Subgroups — #11786338

Order 1152.

487 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	#10592623	C₂	1	1	yes	maximal, characteristic
2	C₃×#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic
4	C₂×#63577	C₂	1	4	yes	maximal
5	#70519	—	3	1	no	maximal
6	#69842	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#3748	C₂²	1	2	yes	—
9	C₂×C₃×#208	C₂²	1	2	yes	—
10	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
13	C₃×#803	C₄	1	2	yes	—
14	C₃×#803	C₄	1	2	yes	—
15	#63577	—	2	8	no	—
16	C₃×#423	—	3	1	no	—
17	C₂×C₃×#97	—	3	1	no	—
18	C₂×#822	—	3	4	no	—
19	C₂×#819	—	3	4	no	—
20	#4924	—	3	1	no	—
21	#4924	—	3	1	no	—
22	#4924	—	6	1	no	—
23	#4924	D₃	1	1	yes	characteristic
24	C₂×#803	D₃	1	1	yes	characteristic
25	C₂²×C₃²×C₄	Q₈	1	2	yes	—
26	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
27	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄²	D₄	1	2	yes	—
30	#3748	—	2	4	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
32	C₃×#208	—	2	4	no	—
33	C₂×C₃×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₃⋊C₈	D₄	1	4	yes	—
36	C₂×C₃×C₃⋊C₈	Q₈	1	4	yes	—
37	#1849	—	9	1	no	Sylow
38	C₂×C₃×M₁₆	—	3	2	no	—
39	C₂²×C₃×C₈	—	3	2	no	—
40	C₂×C₃×C₄²	—	2	1	no	—
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×#208	—	3	2	no	—
44	C₂×#208	—	3	2	no	—
45	C₂×#208	—	6	2	no	—
46	C₂×#208	C₂×D₃	1	2	yes	—
47	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
48	C₃×C₄⋊C₈	—	6	2	no	—
49	C₃×#97	—	3	2	no	—
50	C₃×#97	—	3	2	no	—
51	#822	—	6	8	no	—
52	#819	—	6	8	no	—
53	C₄⋊(C₃⋊C₈)	—	2	2	no	—
54	C₄⋊(C₃⋊C₈)	—	6	4	no	—
55	C₄⋊(C₃⋊C₈)	—	6	2	no	—
56	C₄⋊(C₃⋊C₈)	—	6	2	no	—
57	#803	Dic₃	1	2	yes	—
58	#803	Dic₃	1	2	yes	—
59	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₃²×C₄	—	4	2	no	—
61	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
62	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
63	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
67	C₃⋊(C₃⋊C₈)	—	4	2	no	—
68	C₃×C₃⋊C₈	—	4	2	no	—
69	C₃×C₃⋊C₈	C₄⋊C₄	1	8	yes	—
70	C₂×#108	—	9	4	no	—
71	#423	—	3	1	no	—
72	#423	—	9	1	no	—
73	C₂×#97	—	3	1	no	—
74	C₂²×C₃×C₄	—	2	2	no	—
75	C₂²×C₃×C₄	—	2	1	no	—
76	C₂²×C₃×C₄	Dic₆	1	2	yes	—
77	C₂²×C₃×C₄	Dic₆	1	2	yes	—
78	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₃×M₁₆	—	6	4	no	—
81	C₂×C₃×C₈	—	3	4	no	—
82	C₂×C₃×C₈	—	3	4	no	—
83	C₂×C₃×C₈	—	3	4	no	—
84	C₂×C₃×C₈	—	6	2	no	—
85	C₃×C₄²	—	2	1	no	—
86	C₃×C₄²	—	2	1	no	—
87	C₃×C₄²	—	2	2	no	—
88	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
89	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
90	C₃×C₄²	D₁₂	1	2	yes	—
91	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
92	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
93	C₃×C₄²	#67	1	2	yes	—
94	#208	—	2	4	no	—
95	#208	—	6	4	no	—
96	#208	—	6	4	no	—
97	#208	—	12	4	no	—
98	C₂×C₃⋊C₈	—	2	2	no	—
99	C₂×C₃⋊C₈	—	6	4	no	—
100	C₂×C₃⋊C₈	—	6	2	no	—
101	C₂×C₃⋊C₈	—	6	2	no	—
102	C₂×C₃⋊C₈	Dic₆	1	4	yes	—
103	C₂×C₃⋊C₈	D₁₂	1	4	yes	—
104	C₂×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
105	C₂²×C₃²	—	2	2	no	—
106	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
108	C₃²×C₄	—	2	2	no	—
109	C₃²×C₄	—	2	2	no	—
110	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
111	C₃²×C₄	#127	1	1	yes	characteristic
112	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
113	C₂×M₁₆	—	3	2	no	—
114	C₂×M₁₆	—	9	2	no	—
115	C₂²×C₈	—	3	2	no	—
116	C₂×C₄²	D₃²	1	1	yes	characteristic
117	#108	—	18	8	no	—
118	C₄⋊C₈	—	6	2	no	—
