Subgroups — #11784387

Order 1152.

521 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592805	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10591059	C₂	1	2	yes	maximal
5	#10589162	C₂	1	2	yes	maximal
6	#70702	—	3	1	no	maximal
7	#69779	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	#64135	C₂²	1	2	yes	—
10	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	#63645	C₄	1	2	yes	complemented
14	#63644	C₄	1	2	yes	complemented
15	C₃×#819	C₄	1	4	yes	complemented
16	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
17	#63555	—	2	2	no	—
18	#63553	—	2	2	no	—
19	C₂×C₃×#108	—	3	1	no	—
20	C₂×C₃×C₄×C₈	—	3	1	no	—
21	#5122	—	3	2	no	—
22	C₂×#819	D₃	1	1	yes	characteristic
23	#5106	—	3	1	no	—
24	#5106	—	3	1	no	—
25	#5106	—	6	1	no	—
26	#5101	—	3	2	no	—
27	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
28	#4555	—	3	2	no	—
29	#4523	—	3	2	no	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	—	2	1	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
36	#3757	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
38	C₄×C₃⋊Dic₃	—	2	2	no	—
39	C₃×C₄×Dic₃	—	2	1	no	—
40	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
41	C₃×#208	—	2	2	no	—
42	C₃×#208	C₂×C₄	1	4	yes	—
43	C₂×C₃×C₃⋊C₈	—	2	2	no	—
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
46	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
48	#1772	—	9	1	no	Sylow
49	C₂×C₃×M₁₆	—	3	2	no	—
50	C₂²×C₃×C₈	—	2	1	no	—
51	C₂²×C₃×C₈	—	3	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	—	3	1	no	—
55	#926	—	3	2	no	—
56	#926	—	3	2	no	—
57	#926	—	6	2	no	—
58	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
59	C₂×#208	C₂×D₃	1	2	yes	—
60	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
61	C₃×#108	—	3	4	no	—
62	C₃×C₄×C₈	—	3	4	no	—
63	#819	Dic₃	1	4	yes	complemented
64	#818	—	3	2	no	—
65	#818	—	3	2	no	—
66	#818	—	6	2	no	—
67	#817	—	3	2	no	—
68	#817	—	3	2	no	—
69	#817	—	6	2	no	—
70	#815	—	6	2	no	—
71	#815	—	6	2	no	—
72	C₈×Dic₃	—	6	2	no	—
73	C₈×Dic₃	—	6	2	no	—
74	C₈×Dic₃	Dic₃	1	4	yes	complemented
75	C₂³×C₃²	#41	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×C₄	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×C₃⋊Dic₃	—	2	4	no	—
83	C₂×C₃×Dic₃	—	2	4	no	—
84	C₂×C₃×Dic₃	—	2	2	no	—
85	C₃²×C₈	C₄²	1	2	yes	—
86	C₃²×C₈	C₄²	1	2	yes	—
87	C₃×C₃⋊C₈	—	2	4	no	—
88	C₃×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
89	C₃×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
90	C₂×#108	—	3	1	no	—
91	#427	—	9	1	no	—
92	#405	—	9	2	no	—
93	C₂×C₄×C₈	—	3	1	no	—
94	#339	—	9	2	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	6	1	no	—
97	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂²×Dic₃	—	2	1	no	—
100	C₃×M₁₆	—	3	4	no	—
101	C₃×M₁₆	—	6	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	—	3	1	no	—
108	C₂×C₃×C₈	—	3	1	no	—
109	C₂×C₃×C₈	—	3	2	no	—
110	C₂×C₃×C₈	—	6	2	no	—
111	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
112	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
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	C₂×Dic₃	1	2	yes	—
118	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
119	C₃×C₄²	—	3	2	no	—
120	C₃×C₄²	—	6	1	no	—
121	#217	—	6	2	no	—
122	#217	—	6	2	no	—
123	#217	—	12	2	no	—
124	C₄⋊Dic₃	—	3	4	no	—
125	C₄⋊Dic₃	—	3	4	no	—
126	C₄⋊Dic₃	—	6	4	no	—
127	C₄×Dic₃	—	2	1	no	—
128	C₄×Dic₃	—	6	2	no	—
129	C₄×Dic₃	—	6	2	no	—
130	C₄×Dic₃	—	12	2	no	—
131	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
132	#208	—	2	2	no	—
133	#208	C₂×Dic₃	1	4	yes	—
134	C₂×C₃⋊C₈	—	2	2	no	—
135	C₂×C₃⋊C₈	Dic₆	1	2	yes	—
136	C₂×C₃⋊C₈	D₁₂	1	1	yes	characteristic
137	C₂×C₃⋊C₈	D₁₂	1	1	yes	characteristic
138	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
139	C₂²×C₃²	—	2	2	no	—
140	C₂²×C₃²	#117	1	1	yes	characteristic
141	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
142	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
143	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
144	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
146	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
147	C₃⋊Dic₃	—	4	4	no	—
148	C₃×Dic₃	—	2	4	no	—
149	C₂×M₁₆	—	3	2	no	—
150	C₂²×C₈	—	3	1	no	—
151	C₂²×C₈	D₃²	1	1	yes	characteristic
152	#117	—	9	2	no	—
153	C₂×C₄²	—	3	1	no	—
154	#108	—	3	4	no	—
155	#107	—	9	2	no	—
156	#106	—	9	2	no	—
157	#97	—	18	2	no	—
158	C₄×C₈	—	3	4	no	—
