Subgroups — #11785542

Order 1152.

768 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	#10592935	C₂	1	2	yes	maximal
3	C₂×#63646	C₂	1	1	yes	maximal, characteristic, complemented
4	#10592805	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589425	C₂	1	2	yes	maximal, complemented
6	#72013	—	3	4	no	maximal
7	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
8	C₃²×#131	—	2	1	no	—
9	C₃²×#131	—	2	1	no	—
10	C₃²×#131	C₄	1	2	yes	complemented
11	C₃²×#131	C₄	1	2	yes	complemented
12	C₂×C₃²×M₁₆	—	2	1	no	—
13	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃²×C₈	—	2	1	no	—
15	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
16	#64135	C₂²	1	2	yes	—
17	C₂×#3748	C₂²	1	2	yes	—
18	#63664	—	2	4	no	—
19	#63646	—	2	1	no	—
20	#63646	—	2	1	no	—
21	#63645	—	2	1	no	—
22	#63644	—	2	1	no	—
23	C₂×C₃×#131	D₃	1	4	yes	—
24	#5236	—	3	8	no	—
25	C₂×#819	—	3	4	no	—
26	#5106	—	3	4	no	—
27	#4559	—	3	8	no	—
28	C₃²×#41	—	2	1	no	—
29	C₃²×#41	—	2	1	no	—
30	C₃²×#41	C₂×C₄	1	2	yes	—
31	C₃²×#41	D₄	1	2	yes	—
32	C₂×C₃²×Q₈	Q₈	1	1	yes	characteristic
33	C₂×C₃²×D₄	—	2	1	no	—
34	C₂×C₃²×D₄	Q₈	1	1	yes	characteristic
35	C₂²×C₃²×C₄	—	2	1	no	—
36	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
37	C₃²×M₁₆	—	2	1	no	—
38	C₃²×M₁₆	—	2	2	no	—
39	C₃²×M₁₆	—	2	2	no	—
40	C₃²×M₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₃²×C₈	—	2	1	no	—
42	C₂×C₃²×C₈	—	2	2	no	—
43	C₂×C₃²×C₈	—	2	2	no	—
44	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
47	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
48	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
49	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
50	#3757	—	2	2	no	—
51	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
52	C₄×C₃⋊Dic₃	—	2	2	no	—
53	#3748	—	2	2	no	—
54	#3748	—	2	2	no	—
55	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
56	#1814	—	9	1	no	Sylow
57	C₂×C₃×#41	C₂×D₃	1	4	yes	—
58	C₃×#131	—	2	4	no	—
59	C₃×#131	—	2	4	no	—
60	C₃×#131	Dic₃	1	8	yes	complemented
61	C₃×#131	Dic₃	1	8	yes	complemented
62	C₂×C₃×M₁₆	—	2	4	no	—
63	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
64	C₂²×C₃×C₈	—	2	4	no	—
65	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
66	#926	—	3	8	no	—
67	C₂×#208	—	3	8	no	—
68	#837	—	6	16	no	—
69	#819	—	6	4	no	—
70	#819	—	6	4	no	—
71	#818	—	6	4	no	—
72	#817	—	6	4	no	—
73	C₂³×C₃²	—	2	1	no	—
74	C₂³×C₃²	#41	1	1	yes	characteristic
75	C₃²×Q₈	—	2	1	no	—
76	C₃²×Q₈	C₄⋊C₄	1	2	yes	—
77	C₃²×D₄	—	2	2	no	—
78	C₃²×D₄	—	2	2	no	—
79	C₃²×D₄	—	2	1	no	—
80	C₃²×D₄	C₄⋊C₄	1	2	yes	—
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₃²×C₄	#41	1	1	yes	characteristic
90	C₂×C₃⋊Dic₃	—	4	2	no	—
91	C₃²×C₈	—	2	2	no	—
92	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₃⋊(C₃⋊C₈)	—	4	2	no	—
97	C₂×#131	C₃⋊D₃	1	1	yes	characteristic
98	C₂×#108	—	9	1	no	—
99	#427	—	9	1	no	—
100	#421	—	9	2	no	—
101	#344	—	9	2	no	—
102	C₃×#41	—	2	4	no	—
103	C₃×#41	—	2	4	no	—
104	C₃×#41	D₁₂	1	8	yes	—
105	C₃×#41	C₂×Dic₃	1	8	yes	—
106	C₂×C₃×Q₈	Dic₆	1	4	yes	—
107	C₂×C₃×D₄	—	2	4	no	—
108	C₂×C₃×D₄	Dic₆	1	4	yes	—
109	C₂²×C₃×C₄	—	2	4	no	—
110	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
111	C₃×M₁₆	—	2	4	no	—
112	C₃×M₁₆	—	2	8	no	—
113	C₃×M₁₆	—	2	8	no	—
114	C₃×M₁₆	C₂×Dic₃	1	8	yes	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	—	2	8	no	—
117	C₂×C₃×C₈	—	2	8	no	—
118	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
119	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
120	C₂×C₃×C₈	#67	1	4	yes	—
121	C₂×C₃×C₈	#67	1	4	yes	—
122	C₂×C₃×C₈	#67	1	4	yes	—
123	C₂×C₃×C₈	#67	1	4	yes	—
124	#217	—	6	8	no	—
125	C₄⋊Dic₃	—	6	8	no	—
126	C₄×Dic₃	—	6	8	no	—
127	#208	—	6	8	no	—
128	#208	—	6	8	no	—
129	C₂×C₃⋊C₈	—	6	8	no	—
130	C₂²×C₃²	—	2	2	no	—
131	C₂²×C₃²	—	4	1	no	—
132	C₂²×C₃²	—	4	2	no	—
133	C₂²×C₃²	#117	1	1	yes	characteristic
134	C₂²×C₃²	#121	1	1	yes	characteristic
135	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
136	C₃²×C₄	—	2	2	no	—
137	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₃²×C₄	#121	1	1	yes	characteristic
140	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
141	C₃⋊Dic₃	—	8	2	no	—
142	C₂×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
