Subgroups — #11786017

Order 1152.

409 conjugacy classes of subgroups.

The subgroup lattice has 310 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×#819	C₂	1	1	yes	maximal, characteristic
3	C₃×#5106	C₂	1	1	yes	maximal, characteristic
4	#10589170	C₂	1	2	yes	maximal
5	#10589158	C₂	1	2	yes	maximal
6	#70698	—	3	1	no	maximal
7	#70657	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	#64135	C₂²	1	1	yes	characteristic
10	C₂×#3748	C₂²	1	1	yes	characteristic
11	C₃×#926	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	2	yes	—
13	#63642	C₄	1	2	yes	complemented
14	C₈×C₃⋊Dic₃	C₄	1	2	yes	complemented
15	C₃×#819	—	2	1	no	—
16	C₃×#819	—	2	1	no	—
17	C₃×#818	—	2	1	no	—
18	C₃×#817	—	2	1	no	—
19	C₂×C₃×#108	—	3	1	no	—
20	C₃×#427	—	3	1	no	—
21	C₂×#819	D₃	1	1	yes	characteristic
22	#5106	D₃	1	1	yes	characteristic
23	#5101	—	3	1	no	—
24	#5101	—	3	1	no	—
25	#5101	—	6	1	no	—
26	#4559	—	3	2	no	—
27	#4557	—	3	2	no	—
28	#4533	—	3	2	no	—
29	#4531	—	3	2	no	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
36	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
37	#3757	—	2	1	no	—
38	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
39	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
40	#3748	—	2	1	no	—
41	#3748	—	2	1	no	—
42	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
43	C₃×#217	—	2	2	no	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄×Dic₃	—	2	2	no	—
46	C₃×#208	—	2	2	no	—
47	C₃×#208	—	2	2	no	—
48	C₂×C₃×C₃⋊C₈	—	2	2	no	—
49	#1833	—	9	1	no	Sylow
50	C₂×C₃×M₁₆	—	3	2	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₃×#117	—	3	2	no	—
55	#926	—	3	1	no	—
56	#926	—	3	1	no	—
57	#926	—	6	1	no	—
58	#926	C₂×D₃	1	2	yes	—
59	C₂×#208	—	3	1	no	—
60	C₂×#208	—	3	1	no	—
61	C₂×#208	—	6	1	no	—
62	C₂×#208	C₂×D₃	1	2	yes	—
63	C₃×#108	—	6	1	no	—
64	C₃×#108	—	6	1	no	—
65	C₃×#107	—	6	1	no	—
66	C₃×#106	—	6	1	no	—
67	#819	—	2	1	no	—
68	#819	—	2	1	no	—
69	#818	—	2	1	no	—
70	#817	—	2	1	no	—
71	#815	—	3	2	no	—
72	#815	—	3	2	no	—
73	#815	—	6	2	no	—
74	C₈×Dic₃	—	3	2	no	—
75	C₈×Dic₃	—	3	2	no	—
76	C₈×Dic₃	—	6	2	no	—
77	C₂³×C₃²	#41	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₃²×C₄	#41	1	1	yes	characteristic
82	C₂×C₃²×C₄	#41	1	1	yes	characteristic
83	C₂×C₃²×C₄	#41	1	1	yes	characteristic
84	C₂×C₃⋊Dic₃	—	2	2	no	—
85	C₂×C₃×Dic₃	—	4	2	no	—
86	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
90	C₃⋊(C₃⋊C₈)	—	2	2	no	—
91	C₃×C₃⋊C₈	—	4	2	no	—
92	C₂×#108	—	3	1	no	—
93	#427	—	3	1	no	—
94	#405	—	9	1	no	—
95	#344	—	9	2	no	—
96	H(3,ℤ/4)	—	9	2	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₃×M₁₆	—	6	2	no	—
101	C₃×M₁₆	—	6	2	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	6	2	no	—
109	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
110	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
111	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
112	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
113	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
114	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
