Subgroups — #11795898

Order 1152.

433 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592800	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#10589104	C₂	1	2	yes	maximal, complemented
5	#10589102	C₂	1	2	yes	maximal, complemented
6	#71419	—	3	1	no	maximal
7	#71399	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	#64135	C₄	1	1	yes	characteristic
10	C₂×#3748	C₄	1	1	yes	characteristic
11	#63642	C₄	1	2	yes	—
12	C₈×C₃⋊Dic₃	C₂²	1	2	yes	—
13	C₃×#812	C₄	1	4	yes	complemented
14	C₂×C₃×C₃⋊C₁₆	C₄	1	4	yes	complemented
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×M₃₂	—	3	1	no	—
19	C₂²×C₃×C₁₆	—	3	1	no	—
20	#5101	—	3	1	no	—
21	#5101	—	3	1	no	—
22	#5101	—	6	1	no	—
23	C₂×#812	D₃	1	1	yes	characteristic
24	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic
25	#4502	—	3	2	no	—
26	#4502	—	3	2	no	—
27	Dic₃×C₁₆	—	3	2	no	—
28	Dic₃×C₁₆	—	3	2	no	—
29	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
34	#3757	C₈	1	2	yes	—
35	C₄⋊(C₃⋊Dic₃)	C₈	1	2	yes	—
36	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
37	#3748	C₈	1	4	yes	—
38	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
39	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
40	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
41	C₃×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
42	#2108	—	9	1	no	Sylow
43	C₂²×C₃×C₈	—	2	1	no	—
44	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
46	#926	—	3	1	no	—
47	#926	—	3	1	no	—
48	#926	—	6	1	no	—
49	C₂×#208	—	3	1	no	—
50	C₂×#208	—	3	1	no	—
51	C₂×#208	—	6	1	no	—
52	C₃×M₃₂	—	3	4	no	—
53	C₂×C₃×C₁₆	—	3	4	no	—
54	C₂×C₃×C₁₆	—	3	1	no	—
55	C₂×C₃×C₁₆	—	3	1	no	—
56	C₂×C₃×C₁₆	—	3	2	no	—
57	#815	—	3	2	no	—
58	#815	—	3	2	no	—
59	#815	—	6	2	no	—
60	C₈×Dic₃	—	3	2	no	—
61	C₈×Dic₃	—	3	2	no	—
62	C₈×Dic₃	—	6	2	no	—
63	#812	Dic₃	1	4	yes	complemented
64	C₂×C₃⋊C₁₆	Dic₃	1	4	yes	complemented
65	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
67	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
68	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₄²	1	2	yes	—
70	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃⋊Dic₃	C₂×C₈	1	4	yes	—
74	C₃²×C₈	C₄²	1	2	yes	—
75	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₃⋊(C₃⋊C₈)	C₂×C₈	1	4	yes	—
78	C₂×M₃₂	—	3	1	no	—
79	C₂²×C₁₆	—	3	1	no	—
80	#405	—	9	1	no	—
81	#346	—	9	2	no	—
82	C₄×C₁₆	—	9	2	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
91	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
92	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
95	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
96	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
98	#217	—	3	2	no	—
99	#217	—	3	2	no	—
100	#217	—	6	2	no	—
101	C₄⋊Dic₃	—	3	2	no	—
102	C₄⋊Dic₃	—	3	2	no	—
103	C₄⋊Dic₃	—	6	2	no	—
104	C₄×Dic₃	—	3	2	no	—
105	C₄×Dic₃	—	3	2	no	—
106	C₄×Dic₃	—	6	2	no	—
107	#208	—	3	4	no	—
108	#208	—	3	4	no	—
109	#208	—	6	4	no	—
110	C₂×C₃⋊C₈	—	3	2	no	—
111	C₂×C₃⋊C₈	—	3	2	no	—
112	C₂×C₃⋊C₈	—	6	2	no	—
113	C₃×C₁₆	—	3	2	no	—
114	C₃×C₁₆	—	3	2	no	—
115	C₃×C₁₆	—	3	4	no	—
116	C₃⋊C₁₆	C₂×Dic₃	1	2	yes	—
117	C₃⋊C₁₆	C₂×Dic₃	1	2	yes	—
118	C₃⋊C₁₆	C₂×Dic₃	1	4	yes	—
119	C₂²×C₃²	—	2	2	no	—
120	C₂²×C₃²	C₄×C₈	1	2	yes	—
121	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
122	C₃²×C₄	C₄×C₈	1	2	yes	—
123	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
124	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
125	C₃⋊Dic₃	—	2	4	no	—
126	C₂×M₁₆	—	9	1	no	—
127	C₂²×C₈	D₃²	1	1	yes	characteristic
128	#117	—	9	1	no	—
129	M₃₂	—	3	4	no	—
130	C₂×C₁₆	—	3	4	no	—
131	C₂×C₁₆	—	3	1	no	—
132	C₂×C₁₆	—	3	2	no	—
133	C₂×C₁₆	—	3	1	no	—
134	#97	—	9	2	no	—
135	C₄×C₈	—	9	2	no	—
