Subgroups — #11790536

Order 1152.

433 conjugacy classes of subgroups.

The subgroup lattice has 286 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	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
4	#10591141	C₂	1	2	yes	maximal
5	#10589171	C₂	1	2	yes	maximal
6	#70707	—	3	1	no	maximal
7	#70706	—	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	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	2	yes	—
13	#63642	—	2	1	no	—
14	C₈×C₃⋊Dic₃	—	2	1	no	—
15	C₃×#819	—	2	1	no	—
16	C₃×#819	—	2	1	no	—
17	C₃×#819	C₄	1	4	yes	complemented
18	#63577	—	2	4	no	—
19	C₂×C₃×#108	—	3	1	no	—
20	C₂×C₃×#108	—	3	1	no	—
21	#5125	—	3	2	no	—
22	#5124	—	3	2	no	—
23	C₂×#819	D₃	1	1	yes	characteristic
24	C₂×#819	D₃	1	1	yes	characteristic
25	#5101	—	3	1	no	—
26	#5101	—	3	1	no	—
27	#5101	—	6	1	no	—
28	#4559	—	3	2	no	—
29	#4556	—	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	#3757	Q₈	1	2	yes	—
36	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
37	C₄×C₃⋊Dic₃	—	2	1	no	—
38	#3748	—	2	1	no	—
39	#3748	D₄	1	2	yes	—
40	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
41	C₃×#208	—	2	2	no	—
42	C₃×#208	—	2	2	no	—
43	C₃×#208	—	2	2	no	—
44	C₃×#208	C₂×C₄	1	4	yes	—
45	C₂×C₃×C₃⋊C₈	—	2	2	no	—
46	C₂×C₃×C₃⋊C₈	—	2	2	no	—
47	#1953	—	9	1	no	Sylow
48	C₂×C₃×M₁₆	—	3	2	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	#926	—	3	1	no	—
54	#926	—	3	1	no	—
55	#926	—	6	1	no	—
56	C₂×#208	—	3	1	no	—
57	C₂×#208	—	3	1	no	—
58	C₂×#208	—	6	1	no	—
59	C₂×#208	C₂×D₃	1	2	yes	—
60	C₂×#208	C₂×D₃	1	2	yes	—
61	C₃×#108	—	3	4	no	—
62	C₃×#108	—	6	1	no	—
63	C₃×#108	—	6	1	no	—
64	#822	—	6	4	no	—
65	#819	—	2	1	no	—
66	#819	—	2	1	no	—
67	#819	—	6	4	no	—
68	#819	Dic₃	1	4	yes	complemented
69	#815	—	6	2	no	—
70	#815	—	6	1	no	—
71	#815	—	6	1	no	—
72	C₈×Dic₃	—	6	2	no	—
73	C₈×Dic₃	—	6	1	no	—
74	C₈×Dic₃	—	6	1	no	—
75	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₂×C₃²×C₄	#41	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	—	2	2	no	—
83	C₃²×C₈	C₄⋊C₄	1	2	yes	—
84	C₃²×C₈	C₄⋊C₄	1	2	yes	—
85	C₃⋊(C₃⋊C₈)	—	2	2	no	—
86	C₃×C₃⋊C₈	—	2	4	no	—
87	C₃×C₃⋊C₈	—	4	2	no	—
88	#430	—	9	2	no	—
89	C₂×#108	—	3	1	no	—
90	C₂×#108	—	3	1	no	—
91	#405	—	9	1	no	—
92	#344	—	9	2	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₃×M₁₆	—	3	4	no	—
97	C₃×M₁₆	—	6	2	no	—
98	C₃×M₁₆	—	6	2	no	—
99	C₃×M₁₆	—	6	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	1	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	6	2	no	—
105	C₂×C₃×C₈	—	6	2	no	—
106	C₂×C₃×C₈	Dic₆	1	2	yes	—
107	C₂×C₃×C₈	Dic₆	1	2	yes	—
108	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
109	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
110	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
112	C₂×C₃×C₈	#67	1	1	yes	characteristic
113	C₂×C₃×C₈	#67	1	1	yes	characteristic
114	#217	—	3	2	no	—
115	#217	—	3	2	no	—
116	#217	—	6	2	no	—
117	C₄⋊Dic₃	—	6	2	no	—
118	C₄⋊Dic₃	—	6	1	no	—
119	C₄⋊Dic₃	—	6	1	no	—
120	C₄×Dic₃	—	6	2	no	—
121	C₄×Dic₃	—	6	1	no	—
122	C₄×Dic₃	—	6	1	no	—
123	#208	—	2	2	no	—
124	#208	—	2	2	no	—
125	#208	—	2	2	no	—
126	#208	—	3	2	no	—
127	#208	—	3	2	no	—
128	#208	—	6	2	no	—
129	#208	—	6	2	no	—
130	#208	—	6	1	no	—
131	#208	—	6	1	no	—
132	#208	C₂×Dic₃	1	4	yes	—
133	C₂×C₃⋊C₈	—	2	2	no	—
134	C₂×C₃⋊C₈	—	2	2	no	—
135	C₂×C₃⋊C₈	—	6	2	no	—
136	C₂×C₃⋊C₈	—	6	1	no	—
137	C₂×C₃⋊C₈	—	6	1	no	—
138	C₂²×C₃²	—	4	1	no	—
139	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
140	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
142	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
144	C₃²×C₄	#121	1	1	yes	characteristic
145	C₃²×C₄	#125	1	1	yes	characteristic
146	C₃⋊Dic₃	—	4	2	no	—
147	C₂×M₁₆	—	3	2	no	—
148	C₂×M₁₆	—	3	2	no	—
149	C₂×M₁₆	—	9	1	no	—
150	C₂²×C₈	D₃²	1	1	yes	characteristic
151	#117	—	9	1	no	—
152	#108	—	3	4	no	—
153	#108	—	6	1	no	—
154	#108	—	6	1	no	—
