Subgroups — #11791633

Order 1152.

681 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589164	C₂	1	4	yes	maximal
5	#71780	—	3	1	no	maximal
6	#70460	—	3	1	no	maximal
7	C₂²×C₃×Dic₆	—	2	1	no	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×C₄×C₃⋊Dic₃	—	2	1	no	—
11	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
14	C₃×#869	—	2	2	no	—
15	C₃×#869	C₄	1	4	yes	complemented
16	#63658	—	2	1	no	—
17	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
18	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
19	C₂×#869	D₃	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₈	—	3	1	no	—
21	C₄×C₄⋊Dic₃	—	3	1	no	—
22	C₄×C₄⋊Dic₃	—	3	1	no	—
23	C₄×C₄⋊Dic₃	—	6	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
25	#4551	—	3	4	no	—
26	#4529	—	3	4	no	—
27	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	—	2	2	no	—
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂×C₃×Dic₆	—	2	4	no	—
33	C₂×C₃×Dic₆	—	4	2	no	—
34	C₂×C₃×Dic₆	—	4	2	no	—
35	C₃²×C₄²	C₂×C₄	1	2	yes	—
36	C₃²×C₄²	D₄	1	2	yes	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₄⋊(C₃⋊Dic₃)	Q₈	1	4	yes	complemented
39	C₄×C₃⋊Dic₃	—	4	2	no	—
40	C₃×C₄⋊Dic₃	—	2	4	no	—
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
43	C₂×C₃×C₃⋊C₈	—	2	2	no	—
44	C₂×C₃×C₃⋊C₈	—	2	4	no	—
45	#1988	—	9	1	no	Sylow
46	C₂²×C₃×Q₈	—	6	1	no	—
47	C₂²×Dic₆	—	2	1	no	—
48	C₂²×C₃×C₈	—	3	2	no	—
49	C₃×C₄⋊Q₈	—	3	4	no	—
50	C₃×C₄⋊Q₈	—	6	2	no	—
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₃×C₄⋊C₄	—	6	1	no	—
53	C₂×C₃×C₄²	—	2	1	no	—
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×C₄⋊Dic₃	—	2	1	no	—
57	C₂×C₄⋊Dic₃	—	3	2	no	—
58	C₂×C₄⋊Dic₃	—	3	2	no	—
59	C₂×C₄⋊Dic₃	—	6	2	no	—
60	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
61	C₂×C₄×Dic₃	—	6	2	no	—
62	C₂×C₄×Dic₃	—	6	1	no	—
63	C₂×C₄×Dic₃	—	6	1	no	—
64	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
65	#869	—	2	2	no	—
66	#869	Dic₃	1	4	yes	complemented
67	C₃×C₄⋊C₈	—	6	2	no	—
68	C₃⋊#95	—	6	2	no	—
69	C₃⋊#95	—	6	1	no	—
70	C₃⋊#95	—	6	1	no	—
71	C₄⋊(C₃⋊C₈)	—	2	2	no	—
72	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	—	2	2	no	—
74	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
75	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
76	C₂×C₃²×C₄	D₈	1	2	yes	—
77	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
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	2	yes	—
81	C₂×C₃²×C₄	#41	1	2	yes	—
82	C₂×C₃⋊Dic₃	—	2	4	no	—
83	C₂×C₃⋊Dic₃	—	4	2	no	—
84	C₂×C₃⋊Dic₃	—	4	2	no	—
85	C₂×C₃×Dic₃	—	2	4	no	—
86	C₂×C₃×Dic₃	—	4	2	no	—
87	C₂×C₃×Dic₃	—	4	2	no	—
88	C₃×Dic₆	—	4	8	no	—
89	C₃×C₃⋊C₈	—	4	2	no	—
90	C₂×C₄⋊Q₈	—	3	1	no	—
91	C₂×C₄⋊C₈	—	3	1	no	—
92	C₄×C₄⋊C₄	—	9	1	no	—
93	#340	—	9	4	no	—
94	C₂×C₃×Q₈	—	6	4	no	—
95	C₂×C₃×Q₈	—	12	2	no	—
96	C₂×C₃×Q₈	—	12	2	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	—	2	1	no	—
100	C₂²×C₃×C₄	—	6	2	no	—
101	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
102	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
103	C₂²×C₃×C₄	#67	1	1	yes	characteristic
104	C₂²×C₃×C₄	#67	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂²×Dic₃	—	2	2	no	—
108	C₂²×Dic₃	—	6	4	no	—
109	C₂²×Dic₃	—	6	2	no	—
