Subgroups — #11790209

Order 1152.

553 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63642	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
4	#10591357	C₂	1	2	yes	maximal
5	#10589163	C₂	1	2	yes	maximal
6	#70614	—	3	1	no	maximal
7	#70562	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×Dic₃⋊Dic₃	—	2	2	no	—
14	#63642	—	2	2	no	—
15	C₃×#818	—	2	2	no	—
16	C₃×#817	C₄	1	4	yes	complemented
17	#63579	—	2	2	no	—
18	C₂×C₃×#107	—	3	1	no	—
19	C₂×C₃×#106	—	3	1	no	—
20	C₂×#818	D₃	1	1	yes	characteristic
21	C₂×#817	D₃	1	1	yes	characteristic
22	C₂×#815	—	3	1	no	—
23	C₂×#815	—	3	1	no	—
24	C₂×#815	—	6	1	no	—
25	#4984	—	3	2	no	—
26	#4972	—	3	2	no	—
27	#4529	—	3	2	no	—
28	#4529	—	3	2	no	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	D₄	1	1	yes	characteristic
31	C₂²×C₃⋊Dic₃	D₄	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	—	2	2	no	—
33	C₂²×C₃×Dic₃	—	2	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	2	yes	—
36	C₂×C₃²×C₈	Q₈	1	2	yes	—
37	C₄×C₃⋊Dic₃	—	2	1	no	—
38	C₄×C₃⋊Dic₃	—	2	1	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
41	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
42	C₃×C₄⋊Dic₃	—	2	2	no	—
43	C₃×C₄⋊Dic₃	—	2	2	no	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
46	Dic₃⋊Dic₃	—	4	4	no	—
47	#1944	—	9	1	no	Sylow
48	C₂²×C₃×C₈	—	2	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₃×C₄⋊C₄	—	3	2	no	—
53	C₂×C₄⋊Dic₃	—	6	2	no	—
54	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
55	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
56	C₂×Dic₃⋊C₄	—	6	2	no	—
57	C₂×C₄×Dic₃	—	3	1	no	—
58	C₂×C₄×Dic₃	—	3	1	no	—
59	C₂×C₄×Dic₃	—	6	1	no	—
60	C₂²×C₃⋊C₈	—	3	1	no	—
61	C₂²×C₃⋊C₈	—	3	1	no	—
62	C₂²×C₃⋊C₈	—	6	1	no	—
63	C₃×#107	—	6	2	no	—
64	C₃×#106	—	3	4	no	—
65	C₃⋊#95	—	6	2	no	—
66	C₃⋊#95	—	6	2	no	—
67	#818	—	2	2	no	—
68	#817	Dic₃	1	4	yes	complemented
69	#815	—	6	4	no	—
70	#815	—	6	2	no	—
71	#815	—	6	2	no	—
72	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃²×C₄	#41	1	1	yes	characteristic
78	C₂×C₃²×C₄	#41	1	1	yes	characteristic
79	C₂×C₃⋊Dic₃	—	2	1	no	—
80	C₂×C₃⋊Dic₃	—	2	1	no	—
81	C₂×C₃⋊Dic₃	—	2	2	no	—
82	C₂×C₃⋊Dic₃	—	2	2	no	—
83	C₂×C₃×Dic₃	—	2	4	no	—
84	C₂×C₃×Dic₃	—	4	2	no	—
85	C₂×C₃×Dic₃	—	4	2	no	—
86	C₂×C₃×Dic₃	—	4	2	no	—
87	C₂×C₃×Dic₃	—	4	2	no	—
88	C₂×C₃×Dic₃	—	4	2	no	—
89	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
90	C₃⋊(C₃⋊C₈)	—	2	2	no	—
91	C₂×#107	—	3	1	no	—
92	C₂×#106	—	3	1	no	—
93	C₂×#97	—	9	1	no	—
94	#389	—	9	2	no	—
95	#340	—	9	2	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	—	6	2	no	—
98	C₂²×C₃×C₄	—	6	2	no	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	2	2	no	—
102	C₂²×Dic₃	—	2	2	no	—
103	C₂²×Dic₃	—	3	1	no	—
104	C₂²×Dic₃	—	3	1	no	—
105	C₂²×Dic₃	—	3	1	no	—
106	C₂²×Dic₃	—	3	1	no	—
107	C₂²×Dic₃	—	6	1	no	—
108	C₂²×Dic₃	—	6	1	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	Dic₆	1	2	yes	—
113	C₂×C₃×C₈	Dic₆	1	2	yes	—
114	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
115	C₂×C₃×C₈	D₁₂	1	2	yes	—
116	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
117	C₂×C₃×C₈	#67	1	2	yes	—
118	C₃×C₄⋊C₄	—	3	4	no	—
119	C₃×C₄⋊C₄	—	6	2	no	—
120	C₃×C₄⋊C₄	—	6	2	no	—
121	C₃×C₄⋊C₄	—	6	2	no	—
122	C₄⋊Dic₃	—	2	2	no	—
123	C₄⋊Dic₃	—	2	2	no	—
124	C₄⋊Dic₃	—	2	2	no	—
