Subgroups — #11790535

Order 1152.

421 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63646	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₃×#5101	C₂	1	1	yes	maximal, characteristic
4	#10591141	C₂	1	2	yes	maximal
5	#10589170	C₂	1	2	yes	maximal
6	#70707	—	3	1	no	maximal
7	#69901	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3748	C₂²	1	2	yes	—
10	C₃×#926	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	#63646	—	2	1	no	—
14	#63646	—	2	1	no	—
15	C₃×#819	C₄	1	4	yes	complemented
16	C₃×#815	—	2	1	no	—
17	C₃×C₈×Dic₃	—	2	1	no	—
18	#63577	—	2	4	no	—
19	C₂×C₃×#108	—	3	1	no	—
20	C₃×#405	—	3	1	no	—
21	#5125	—	3	2	no	—
22	#5124	—	3	2	no	—
23	C₂×#819	—	3	1	no	—
24	C₂×#819	—	3	1	no	—
25	C₂×#819	—	6	1	no	—
26	C₂×#819	D₃	1	1	yes	characteristic
27	#5101	D₃	1	1	yes	characteristic
28	#4559	—	3	2	no	—
29	#4533	—	3	2	no	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	Q₈	1	2	yes	—
35	#3748	—	2	2	no	—
36	#3748	—	2	2	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
38	C₃×#217	Q₈	1	2	yes	—
39	C₃×C₄⋊Dic₃	—	2	1	no	—
40	C₃×C₄×Dic₃	—	2	1	no	—
41	C₃×#208	—	2	2	no	—
42	C₃×#208	—	2	1	no	—
43	C₃×#208	C₂×C₄	1	4	yes	—
44	C₃×#208	D₄	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	—	2	2	no	—
46	C₂×C₃×C₃⋊C₈	—	2	1	no	—
47	#1953	—	9	1	no	Sylow
48	C₂×C₃×M₁₆	—	3	1	no	—
49	C₂×C₃×M₁₆	—	3	2	no	—
50	C₂²×C₃×C₈	—	2	1	no	—
51	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₃×#117	—	3	1	no	—
54	#926	C₂×D₃	1	1	yes	characteristic
55	C₂×#208	—	3	2	no	—
56	C₂×#208	—	3	2	no	—
57	C₂×#208	—	6	2	no	—
58	C₂×#208	C₂×D₃	1	1	yes	characteristic
59	C₂×#208	C₂×D₃	1	2	yes	—
60	C₃×#108	—	3	4	no	—
61	C₃×#97	—	6	1	no	—
62	C₃×C₄×C₈	—	6	1	no	—
63	#822	—	6	4	no	—
64	#819	—	6	2	no	—
65	#819	—	6	2	no	—
66	#819	—	6	1	no	—
67	#819	—	6	1	no	—
68	#819	—	6	1	no	—
69	#819	—	6	1	no	—
70	#819	—	6	4	no	—
71	#819	Dic₃	1	4	yes	complemented
72	#815	—	2	1	no	—
73	C₈×Dic₃	—	2	1	no	—
74	C₂³×C₃²	C₂×Q₈	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×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₃²×C₄	#41	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₂×C₃×Dic₃	—	2	2	no	—
82	C₃²×C₈	C₄⋊C₄	1	2	yes	—
83	C₃²×C₈	C₄⋊C₄	1	2	yes	—
84	C₃⋊(C₃⋊C₈)	—	4	2	no	—
85	C₃×C₃⋊C₈	—	2	4	no	—
86	C₃×C₃⋊C₈	—	2	2	no	—
87	#430	—	9	2	no	—
88	C₂×#108	—	3	1	no	—
89	C₂×#108	—	9	1	no	—
90	#405	—	3	1	no	—
91	#344	—	9	2	no	—
92	C₂²×C₃×C₄	—	2	1	no	—
93	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₃×M₁₆	—	3	4	no	—
96	C₃×M₁₆	—	3	2	no	—
97	C₃×M₁₆	—	6	2	no	—
98	C₃×M₁₆	—	6	1	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	2	1	no	—
101	C₂×C₃×C₈	—	2	1	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	6	2	no	—
104	C₂×C₃×C₈	—	6	1	no	—
105	C₂×C₃×C₈	Dic₆	1	2	yes	—
106	C₂×C₃×C₈	Dic₆	1	2	yes	—
107	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
108	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
109	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
111	C₂×C₃×C₈	#67	1	1	yes	characteristic
112	C₂×C₃×C₈	#67	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	6	1	no	—
114	C₃×C₂²⋊C₄	—	3	2	no	—
115	C₃×C₄²	—	6	1	no	—
116	#217	Dic₆	1	2	yes	—
117	C₄⋊Dic₃	—	2	1	no	—
118	C₄×Dic₃	—	2	1	no	—
119	#208	—	2	1	no	—
120	#208	—	2	2	no	—
121	#208	—	6	4	no	—
122	#208	—	6	2	no	—
123	#208	—	6	2	no	—
124	#208	—	6	2	no	—
125	#208	—	6	2	no	—
126	#208	—	12	2	no	—
127	#208	D₁₂	1	2	yes	—
128	#208	C₂×Dic₃	1	4	yes	—
129	C₂×C₃⋊C₈	—	2	2	no	—
130	C₂×C₃⋊C₈	—	2	1	no	—
131	C₂×C₃⋊C₈	—	6	2	no	—
132	C₂×C₃⋊C₈	—	6	2	no	—
133	C₂×C₃⋊C₈	—	12	2	no	—
134	C₂²×C₃²	—	4	1	no	—
135	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
136	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
138	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
140	C₃²×C₄	#121	1	1	yes	characteristic
141	C₃²×C₄	#125	1	1	yes	characteristic
142	C₃×Dic₃	—	4	2	no	—
143	C₂×M₁₆	—	3	2	no	—
144	C₂×M₁₆	—	3	1	no	—
145	C₂×M₁₆	—	9	2	no	—
146	C₂²×C₈	D₃²	1	1	yes	characteristic
147	#117	—	3	1	no	—
148	#108	—	3	4	no	—
149	#108	—	18	4	no	—
