Subgroups — #11860677

Order 1152.

1584 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3745	C₂	1	1	yes	maximal, characteristic
2	#10595992	C₂	1	2	yes	maximal, complemented
3	C₃²×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	#10592830	C₂	1	2	yes	maximal, complemented
5	#10592805	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592410	C₂	1	4	yes	maximal, complemented
7	C₄×#3745	C₂	1	4	yes	maximal, complemented
8	#78561	—	3	4	no	maximal
9	C₂²×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
10	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
12	C₃²×#117	C₂²	1	1	yes	characteristic
13	#64135	C₂²	1	2	yes	—
14	C₂×#3745	C₄	1	8	yes	complemented
15	C₂×#3745	C₂²	1	2	yes	—
16	C₂×#3745	C₂²	1	2	yes	—
17	#64102	—	2	4	no	—
18	C₂²⋊C₄×C₃⋊D₃	—	2	4	no	—
19	C₄×C₄⋊(C₃⋊D₃)	C₂²	1	8	yes	complemented
20	C₃²×#97	C₂²	1	2	yes	complemented
21	C₃²×C₄×C₈	C₂²	1	2	yes	complemented
22	#63648	C₂²	1	8	yes	complemented
23	#63645	C₂²	1	2	yes	complemented
24	#63644	C₂²	1	2	yes	complemented
25	C₂²×D₂₄	—	3	4	no	—
26	#5521	—	3	8	no	—
27	C₃×#405	D₃	1	4	yes	complemented
28	#5131	—	3	8	no	—
29	#5106	—	3	4	no	—
30	#4711	—	3	16	no	—
31	C₄×D₂₄	—	3	16	no	—
32	C₂³×C₃⋊D₃	—	2	2	no	—
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
35	C₂×C₄⋊(C₃⋊D₃)	—	2	4	no	—
36	C₂×C₄⋊(C₃⋊D₃)	C₂×C₄	1	8	yes	complemented
37	C₂×C₄⋊(C₃⋊D₃)	C₂³	1	4	yes	—
38	C₂×C₄×C₃⋊D₃	—	2	8	no	—
39	C₃²×M₁₆	—	2	2	no	—
40	C₂×C₃²×C₈	C₂×C₄	1	4	yes	complemented
41	C₂×C₃²×C₈	C₂³	1	2	yes	—
42	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
44	C₃²×C₄⋊C₄	D₄	1	2	yes	—
45	C₃²×C₂²⋊C₄	D₄	1	2	yes	—
46	C₃²×C₄²	C₂³	1	2	yes	—
47	#3757	—	2	2	no	—
48	#3752	—	2	8	no	—
49	C₄⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
50	C₄×C₃⋊Dic₃	—	2	2	no	—
51	#3745	C₂×C₄	1	8	yes	complemented
52	#3745	C₂×C₄	1	8	yes	complemented
53	#2948	—	9	1	no	Sylow
54	C₂²×D₁₂	—	3	8	no	—
55	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
56	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
57	C₃×#117	C₂×D₃	1	4	yes	—
58	#926	—	3	8	no	—
59	C₂×D₂₄	—	3	32	no	—
60	C₂×D₂₄	—	3	8	no	—
61	C₂×D₂₄	—	3	8	no	—
62	#893	—	6	16	no	—
63	D₃×C₂²⋊C₄	—	6	16	no	—
64	C₄×D₁₂	—	3	32	no	—
65	C₃×#97	C₂×D₃	1	8	yes	complemented
66	C₃×C₄×C₈	C₂×D₃	1	8	yes	complemented
67	#821	—	3	32	no	—
68	#818	—	3	8	no	—
69	#817	—	3	8	no	—
70	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃⋊D₃	—	2	4	no	—
72	C₂²×C₃⋊D₃	—	2	8	no	—
73	C₂²×C₃⋊D₃	—	8	4	no	—
74	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
75	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
76	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	#41	1	2	yes	—
78	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
79	C₂×C₃⋊Dic₃	—	2	4	no	—
80	C₄⋊(C₃⋊D₃)	—	2	8	no	—
81	C₄⋊(C₃⋊D₃)	—	2	4	no	—
82	C₄⋊(C₃⋊D₃)	C₂²×C₄	1	8	yes	—
83	C₄×C₃⋊D₃	—	8	8	no	—