143	#131	—	2	1	no	—
144	#131	—	2	1	no	—
145	#131	C₃⋊Dic₃	1	2	yes	complemented
146	#131	C₃⋊Dic₃	1	2	yes	complemented
147	C₂×M₁₆	—	2	1	no	—
148	C₂×M₁₆	—	9	2	no	—
149	C₂×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
152	#117	—	9	2	no	—
153	#108	—	18	1	no	—
154	#108	—	18	1	no	—
155	#107	—	18	1	no	—
156	#106	—	18	1	no	—
157	#104	—	18	4	no	—
158	C₂³×C₃	—	2	4	no	—
159	C₂³×C₃	#235	1	4	yes	—
160	C₃×Q₈	—	2	4	no	—
161	C₃×Q₈	C₄⋊Dic₃	1	8	yes	—
162	C₃×D₄	—	2	8	no	—
163	C₃×D₄	—	2	8	no	—
164	C₃×D₄	—	2	4	no	—
165	C₃×D₄	C₄⋊Dic₃	1	8	yes	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	8	no	—
168	C₂×C₃×C₄	—	2	8	no	—
169	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
170	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
171	C₂×C₃×C₄	#235	1	4	yes	—
172	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
173	C₂×C₃×C₄	C₂×#67	1	4	yes	—
174	C₂×C₃×C₄	C₂×#67	1	4	yes	—
175	C₂×Dic₃	—	12	8	no	—
176	C₃×C₈	—	2	8	no	—
177	C₃×C₈	#217	1	4	yes	—
178	C₃×C₈	#217	1	4	yes	—
179	C₃×C₈	#217	1	4	yes	—
180	C₃×C₈	#217	1	4	yes	—
181	C₃⋊C₈	—	12	8	no	—
182	C₂×C₃²	—	2	1	no	—
183	C₂×C₃²	—	2	1	no	—
184	C₂×C₃²	—	2	1	no	—
185	C₂×C₃²	—	4	2	no	—
186	C₂×C₃²	#380	1	1	yes	characteristic
187	#41	—	2	1	no	—
188	#41	—	2	1	no	—
189	#41	C₄⋊(C₃⋊D₃)	1	2	yes	—
190	#41	C₂×C₃⋊Dic₃	1	2	yes	—
191	C₂×Q₈	#634	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	#634	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
196	M₁₆	—	2	1	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	18	2	no	—
200	M₁₆	—	18	2	no	—
201	M₁₆	C₂×C₃⋊Dic₃	1	2	yes	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	18	2	no	—
206	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
208	C₂×C₈	#638	1	1	yes	characteristic
209	C₂×C₈	#638	1	1	yes	characteristic
210	C₂×C₈	#638	1	1	yes	characteristic
211	C₂×C₈	#638	1	1	yes	characteristic
212	C₄⋊C₄	—	18	2	no	—
213	C₂²⋊C₄	—	18	2	no	—
214	C₄²	—	18	2	no	—
215	C₂²×C₃	—	2	8	no	—
216	C₂²×C₃	—	4	4	no	—
217	C₂²×C₃	—	4	8	no	—
218	C₂²×C₃	#924	1	4	yes	—
219	C₂²×C₃	#926	1	4	yes	—
220	C₂²×C₃	#930	1	4	yes	—
221	C₃×C₄	—	2	8	no	—
222	C₃×C₄	#924	1	4	yes	—
223	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
224	C₃×C₄	#930	1	4	yes	—
225	C₃×C₄	C₂×#217	1	4	yes	—
226	Dic₃	—	24	8	no	—
227	C₃²	#1814	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	2	1	no	—
229	C₂³	#3828	1	1	yes	characteristic
230	Q₈	—	2	1	no	—
231	Q₈	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	36	2	no	—
240	C₂×C₄	C₂×#634	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
242	C₂×C₄	#3828	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#638	1	1	yes	characteristic
245	C₂×C₄	C₂×#638	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	36	2	no	—
248	C₈	#3757	1	1	yes	characteristic
249	C₈	#3757	1	1	yes	characteristic
250	C₈	#3757	1	1	yes	characteristic
251	C₈	#3757	1	1	yes	characteristic
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	—	2	4	no	—
255	C₂×C₃	—	4	8	no	—
256	C₂×C₃	#5212	1	4	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#64133	1	1	yes	characteristic
261	C₂²	#64135	1	1	yes	characteristic
262	C₂²	#64139	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	72	2	no	—
265	C₄	#64133	1	1	yes	characteristic
266	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
267	C₄	#64139	1	1	yes	characteristic
268	C₄	C₂×#3757	1	1	yes	characteristic
269	C₃	#72013	1	4	yes	complemented
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	#10592911	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#64139
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#131	C₂
socle	C₂×C₃²	#380
prime radical	C₂×C₃²×#41	C₂²
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#131	C₃⋊D₃
hypocenter	C₃²	#1814
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
3. C₂×#63646	C₂	semidirect
4. #10592805	C₂	semidirect
5. #10589425	C₂	semidirect
10. C₃²×#131	C₄	semidirect
11. C₃²×#131	C₄	semidirect
60. C₃×#131	Dic₃	semidirect
61. C₃×#131	Dic₃	semidirect
145. #131	C₃⋊Dic₃	semidirect
146. #131	C₃⋊Dic₃	semidirect
227. C₃²	#1814	semidirect
269. C₃	#72013	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
#10592935	2
C₂×#63646	2
#10592805	2
#10589425	2
#72013	3