110	C₂²×Dic₃	—	6	2	no	—
111	C₂×Dic₆	—	2	4	no	—
112	C₂×Dic₆	—	4	2	no	—
113	C₂×Dic₆	—	4	2	no	—
114	C₂×C₃×C₈	—	6	4	no	—
115	C₂×C₃×C₈	—	6	2	no	—
116	C₃×C₄⋊C₄	—	3	4	no	—
117	C₃×C₄⋊C₄	—	6	4	no	—
118	C₃×C₄⋊C₄	—	6	2	no	—
119	C₃×C₄²	—	2	2	no	—
120	C₃×C₄²	—	2	2	no	—
121	C₃×C₄²	C₄×D₃	1	2	yes	—
122	C₃×C₄²	C₂×Dic₃	1	2	yes	—
123	C₃×C₄²	#67	1	2	yes	—
124	C₃×C₄²	#67	1	2	yes	—
125	C₄⋊Dic₃	—	2	4	no	—
126	C₄⋊Dic₃	—	2	2	no	—
127	C₄⋊Dic₃	—	3	4	no	—
128	C₄⋊Dic₃	—	3	4	no	—
129	C₄⋊Dic₃	—	6	4	no	—
130	C₄⋊Dic₃	—	6	4	no	—
131	C₄⋊Dic₃	—	6	2	no	—
132	C₄⋊Dic₃	—	6	2	no	—
133	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
134	C₄×Dic₃	—	12	4	no	—
135	C₄×Dic₃	—	12	2	no	—
136	C₄×Dic₃	—	12	2	no	—
137	C₂×C₃⋊C₈	—	2	4	no	—
138	C₂×C₃⋊C₈	—	2	2	no	—
139	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
141	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
142	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
143	C₂²×C₃²	#137	1	2	yes	—
144	C₃²×C₄	—	2	2	no	—
145	C₃²×C₄	#102	1	4	yes	—
146	C₃²×C₄	C₄×Q₈	1	2	yes	—
147	C₃²×C₄	#124	1	1	yes	characteristic
148	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
149	C₃⋊Dic₃	—	4	4	no	—
150	C₃×Dic₃	—	4	4	no	—
151	C₂²×Q₈	—	6	1	no	—
152	C₂²×C₈	—	3	2	no	—
153	C₄⋊Q₈	—	3	4	no	—
154	C₄⋊Q₈	—	6	2	no	—
155	C₂×C₄⋊C₄	—	3	2	no	—
156	C₂×C₄⋊C₄	—	6	1	no	—
157	C₂×C₄⋊C₄	—	9	2	no	—
158	C₂×C₄²	—	18	1	no	—
159	C₂×C₄²	D₃²	1	1	yes	characteristic
160	C₄⋊C₈	—	6	2	no	—
161	#95	—	18	1	no	—
162	C₂³×C₃	—	2	1	no	—
163	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
164	C₂³×C₃	C₂×#67	1	1	yes	characteristic
165	C₃×Q₈	—	12	8	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	6	4	no	—
178	C₂×C₃×C₄	—	12	2	no	—
179	C₂×C₃×C₄	—	12	2	no	—
180	C₂×C₃×C₄	#204	1	2	yes	—
181	C₂×C₃×C₄	D₂₄	1	2	yes	—
182	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
183	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
184	C₂×C₃×C₄	#213	1	2	yes	—
185	C₂×C₃×C₄	#214	1	2	yes	—
186	C₂×C₃×C₄	#217	1	2	yes	—
187	C₂×C₃×C₄	#217	1	2	yes	—
188	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
189	C₂×C₃×C₄	#237	1	2	yes	—
190	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
191	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
192	C₂×C₃×C₄	#239	1	2	yes	—
193	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
196	C₂×Dic₃	—	2	4	no	—
197	C₂×Dic₃	—	4	2	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	—	6	8	no	—
200	C₂×Dic₃	—	6	4	no	—
201	C₂×Dic₃	—	6	4	no	—
202	C₂×Dic₃	—	12	4	no	—
203	C₂×Dic₃	—	12	4	no	—
204	C₂×Dic₃	—	12	2	no	—
205	C₂×Dic₃	—	12	2	no	—
206	C₂×Dic₃	—	12	2	no	—
207	C₂×Dic₃	—	12	2	no	—
208	Dic₆	—	4	8	no	—
209	C₃×C₈	—	12	2	no	—
210	C₃⋊C₈	—	4	2	no	—
211	C₂×C₃²	#391	1	1	yes	characteristic
212	C₂×C₃²	C₂×#102	1	1	yes	characteristic
213	C₂×C₃²	#419	1	1	yes	characteristic
214	C₂×C₃²	#474	1	2	yes	—
215	C₂×C₃²	#475	1	2	yes	—
216	C₂×Q₈	—	6	4	no	—
217	C₂×Q₈	—	12	2	no	—
218	C₂×Q₈	—	12	2	no	—
219	C₂²×C₄	—	6	2	no	—
220	C₂²×C₄	—	18	2	no	—
221	C₂²×C₄	#626	1	1	yes	characteristic
222	C₂²×C₄	#626	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
224	C₂×C₈	—	6	4	no	—
225	C₂×C₈	—	6	2	no	—
226	C₄⋊C₄	—	3	4	no	—
227	C₄⋊C₄	—	6	4	no	—
