Subgroups — #11786183

Order 1152.

669 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃⋊(C₃⋊C₈)	C₂	1	4	yes	maximal
5	#70445	—	3	1	no	maximal
6	#69394	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₃	C₄	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
13	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
14	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
15	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
16	Dic₃⋊(C₃⋊C₈)	—	2	8	no	—
17	C₂×C₃×C₄⋊C₈	—	3	1	no	—
18	C₃×C₄³	—	3	1	no	—
19	C₂×Dic₃⋊C₈	—	3	4	no	—
20	C₄²×Dic₃	D₃	1	1	yes	characteristic
21	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
22	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
23	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
24	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
25	C₂×C₄⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
30	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
31	C₃²×C₄²	D₄	1	2	yes	—
32	C₃²×C₄²	Q₈	1	2	yes	—
33	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
35	C₃×C₄×Dic₃	—	2	2	no	—
36	C₃×C₄×Dic₃	—	2	4	no	—
37	C₃×C₄×Dic₃	C₈	1	4	yes	complemented
38	C₃×C₄×Dic₃	D₄	1	4	yes	—
39	C₃×C₄×Dic₃	Q₈	1	4	yes	—
40	C₂×C₃×C₃⋊C₈	—	2	4	no	—
41	C₂×C₃×C₃⋊C₈	—	2	2	no	—
42	#1842	—	9	1	no	Sylow
43	C₂²×C₃×C₈	—	3	2	no	—
44	C₂×C₃×C₄²	—	2	1	no	—
45	C₂×C₃×C₄²	—	3	1	no	—
46	C₂×C₃×C₄²	—	3	2	no	—
47	C₂×C₃×C₄²	—	3	2	no	—
48	C₂×C₃×C₄²	—	3	1	no	—
49	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
52	C₂×C₄×Dic₃	Dic₃	1	2	yes	—
53	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
54	C₂×C₄×Dic₃	C₂×D₃	1	2	yes	—
55	C₂²×C₃⋊C₈	—	3	2	no	—
56	C₂²×C₃⋊C₈	—	3	2	no	—
57	C₂²×C₃⋊C₈	—	6	2	no	—
58	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
59	C₃×C₄⋊C₈	—	6	2	no	—
60	Dic₃⋊C₈	—	6	8	no	—
61	C₄⋊(C₃⋊C₈)	—	2	2	no	—
62	C₄⋊(C₃⋊C₈)	—	6	8	no	—
63	C₄⋊(C₃⋊C₈)	—	6	4	no	—
64	C₄⋊(C₃⋊C₈)	—	6	2	no	—
65	C₄⋊(C₃⋊C₈)	—	6	2	no	—
66	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	—	2	2	no	—
68	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
69	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
70	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
71	C₂×C₃²×C₄	M₁₆	1	2	yes	—
72	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
76	C₂×C₃×Dic₃	—	2	2	no	—
77	C₂×C₃×Dic₃	—	2	2	no	—
78	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
79	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
80	C₂×C₃×Dic₃	C₂×C₈	1	4	yes	—
81	C₂×C₃×Dic₃	M₁₆	1	4	yes	—
82	C₃⋊(C₃⋊C₈)	—	4	2	no	—
83	C₃×C₃⋊C₈	—	4	2	no	—
84	C₂×C₄⋊C₈	—	3	1	no	—
85	C₂×C₄⋊C₈	—	9	4	no	—
86	C₂×C₄⋊C₈	—	9	1	no	—
87	C₄³	—	3	1	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	2	1	no	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	—	3	2	no	—
92	C₂²×C₃×C₄	—	3	2	no	—
93	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
100	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
101	C₂×C₃×C₈	—	6	4	no	—
102	C₂×C₃×C₈	—	6	2	no	—
103	C₃×C₄²	—	2	2	no	—
104	C₃×C₄²	—	2	2	no	—
105	C₃×C₄²	—	3	4	no	—
106	C₃×C₄²	—	3	4	no	—
107	C₃×C₄²	—	3	4	no	—
108	C₃×C₄²	—	6	4	no	—
109	C₃×C₄²	—	6	2	no	—
110	C₃×C₄²	Dic₆	1	2	yes	—
111	C₃×C₄²	Dic₆	1	2	yes	—
112	C₃×C₄²	D₁₂	1	2	yes	—
