Subgroups — #11784365

Order 1152.

696 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#63649	C₂	1	1	yes	maximal, characteristic
3	#10592800	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
5	#10592697	C₂	1	1	yes	maximal, characteristic
6	#10589421	C₂	1	1	yes	maximal, characteristic
7	#10589398	C₂	1	1	yes	maximal, characteristic
8	#69889	—	3	4	no	maximal
9	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃²×#117	C₂²	1	1	yes	characteristic
12	#64135	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×#3748	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₃²×#97	—	2	1	no	—
17	C₃²×C₄×C₈	—	2	1	no	—
18	#63649	C₄	1	4	yes	complemented
19	#63642	—	2	1	no	—
20	C₈×C₃⋊Dic₃	—	2	1	no	—
21	C₈×C₃⋊Dic₃	C₄	1	4	yes	complemented
22	#63635	C₄	1	2	yes	complemented
23	#63634	C₄	1	2	yes	complemented
24	C₃×#405	D₃	1	4	yes	—
25	C₂×#822	—	3	4	no	—
26	#5101	—	3	4	no	—
27	C₂×C₈×Dic₃	—	3	4	no	—
28	#4998	—	3	4	no	—
29	#4555	—	3	4	no	—
30	#4532	—	3	4	no	—
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃⋊Dic₃	—	2	1	no	—
33	C₃²×M₁₆	—	2	1	no	—
34	C₃²×M₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	—	2	1	no	—
36	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	Q₈	1	2	yes	—
40	C₃²×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₃²×C₂²⋊C₄	—	2	1	no	—
42	C₃²×C₄²	—	2	1	no	—
43	#3757	—	2	1	no	—
44	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
45	C₄×C₃⋊Dic₃	—	2	1	no	—
46	C₄×C₃⋊Dic₃	—	2	1	no	—
47	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
48	#3748	—	2	1	no	—
49	#3748	C₂×C₄	1	2	yes	—
50	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
51	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
53	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
54	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
55	#1772	—	9	1	no	Sylow
56	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
57	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
58	C₃×#117	C₂×D₃	1	4	yes	—
59	#926	—	3	4	no	—
60	C₂×C₄×Dic₃	—	3	4	no	—
61	C₂×#208	—	3	4	no	—
62	C₂²×C₃⋊C₈	—	3	4	no	—
63	C₃×#97	—	2	4	no	—
64	C₃×C₄×C₈	—	2	4	no	—
65	#822	—	3	16	no	—
66	#815	—	6	4	no	—
67	C₈×Dic₃	—	3	16	no	—
68	C₈×Dic₃	—	6	4	no	—
69	#808	—	3	8	no	—
70	#807	—	3	8	no	—
71	C₂³×C₃²	#41	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃²×C₄	#41	1	1	yes	characteristic
79	C₂×C₃⋊Dic₃	—	2	4	no	—
80	C₂×C₃⋊Dic₃	—	2	2	no	—
81	C₂×C₃⋊Dic₃	—	2	2	no	—
82	C₃²×C₈	—	2	2	no	—
83	C₃²×C₈	C₄⋊C₄	1	2	yes	—
84	C₃²×C₈	C₄⋊C₄	1	2	yes	—
85	C₃⋊(C₃⋊C₈)	—	2	2	no	—
86	C₃⋊(C₃⋊C₈)	C₄²	1	2	yes	—
87	C₃⋊(C₃⋊C₈)	C₄²	1	2	yes	—
88	C₂×#108	—	9	1	no	—
89	#427	—	9	1	no	—
90	#405	—	9	1	no	—
91	#405	C₃⋊D₃	1	1	yes	characteristic
92	C₂×C₄×C₈	—	9	1	no	—
93	#339	—	9	1	no	—
94	#339	—	9	1	no	—
95	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
96	C₂²×Dic₃	—	6	4	no	—
97	C₃×M₁₆	—	2	4	no	—
98	C₃×M₁₆	C₄×D₃	1	8	yes	—
99	C₂×C₃×C₈	—	2	4	no	—
100	C₂×C₃×C₈	Dic₆	1	8	yes	—
101	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
102	C₂×C₃×C₈	#67	1	4	yes	—
103	C₂×C₃×C₈	#67	1	4	yes	—
104	C₃×C₄⋊C₄	C₄×D₃	1	8	yes	—
105	C₃×C₂²⋊C₄	—	2	4	no	—
106	C₃×C₄²	—	2	4	no	—
107	#217	—	6	4	no	—
108	C₄⋊Dic₃	—	3	8	no	—
109	C₄×Dic₃	—	3	8	no	—
110	C₄×Dic₃	—	6	4	no	—
111	C₄×Dic₃	—	6	4	no	—
112	#208	—	3	8	no	—
113	#208	—	6	4	no	—
114	C₂×C₃⋊C₈	—	3	4	no	—
115	C₂×C₃⋊C₈	—	3	4	no	—
116	C₂×C₃⋊C₈	—	3	8	no	—
117	C₂×C₃⋊C₈	—	3	8	no	—
118	C₂×C₃⋊C₈	—	6	4	no	—
119	C₂²×C₃²	—	2	2	no	—
120	C₂²×C₃²	#117	1	1	yes	characteristic
121	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
122	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
123	C₃²×C₄	—	4	2	no	—
124	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
125	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
127	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
