Subgroups — #11797134

Order 1152.

345 conjugacy classes of subgroups.

The subgroup lattice has 265 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592805	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
4	#10589137	C₂	1	4	yes	maximal, complemented
5	#71427	—	3	1	no	maximal
6	#70685	—	3	1	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	#64135	—	2	1	no	—
9	C₂×C₃×#208	—	2	1	no	—
10	#63645	—	2	1	no	—
11	#63644	C₂²	1	2	yes	—
12	C₃×#819	—	2	1	no	—
13	C₃×#819	C₂²	1	2	yes	—
14	C₃×#812	C₄	1	2	yes	complemented
15	C₃×#812	C₂²	1	2	yes	—
16	C₂×C₃×C₃⋊C₁₆	C₄	1	2	yes	complemented
17	C₂×C₃×M₃₂	—	3	1	no	—
18	C₂×C₃×#108	—	3	1	no	—
19	C₂×#819	D₃	1	1	yes	characteristic
20	#5106	—	3	1	no	—
21	#5106	—	3	1	no	—
22	#5106	—	6	1	no	—
23	C₂×#812	D₃	1	1	yes	characteristic
24	#4513	—	3	4	no	—
25	#4494	—	3	4	no	—
26	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
33	#3757	—	4	1	no	—
34	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
35	C₄×C₃⋊Dic₃	—	4	1	no	—
36	C₃×#208	—	2	2	no	—
37	C₃×#208	—	4	1	no	—
38	C₂×C₃×C₃⋊C₈	—	4	1	no	—
39	C₃×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
40	#2155	—	9	1	no	Sylow
41	C₂×C₃×M₁₆	—	6	1	no	—
42	C₂²×C₃×C₈	—	2	1	no	—
43	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
45	#926	—	6	2	no	—
46	#926	—	6	1	no	—
47	#926	—	6	1	no	—
48	C₂×#208	—	2	1	no	—
49	C₃×M₃₂	—	3	2	no	—
50	C₃×M₃₂	—	3	2	no	—
51	C₂×C₃×C₁₆	—	3	2	no	—
52	C₃×#108	—	3	2	no	—
53	C₃×#108	—	6	1	no	—
54	#819	—	2	1	no	—
55	#819	C₂×D₃	1	2	yes	—
56	#818	—	6	2	no	—
57	#818	—	6	1	no	—
58	#818	—	6	1	no	—
59	#817	—	3	2	no	—
60	#817	—	3	2	no	—
61	#817	—	6	2	no	—
62	#812	Dic₃	1	2	yes	complemented
63	#812	C₂×D₃	1	2	yes	—
64	C₂×C₃⋊C₁₆	Dic₃	1	2	yes	complemented
65	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
69	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
70	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃⋊Dic₃	—	4	2	no	—
73	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₃×C₃⋊C₈	—	4	2	no	—
78	C₂×M₃₂	—	3	1	no	—
79	C₂×#108	—	3	1	no	—
80	#427	—	9	1	no	—
81	#365	—	9	4	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
84	C₂²×C₃×C₄	#67	1	1	yes	characteristic
85	C₃×M₁₆	—	6	2	no	—
86	C₃×M₁₆	—	12	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	12	1	no	—
94	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
95	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
96	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
97	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
98	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₃×C₈	#67	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
106	#217	—	12	2	no	—
107	#217	—	12	1	no	—
108	#217	—	12	1	no	—
109	C₄⋊Dic₃	—	6	4	no	—
110	C₄⋊Dic₃	—	6	2	no	—
111	C₄⋊Dic₃	—	6	2	no	—
112	C₄×Dic₃	—	12	2	no	—
113	C₄×Dic₃	—	12	1	no	—
114	C₄×Dic₃	—	12	1	no	—
115	#208	—	2	2	no	—
116	#208	—	4	1	no	—
117	C₂×C₃⋊C₈	—	4	1	no	—
118	C₃×C₁₆	—	3	4	no	—
119	C₃⋊C₁₆	C₂×Dic₃	1	4	yes	—
120	C₂²×C₃²	—	4	1	no	—
121	C₂²×C₃²	#106	1	1	yes	characteristic
122	C₂²×C₃²	#106	1	1	yes	characteristic
123	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
124	C₃²×C₄	#106	1	1	yes	characteristic
125	C₃²×C₄	#106	1	1	yes	characteristic
126	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
128	C₃⋊Dic₃	—	8	2	no	—
129	C₂×M₁₆	—	6	1	no	—
130	C₂²×C₈	D₃²	1	1	yes	characteristic
131	#117	—	18	1	no	—
132	M₃₂	—	3	2	no	—
133	M₃₂	—	3	2	no	—
134	C₂×C₁₆	—	3	2	no	—
135	#108	—	3	2	no	—
136	#108	—	6	1	no	—
137	#107	—	18	1	no	—
138	#106	—	9	2	no	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	#204	1	1	yes	characteristic
141	C₂³×C₃	#215	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	#204	1	1	yes	characteristic
149	C₂×C₃×C₄	#204	1	1	yes	characteristic
150	C₂×C₃×C₄	#204	1	1	yes	characteristic
151	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
155	C₂×C₃×C₄	#214	1	1	yes	characteristic
156	C₂×C₃×C₄	#214	1	1	yes	characteristic
157	C₂×C₃×C₄	#215	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
160	C₂×Dic₃	—	12	4	no	—