113	C₃×C₄²	#67	1	2	yes	—
114	C₄×Dic₃	—	2	2	no	—
115	C₄×Dic₃	—	2	4	no	—
116	C₄×Dic₃	C₃⋊C₈	1	4	yes	complemented
117	C₄×Dic₃	Dic₆	1	4	yes	—
118	C₄×Dic₃	D₁₂	1	4	yes	—
119	C₂×C₃⋊C₈	—	2	4	no	—
120	C₂×C₃⋊C₈	—	2	2	no	—
121	C₂×C₃⋊C₈	—	6	4	no	—
122	C₂×C₃⋊C₈	—	6	2	no	—
123	C₂×C₃⋊C₈	—	6	4	no	—
124	C₂×C₃⋊C₈	—	6	4	no	—
125	C₂×C₃⋊C₈	—	6	2	no	—
126	C₂×C₃⋊C₈	—	12	4	no	—
127	C₂²×C₃²	C₂×C₄⋊C₄	1	2	yes	—
128	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
130	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
131	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
132	C₃²×C₄	—	2	2	no	—
133	C₃²×C₄	C₄⋊C₈	1	4	yes	complemented
134	C₃²×C₄	#127	1	1	yes	characteristic
135	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
136	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
137	C₃×Dic₃	—	2	4	no	—
138	C₃×Dic₃	C₄⋊C₈	1	8	yes	complemented
139	C₂²×C₈	—	3	2	no	—
140	C₂²×C₈	—	9	2	no	—
141	C₂×C₄²	—	3	1	no	—
142	C₂×C₄²	—	3	2	no	—
143	C₂×C₄²	—	3	2	no	—
144	C₂×C₄²	—	3	1	no	—
145	C₂×C₄²	D₃²	1	1	yes	characteristic
146	C₄⋊C₈	—	6	2	no	—
147	C₄⋊C₈	—	18	8	no	—
148	C₄⋊C₈	—	18	2	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	3	4	no	—
163	C₂×C₃×C₄	—	3	4	no	—
164	C₂×C₃×C₄	—	3	4	no	—
165	C₂×C₃×C₄	—	3	4	no	—
166	C₂×C₃×C₄	—	4	2	no	—
167	C₂×C₃×C₄	—	6	2	no	—
168	C₂×C₃×C₄	—	6	2	no	—
169	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
170	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
172	C₂×C₃×C₄	#208	1	2	yes	—
173	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
174	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
175	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
176	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
177	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
178	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
181	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
182	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
183	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
184	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
185	C₂×Dic₃	—	2	2	no	—
186	C₂×Dic₃	—	2	2	no	—
187	C₂×Dic₃	C₂×C₃⋊C₈	1	4	yes	—
188	C₂×Dic₃	#208	1	4	yes	—
189	C₂×Dic₃	C₄⋊Dic₃	1	4	yes	—
190	C₂×Dic₃	C₄⋊Dic₃	1	4	yes	—
191	C₃×C₈	—	12	2	no	—
192	C₃⋊C₈	—	4	2	no	—
193	C₃⋊C₈	—	12	4	no	—
194	C₃⋊C₈	—	12	2	no	—
195	C₃⋊C₈	—	12	2	no	—
196	C₂×C₃²	#384	1	1	yes	characteristic
197	C₂×C₃²	C₂×C₄⋊C₈	1	2	yes	—
198	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
199	C₂×C₃²	#423	1	2	yes	—
200	C₂×C₃²	#423	1	1	yes	characteristic
201	C₂²×C₄	—	3	2	no	—
202	C₂²×C₄	—	3	2	no	—
203	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
204	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
206	C₂×C₈	—	6	4	no	—
207	C₂×C₈	—	6	2	no	—
208	C₂×C₈	—	18	4	no	—
209	C₂×C₈	—	18	2	no	—
210	C₄²	—	3	4	no	—
211	C₄²	—	3	4	no	—
212	C₄²	—	3	4	no	—
213	C₄²	—	6	4	no	—
214	C₄²	—	6	2	no	—
215	C₄²	#626	1	2	yes	—
216	C₄²	#627	1	2	yes	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	D₃×C₄⋊C₄	1	2	yes	—
223	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
224	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
225	C₂²×C₃	D₃×M₁₆	1	2	yes	—