128	C₃⋊Dic₃	—	2	4	no	—
129	C₃⋊Dic₃	—	4	2	no	—
130	C₂×M₁₆	—	9	1	no	—
131	C₂×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
132	C₂²×C₈	—	9	1	no	—
133	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
134	#117	—	9	1	no	—
135	#117	C₂×C₃⋊D₃	1	1	yes	characteristic
136	C₂×C₄²	—	9	1	no	—
137	#108	—	9	4	no	—
138	#107	—	9	2	no	—
139	#106	—	9	2	no	—
140	#97	—	2	1	no	—
141	#97	—	18	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	9	4	no	—
144	C₄×C₈	—	18	1	no	—
145	C₂³×C₃	#235	1	4	yes	—
146	C₂×C₃×C₄	—	2	8	no	—
147	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
148	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
149	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
150	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
151	C₂×C₃×C₄	#235	1	4	yes	—
152	C₂×C₃×C₄	C₂×#67	1	4	yes	—
153	C₂×Dic₃	—	6	16	no	—
154	C₂×Dic₃	—	6	8	no	—
155	C₂×Dic₃	—	6	8	no	—
156	C₃×C₈	—	2	8	no	—
157	C₃×C₈	Dic₃⋊C₄	1	8	yes	—
158	C₃×C₈	Dic₃⋊C₄	1	8	yes	—
159	C₃⋊C₈	—	3	8	no	—
160	C₃⋊C₈	—	3	8	no	—
161	C₃⋊C₈	—	6	8	no	—
162	C₂×C₃²	—	2	1	no	—
163	C₂×C₃²	—	2	1	no	—
164	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₃²	#443	1	2	yes	—
166	C₂²×C₄	—	18	1	no	—
167	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
168	M₁₆	—	2	1	no	—
169	M₁₆	—	9	2	no	—
170	M₁₆	—	18	1	no	—
171	M₁₆	C₄×C₃⋊D₃	1	2	yes	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	9	2	no	—
174	C₂×C₈	—	9	1	no	—
175	C₂×C₈	—	9	1	no	—
176	C₂×C₈	—	9	2	no	—
177	C₂×C₈	—	18	1	no	—
178	C₂×C₈	#634	1	2	yes	—
179	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
180	C₂×C₈	#638	1	1	yes	characteristic
181	C₂×C₈	#638	1	1	yes	characteristic
182	C₄⋊C₄	—	9	2	no	—
183	C₄⋊C₄	C₄×C₃⋊D₃	1	2	yes	—
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	—	18	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	9	2	no	—
188	C₄²	—	18	1	no	—
189	C₄²	—	18	1	no	—
190	C₂²×C₃	—	2	8	no	—
191	C₂²×C₃	C₄×Dic₆	1	4	yes	—
192	C₂²×C₃	#872	1	4	yes	—
193	C₂²×C₃	C₄×#67	1	4	yes	—
194	C₃×C₄	—	4	8	no	—
195	C₃×C₄	C₄×Dic₆	1	4	yes	—
196	C₃×C₄	C₄²×D₃	1	4	yes	—
197	C₃×C₄	C₂×Dic₃⋊C₄	1	4	yes	—
198	C₃×C₄	C₄×#67	1	4	yes	—
199	Dic₃	—	6	16	no	—
200	Dic₃	—	12	8	no	—
201	C₃²	#1772	1	1	yes	characteristic, complemented, Sylow
202	C₂³	#3828	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	18	4	no	—
205	C₂×C₄	—	18	2	no	—
206	C₂×C₄	—	18	2	no	—
207	C₂×C₄	C₂×#634	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
211	C₂×C₄	#3828	1	1	yes	characteristic
212	C₂×C₄	C₂×#638	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	9	2	no	—
215	C₈	—	9	2	no	—
216	C₈	—	18	2	no	—
217	C₈	#3750	1	2	yes	—
218	C₈	#3750	1	2	yes	—
219	C₂×C₃	—	2	4	no	—
220	C₂×C₃	—	2	4	no	—
221	C₂×C₃	C₄×Dic₃⋊C₄	1	4	yes	—
222	C₂×C₃	#5144	1	8	yes	—
223	C₂²	—	2	2	no	—
224	C₂²	C₄×#634	1	1	yes	characteristic
225	C₂²	#64081	1	1	yes	characteristic
226	C₂²	C₄×#638	1	1	yes	characteristic
227	C₄	—	4	2	no	—
228	C₄	—	18	4	no	—
229	C₄	—	36	2	no	—
230	C₄	C₄×#634	1	1	yes	characteristic
231	C₄	C₄²×C₃⋊D₃	1	1	yes	characteristic
232	C₄	C₂×#3750	1	1	yes	characteristic
233	C₄	C₄×#638	1	1	yes	characteristic
234	C₃	#69889	1	4	yes	complemented
235	C₂	—	2	1	no	—
236	C₂	—	2	1	no	—
237	C₂	C₄×#3750	1	1	yes	characteristic
238	C₂	#10592843	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#638
commutator	C₃²×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#405	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	#405	C₃⋊D₃
hypocenter	C₃²	#1772
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
18. #63649	C₄	semidirect
21. C₈×C₃⋊Dic₃	C₄	semidirect
22. #63635	C₄	semidirect
23. #63634	C₄	semidirect
201. C₃²	#1772	semidirect
234. C₃	#69889	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#405	2
C₂×#63649	2
#10592800	2
C₂×C₈×C₃⋊Dic₃	2
#10592697	2
#10589421	2
#10589398	2
#69889	3