150	#108	—	18	1	no	—
151	#108	—	18	1	no	—
152	#97	—	6	1	no	—
153	C₄×C₈	—	6	1	no	—
154	C₂³×C₃	—	2	1	no	—
155	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
156	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	6	2	no	—
164	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
169	C₂×C₃×C₄	#235	1	1	yes	characteristic
170	C₂×C₃×C₄	#235	1	1	yes	characteristic
171	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	#237	1	1	yes	characteristic
173	C₂×C₃×C₄	#239	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
176	C₂×Dic₃	—	2	2	no	—
177	C₃×C₈	—	2	2	no	—
178	C₃×C₈	—	2	2	no	—
179	C₃×C₈	—	6	4	no	—
180	C₃×C₈	—	6	2	no	—
181	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
182	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
183	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
184	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
185	C₃⋊C₈	—	2	2	no	—
186	C₃⋊C₈	—	2	4	no	—
187	C₃⋊C₈	—	12	4	no	—
188	C₃⋊C₈	—	12	2	no	—
189	C₃⋊C₈	—	12	2	no	—
190	C₂×C₃²	—	2	1	no	—
191	C₂×C₃²	—	2	1	no	—
192	C₂×C₃²	—	2	1	no	—
193	C₂×C₃²	#389	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
195	M₁₆	—	3	2	no	—
196	M₁₆	—	3	4	no	—
197	M₁₆	—	6	2	no	—
198	M₁₆	—	6	1	no	—
199	M₁₆	—	18	2	no	—
200	M₁₆	—	18	2	no	—
201	C₂×C₈	—	6	2	no	—
202	C₂×C₈	—	6	1	no	—
203	C₂×C₈	—	18	2	no	—
204	C₂×C₈	D₃×Dic₃	1	2	yes	—
205	C₂×C₈	#626	1	1	yes	characteristic
206	C₂×C₈	#626	1	1	yes	characteristic
207	C₂×C₈	#627	1	2	yes	—
208	C₄⋊C₄	—	6	1	no	—
209	C₂²⋊C₄	—	3	2	no	—
210	C₄²	—	6	1	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	—	4	2	no	—
217	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
218	C₂²×C₃	#869	1	1	yes	characteristic
219	C₂²×C₃	#888	1	1	yes	characteristic
220	C₂²×C₃	#897	1	1	yes	characteristic
221	C₂²×C₃	#924	1	1	yes	characteristic
222	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	12	2	no	—
228	C₃×C₄	#870	1	1	yes	characteristic
229	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
230	C₃×C₄	#890	1	1	yes	characteristic
231	C₃×C₄	#895	1	1	yes	characteristic
232	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
233	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
234	C₃×C₄	#930	1	1	yes	characteristic
235	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
236	Dic₃	—	4	2	no	—
237	C₃²	#1953	1	1	yes	characteristic, complemented, Sylow
238	C₂³	D₃×Dic₆	1	1	yes	characteristic
239	C₂×C₄	—	6	2	no	—
240	C₂×C₄	#3795	1	1	yes	characteristic
241	C₂×C₄	#3799	1	1	yes	characteristic
242	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
243	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#626	1	1	yes	characteristic
245	C₂×C₄	C₂×#627	1	1	yes	characteristic
246	C₈	—	6	4	no	—
247	C₈	—	6	2	no	—
248	C₈	—	36	2	no	—
249	C₈	Dic₃⋊Dic₃	1	2	yes	—
250	C₈	Dic₃⋊Dic₃	1	2	yes	—
251	C₂×C₃	—	2	2	no	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	4	1	no	—
260	C₂×C₃	—	4	1	no	—
261	C₂×C₃	#4928	1	1	yes	characteristic
262	C₂×C₃	#4984	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	Dic₃×Dic₆	1	1	yes	characteristic
265	C₂²	#63857	1	1	yes	characteristic
266	C₂²	#63918	1	1	yes	characteristic
267	C₄	—	12	2	no	—
268	C₄	#63869	1	1	yes	characteristic
269	C₄	Dic₃×D₁₂	1	1	yes	characteristic
270	C₄	#63909	1	1	yes	characteristic
271	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
272	C₃	—	2	1	no	—
273	C₃	#69901	1	1	yes	characteristic, complemented
274	C₃	#70707	1	1	yes	characteristic, complemented
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	#10591352	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63909
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂×C₃²	#389
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1953
LCS	G ▷ C₂×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
15. C₃×#819	C₄	semidirect
71. #819	Dic₃	semidirect
237. C₃²	#1953	semidirect
273. C₃	#69901	semidirect
274. C₃	#70707	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63646	2
C₂×C₃×#819	2
C₃×#5101	2
#10591141	2
#10589170	2
#70707	3
#69901	3