228	C₄⋊C₄	—	6	2	no	—
229	C₄⋊C₄	—	9	4	no	—
230	C₄⋊C₄	—	18	2	no	—
231	C₄²	—	36	2	no	—
232	C₄²	D₃×Dic₃	1	2	yes	—
233	C₄²	#625	1	2	yes	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	#897	1	2	yes	—
240	C₂²×C₃	C₂×#204	1	1	yes	characteristic
241	C₂²×C₃	C₂×D₂₄	1	1	yes	characteristic
242	C₂²×C₃	#909	1	2	yes	—
243	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
244	C₂²×C₃	C₂×#213	1	1	yes	characteristic
245	C₂²×C₃	C₂×#214	1	1	yes	characteristic
246	C₂²×C₃	#942	1	2	yes	—
247	C₂²×C₃	#946	1	2	yes	—
248	C₂²×C₃	C₂×#217	1	1	yes	characteristic
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	4	2	no	—
256	C₃×C₄	—	12	4	no	—
257	C₃×C₄	#821	1	4	yes	—
258	C₃×C₄	#832	1	4	yes	—
259	C₃×C₄	#887	1	2	yes	—
260	C₃×C₄	#936	1	1	yes	characteristic
261	C₃×C₄	#944	1	1	yes	characteristic
262	C₃×C₄	#944	1	1	yes	characteristic
263	C₃×C₄	Q₈×Dic₃	1	2	yes	—
264	C₃×C₄	#947	1	1	yes	characteristic
265	Dic₃	—	4	4	no	—
266	Dic₃	—	12	8	no	—
267	Dic₃	—	12	4	no	—
268	Dic₃	—	12	4	no	—
269	C₃²	#1988	1	1	yes	characteristic, complemented, Sylow
270	C₂³	C₂×#626	1	1	yes	characteristic
271	Q₈	—	12	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	6	4	no	—
274	C₂×C₄	—	12	2	no	—
275	C₂×C₄	—	12	2	no	—
276	C₂×C₄	—	18	4	no	—
277	C₂×C₄	—	36	2	no	—
278	C₂×C₄	—	36	2	no	—
279	C₂×C₄	#3714	1	2	yes	—
280	C₂×C₄	#3716	1	2	yes	—
281	C₂×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
282	C₂×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
283	C₂×C₄	#3796	1	2	yes	—
284	C₂×C₄	#3797	1	2	yes	—
285	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#625	1	1	yes	characteristic
287	C₈	—	12	2	no	—
288	C₂×C₃	—	2	2	no	—
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	#4846	1	2	yes	—
294	C₂×C₃	#4851	1	2	yes	—
295	C₂×C₃	#4973	1	1	yes	characteristic
296	C₂×C₃	#5087	1	2	yes	—
297	C₂×C₃	#5094	1	2	yes	—
298	C₂×C₃	C₂×#821	1	1	yes	characteristic
299	C₂×C₃	#5120	1	1	yes	characteristic
300	C₂×C₃	C₂×#832	1	1	yes	characteristic
301	C₂×C₃	#5225	1	1	yes	characteristic
302	C₂×C₃	#5229	1	1	yes	characteristic
303	C₂²	C₂×#3714	1	1	yes	characteristic
304	C₂²	C₂×#3716	1	1	yes	characteristic
305	C₂²	#63833	1	2	yes	—
306	C₂²	#63871	1	2	yes	—
307	C₂²	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	12	4	no	—
310	C₄	—	36	4	no	—
311	C₄	#63563	1	4	yes	—
312	C₄	#63843	1	2	yes	—
313	C₄	#63863	1	1	yes	characteristic
314	C₄	#63873	1	1	yes	characteristic
315	C₃	—	2	1	no	—
316	C₃	#70460	1	1	yes	characteristic, complemented
317	C₃	#71780	1	1	yes	characteristic, complemented
318	C₂	#10590307	1	2	yes	—
319	C₂	#10590313	1	2	yes	—
320	C₂	C₂×#63563	1	1	yes	characteristic
321	C₂	#10591100	1	1	yes	characteristic
322	C₂	#10591221	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#626
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1988
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
15. C₃×#869	C₄	semidirect
38. C₄⋊(C₃⋊Dic₃)	Q₈	semidirect
66. #869	Dic₃	semidirect
269. C₃²	#1988	semidirect
316. C₃	#70460	semidirect
317. C₃	#71780	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#869	2
C₄×C₄⋊(C₃⋊Dic₃)	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
#10589164	2
#71780	3
#70460	3