Subgroups — #11790025

Order 1152.

780 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63641	C₂	1	2	yes	maximal
3	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
5	#10589416	C₂	1	2	yes	maximal
6	#69786	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
11	C₂×C₄⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
12	C₂×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
13	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃²×C₄×C₈	—	2	2	no	—
15	#63658	C₄	1	2	yes	—
16	#63641	—	2	4	no	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
18	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
19	C₂×Dic₃⋊C₈	—	3	8	no	—
20	C₄×C₄⋊Dic₃	—	3	4	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
22	#4550	—	3	8	no	—
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
27	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
28	C₂×C₃²×C₈	—	2	1	no	—
29	C₂×C₃²×C₈	—	2	1	no	—
30	C₂×C₃²×C₈	—	2	1	no	—
31	C₂×C₃²×C₈	—	2	1	no	—
32	C₂×C₃²×C₈	D₄	1	2	yes	—
33	C₂×C₃²×C₈	Q₈	1	2	yes	—
34	C₃²×C₄²	D₄	1	2	yes	—
35	C₃²×C₄²	Q₈	1	2	yes	—
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	C₈	1	4	yes	complemented
38	C₄×C₃⋊Dic₃	—	2	2	no	—
39	C₄×C₃⋊Dic₃	—	2	2	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
41	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
43	#1939	—	9	1	no	Sylow
44	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
45	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
46	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
47	C₂×C₄⋊Dic₃	—	3	4	no	—
48	C₂×C₄⋊Dic₃	—	3	4	no	—
49	C₂×C₄×Dic₃	—	3	8	no	—
50	C₂²×C₃⋊C₈	—	3	8	no	—
51	C₃×C₄×C₈	—	2	8	no	—
52	C₃⋊#95	—	3	8	no	—
53	Dic₃⋊C₈	—	6	16	no	—
54	C₄⋊(C₃⋊C₈)	—	6	8	no	—
55	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	—	2	1	no	—
57	C₂×C₃²×C₄	—	2	1	no	—
58	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
59	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
60	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
61	C₂×C₃²×C₄	M₁₆	1	2	yes	—
62	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃²×C₄	#41	1	1	yes	characteristic
67	C₂×C₃²×C₄	#41	1	1	yes	characteristic
68	C₂×C₃⋊Dic₃	—	2	2	no	—
69	C₂×C₃⋊Dic₃	—	2	2	no	—
70	C₂×C₃⋊Dic₃	—	2	2	no	—
71	C₂×C₃⋊Dic₃	—	2	2	no	—
72	C₂×C₃⋊Dic₃	—	2	2	no	—
73	C₂×C₃⋊Dic₃	C₂×C₈	1	4	yes	—
74	C₃²×C₈	—	2	2	no	—
75	C₃²×C₈	—	2	2	no	—
76	C₃⋊(C₃⋊C₈)	—	4	2	no	—
77	C₂×C₄⋊C₈	—	9	2	no	—
78	C₂×C₄⋊C₈	—	9	1	no	—
79	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
80	C₄×C₄⋊C₄	—	9	1	no	—
81	#336	—	9	2	no	—
82	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
83	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
84	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
85	C₂²×Dic₃	—	3	8	no	—
86	C₂²×Dic₃	—	3	8	no	—
87	C₂×C₃×C₈	—	2	4	no	—
88	C₂×C₃×C₈	—	2	4	no	—
89	C₂×C₃×C₈	—	2	4	no	—
90	C₂×C₃×C₈	—	2	4	no	—
91	C₂×C₃×C₈	Dic₆	1	8	yes	—
92	C₂×C₃×C₈	D₁₂	1	8	yes	—
93	C₃×C₄²	Dic₆	1	8	yes	—
94	C₃×C₄²	#67	1	8	yes	—
95	C₄⋊Dic₃	—	3	16	no	—
96	C₄⋊Dic₃	—	6	8	no	—
97	C₄×Dic₃	—	6	8	no	—
98	C₄×Dic₃	—	6	8	no	—
99	C₂×C₃⋊C₈	—	6	8	no	—
100	C₂×C₃⋊C₈	—	6	8	no	—
101	C₂×C₃⋊C₈	—	6	8	no	—
102	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
105	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
106	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
107	C₂²×C₃²	#131	1	1	yes	characteristic
108	C₂²×C₃²	#131	1	1	yes	characteristic
109	C₃²×C₄	—	2	2	no	—
110	C₃²×C₄	C₄⋊C₈	1	4	yes	complemented
111	C₃²×C₄	#121	1	1	yes	characteristic
112	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
113	C₃²×C₄	#125	1	1	yes	characteristic
114	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
115	C₃⋊Dic₃	—	2	4	no	—
116	C₃⋊Dic₃	—	4	2	no	—
117	C₂²×C₈	—	9	2	no	—
118	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	9	1	no	—
121	C₂×C₄⋊C₄	—	9	1	no	—
122	C₂×C₄²	—	9	2	no	—
123	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
124	C₄⋊C₈	—	18	4	no	—
125	C₄⋊C₈	—	18	2	no	—
126	C₄×C₈	—	2	2	no	—
127	#95	—	9	2	no	—
128	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	D₃×C₈	1	8	yes	—
