Subgroups — #29578804

Order 15616.

530 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445936	C₂	1	1	yes	maximal, characteristic
2	#21445565	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#92418	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	C₂²	1	2	yes	—
6	C₂×C₄×#92418	C₂²	1	2	yes	—
7	#16839517	—	2	2	no	—
8	#16839517	—	2	2	no	—
9	Q₈⋊#92418	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#391	C₄	1	2	yes	—
13	C₆₁×#389	—	2	2	no	—
14	C₆₁×#384	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
16	C₄³×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×#92418	—	2	2	no	—
18	C₂²×#92418	—	2	2	no	—
19	C₄⋊#92418	—	2	4	no	—
20	C₄×#92418	—	2	2	no	—
21	C₄×#92418	D₄	1	4	yes	—
22	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₄⋊Q₈×C₆₁	—	2	2	no	—
24	C₄⋊Q₈×C₆₁	C₈	1	4	yes	complemented
25	C₄×Q₈×C₆₁	—	2	4	no	—
26	C₄×Q₈×C₆₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₁	—	2	2	no	—
32	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	D₄	1	2	yes	—
35	C₂×C₄²×C₆₁	D₄	1	2	yes	—
36	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	#95×C₆₁	—	2	4	no	—
38	C₂×Q₈×C₆₁	—	2	2	no	—
39	C₂×Q₈×C₆₁	—	2	4	no	—
40	C₂×Q₈×C₆₁	—	2	4	no	—
41	C₂×Q₈×C₆₁	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₆₁	—	2	2	no	—
43	C₂²×C₄×C₆₁	—	2	2	no	—
44	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₄⋊C₄×C₆₁	—	2	8	no	—
50	C₄⋊C₄×C₆₁	—	2	4	no	—
51	C₄⋊C₄×C₆₁	—	4	4	no	—
52	C₄⋊C₄×C₆₁	—	4	4	no	—
53	C₄²×C₆₁	—	2	2	no	—
54	C₄²×C₆₁	—	2	4	no	—
55	C₄²×C₆₁	—	4	2	no	—
56	C₄²×C₆₁	—	4	2	no	—
57	C₄²×C₆₁	—	4	2	no	—
58	C₄²×C₆₁	C₂×C₈	1	2	yes	—
59	C₄²×C₆₁	M₁₆	1	2	yes	—
60	C₄²×C₆₁	C₂×D₄	1	2	yes	—
61	C₄²×C₆₁	#41	1	2	yes	—
62	C₂×#92418	—	2	4	no	—
63	C₂×#92418	—	4	4	no	—
64	C₂×#92418	—	4	2	no	—
65	C₂×#92418	—	4	4	no	—
66	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	Q₈×C₆₁	—	2	8	no	—
68	Q₈×C₆₁	—	4	4	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	4	no	—
73	C₂×C₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	4	2	no	—
77	C₂×C₄×C₆₁	—	4	2	no	—
78	C₂×C₄×C₆₁	—	4	2	no	—
79	C₂×C₄×C₆₁	—	4	4	no	—
80	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
81	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₆₁	#121	1	2	yes	—
83	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#131	1	2	yes	—
88	C₂×C₄×C₆₁	#137	1	4	yes	—
89	#92418	—	4	4	no	—
90	#92418	—	8	2	no	—
91	#9641	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	#390	1	1	yes	characteristic
93	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
94	C₂²×C₆₁	#408	1	1	yes	characteristic
95	C₂²×C₆₁	#419	1	2	yes	—
96	C₂²×C₆₁	#421	1	2	yes	—
97	C₄×C₆₁	—	2	2	no	—
98	C₄×C₆₁	—	2	4	no	—
99	C₄×C₆₁	—	4	4	no	—
100	C₄×C₆₁	—	4	2	no	—
101	C₄×C₆₁	—	8	2	no	—
102	C₄×C₆₁	C₈×D₄	1	2	yes	—
103	C₄×C₆₁	#436	1	2	yes	—
104	C₄×C₆₁	#491	1	2	yes	—
105	C₄×C₆₁	#497	1	2	yes	—
106	C₄×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
107	#1831	—	61	1	no	—
108	#1744	—	61	1	no	—
109	C₂×Q₈⋊C₈	—	61	4	no	—
110	C₂×C₆₁	#1479	1	2	yes	—
111	C₂×C₆₁	#1497	1	2	yes	—
112	C₂×C₆₁	#1954	1	1	yes	characteristic
113	C₂×C₆₁	#1960	1	1	yes	characteristic