226	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
227	C₂²×C₃	C₂×#208	1	2	yes	—
228	C₂²×C₃	C₂×#208	1	1	yes	characteristic
229	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
230	C₂²×C₃	C₂×C₄⋊Dic₃	1	2	yes	—
231	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	4	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	3	8	no	—
239	C₃×C₄	—	4	2	no	—
240	C₃×C₄	—	6	4	no	—
241	C₃×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	complemented
242	C₃×C₄	Dic₃⋊C₈	1	4	yes	complemented
243	C₃×C₄	#869	1	1	yes	characteristic
244	C₃×C₄	#869	1	2	yes	—
245	C₃×C₄	#874	1	1	yes	characteristic
246	C₃×C₄	#888	1	2	yes	—
247	C₃×C₄	#940	1	1	yes	characteristic
248	C₃×C₄	#944	1	1	yes	characteristic
249	Dic₃	—	2	4	no	—
250	Dic₃	C₄⋊(C₃⋊C₈)	1	8	yes	complemented
251	C₃²	#1842	1	1	yes	characteristic, complemented, Sylow
252	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	3	4	no	—
255	C₂×C₄	—	3	4	no	—
256	C₂×C₄	—	3	4	no	—
257	C₂×C₄	—	3	4	no	—
258	C₂×C₄	—	6	2	no	—
259	C₂×C₄	—	6	2	no	—
260	C₂×C₄	D₃×C₃⋊C₈	1	2	yes	—
261	C₂×C₄	#3711	1	2	yes	—
262	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
263	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
264	C₂×C₄	D₃×Dic₆	1	2	yes	—
265	C₂×C₄	D₃×D₁₂	1	2	yes	—
266	C₂×C₄	C₂×#626	1	1	yes	characteristic
267	C₂×C₄	C₂×#627	1	1	yes	characteristic
268	C₈	—	12	2	no	—
269	C₈	—	36	2	no	—
270	C₂×C₃	—	2	2	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	D₃×C₄⋊C₈	1	2	yes	—
276	C₂×C₃	#4837	1	2	yes	—
277	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
278	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
279	C₂×C₃	#4924	1	2	yes	—
280	C₂×C₃	#4924	1	1	yes	characteristic
281	C₂×C₃	#4935	1	1	yes	characteristic
282	C₂×C₃	#4975	1	1	yes	characteristic
283	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
284	C₂×C₃	#5118	1	1	yes	characteristic
285	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
286	C₂²	D₃×#208	1	2	yes	—
287	C₂²	C₂×#3711	1	1	yes	characteristic
288	C₂²	D₃×C₄⋊Dic₃	1	2	yes	—
289	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	3	8	no	—
292	C₄	—	6	4	no	—
293	C₄	Dic₃⋊(C₃⋊C₈)	1	4	yes	complemented
294	C₄	#63866	1	1	yes	characteristic
295	C₄	#63911	1	1	yes	characteristic
296	C₄	#63918	1	2	yes	—
297	C₃	—	2	1	no	—
298	C₃	#69394	1	1	yes	characteristic, complemented
299	C₃	#70445	1	1	yes	characteristic, complemented
300	C₂	D₃×C₄⋊(C₃⋊C₈)	1	2	yes	—
301	C₂	#10590336	1	2	yes	—
302	C₂	C₂×Dic₃⋊(C₃⋊C₈)	1	1	yes	characteristic
303	C₂	#10591115	1	1	yes	characteristic
304	C₂	#10591349	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃²
commutator	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1842
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
37. C₃×C₄×Dic₃	C₈	semidirect
116. C₄×Dic₃	C₃⋊C₈	semidirect
133. C₃²×C₄	C₄⋊C₈	semidirect
138. C₃×Dic₃	C₄⋊C₈	semidirect
241. C₃×C₄	C₄⋊(C₃⋊C₈)	semidirect
242. C₃×C₄	Dic₃⋊C₈	semidirect
250. Dic₃	C₄⋊(C₃⋊C₈)	semidirect
251. C₃²	#1842	semidirect
293. C₄	Dic₃⋊(C₃⋊C₈)	semidirect
298. C₃	#69394	semidirect
299. C₃	#70445	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×C₄²×Dic₃	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₂×Dic₃⋊(C₃⋊C₈)	2
#70445	3
#69394	3