Subgroups — #13179751

Order 1728.

3276 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11723988	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11723971	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
6	#11723939	C₂	1	2	yes	maximal, complemented
7	#11723934	C₂	1	1	yes	maximal, characteristic, complemented
8	#11723933	C₂	1	2	yes	maximal, complemented
9	#11723931	C₂	1	1	yes	maximal, characteristic, complemented
10	#11723927	C₂	1	2	yes	maximal, complemented
11	#11723922	C₂	1	2	yes	maximal, complemented
12	#10596095	—	3	13	no	maximal
13	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
15	C₂×#6312	C₂²	1	2	yes	—
16	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
17	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
20	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
23	#88859	C₂²	1	2	yes	—
24	#88859	C₂²	1	2	yes	—
25	#88859	C₂²	1	2	yes	—
26	#88859	C₂²	1	2	yes	—
27	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
28	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
29	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
30	#88857	C₂²	1	2	yes	—
31	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
32	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
33	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
34	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
36	C₃²×C₂²⋊Q₈	D₃	1	13	yes	complemented
37	#64156	—	3	13	no	—
38	Q₈×C₃⋊Dic₃	—	3	13	no	—
39	#64139	—	3	13	no	—
40	C₂×C₄×C₃⋊Dic₃	—	3	13	no	—
41	#64107	—	3	26	no	—
42	#64102	—	3	13	no	—
43	#64101	—	3	26	no	—
44	#64099	—	3	13	no	—
45	#64095	—	3	26	no	—
46	#64090	—	3	26	no	—
47	C₂³×C₃³	C₂³	1	1	yes	characteristic
48	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	2	yes	—
49	C₃³×Q₈	—	2	2	no	—
50	C₂×C₃³×C₄	—	2	2	no	—
51	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
52	C₂×C₃³×C₄	C₂³	1	2	yes	—
53	C₂×C₃³×C₄	C₂³	1	2	yes	—
54	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
55	#6312	—	2	4	no	—
56	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
57	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
58	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
59	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
60	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
61	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
62	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
63	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	2	no	—
64	C₄×C₃⋊(C₃⋊D₃)	—	2	4	no	—
65	#5624	—	9	13	no	—
66	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
67	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
68	C₂×#638	—	3	26	no	—
69	C₂²×C₃⋊Dic₃	—	3	13	no	—
70	C₂²×C₃⋊Dic₃	—	3	13	no	—
71	C₂×C₄⋊(C₃⋊D₃)	—	3	13	no	—
72	C₂×C₄×C₃⋊D₃	—	3	26	no	—
73	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
74	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
75	C₃²×C₂²⋊C₄	C₂×D₃	1	26	yes	—
76	#3752	—	3	26	no	—
77	#3752	—	3	26	no	—
78	#3752	—	3	26	no	—
79	#3752	—	3	26	no	—
80	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
81	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
82	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
83	#3750	—	3	26	no	—
84	C₄×C₃⋊Dic₃	—	3	26	no	—
85	C₄×C₃⋊Dic₃	—	3	13	no	—
86	C₄×C₃⋊Dic₃	—	3	13	no	—
87	C₄×C₃⋊Dic₃	—	3	13	no	—
88	C₄×C₃⋊Dic₃	—	3	13	no	—
89	C₂²×C₃³	—	2	1	no	—
90	C₂²×C₃³	—	2	1	no	—
91	C₂²×C₃³	#41	1	2	yes	—
92	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
93	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
94	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
95	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
96	C₃³×C₄	—	2	2	no	—
97	C₃³×C₄	—	2	1	no	—
98	C₃³×C₄	—	2	2	no	—
99	C₃³×C₄	#41	1	2	yes	—
100	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
101	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
102	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
103	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
104	C₃⋊(C₃⋊Dic₃)	#41	1	2	yes	—
105	C₃×C₂²⋊Q₈	C₃⋊D₃	1	13	yes	complemented
106	#947	—	9	13	no	—
107	Q₈×Dic₃	—	9	13	no	—
108	#930	—	9	13	no	—
109	C₂×C₄×Dic₃	—	9	13	no	—
110	#898	—	9	26	no	—
111	#893	—	9	13	no	—
112	#892	—	9	26	no	—
113	#890	—	9	13	no	—
114	#886	—	9	26	no	—