132	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	8	yes	—
133	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
134	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
135	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
136	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
137	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
138	C₂×C₃×C₄	#235	1	4	yes	—
139	C₂×C₃×C₄	#235	1	4	yes	—
140	C₂×C₃×C₄	C₂×#67	1	4	yes	—
141	C₂×Dic₃	—	3	16	no	—
142	C₂×Dic₃	—	6	8	no	—
143	C₂×Dic₃	—	6	8	no	—
144	C₂×Dic₃	—	6	8	no	—
145	C₂×Dic₃	—	6	8	no	—
146	C₂×Dic₃	—	6	8	no	—
147	C₃×C₈	—	2	8	no	—
148	C₃×C₈	—	2	8	no	—
149	C₃⋊C₈	—	12	8	no	—
150	C₂×C₃²	#389	1	1	yes	characteristic
151	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₃²	#424	1	1	yes	characteristic
153	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
154	C₂×C₃²	#436	1	1	yes	characteristic
155	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
156	C₂×C₃²	#446	1	1	yes	characteristic
157	C₂²×C₄	—	9	2	no	—
158	C₂²×C₄	—	9	2	no	—
159	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
160	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	18	2	no	—
167	C₂×C₈	—	18	2	no	—
168	C₂×C₈	—	18	2	no	—
169	C₂×C₈	#634	1	2	yes	—
170	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
171	C₄⋊C₄	—	9	4	no	—
172	C₄⋊C₄	—	18	2	no	—
173	C₄²	—	18	2	no	—
174	C₄²	—	18	2	no	—
175	C₄²	#634	1	2	yes	—
176	C₄²	#638	1	2	yes	—
177	C₂²×C₃	C₄×Dic₆	1	4	yes	—
178	C₂²×C₃	C₄×D₁₂	1	4	yes	—
179	C₂²×C₃	C₂×D₃×C₈	1	4	yes	—
180	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
181	C₂²×C₃	#901	1	4	yes	—
182	C₂²×C₃	#901	1	4	yes	—
183	C₂²×C₃	C₂×Dic₃⋊C₄	1	4	yes	—
184	C₃×C₄	—	2	8	no	—
185	C₃×C₄	Dic₃⋊C₈	1	16	yes	complemented
186	C₃×C₄	#869	1	4	yes	—
187	C₃×C₄	#870	1	4	yes	—
188	C₃×C₄	#924	1	4	yes	—
189	C₃×C₄	#930	1	4	yes	—
190	Dic₃	—	6	16	no	—
191	Dic₃	—	12	8	no	—
192	C₃²	#1939	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	9	4	no	—
197	C₂×C₄	—	18	2	no	—
198	C₂×C₄	—	18	2	no	—
199	C₂×C₄	—	18	2	no	—
200	C₂×C₄	—	18	2	no	—
201	C₂×C₄	—	18	2	no	—
202	C₂×C₄	C₈×C₃⋊D₃	1	2	yes	—
203	C₂×C₄	#3743	1	2	yes	—
204	C₂×C₄	#3750	1	2	yes	—
205	C₂×C₄	#3750	1	2	yes	—
206	C₂×C₄	C₂×#634	1	1	yes	characteristic
207	C₂×C₄	C₂×#634	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
209	C₂×C₄	#3828	1	1	yes	characteristic
210	C₂×C₄	#3828	1	1	yes	characteristic
211	C₂×C₄	C₂×#638	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	2	no	—
214	C₈	—	36	2	no	—
215	C₂×C₃	C₈×Dic₆	1	4	yes	—
216	C₂×C₃	#4679	1	4	yes	—
217	C₂×C₃	C₈×D₁₂	1	4	yes	—
218	C₂×C₃	#4688	1	4	yes	—
219	C₂×C₃	#4928	1	4	yes	—
220	C₂×C₃	C₂×Dic₃⋊C₈	1	4	yes	—
221	C₂×C₃	#5102	1	4	yes	—
222	C₂²	C₄×#634	1	1	yes	characteristic
223	C₂²	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
224	C₂²	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
225	C₂²	C₂×#3743	1	1	yes	characteristic
226	C₂²	#64110	1	1	yes	characteristic
227	C₂²	#64110	1	1	yes	characteristic
228	C₂²	C₂×#3750	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	18	4	no	—
231	C₄	—	36	2	no	—
232	C₄	#63641	1	4	yes	complemented
233	C₄	#64078	1	1	yes	characteristic
234	C₄	#64079	1	1	yes	characteristic
235	C₄	#64133	1	1	yes	characteristic
236	C₄	#64139	1	1	yes	characteristic
237	C₃	#69786	1	4	yes	complemented
238	C₂	C₈×#634	1	1	yes	characteristic
239	C₂	#10592378	1	1	yes	characteristic
240	C₂	C₈×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
241	C₂	#10592387	1	1	yes	characteristic
242	C₂	#10592627	1	1	yes	characteristic
243	C₂	C₂×#63641	1	1	yes	characteristic
244	C₂	#10592801	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#1939
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
37. C₄⋊(C₃⋊Dic₃)	C₈	semidirect
110. C₃²×C₄	C₄⋊C₈	semidirect
185. C₃×C₄	Dic₃⋊C₈	semidirect
192. C₃²	#1939	semidirect
232. C₄	#63641	semidirect
237. C₃	#69786	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×#63641	2
C₄×C₄⋊(C₃⋊Dic₃)	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10589416	2
#69786	3