155	#108	—	18	4	no	—
156	#97	—	18	1	no	—
157	C₄×C₈	—	18	1	no	—
158	C₂³×C₃	—	2	1	no	—
159	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
160	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
171	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	#237	1	1	yes	characteristic
174	C₂×C₃×C₄	#237	1	1	yes	characteristic
175	C₂×C₃×C₄	#239	1	1	yes	characteristic
176	C₂×C₃×C₄	#239	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
179	C₂×Dic₃	—	6	4	no	—
180	C₂×Dic₃	—	6	2	no	—
181	C₂×Dic₃	—	6	2	no	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	6	4	no	—
185	C₃×C₈	—	12	2	no	—
186	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
187	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
188	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
189	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
190	C₃⋊C₈	—	2	4	no	—
191	C₃⋊C₈	—	4	2	no	—
192	C₃⋊C₈	—	6	4	no	—
193	C₃⋊C₈	—	6	2	no	—
194	C₃⋊C₈	—	6	2	no	—
195	C₂×C₃²	—	2	1	no	—
196	C₂×C₃²	—	2	1	no	—
197	C₂×C₃²	—	2	1	no	—
198	C₂×C₃²	#389	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
200	M₁₆	—	3	4	no	—
201	M₁₆	—	6	2	no	—
202	M₁₆	—	6	2	no	—
203	M₁₆	—	6	2	no	—
204	M₁₆	—	9	2	no	—
205	M₁₆	—	18	1	no	—
206	C₂×C₈	—	6	2	no	—
207	C₂×C₈	—	6	2	no	—
208	C₂×C₈	—	18	1	no	—
209	C₂×C₈	D₃×Dic₃	1	2	yes	—
210	C₂×C₈	#626	1	1	yes	characteristic
211	C₂×C₈	#626	1	1	yes	characteristic
212	C₂×C₈	#627	1	2	yes	—
213	C₄⋊C₄	—	18	1	no	—
214	C₂²⋊C₄	—	9	2	no	—
215	C₄²	—	18	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	4	1	no	—
220	C₂²×C₃	—	4	1	no	—
221	C₂²×C₃	—	4	2	no	—
222	C₂²×C₃	#887	1	1	yes	characteristic
223	C₂²×C₃	#888	1	1	yes	characteristic
224	C₂²×C₃	#888	1	1	yes	characteristic
225	C₂²×C₃	#897	1	1	yes	characteristic
226	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
227	C₂²×C₃	#946	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	#890	1	1	yes	characteristic
233	C₃×C₄	#890	1	1	yes	characteristic
234	C₃×C₄	#893	1	1	yes	characteristic
235	C₃×C₄	#895	1	1	yes	characteristic
236	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
237	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
238	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
239	C₃×C₄	#938	1	1	yes	characteristic
240	Dic₃	—	12	4	no	—
241	Dic₃	—	12	2	no	—
242	Dic₃	—	12	2	no	—
243	C₃²	#1953	1	1	yes	characteristic, complemented, Sylow
244	C₂³	#3797	1	1	yes	characteristic
245	C₂×C₄	—	18	2	no	—
246	C₂×C₄	#3796	1	1	yes	characteristic
247	C₂×C₄	#3796	1	1	yes	characteristic
248	C₂×C₄	#3802	1	1	yes	characteristic
249	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
250	C₂×C₄	C₂×#626	1	1	yes	characteristic
251	C₂×C₄	C₂×#627	1	1	yes	characteristic
252	C₈	—	6	4	no	—
253	C₈	—	12	2	no	—
254	C₈	—	18	2	no	—
255	C₈	Dic₃⋊Dic₃	1	2	yes	—
256	C₈	Dic₃⋊Dic₃	1	2	yes	—
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	4	1	no	—
266	C₂×C₃	—	4	1	no	—
267	C₂×C₃	#4972	1	1	yes	characteristic
268	C₂×C₃	#4984	1	1	yes	characteristic
269	C₂²	—	4	1	no	—
270	C₂²	#63843	1	1	yes	characteristic
271	C₂²	#63871	1	1	yes	characteristic
272	C₂²	#63919	1	1	yes	characteristic
273	C₄	—	36	2	no	—
274	C₄	#63872	1	1	yes	characteristic
275	C₄	#63898	1	1	yes	characteristic
276	C₄	#63916	1	1	yes	characteristic
277	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
278	C₃	—	2	1	no	—
279	C₃	#70706	1	1	yes	characteristic, complemented
280	C₃	#70707	1	1	yes	characteristic, complemented
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	#10591357	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63916
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
17. C₃×#819	C₄	semidirect
68. #819	Dic₃	semidirect
243. C₃²	#1953	semidirect
279. C₃	#70706	semidirect
280. C₃	#70707	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
#10591141	2
#10589171	2
#70707	3
#70706	3