characteristic
61	C₂×#92418	#41	1	1	yes	characteristic
62	#11758187	—	2	2	no	—
63	#11758187	—	2	2	no	—
64	#11758187	—	4	2	no	—
65	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	—	2	2	no	—
73	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₆₁	C₂×C₁₆	1	4	yes	—
76	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
77	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
84	#92418	—	2	2	no	—
85	#92418	—	4	2	no	—
86	#92418	#121	1	1	yes	characteristic
87	#92418	C₂²⋊Q₈	1	1	yes	characteristic
88	#92418	#125	1	1	yes	characteristic
89	#92418	C₄⋊Q₈	1	1	yes	characteristic
90	#13159	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
93	C₂²×C₆₁	C₈×Q₈	1	1	yes	characteristic
94	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
95	C₂²×C₆₁	C₂×M₃₂	1	1	yes	characteristic
96	C₂²×C₆₁	#504	1	1	yes	characteristic
97	C₂²×C₆₁	#504	1	1	yes	characteristic
98	C₄×C₆₁	—	2	2	no	—
99	C₄×C₆₁	—	2	4	no	—
100	C₄×C₆₁	—	4	2	no	—
101	C₄×C₆₁	C₄⋊C₁₆	1	4	yes	complemented
102	C₄×C₆₁	#389	1	1	yes	characteristic
103	C₄×C₆₁	#424	1	1	yes	characteristic
104	C₄×C₆₁	#436	1	1	yes	characteristic
105	C₄×C₆₁	#446	1	1	yes	characteristic
106	C₂×C₄⋊C₁₆	—	61	1	no	—
107	C₂×C₄⋊C₁₆	—	61	2	no	—
108	C₂×C₄×C₁₆	—	61	1	no	—
109	C₈×C₄⋊C₄	—	61	1	no	—
110	#1374	—	61	2	no	—
111	C₂×C₆₁	#1939	1	1	yes	characteristic
112	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
113	C₂×C₆₁	#2151	1	1	yes	characteristic
114	C₂×C₆₁	D₄×C₁₆	1	1	yes	characteristic
115	C₂×C₆₁	#2169	1	1	yes	characteristic
116	C₂×C₆₁	Q₈×C₁₆	1	1	yes	characteristic
117	C₂×C₆₁	#2183	1	1	yes	characteristic
118	C₂²×C₁₆	—	61	1	no	—
119	C₂²×C₁₆	—	61	1	no	—
120	C₂²×C₁₆	—	61	2	no	—
121	C₂×C₄⋊C₈	—	61	1	no	—
122	C₂×C₄×C₈	—	61	1	no	—
123	C₂×C₄×C₈	—	61	2	no	—
124	C₄×C₄⋊C₄	#6923	1	1	yes	characteristic
125	C₄⋊C₁₆	—	122	2	no	—
126	C₄⋊C₁₆	—	122	4	no	—
127	C₄×C₁₆	—	122	2	no	—
128	#336	—	61	2	no	—
129	C₆₁	#13159	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₈	—	61	1	no	—
131	C₂²×C₈	—	61	1	no	—
132	C₂²×C₈	—	61	2	no	—
133	C₂×C₄⋊C₄	#92420	1	1	yes	characteristic
134	C₂×C₄⋊C₄	#92420	1	1	yes	characteristic
135	C₂×C₄²	C₂×#6923	1	2	yes	—
136	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
137	C₂×C₁₆	—	61	2	no	—
138	C₂×C₁₆	—	61	2	no	—
139	C₂×C₁₆	—	122	1	no	—
140	C₂×C₁₆	—	122	1	no	—
141	C₂×C₁₆	—	122	1	no	—
142	C₂×C₁₆	—	122	1	no	—
143	C₂×C₁₆	—	122	2	no	—
144	C₂×C₁₆	—	122	2	no	—
145	C₂×C₁₆	—	122	2	no	—
146	C₄⋊C₈	—	122	2	no	—
147	C₄×C₈	—	61	2	no	—
148	C₄×C₈	—	61	2	no	—
149	C₄×C₈	—	122	2	no	—
150	C₄×C₈	—	122	2	no	—
151	#95	#92420	1	2	yes	—
152	C₂²×C₄	C₂×#92420	1	2	yes	—
153	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
154	C₂²×C₄	C₂×#92420	1	2	yes	—