136	C₂³×C₃	—	2	1	no	—
137	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
138	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
144	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
145	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
146	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
147	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
148	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×Dic₃	—	3	4	no	—
152	C₂×Dic₃	—	3	4	no	—
153	C₂×Dic₃	—	6	4	no	—
154	C₃×C₈	—	2	2	no	—
155	C₃×C₈	—	2	1	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	C₄×Dic₃	1	2	yes	—
158	C₃×C₈	C₄×Dic₃	1	2	yes	—
159	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
160	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
161	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
162	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
163	C₃⋊C₈	—	3	4	no	—
164	C₃⋊C₈	—	3	4	no	—
165	C₃⋊C₈	—	6	4	no	—
166	C₂×C₃²	—	2	2	no	—
167	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
168	C₂×C₃²	#504	1	2	yes	—
169	C₂²×C₄	#624	1	1	yes	characteristic
170	M₁₆	—	9	4	no	—
171	C₂×C₈	—	9	2	no	—
172	C₂×C₈	D₃×Dic₃	1	2	yes	—
173	C₂×C₈	D₃×Dic₃	1	2	yes	—
174	C₂×C₈	#624	1	1	yes	characteristic
175	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
176	C₄⋊C₄	—	9	2	no	—
177	C₂²⋊C₄	—	9	2	no	—
178	C₄²	—	9	2	no	—
179	C₁₆	—	3	2	no	—
180	C₁₆	—	3	2	no	—
181	C₁₆	—	3	4	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	2	1	no	—
186	C₂²×C₃	—	4	2	no	—
187	C₂²×C₃	C₈×Dic₃	1	2	yes	—
188	C₂²×C₃	C₈×Dic₃	1	2	yes	—
189	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
190	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	C₈×Dic₃	1	2	yes	—
195	C₃×C₄	C₈×Dic₃	1	2	yes	—
196	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
197	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
198	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
199	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
200	Dic₃	—	6	4	no	—
201	Dic₃	—	6	4	no	—
202	Dic₃	—	12	4	no	—
203	C₃²	#2108	1	1	yes	characteristic, complemented, Sylow
204	C₂³	#3710	1	1	yes	characteristic
205	C₂×C₄	—	9	4	no	—
206	C₂×C₄	#3710	1	1	yes	characteristic
207	C₂×C₄	#3710	1	2	yes	—
208	C₂×C₄	Dic₃²	1	2	yes	—
209	C₂×C₄	C₂×#624	1	1	yes	characteristic
210	C₈	—	9	4	no	—
211	C₈	Dic₃²	1	2	yes	—
212	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
213	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	—	4	2	no	—
219	C₂×C₃	#4901	1	2	yes	—
220	C₂×C₃	#4907	1	2	yes	—
221	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
222	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
223	C₂²	—	2	2	no	—
224	C₂²	#63553	1	2	yes	—
225	C₂²	C₂×#3710	1	1	yes	characteristic
226	C₄	—	18	4	no	—
227	C₄	#63553	1	2	yes	—
228	C₄	C₂×#3710	1	1	yes	characteristic
229	C₄	C₂×Dic₃²	1	1	yes	characteristic
230	C₃	—	2	1	no	—
231	C₃	#71399	1	1	yes	characteristic, complemented
232	C₃	#71419	1	1	yes	characteristic, complemented
233	C₂	—	2	2	no	—
234	C₂	#10590624	1	2	yes	—
235	C₂	C₂×#63553	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator	C₂×C₃²	C₂×C₄×C₈
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#2108
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. #10589104	C₂	semidirect
5. #10589102	C₂	semidirect
13. C₃×#812	C₄	semidirect
14. C₂×C₃×C₃⋊C₁₆	C₄	semidirect
63. #812	Dic₃	semidirect
64. C₂×C₃⋊C₁₆	Dic₃	semidirect
203. C₃²	#2108	semidirect
231. C₃	#71399	semidirect
232. C₃	#71419	semidirect

Maximal subgroups

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

subgroup	index
#10592800	2
C₂×C₃×#812	2
C₂²×C₃×C₃⋊C₁₆	2
#10589104	2
#10589102	2
#71419	3
#71399	3