161	C₂×Dic₃	—	12	2	no	—
162	C₂×Dic₃	—	12	2	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	2	1	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	12	2	no	—
168	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
169	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
170	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
171	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
172	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
173	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
174	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
175	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
176	C₃⋊C₈	—	4	2	no	—
177	C₂×C₃²	—	2	1	no	—
178	C₂×C₃²	—	2	1	no	—
179	C₂×C₃²	—	2	1	no	—
180	C₂×C₃²	C₂×#106	1	1	yes	characteristic
181	C₂²×C₄	#626	1	1	yes	characteristic
182	M₁₆	—	6	2	no	—
183	M₁₆	—	12	1	no	—
184	C₂×C₈	—	12	1	no	—
185	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
186	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
187	C₂×C₈	#626	1	1	yes	characteristic
188	C₂×C₈	#627	1	1	yes	characteristic
189	C₂×C₈	#627	1	1	yes	characteristic
190	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
191	C₄⋊C₄	—	18	2	no	—
192	C₂²⋊C₄	—	36	1	no	—
193	C₄²	—	36	1	no	—
194	C₁₆	—	3	4	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	1	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	#808	1	1	yes	characteristic
202	C₂²×C₃	#808	1	1	yes	characteristic
203	C₂²×C₃	#817	1	1	yes	characteristic
204	C₂²×C₃	#817	1	1	yes	characteristic
205	C₂²×C₃	C₂×#204	1	1	yes	characteristic
206	C₂²×C₃	C₂×#215	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	#808	1	1	yes	characteristic
212	C₃×C₄	#808	1	1	yes	characteristic
213	C₃×C₄	#817	1	1	yes	characteristic
214	C₃×C₄	#817	1	1	yes	characteristic
215	C₃×C₄	C₂×#204	1	1	yes	characteristic
216	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
217	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
218	C₃×C₄	C₂×#214	1	1	yes	characteristic
219	Dic₃	—	24	4	no	—
220	Dic₃	—	24	2	no	—
221	Dic₃	—	24	2	no	—
222	C₃²	#2155	1	1	yes	characteristic, complemented, Sylow
223	C₂³	#3717	1	1	yes	characteristic
224	C₂×C₄	—	36	2	no	—
225	C₂×C₄	#3716	1	1	yes	characteristic
226	C₂×C₄	#3716	1	1	yes	characteristic
227	C₂×C₄	#3717	1	1	yes	characteristic
228	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
229	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
230	C₂×C₄	C₂×#626	1	1	yes	characteristic
231	C₈	—	12	2	no	—
232	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
233	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
234	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
235	C₈	C₂×#627	1	1	yes	characteristic
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	4	1	no	—
245	C₂×C₃	—	4	1	no	—
246	C₂×C₃	C₂×#808	1	1	yes	characteristic
247	C₂×C₃	C₂×#817	1	1	yes	characteristic
248	C₂²	—	4	1	no	—
249	C₂²	#63573	1	1	yes	characteristic
250	C₂²	#63573	1	1	yes	characteristic
251	C₂²	C₂×#3717	1	1	yes	characteristic
252	C₄	—	72	2	no	—
253	C₄	#63573	1	1	yes	characteristic
254	C₄	#63573	1	1	yes	characteristic
255	C₄	C₂×#3716	1	1	yes	characteristic
256	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
257	C₃	—	2	1	no	—
258	C₃	#70685	1	1	yes	characteristic, complemented
259	C₃	#71427	1	1	yes	characteristic, complemented
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	C₂×#63573	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63573
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂×C₃²	C₂×#106
prime radical	C₂³×C₃²	SD₁₆
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#2155
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. #10589137	C₂	semidirect
14. C₃×#812	C₄	semidirect
16. C₂×C₃×C₃⋊C₁₆	C₄	semidirect
62. #812	Dic₃	semidirect
64. C₂×C₃⋊C₁₆	Dic₃	semidirect
222. C₃²	#2155	semidirect
258. C₃	#70685	semidirect
259. C₃	#71427	semidirect

Maximal subgroups

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

subgroup	index
#10592805	2
C₂×C₃×#819	2
C₂×C₃×#812	2
#10589137	2
#71427	3
#70685	3