125	C₄⋊Dic₃	—	12	4	no	—
126	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
127	Dic₃⋊C₄	—	12	4	no	—
128	C₄×Dic₃	—	6	2	no	—
129	C₄×Dic₃	—	6	2	no	—
130	C₄×Dic₃	—	6	1	no	—
131	C₄×Dic₃	—	6	1	no	—
132	C₄×Dic₃	—	6	1	no	—
133	C₄×Dic₃	—	6	1	no	—
134	C₂×C₃⋊C₈	—	3	2	no	—
135	C₂×C₃⋊C₈	—	3	2	no	—
136	C₂×C₃⋊C₈	—	6	2	no	—
137	C₂×C₃⋊C₈	—	6	2	no	—
138	C₂×C₃⋊C₈	—	6	2	no	—
139	C₂×C₃⋊C₈	—	6	1	no	—
140	C₂×C₃⋊C₈	—	6	1	no	—
141	C₂×C₃⋊C₈	—	6	1	no	—
142	C₂×C₃⋊C₈	—	6	1	no	—
143	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
144	C₂²×C₃²	#121	1	1	yes	characteristic
145	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₃²	AGL(1,ℤ/8)	1	2	yes	—
147	C₂²×C₃²	#137	1	2	yes	—
148	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
150	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
151	C₃²×C₄	#125	1	1	yes	characteristic
152	C₃⋊Dic₃	—	4	1	no	—
153	C₃⋊Dic₃	—	4	1	no	—
154	C₃×Dic₃	—	4	4	no	—
155	C₃×Dic₃	—	8	2	no	—
156	C₂²×C₈	—	9	1	no	—
157	C₂²×C₈	D₃²	1	1	yes	characteristic
158	C₂×C₄⋊C₄	—	3	2	no	—
159	C₂×C₄⋊C₄	—	3	2	no	—
160	C₂×C₄⋊C₄	—	18	2	no	—
161	C₂×C₄²	—	9	1	no	—
162	#107	—	6	2	no	—
163	#106	—	3	4	no	—
164	#97	—	18	2	no	—
165	#95	—	18	2	no	—
166	C₂³×C₃	—	2	1	no	—
167	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
168	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
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	—	6	4	no	—
176	C₂×C₃×C₄	—	12	2	no	—
177	C₂×C₃×C₄	—	12	2	no	—
178	C₂×C₃×C₄	—	12	2	no	—
179	C₂×C₃×C₄	—	12	2	no	—
180	C₂×C₃×C₄	—	12	2	no	—
181	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
185	C₂×C₃×C₄	#237	1	1	yes	characteristic
186	C₂×C₃×C₄	#237	1	1	yes	characteristic
187	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
188	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
189	C₂×C₃×C₄	#239	1	1	yes	characteristic
190	C₂×C₃×C₄	#239	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
193	C₂×Dic₃	—	2	4	no	—
194	C₂×Dic₃	—	4	2	no	—
195	C₂×Dic₃	—	4	2	no	—
196	C₂×Dic₃	—	4	2	no	—
197	C₂×Dic₃	—	4	2	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	—	6	2	no	—
200	C₂×Dic₃	—	6	2	no	—
201	C₂×Dic₃	—	6	1	no	—
202	C₂×Dic₃	—	6	1	no	—
203	C₂×Dic₃	—	6	2	no	—
204	C₂×Dic₃	—	6	2	no	—
205	C₂×Dic₃	—	6	2	no	—
206	C₂×Dic₃	—	6	2	no	—
207	C₂×Dic₃	—	6	1	no	—
208	C₂×Dic₃	—	6	1	no	—
209	C₂×Dic₃	—	12	2	no	—
210	C₂×Dic₃	—	12	2	no	—
211	C₃×C₈	—	2	4	no	—
212	C₃×C₈	Dic₃⋊C₄	1	4	yes	complemented
213	C₃×C₈	C₄⋊Dic₃	1	4	yes	complemented
214	C₃⋊C₈	—	6	4	no	—
215	C₃⋊C₈	—	6	2	no	—
216	C₃⋊C₈	—	6	2	no	—
217	C₂×C₃²	#389	1	1	yes	characteristic
218	C₂×C₃²	#440	1	2	yes	—
219	C₂×C₃²	#469	1	1	yes	characteristic
220	C₂×C₃²	#470	1	1	yes	characteristic
221	C₂×C₃²	#501	1	2	yes	—
222	C₂²×C₄	—	6	2	no	—
223	C₂²×C₄	—	6	2	no	—
224	C₂²×C₄	—	9	1	no	—
225	C₂²×C₄	—	9	1	no	—
226	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
227	C₂×C₈	—	9	2	no	—
228	C₂×C₈	—	18	1	no	—
229	C₂×C₈	—	18	1	no	—
230	C₂×C₈	D₃×Dic₃	1	2	yes	—
231	C₂×C₈	#626	1	2	yes	—
232	C₂×C₈	#627	1	2	yes	—
233	C₄⋊C₄	—	3	4	no	—
234	C₄⋊C₄	—	6	2	no	—
235	C₄⋊C₄	—	6	2	no	—
236	C₄⋊C₄	—	6	2	no	—
237	C₄⋊C₄	—	36	4	no	—
238	C₄²	—	18	1	no	—
239	C₄²	—	18	1	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	#888	1	1	yes	characteristic
246	C₂²×C₃	#888	1	1	yes	characteristic