119	C₄⋊C₈	—	18	2	no	—
120	#97	—	3	2	no	—
121	#97	—	3	2	no	—
122	C₂³×C₃	—	2	1	no	—
123	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
124	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	—	4	2	no	—
133	C₂×C₃×C₄	—	4	4	no	—
134	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
135	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
136	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
137	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
138	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
139	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
141	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
142	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
143	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
144	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
146	C₃×C₈	—	3	8	no	—
147	C₃×C₈	—	12	2	no	—
148	C₃⋊C₈	—	4	2	no	—
149	C₃⋊C₈	—	12	4	no	—
150	C₃⋊C₈	—	12	2	no	—
151	C₃⋊C₈	—	12	2	no	—
152	C₃⋊C₈	C₄⋊Dic₃	1	8	yes	—
153	C₂×C₃²	—	2	2	no	—
154	C₂×C₃²	#384	1	1	yes	characteristic
155	C₂×C₃²	#430	1	2	yes	—
156	C₂²×C₄	#627	1	2	yes	—
157	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
158	M₁₆	—	6	4	no	—
159	M₁₆	—	18	4	no	—
160	C₂×C₈	—	3	4	no	—
161	C₂×C₈	—	3	4	no	—
162	C₂×C₈	—	3	4	no	—
163	C₂×C₈	—	6	2	no	—
164	C₂×C₈	—	18	2	no	—
165	C₄²	D₃×Dic₃	1	1	yes	characteristic
166	C₄²	D₃×Dic₃	1	1	yes	characteristic
167	C₄²	#626	1	2	yes	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	4	2	no	—
173	C₂²×C₃	#869	1	2	yes	—
174	C₂²×C₃	#888	1	2	yes	—
175	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
176	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	4	2	no	—
186	C₃×C₄	#869	1	1	yes	characteristic
187	C₃×C₄	#874	1	1	yes	characteristic
188	C₃×C₄	D₃×C₄⋊C₄	1	2	yes	—
189	C₃×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
190	C₃×C₄	#940	1	1	yes	characteristic
191	C₃×C₄	#944	1	1	yes	characteristic
192	C₃²	#1849	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂×#627	1	1	yes	characteristic
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
196	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
197	C₂×C₄	D₃×Dic₆	1	2	yes	—
198	C₂×C₄	D₃×D₁₂	1	2	yes	—
199	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
200	C₂×C₄	C₂×#626	1	1	yes	characteristic
201	C₈	—	3	8	no	—
202	C₈	—	12	2	no	—
203	C₈	—	36	2	no	—
204	C₂×C₃	—	2	2	no	—
205	C₂×C₃	—	2	2	no	—
206	C₂×C₃	—	2	1	no	—
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	—	4	2	no	—
209	C₂×C₃	#4935	1	1	yes	characteristic
210	C₂×C₃	#4975	1	1	yes	characteristic
211	C₂×C₃	#5124	1	2	yes	—
212	C₂×C₃	#5125	1	2	yes	—
213	C₂²	—	2	2	no	—
214	C₂²	#63918	1	2	yes	—
215	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	#63866	1	1	yes	characteristic
219	C₄	D₃×C₄⋊Dic₃	1	2	yes	—
220	C₄	#63911	1	1	yes	characteristic
221	C₃	—	2	1	no	—
222	C₃	#69842	1	1	yes	characteristic, complemented
223	C₃	#70519	1	1	yes	characteristic, complemented
224	C₂	—	2	2	no	—
225	C₂	#10591141	1	2	yes	—
226	C₂	#10591349	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#626
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1849
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
192. C₃²	#1849	semidirect
222. C₃	#69842	semidirect
223. C₃	#70519	semidirect

Maximal subgroups

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

subgroup	index
#10592623	2
C₃×#4924	2
C₂×C₃×#803	2
C₂×#63577	2
#70519	3
#69842	3