84	C₃²×C₈	—	2	2	no	—
85	C₃²×C₈	C₂²×C₄	1	2	yes	—
86	C₃²×C₈	C₂²×C₄	1	2	yes	—
87	C₂²×D₈	—	9	1	no	—
88	#518	—	9	2	no	—
89	#442	—	9	4	no	—
90	C₄×D₈	—	9	4	no	—
91	#427	—	9	1	no	—
92	#418	—	9	2	no	—
93	#405	C₃⋊D₃	1	1	yes	characteristic, complemented
94	C₂³×D₃	—	6	8	no	—
95	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
96	C₂×D₁₂	—	3	32	no	—
97	C₂×D₁₂	—	3	16	no	—
98	C₂×D₁₂	—	6	8	no	—
99	C₂×D₁₂	—	6	16	no	—
100	C₂×C₄×D₃	—	6	32	no	—
101	C₃×M₁₆	—	2	8	no	—
102	C₂×C₃×C₈	C₄×D₃	1	16	yes	complemented
103	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
104	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
105	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
106	C₃×C₄⋊C₄	D₁₂	1	8	yes	—
107	C₃×C₂²⋊C₄	D₁₂	1	8	yes	—
108	C₃×C₄²	C₂²×D₃	1	8	yes	—
109	#217	—	6	8	no	—
110	(C₂×D₃)⋊C₄	—	6	32	no	—
111	C₄⋊Dic₃	—	3	16	no	—
112	C₄×Dic₃	—	6	8	no	—
113	D₂₄	—	3	32	no	—
114	D₂₄	—	3	32	no	—
115	C₂²×C₃²	—	2	2	no	—
116	C₂²×C₃²	C₄×D₄	1	2	yes	—
117	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₃⋊D₃	—	2	8	no	—
119	C₂×C₃⋊D₃	—	4	8	no	—
120	C₂×C₃⋊D₃	—	8	8	no	—
121	C₃²×C₄	—	2	4	no	—
122	C₃²×C₄	C₄×D₄	1	2	yes	—
123	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
124	C₃²×C₄	C₂×#41	1	1	yes	characteristic
125	C₃⋊Dic₃	—	4	4	no	—
126	C₂²×D₄	—	9	2	no	—
127	C₂×D₈	—	9	8	no	—
128	C₂×D₈	—	9	2	no	—
129	C₂×D₈	—	9	2	no	—
130	C₂×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
132	C₄×D₄	—	9	8	no	—
133	C₄×D₄	—	18	4	no	—
134	#117	—	9	2	no	—
135	#117	C₂×C₃⋊D₃	1	1	yes	characteristic
136	C₂×C₂²⋊C₄	—	18	4	no	—
137	#107	—	9	2	no	—
138	#106	—	9	2	no	—
139	#102	—	9	8	no	—
140	#97	C₂×C₃⋊D₃	1	2	yes	complemented
141	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	complemented
142	C₂³×C₃	C₂×D₁₂	1	4	yes	—
143	C₂²×D₃	—	6	16	no	—
144	C₂²×D₃	—	6	32	no	—
145	C₂²×D₃	—	24	16	no	—
146	C₂×C₃×C₄	C₂×C₄×D₃	1	8	yes	—
147	C₂×C₃×C₄	C₂×D₁₂	1	16	yes	—
148	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
149	C₂×C₃×C₄	#235	1	8	yes	—
150	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
151	C₂×Dic₃	—	6	16	no	—
152	D₁₂	—	3	32	no	—
153	D₁₂	—	6	32	no	—
154	D₁₂	—	6	16	no	—
155	C₄×D₃	—	24	32	no	—
156	C₃×C₈	—	2	8	no	—
157	C₃×C₈	C₂×C₄×D₃	1	8	yes	—
158	C₃×C₈	C₂×C₄×D₃	1	8	yes	—
159	C₂×C₃²	—	2	2	no	—
160	C₂×C₃²	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×C₃²	#576	1	2	yes	—
162	C₃⋊D₃	—	4	8	no	—
163	C₂⁴	—	18	2	no	—
164	C₂×D₄	—	9	4	no	—
165	C₂×D₄	—	9	8	no	—
166	C₂×D₄	—	18	2	no	—
167	C₂×D₄	—	18	4	no	—
168	C₂²×C₄	—	18	8	no	—
169	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
170	D₈	—	9	8	no	—
171	D₈	—	9	8	no	—
172	M₁₆	—	2	2	no	—
173	C₂×C₈	C₄×C₃⋊D₃	1	4	yes	complemented
174	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
175	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
176	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