115	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
116	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
117	C₂×C₃×C₈	#67	1	1	yes	characteristic
118	C₂×C₃×C₈	#67	1	1	yes	characteristic
119	C₂×C₃×C₈	#67	1	1	yes	characteristic
120	C₂×C₃×C₈	#67	1	1	yes	characteristic
121	C₃×C₄⋊C₄	—	6	2	no	—
122	C₃×C₂²⋊C₄	—	6	2	no	—
123	C₃×C₄²	—	6	2	no	—
124	#217	—	2	2	no	—
125	#217	—	6	2	no	—
126	#217	—	6	1	no	—
127	#217	—	6	1	no	—
128	C₄⋊Dic₃	—	2	2	no	—
129	C₄⋊Dic₃	—	6	2	no	—
130	C₄⋊Dic₃	—	6	1	no	—
131	C₄⋊Dic₃	—	6	1	no	—
132	C₄×Dic₃	—	2	2	no	—
133	C₄×Dic₃	—	3	2	no	—
134	C₄×Dic₃	—	3	2	no	—
135	C₄×Dic₃	—	6	2	no	—
136	#208	—	2	2	no	—
137	#208	—	2	2	no	—
138	#208	—	6	2	no	—
139	#208	—	6	2	no	—
140	#208	—	6	1	no	—
141	#208	—	6	1	no	—
142	#208	—	6	1	no	—
143	#208	—	6	1	no	—
144	C₂×C₃⋊C₈	—	2	2	no	—
145	C₂×C₃⋊C₈	—	3	2	no	—
146	C₂×C₃⋊C₈	—	3	2	no	—
147	C₂×C₃⋊C₈	—	6	2	no	—
148	C₂²×C₃²	—	4	1	no	—
149	C₂²×C₃²	#117	1	1	yes	characteristic
150	C₂²×C₃²	#124	1	1	yes	characteristic
151	C₂²×C₃²	#125	1	1	yes	characteristic
152	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₃²×C₄	#117	1	1	yes	characteristic
154	C₃²×C₄	#124	1	1	yes	characteristic
155	C₃²×C₄	#125	1	1	yes	characteristic
156	C₃⋊Dic₃	—	4	2	no	—
157	C₃×Dic₃	—	8	2	no	—
158	C₂×M₁₆	—	3	2	no	—
159	C₂×M₁₆	—	9	1	no	—
160	C₂²×C₈	D₃²	1	1	yes	characteristic
161	#117	—	3	2	no	—
162	#117	—	9	1	no	—
163	#108	—	6	1	no	—
164	#108	—	6	1	no	—
165	#107	—	6	1	no	—
166	#106	—	6	1	no	—
167	#97	—	9	2	no	—
168	C₄×C₈	—	9	2	no	—
169	C₂³×C₃	—	2	1	no	—
170	C₂³×C₃	#237	1	1	yes	characteristic
171	C₂³×C₃	#239	1	1	yes	characteristic
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	12	2	no	—
179	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₃×C₄	#237	1	1	yes	characteristic
184	C₂×C₃×C₄	#237	1	1	yes	characteristic
185	C₂×C₃×C₄	#237	1	1	yes	characteristic
186	C₂×C₃×C₄	#239	1	1	yes	characteristic
187	C₂×C₃×C₄	#239	1	1	yes	characteristic
188	C₂×C₃×C₄	#239	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
191	C₂×Dic₃	—	4	2	no	—
192	C₂×Dic₃	—	6	4	no	—
193	C₂×Dic₃	—	6	2	no	—
194	C₂×Dic₃	—	6	2	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	—	2	1	no	—
197	C₃×C₈	—	2	1	no	—
198	C₃×C₈	—	2	1	no	—
199	C₃×C₈	—	12	2	no	—
200	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
201	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
202	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
203	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
204	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
205	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
206	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
207	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
208	C₃⋊C₈	—	4	2	no	—
209	C₃⋊C₈	—	6	4	no	—
210	C₃⋊C₈	—	6	2	no	—
211	C₃⋊C₈	—	6	2	no	—
212	C₂×C₃²	—	2	1	no	—
213	C₂×C₃²	—	2	1	no	—
214	C₂×C₃²	—	2	1	no	—
215	C₂×C₃²	#382	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
217	M₁₆	—	6	2	no	—
218	M₁₆	—	6	2	no	—
219	M₁₆	—	18	1	no	—