155	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
156	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
157	C₂×C₈	—	61	1	no	—
158	C₂×C₈	—	61	1	no	—
159	C₂×C₈	—	61	1	no	—
160	C₂×C₈	—	61	1	no	—
161	C₂×C₈	—	61	1	no	—
162	C₂×C₈	—	61	1	no	—
163	C₂×C₈	—	61	2	no	—
164	C₂×C₈	—	122	1	no	—
165	C₂×C₈	—	122	1	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	—	122	2	no	—
168	C₂×C₈	—	122	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	#11758187	1	4	yes	complemented
171	C₄²	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	#11758213	1	2	yes	—
174	C₄²	C₄⋊#6923	1	2	yes	—
175	C₁₆	—	122	2	no	—
176	C₁₆	—	122	2	no	—
177	C₁₆	—	244	2	no	—
178	C₂³	C₂²×#92420	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₄⋊#92420	1	2	yes	—
187	C₂×C₄	C₄⋊#92420	1	2	yes	—
188	C₂×C₄	C₂×#11758187	1	4	yes	—
189	C₂×C₄	C₂×#11758187	1	2	yes	—
190	C₂×C₄	#14519851	1	2	yes	—
191	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
192	C₂×C₄	D₄×#6923	1	1	yes	characteristic
193	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
194	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
195	C₂×C₄	#14519864	1	1	yes	characteristic
196	C₂×C₄	#14519865	1	1	yes	characteristic
197	C₈	—	61	1	no	—
198	C₈	—	61	1	no	—
199	C₈	—	61	1	no	—
200	C₈	—	61	1	no	—
201	C₈	—	122	2	no	—
202	C₈	—	244	2	no	—
203	C₂²	C₂×C₄⋊#92420	1	1	yes	characteristic
204	C₂²	D₄×#92420	1	1	yes	characteristic
205	C₂²	Q₈×#92420	1	1	yes	characteristic
206	C₂²	C₂²×#11758187	1	1	yes	characteristic
207	C₂²	C₂×#14519851	1	1	yes	characteristic
208	C₂²	#16840820	1	1	yes	characteristic
209	C₂²	#16840820	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	4	no	—
212	C₄	—	4	2	no	—
213	C₄	C₄⋊#11758187	1	4	yes	complemented
214	C₄	#16840713	1	1	yes	characteristic
215	C₄	#16840769	1	1	yes	characteristic
216	C₄	#16840796	1	1	yes	characteristic
217	C₄	#16840812	1	1	yes	characteristic
218	C₂	#21461916	1	1	yes	characteristic
219	C₂	C₂×C₄⋊#11758187	1	1	yes	characteristic
220	C₂	#21462154	1	1	yes	characteristic
221	C₂	D₄×#11758187	1	1	yes	characteristic
222	C₂	#21462203	1	1	yes	characteristic
223	C₂	Q₈×#11758187	1	1	yes	characteristic
224	C₂	#21462234	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂²×C₁₆
Fitting	C₄×C₄⋊C₄×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	C₄×C₄⋊C₄	#6923
hypocenter	C₆₁	#13159
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
45. C₄⋊C₄×C₆₁	C₁₆	semidirect
101. C₄×C₆₁	C₄⋊C₁₆	semidirect
129. C₆₁	#13159	semidirect
170. C₄⋊C₄	#11758187	semidirect
213. C₄	C₄⋊#11758187	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11758187	2
C₂×C₄×#11758187	2
#21460857	2
C₄⋊C₄×#92418	2
#13159	61