177	C₄⋊C₄	—	9	4	no	—
178	C₄⋊C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
179	C₂²⋊C₄	—	18	2	no	—
180	C₂²⋊C₄	—	18	8	no	—
181	C₂²⋊C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
182	C₄²	—	18	2	no	—
183	C₄²	C₂²×C₃⋊D₃	1	2	yes	—
184	C₂²×C₃	—	2	8	no	—
185	C₂²×C₃	C₄×D₁₂	1	8	yes	—
186	C₂²×C₃	C₂²×D₁₂	1	4	yes	—
187	C₂×D₃	—	6	32	no	—
188	C₂×D₃	—	12	32	no	—
189	C₂×D₃	—	24	32	no	—
190	C₃×C₄	—	2	16	no	—
191	C₃×C₄	C₄×D₁₂	1	8	yes	—
192	C₃×C₄	C₂²×C₄×D₃	1	4	yes	—
193	C₃×C₄	C₂×#235	1	4	yes	—
194	Dic₃	—	12	16	no	—
195	C₃²	#2948	1	1	yes	characteristic, complemented, Sylow
196	C₂³	—	18	4	no	—
197	C₂³	—	18	8	no	—
198	C₂³	—	72	4	no	—
199	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
200	D₄	—	9	8	no	—
201	D₄	—	18	8	no	—
202	D₄	—	18	4	no	—
203	C₂×C₄	—	18	4	no	—
204	C₂×C₄	—	72	8	no	—
205	C₂×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
208	C₂×C₄	#3828	1	2	yes	—
209	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	C₂×C₄×C₃⋊D₃	1	2	yes	—
212	C₈	C₂×C₄×C₃⋊D₃	1	2	yes	—
213	C₂×C₃	—	2	8	no	—
214	C₂×C₃	C₂×C₄×D₁₂	1	4	yes	—
215	C₂×C₃	#5752	1	8	yes	—
216	D₃	—	12	32	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	18	8	no	—
219	C₂²	—	36	8	no	—
220	C₂²	—	72	8	no	—
221	C₂²	C₄×C₄⋊(C₃⋊D₃)	1	2	yes	—
222	C₂²	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
223	C₄	—	2	4	no	—
224	C₄	—	36	4	no	—
225	C₄	C₄×C₄⋊(C₃⋊D₃)	1	2	yes	—
226	C₄	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
227	C₄	C₂×#3828	1	1	yes	characteristic
228	C₃	#78561	1	4	yes	complemented
229	C₂	—	2	2	no	—
230	C₂	—	36	8	no	—
231	C₂	C₂×C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
232	C₂	#10596223	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₃²×#405	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂²×#3745	C₂
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	#405	C₃⋊D₃
hypocenter	C₃²	#2948
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #10595992	C₂	semidirect
3. C₃²×#405	C₂	semidirect
4. #10592830	C₂	semidirect
5. #10592805	C₂	semidirect
6. #10592410	C₂	semidirect
7. C₄×#3745	C₂	semidirect
14. C₂×#3745	C₄	semidirect
19. C₄×C₄⋊(C₃⋊D₃)	C₂²	semidirect
20. C₃²×#97	C₂²	semidirect
21. C₃²×C₄×C₈	C₂²	semidirect
22. #63648	C₂²	semidirect
23. #63645	C₂²	semidirect
24. #63644	C₂²	semidirect
27. C₃×#405	D₃	semidirect
36. C₂×C₄⋊(C₃⋊D₃)	C₂×C₄	semidirect
40. C₂×C₃²×C₈	C₂×C₄	semidirect
51. #3745	C₂×C₄	semidirect
52. #3745	C₂×C₄	semidirect
65. C₃×#97	C₂×D₃	semidirect
66. C₃×C₄×C₈	C₂×D₃	semidirect
93. #405	C₃⋊D₃	semidirect
102. C₂×C₃×C₈	C₄×D₃	semidirect
140. #97	C₂×C₃⋊D₃	semidirect
141. C₄×C₈	C₂×C₃⋊D₃	semidirect
173. C₂×C₈	C₄×C₃⋊D₃	semidirect
195. C₃²	#2948	semidirect
228. C₃	#78561	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#3745	2
#10595992	2
C₃²×#405	2
#10592830	2
#10592805	2
#10592410	2
C₄×#3745	2
#78561	3