114	C₂×C₆₁	#1963	1	1	yes	characteristic
115	C₂×C₄⋊Q₈	Dic₆₁	1	1	yes	characteristic
116	C₂×C₄×Q₈	C₂×D₆₁	1	2	yes	—
117	C₂×C₄⋊C₈	—	61	2	no	—
118	C₂×C₄×C₈	—	61	2	no	—
119	#391	Dic₆₁	1	2	yes	—
120	#389	—	2	2	no	—
121	#384	Dic₆₁	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	2	2	no	—
123	C₄³	C₂×D₆₁	1	1	yes	characteristic
124	#336	—	122	2	no	—
125	#336	—	122	2	no	—
126	Q₈⋊C₈	—	122	8	no	—
127	C₆₁	#9641	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	C₂×Dic₆₁	1	2	yes	—
129	C₂²×C₈	—	122	2	no	—
130	C₂²×C₈	—	122	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	#92418	1	4	yes	complemented
133	C₄×Q₈	—	2	4	no	—
134	C₄×Q₈	—	2	4	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
141	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
143	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
144	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
145	C₄⋊C₈	—	122	4	no	—
146	C₄×C₈	—	61	4	no	—
147	C₄×C₈	—	122	2	no	—
148	#95	—	2	4	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	C₂×#92418	1	4	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
156	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
160	C₂×C₈	—	122	4	no	—
161	C₂×C₈	—	244	4	no	—
162	C₂×C₈	—	244	2	no	—
163	C₂×C₈	—	244	4	no	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄²	—	2	4	no	—
169	C₄²	—	2	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	4	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	C₂×#92418	1	2	yes	—
174	C₄²	#11758194	1	2	yes	—
175	C₄²	D₄×D₆₁	1	2	yes	—
176	C₄²	#11758224	1	2	yes	—
177	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
178	Q₈	—	2	8	no	—
179	Q₈	—	4	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₂²⋊#92418	1	4	yes	—
192	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
193	C₂×C₄	#14519748	1	2	yes	—
194	C₂×C₄	#14519764	1	1	yes	characteristic
195	C₂×C₄	#14519773	1	1	yes	characteristic
196	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
197	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
198	C₂×C₄	#14519784	1	2	yes	—
199	C₂×C₄	#14519808	1	4	yes	—
200	C₈	—	488	2	no	—
201	C₈	—	244	4	no	—
202	C₂²	#16839665	1	1	yes	characteristic
203	C₂²	C₂×C₂²⋊#92418	1	1	yes	characteristic
204	C₂²	#16839729	1	1	yes	characteristic
205	C₂²	#16839771	1	2	yes	—
206	C₂²	#16839779	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	4	no	—
211	C₄	—	8	2	no	—
212	C₄	D₄×#92418	1	2	yes	—
213	C₄	#16839840	1	2	yes	—
214	C₄	#16840130	1	2	yes	—
215	C₄	#16840154	1	2	yes	—
216	C₂	#21444214	1	2	yes	—
217	C₂	#21444292	1	2	yes	—
218	C₂	#21446614	1	1	yes	characteristic
219	C₂	#21446640	1	1	yes	characteristic
220	C₂	#21446652	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×C₄⋊Q₈×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#9641
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₄⋊Q₈×C₆₁	C₈	semidirect
127. C₆₁	#9641	semidirect
132. C₄⋊Q₈	#92418	semidirect

Maximal subgroups

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

subgroup	index
#21445936	2
#21445565	2
C₂×Q₈⋊#92418	2
C₄×C₄⋊Q₈×C₆₁	2
#9641	61