Subgroups — #11786057

Order 1152.

509 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592635	C₂	1	1	yes	maximal, characteristic
2	C₃×#4936	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10589164	C₂	1	4	yes	maximal
5	#69763	—	3	1	no	maximal
6	#69547	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
10	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
11	#63658	—	2	2	no	—
12	C₃×C₃⋊#95	—	2	2	no	—
13	C₃×C₄×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃×C₄×C₈	—	3	1	no	—
15	C₃×#382	—	3	1	no	—
16	#4936	—	3	1	no	—
17	#4936	—	3	1	no	—
18	#4936	—	6	1	no	—
19	#4936	D₃	1	1	yes	characteristic
20	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
21	#4551	—	3	4	no	—
22	#4529	—	3	4	no	—
23	C₂²×C₃²×C₄	D₄	1	2	yes	—
24	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂²×C₃⋊Dic₃	—	2	2	no	—
26	C₂²×C₃×Dic₃	—	2	2	no	—
27	C₃²×C₄²	C₂×C₄	1	2	yes	—
28	C₃²×C₄²	Q₈	1	2	yes	—
29	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
30	C₃×C₄⋊Dic₃	—	2	4	no	—
31	C₂×C₃×C₃⋊C₈	—	2	2	no	—
32	C₂×C₃×C₃⋊C₈	—	2	2	no	—
33	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
34	#1835	—	9	1	no	Sylow
35	C₂²×C₃×C₈	—	3	2	no	—
36	C₂×C₃×C₄⋊C₄	—	3	2	no	—
37	C₂×C₃×C₄²	—	2	1	no	—
38	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂×C₄⋊Dic₃	—	3	2	no	—
41	C₂×C₄⋊Dic₃	—	3	2	no	—
42	C₂×C₄⋊Dic₃	—	6	2	no	—
43	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
44	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
45	C₃×C₄×C₈	—	3	4	no	—
46	C₃×#95	—	6	2	no	—
47	C₃⋊#95	—	2	2	no	—
48	C₃⋊#95	—	6	4	no	—
49	C₃⋊#95	—	6	2	no	—
50	C₃⋊#95	—	6	2	no	—
51	C₄×C₃⋊C₈	Dic₃	1	4	yes	complemented
52	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	—	2	4	no	—
54	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
55	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₃²×C₄	#41	1	2	yes	—
58	C₂×C₃²×C₄	#41	1	2	yes	—
59	C₂×C₃⋊Dic₃	—	4	2	no	—
60	C₂×C₃⋊Dic₃	—	4	4	no	—
61	C₂×C₃×Dic₃	—	4	2	no	—
62	C₂×C₃×Dic₃	—	4	4	no	—
63	C₃×C₃⋊C₈	—	2	4	no	—
64	C₂×C₄×C₈	—	3	1	no	—
65	#382	—	3	1	no	—
66	#382	—	9	1	no	—
67	#340	—	9	4	no	—
68	C₂²×C₃×C₄	—	2	2	no	—
69	C₂²×C₃×C₄	—	2	1	no	—
70	C₂²×C₃×C₄	—	6	2	no	—
71	C₂²×C₃×C₄	D₁₂	1	2	yes	—
72	C₂²×C₃×C₄	#67	1	2	yes	—
73	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂²×Dic₃	—	2	2	no	—
76	C₂²×Dic₃	—	6	4	no	—
77	C₂²×Dic₃	—	6	2	no	—
78	C₂²×Dic₃	—	6	2	no	—
79	C₂×C₃×C₈	—	3	4	no	—
80	C₂×C₃×C₈	—	6	2	no	—
81	C₂×C₃×C₈	—	6	2	no	—
82	C₃×C₄⋊C₄	—	6	4	no	—
83	C₃×C₄²	—	2	2	no	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	Dic₆	1	2	yes	—
86	C₃×C₄²	Dic₆	1	2	yes	—
87	C₃×C₄²	C₄×D₃	1	2	yes	—
88	C₃×C₄²	C₂×Dic₃	1	2	yes	—
89	C₄⋊Dic₃	—	2	4	no	—
90	C₄⋊Dic₃	—	6	4	no	—
91	C₄⋊Dic₃	—	6	4	no	—
92	C₄⋊Dic₃	—	12	4	no	—
93	C₂×C₃⋊C₈	—	2	2	no	—
94	C₂×C₃⋊C₈	—	2	2	no	—
95	C₂×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
96	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₃²	#124	1	1	yes	characteristic
98	C₂²×C₃²	#124	1	1	yes	characteristic
99	C₂²×C₃²	#135	1	4	yes	—
100	C₃²×C₄	—	2	4	no	—
101	C₃²×C₄	#117	1	2	yes	—
102	C₃²×C₄	#125	1	2	yes	—
103	C₃⋊Dic₃	—	8	2	no	—
104	C₃×Dic₃	—	8	2	no	—
105	C₂²×C₈	—	3	2	no	—
106	C₂×C₄⋊C₄	—	3	2	no	—
107	C₂×C₄⋊C₄	—	9	2	no	—
108	C₂×C₄²	D₃²	1	1	yes	characteristic
109	C₄×C₈	—	3	4	no	—
110	#95	—	6	2	no	—
111	#95	—	18	2	no	—
112	C₂³×C₃	—	2	1	no	—
113	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
114	C₂³×C₃	C₂×#67	1	1	yes	characteristic
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	4	4	no	—
123	C₂×C₃×C₄	—	12	2	no	—