159	C₄×C₈	—	18	2	no	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	#235	1	1	yes	characteristic
162	C₂³×C₃	#239	1	1	yes	characteristic
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	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	6	4	no	—
170	C₂×C₃×C₄	—	6	2	no	—
171	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
177	C₂×C₃×C₄	#235	1	1	yes	characteristic
178	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	#237	1	1	yes	characteristic
180	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
183	C₂×Dic₃	—	2	4	no	—
184	C₂×Dic₃	—	2	2	no	—
185	C₂×Dic₃	—	6	4	no	—
186	C₂×Dic₃	—	6	4	no	—
187	C₂×Dic₃	—	12	4	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	3	2	no	—
191	C₃×C₈	—	3	2	no	—
192	C₃×C₈	—	6	4	no	—
193	C₃×C₈	C₄×Dic₃	1	2	yes	—
194	C₃×C₈	C₄×Dic₃	1	2	yes	—
195	C₃×C₈	C₄×Dic₃	1	2	yes	—
196	C₃×C₈	C₄×Dic₃	1	2	yes	—
197	C₃⋊C₈	—	2	4	no	—
198	C₃⋊C₈	C₄⋊Dic₃	1	2	yes	—
199	C₃⋊C₈	C₄⋊Dic₃	1	2	yes	—
200	C₂×C₃²	—	2	1	no	—
201	C₂×C₃²	—	2	1	no	—
202	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₃²	#443	1	2	yes	—
204	C₂²×C₄	—	6	1	no	—
205	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
206	M₁₆	—	3	4	no	—
207	M₁₆	—	6	2	no	—
208	C₂×C₈	—	3	2	no	—
209	C₂×C₈	—	3	1	no	—
210	C₂×C₈	—	3	1	no	—
211	C₂×C₈	—	3	2	no	—
212	C₂×C₈	—	6	2	no	—
213	C₂×C₈	D₃×Dic₃	1	2	yes	—
214	C₂×C₈	D₃×Dic₃	1	2	yes	—
215	C₂×C₈	#624	1	1	yes	characteristic
216	C₂×C₈	#624	1	1	yes	characteristic
217	C₄⋊C₄	—	9	4	no	—
218	C₂²⋊C₄	—	18	2	no	—
219	C₄²	—	3	2	no	—
220	C₄²	—	6	1	no	—
221	C₄²	—	18	2	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
229	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
230	C₂²×C₃	#892	1	1	yes	characteristic
231	C₂²×C₃	#893	1	1	yes	characteristic
232	C₂²×C₃	#926	1	1	yes	characteristic
233	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	6	4	no	—
239	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
240	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
241	C₃×C₄	#887	1	1	yes	characteristic
242	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
243	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
244	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
245	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
246	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
247	Dic₃	—	2	4	no	—
248	Dic₃	—	12	4	no	—
249	Dic₃	—	12	4	no	—
250	Dic₃	—	24	4	no	—
251	C₃²	#1772	1	1	yes	characteristic, complemented, Sylow
252	C₂³	#3799	1	1	yes	characteristic
253	C₂×C₄	—	6	4	no	—
254	C₂×C₄	—	6	2	no	—
255	C₂×C₄	—	18	4	no	—
256	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
257	C₂×C₄	#3795	1	1	yes	characteristic
258	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
259	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
260	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#624	1	1	yes	characteristic
262	C₈	—	3	2	no	—
263	C₈	—	3	2	no	—
264	C₈	—	6	4	no	—
265	C₈	Dic₃²	1	2	yes	—
266	C₈	Dic₃²	1	2	yes	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	4	1	no	—
274	C₂×C₃	—	4	1	no	—
275	C₂×C₃	#4700	1	2	yes	—
276	C₂×C₃	#4895	1	2	yes	—
277	C₂×C₃	C₄×C₄⋊Dic₃	1	1	yes	characteristic
278	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
279	C₂²	—	2	2	no	—
280	C₂²	#63841	1	1	yes	characteristic
281	C₂²	Dic₃×Dic₆	1	1	yes	characteristic
282	C₂²	#63894	1	1	yes	characteristic
283	C₄	—	6	4	no	—
284	C₄	—	36	4	no	—
285	C₄	#63844	1	1	yes	characteristic
286	C₄	D₃×C₄⋊Dic₃	1	1	yes	characteristic
287	C₄	Dic₃×D₁₂	1	1	yes	characteristic
288	C₄	C₂×Dic₃²	1	1	yes	characteristic
289	C₃	—	2	1	no	—
290	C₃	#69779	1	1	yes	characteristic, complemented
291	C₃	#70702	1	1	yes	characteristic, complemented
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	#10590917	1	2	yes	—
295	C₂	Dic₃×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₁₂
commutator	C₃²×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1772
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
13. #63645	C₄	semidirect
14. #63644	C₄	semidirect
15. C₃×#819	C₄	semidirect
16. C₃×C₈×Dic₃	C₄	semidirect
63. #819	Dic₃	semidirect
74. C₈×Dic₃	Dic₃	semidirect
251. C₃²	#1772	semidirect
290. C₃	#69779	semidirect
291. C₃	#70702	semidirect

Maximal subgroups

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

subgroup	index
#10592805	2
C₂×C₃×#819	2
C₂×C₃×C₈×Dic₃	2
#10591059	2
#10589162	2
#70702	3
#69779	3