247	C₂²×C₃	#893	1	1	yes	characteristic
248	C₂²×C₃	#895	1	1	yes	characteristic
249	C₂²×C₃	#911	1	2	yes	—
250	C₂²×C₃	#914	1	2	yes	—
251	C₂²×C₃	#915	1	2	yes	—
252	C₂²×C₃	#918	1	2	yes	—
253	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
254	C₂²×C₃	#938	1	1	yes	characteristic
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	12	4	no	—
260	C₃×C₄	—	24	2	no	—
261	C₃×C₄	#887	1	1	yes	characteristic
262	C₃×C₄	#890	1	1	yes	characteristic
263	C₃×C₄	#890	1	1	yes	characteristic
264	C₃×C₄	#897	1	1	yes	characteristic
265	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
266	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
267	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
268	C₃×C₄	#946	1	1	yes	characteristic
269	Dic₃	—	4	4	no	—
270	Dic₃	—	8	2	no	—
271	Dic₃	—	12	2	no	—
272	Dic₃	—	12	2	no	—
273	Dic₃	—	12	1	no	—
274	Dic₃	—	12	1	no	—
275	Dic₃	—	12	1	no	—
276	Dic₃	—	12	1	no	—
277	C₃²	#1944	1	1	yes	characteristic, complemented, Sylow
278	C₂³	#3802	1	1	yes	characteristic
279	C₂×C₄	—	6	4	no	—
280	C₂×C₄	—	12	2	no	—
281	C₂×C₄	—	12	2	no	—
282	C₂×C₄	—	12	2	no	—
283	C₂×C₄	—	12	2	no	—
284	C₂×C₄	—	12	2	no	—
285	C₂×C₄	—	18	2	no	—
286	C₂×C₄	—	18	2	no	—
287	C₂×C₄	—	18	1	no	—
288	C₂×C₄	—	18	1	no	—
289	C₂×C₄	#3796	1	1	yes	characteristic
290	C₂×C₄	#3796	1	1	yes	characteristic
291	C₂×C₄	#3797	1	1	yes	characteristic
292	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
293	C₂×C₄	C₂×#626	1	1	yes	characteristic
294	C₂×C₄	C₂×#627	1	1	yes	characteristic
295	C₈	—	18	2	no	—
296	C₈	Dic₃⋊Dic₃	1	4	yes	complemented
297	C₂×C₃	—	2	2	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	2	no	—
302	C₂×C₃	#4856	1	2	yes	—
303	C₂×C₃	#4865	1	1	yes	characteristic
304	C₂×C₃	#4867	1	1	yes	characteristic
305	C₂×C₃	#4876	1	2	yes	—
306	C₂×C₃	#4889	1	2	yes	—
307	C₂×C₃	#4972	1	1	yes	characteristic
308	C₂×C₃	#4984	1	1	yes	characteristic
309	C₂×C₃	#5159	1	1	yes	characteristic
310	C₂×C₃	#5164	1	2	yes	—
311	C₂×C₃	#5189	1	1	yes	characteristic
312	C₂²	#63798	1	2	yes	—
313	C₂²	#63811	1	2	yes	—
314	C₂²	#63898	1	1	yes	characteristic
315	C₂²	#63916	1	1	yes	characteristic
316	C₂²	#63919	1	1	yes	characteristic
317	C₄	—	12	4	no	—
318	C₄	—	24	2	no	—
319	C₄	—	36	1	no	—
320	C₄	—	36	1	no	—
321	C₄	#63843	1	1	yes	characteristic
322	C₄	#63871	1	1	yes	characteristic
323	C₄	#63872	1	1	yes	characteristic
324	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
325	C₃	—	2	1	no	—
326	C₃	#70562	1	1	yes	characteristic, complemented
327	C₃	#70614	1	1	yes	characteristic, complemented
328	C₂	#10590779	1	2	yes	—
329	C₂	#10590846	1	1	yes	characteristic
330	C₂	#10590858	1	1	yes	characteristic
331	C₂	#10590903	1	2	yes	—
332	C₂	#10591357	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3802
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1944
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
16. C₃×#817	C₄	semidirect
68. #817	Dic₃	semidirect
89. C₃²×C₈	C₄⋊C₄	semidirect
212. C₃×C₈	Dic₃⋊C₄	semidirect
213. C₃×C₈	C₄⋊Dic₃	semidirect
277. C₃²	#1944	semidirect
296. C₈	Dic₃⋊Dic₃	semidirect
326. C₃	#70562	semidirect
327. C₃	#70614	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63642	2
C₂×C₃×#818	2
C₂×C₃×#817	2
#10591357	2
#10589163	2
#70614	3
#70562	3