124	C₂×C₃×C₄	—	12	4	no	—
125	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
126	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
127	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
130	C₂×C₃×C₄	#235	1	2	yes	—
131	C₂×C₃×C₄	#235	1	2	yes	—
132	C₂×C₃×C₄	#237	1	2	yes	—
133	C₂×C₃×C₄	#239	1	2	yes	—
134	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
135	C₂×Dic₃	—	4	2	no	—
136	C₂×Dic₃	—	4	4	no	—
137	C₂×Dic₃	—	12	4	no	—
138	C₂×Dic₃	—	12	8	no	—
139	C₂×Dic₃	—	12	2	no	—
140	C₂×Dic₃	—	12	4	no	—
141	C₂×Dic₃	—	12	2	no	—
142	C₂×Dic₃	—	12	4	no	—
143	C₃×C₈	—	6	4	no	—
144	C₃⋊C₈	—	2	4	no	—
145	C₂×C₃²	#382	1	1	yes	characteristic
146	C₂×C₃²	#427	1	2	yes	—
147	C₂×C₃²	#488	1	2	yes	—
148	C₂×C₃²	#488	1	2	yes	—
149	C₂²×C₄	—	6	2	no	—
150	C₂²×C₄	—	18	2	no	—
151	C₂²×C₄	#626	1	2	yes	—
152	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
153	C₂×C₈	—	3	4	no	—
154	C₂×C₈	—	6	2	no	—
155	C₂×C₈	—	6	2	no	—
156	C₄⋊C₄	—	6	4	no	—
157	C₄⋊C₄	—	18	4	no	—
158	C₄²	D₃×Dic₃	1	2	yes	—
159	C₄²	#627	1	2	yes	—
160	C₂²×C₃	—	2	4	no	—
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	#875	1	1	yes	characteristic
165	C₂²×C₃	#875	1	1	yes	characteristic
166	C₂²×C₃	#904	1	4	yes	—
167	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
168	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
169	C₂²×C₃	#936	1	1	yes	characteristic
170	C₂²×C₃	#947	1	1	yes	characteristic
171	C₂²×C₃	#950	1	4	yes	—
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	2	4	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	4	4	no	—
177	C₃×C₄	—	24	2	no	—
178	C₃×C₄	#870	1	2	yes	—
179	C₃×C₄	#890	1	2	yes	—
180	C₃×C₄	#892	1	2	yes	—
181	C₃×C₄	#926	1	2	yes	—
182	Dic₃	—	8	2	no	—
183	Dic₃	—	24	4	no	—
184	Dic₃	—	24	2	no	—
185	Dic₃	—	24	2	no	—
186	C₃²	#1835	1	1	yes	characteristic, complemented, Sylow
187	C₂³	C₂×#626	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	12	4	no	—
190	C₂×C₄	—	12	2	no	—
191	C₂×C₄	—	36	4	no	—
192	C₂×C₄	—	36	2	no	—
193	C₂×C₄	Dic₃⋊Dic₃	1	4	yes	—
194	C₂×C₄	#3795	1	2	yes	—
195	C₂×C₄	#3799	1	2	yes	—
196	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
197	C₂×C₄	C₂×#627	1	1	yes	characteristic
198	C₈	—	6	4	no	—
199	C₂×C₃	—	2	2	no	—
200	C₂×C₃	—	2	2	no	—
201	C₂×C₃	—	2	2	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	#4697	1	2	yes	—
204	C₂×C₃	#4697	1	2	yes	—
205	C₂×C₃	#4936	1	1	yes	characteristic
206	C₂×C₃	#4977	1	1	yes	characteristic
207	C₂×C₃	#4998	1	2	yes	—
208	C₂×C₃	#5056	1	2	yes	—
209	C₂×C₃	#5068	1	2	yes	—
210	C₂×C₃	#5106	1	2	yes	—
211	C₂²	#63829	1	4	yes	—
212	C₂²	#63860	1	1	yes	characteristic
213	C₂²	#63864	1	1	yes	characteristic
214	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
215	C₄	—	2	4	no	—
216	C₄	—	24	2	no	—
217	C₄	—	72	2	no	—
218	C₄	#63841	1	2	yes	—
219	C₄	#63869	1	2	yes	—
220	C₃	—	2	1	no	—
221	C₃	#69547	1	1	yes	characteristic, complemented
222	C₃	#69763	1	1	yes	characteristic, complemented
223	C₂	#10590274	1	2	yes	—
224	C₂	#10590293	1	2	yes	—
225	C₂	#10591122	1	2	yes	—
226	C₂	#10591224	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#626
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1835
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
13. C₃×C₄×C₃⋊C₈	C₄	semidirect
51. C₄×C₃⋊C₈	Dic₃	semidirect
186. C₃²	#1835	semidirect
221. C₃	#69547	semidirect
222. C₃	#69763	semidirect

Maximal subgroups

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

subgroup	index
#10592635	2
C₃×#4936	2
C₂×C₃×C₄×C₃⋊C₈	2
#10589164	2
#69763	3
#69547	3