220	M₁₆	—	18	1	no	—
221	C₂×C₈	—	6	2	no	—
222	C₂×C₈	—	9	2	no	—
223	C₂×C₈	#624	1	1	yes	characteristic
224	C₂×C₈	#624	1	1	yes	characteristic
225	C₂×C₈	#625	1	1	yes	characteristic
226	C₂×C₈	#625	1	1	yes	characteristic
227	C₂×C₈	#627	1	1	yes	characteristic
228	C₂×C₈	#627	1	1	yes	characteristic
229	C₄⋊C₄	—	6	2	no	—
230	C₄⋊C₄	—	18	1	no	—
231	C₂²⋊C₄	—	6	2	no	—
232	C₂²⋊C₄	—	18	1	no	—
233	C₄²	—	6	2	no	—
234	C₄²	—	9	2	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	4	1	no	—
239	C₂²×C₃	—	4	1	no	—
240	C₂²×C₃	—	4	2	no	—
241	C₂²×C₃	#890	1	1	yes	characteristic
242	C₂²×C₃	#890	1	1	yes	characteristic
243	C₂²×C₃	#892	1	1	yes	characteristic
244	C₂²×C₃	#892	1	1	yes	characteristic
245	C₂²×C₃	#936	1	1	yes	characteristic
246	C₂²×C₃	#947	1	1	yes	characteristic
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	24	2	no	—
252	C₃×C₄	#890	1	1	yes	characteristic
253	C₃×C₄	#890	1	1	yes	characteristic
254	C₃×C₄	#892	1	1	yes	characteristic
255	C₃×C₄	#892	1	1	yes	characteristic
256	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
257	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
258	C₃×C₄	#936	1	1	yes	characteristic
259	C₃×C₄	#947	1	1	yes	characteristic
260	Dic₃	—	8	2	no	—
261	Dic₃	—	12	4	no	—
262	Dic₃	—	12	2	no	—
263	Dic₃	—	12	2	no	—
264	C₃²	#1833	1	1	yes	characteristic, complemented, Sylow
265	C₂³	#3796	1	1	yes	characteristic
266	C₂×C₄	—	12	2	no	—
267	C₂×C₄	—	18	2	no	—
268	C₂×C₄	#3796	1	1	yes	characteristic
269	C₂×C₄	#3796	1	1	yes	characteristic
270	C₂×C₄	#3796	1	1	yes	characteristic
271	C₂×C₄	C₂×#624	1	1	yes	characteristic
272	C₂×C₄	C₂×#625	1	1	yes	characteristic
273	C₂×C₄	C₂×#627	1	1	yes	characteristic
274	C₈	—	12	2	no	—
275	C₈	—	18	2	no	—
276	C₈	#3724	1	1	yes	characteristic
277	C₈	#3724	1	1	yes	characteristic
278	C₈	#3724	1	1	yes	characteristic
279	C₈	#3724	1	1	yes	characteristic
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	4	1	no	—
289	C₂×C₃	—	4	1	no	—
290	C₂×C₃	#4977	1	1	yes	characteristic
291	C₂×C₃	#4977	1	1	yes	characteristic
292	C₂²	—	4	1	no	—
293	C₂²	#63849	1	1	yes	characteristic
294	C₂²	#63863	1	1	yes	characteristic
295	C₂²	#63872	1	1	yes	characteristic
296	C₄	—	24	2	no	—
297	C₄	—	36	2	no	—
298	C₄	#63849	1	1	yes	characteristic
299	C₄	#63863	1	1	yes	characteristic
300	C₄	#63872	1	1	yes	characteristic
301	C₄	C₂×#3724	1	1	yes	characteristic
302	C₃	—	2	1	no	—
303	C₃	#70657	1	1	yes	characteristic, complemented
304	C₃	#70698	1	1	yes	characteristic, complemented
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	#10591230	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63849
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂×C₃²	#382
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1833
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
13. #63642	C₄	semidirect
14. C₈×C₃⋊Dic₃	C₄	semidirect
264. C₃²	#1833	semidirect
303. C₃	#70657	semidirect
304. C₃	#70698	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×#819	2
C₃×#5106	2
#10589170	2
#10589158	2
#70698	3
#70657	3