115	#67⋊C₄	—	9	26	no	—
116	C₂³×C₃²	C₂²×D₃	1	13	yes	—
117	C₂²×C₃⋊D₃	—	3	26	no	—
118	C₃²×Q₈	—	2	26	no	—
119	C₂×C₃²×C₄	—	2	26	no	—
120	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
121	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
123	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
124	#638	—	6	52	no	—
125	C₂×C₃⋊Dic₃	—	3	13	no	—
126	C₂×C₃⋊Dic₃	—	3	13	no	—
127	C₂×C₃⋊Dic₃	—	3	13	no	—
128	C₂×C₃⋊Dic₃	—	3	13	no	—
129	C₂×C₃⋊Dic₃	—	3	26	no	—
130	C₂×C₃⋊Dic₃	—	6	26	no	—
131	C₂×C₃⋊Dic₃	—	6	26	no	—
132	C₄⋊(C₃⋊D₃)	—	6	26	no	—
133	C₄×C₃⋊D₃	—	6	52	no	—
134	#529	—	27	1	no	Sylow
135	C₂×C₃³	—	2	2	no	—
136	C₂×C₃³	C₂×#41	1	1	yes	characteristic
137	C₂×C₃³	C₂×#41	1	1	yes	characteristic
138	C₂×C₃³	C₂×#41	1	1	yes	characteristic
139	C₃⋊(C₃⋊D₃)	—	4	2	no	—
140	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
141	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
142	C₂×#67	—	9	26	no	—
143	C₂²×Dic₃	—	9	13	no	—
144	C₂²×Dic₃	—	9	13	no	—
145	C₂×D₁₂	—	9	13	no	—
146	C₂×C₄×D₃	—	9	26	no	—
147	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
148	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
149	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
150	(C₂×D₃)⋊C₄	—	9	26	no	—
151	(C₂×D₃)⋊C₄	—	9	26	no	—
152	(C₂×D₃)⋊C₄	—	9	26	no	—
153	(C₂×D₃)⋊C₄	—	9	26	no	—
154	C₄⋊Dic₃	—	9	26	no	—
155	C₄⋊Dic₃	—	9	26	no	—
156	C₄⋊Dic₃	—	9	13	no	—
157	Dic₃⋊C₄	—	9	26	no	—
158	C₄×Dic₃	—	9	26	no	—
159	C₄×Dic₃	—	9	13	no	—
160	C₄×Dic₃	—	9	13	no	—
161	C₄×Dic₃	—	9	13	no	—
162	C₄×Dic₃	—	9	13	no	—
163	C₂²×C₃²	—	2	13	no	—
164	C₂²×C₃²	—	2	13	no	—
165	C₂²×C₃²	#239	1	26	yes	—
166	C₂²×C₃²	C₂³×D₃	1	13	yes	—
167	C₂×C₃⋊D₃	—	6	26	no	—
168	C₂×C₃⋊D₃	—	6	26	no	—
169	C₂×C₃⋊D₃	—	6	26	no	—
170	C₃²×C₄	—	2	13	no	—
171	C₃²×C₄	—	2	26	no	—
172	C₃²×C₄	—	2	26	no	—
173	C₃²×C₄	#237	1	26	yes	—
174	C₃⋊Dic₃	—	3	26	no	—
175	C₃⋊Dic₃	—	6	13	no	—
176	C₃⋊Dic₃	—	6	13	no	—
177	C₃⋊Dic₃	—	6	26	no	—
178	C₃⋊Dic₃	—	6	13	no	—
179	#126	—	27	2	no	—
180	#125	—	27	1	no	—
181	#124	—	27	1	no	—
182	#123	—	27	2	no	—
183	C₂²⋊Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
184	#121	—	27	1	no	—
185	C₄×Q₈	—	27	1	no	—
186	C₄×D₄	—	27	1	no	—
187	C₄×D₄	—	27	2	no	—
188	#117	—	27	2	no	—
189	C₂×C₄²	—	27	1	no	—
190	C₃³	#529	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
192	C₂²×D₃	—	9	26	no	—
193	C₃×Q₈	—	2	26	no	—
194	C₂×C₃×C₄	—	2	26	no	—
195	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
196	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
197	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
198	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
199	#67	—	18	52	no	—
200	C₂×Dic₃	—	9	13	no	—
201	C₂×Dic₃	—	9	13	no	—
202	C₂×Dic₃	—	9	13	no	—
203	C₂×Dic₃	—	9	13	no	—
204	C₂×Dic₃	—	9	26	no	—
205	C₂×Dic₃	—	18	26	no	—
206	C₂×Dic₃	—	18	26	no	—
207	D₁₂	—	18	26	no	—
208	C₄×D₃	—	18	52	no	—
209	C₂×C₃²	—	2	26	no	—
210	C₂×C₃²	C₂×#237	1	13	yes	—
211	C₂×C₃²	C₂×#239	1	13	yes	—
212	C₂×C₃²	D₃×#41	1	13	yes	—
213	C₃⋊D₃	—	12	26	no	—
214	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
215	C₂×D₄	—	27	2	no	—
216	C₂×D₄	—	27	1	no	—
217	C₂²×C₄	—	27	1	no	—
218	C₂²×C₄	—	27	1	no	—
219	C₂²×C₄	—	27	2	no	—
220	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
221	C₄⋊C₄	—	27	2	no	—
222	C₄⋊C₄	—	27	2	no	—
223	C₄⋊C₄	—	27	1	no	—
224	C₄⋊C₄	—	27	2	no	—
225	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
226	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
227	C₂²⋊C₄	—	27	2	no	—
228	C₂²⋊C₄	—	27	2	no	—
229	C₂²⋊C₄	—	27	2	no	—
230	C₂²⋊C₄	—	27	2	no	—
231	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
232	C₄²	—	27	1	no	—
233	C₄²	—	27	1	no	—
234	C₄²	—	27	1	no	—
235	C₄²	—	27	1	no	—
236	C₄²	—	27	2	no	—
237	C₂²×C₃	—	2	13	no	—
238	C₂²×C₃	—	2	13	no	—
239	C₂²×C₃	#3832	1	26	yes	—
240	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
241	C₂×D₃	—	18	26	no	—
242	C₂×D₃	—	18	26	no	—
243	C₂×D₃	—	18	26	no	—
244	C₃×C₄	—	2	13	no	—
245	C₃×C₄	—	2	26	no	—
246	C₃×C₄	—	2	26	no	—
247	C₃×C₄	#3830	1	26	yes	—
248	Dic₃	—	9	26	no	—
249	Dic₃	—	18	13	no	—
250	Dic₃	—	18	13	no	—
251	Dic₃	—	18	26	no	—
252	Dic₃	—	18	13	no	—
253	C₃²	#5624	1	13	yes	complemented
254	C₂³	—	27	2	no	—
255	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
256	Q₈	—	2	2	no	—
257	D₄	—	54	4	no	—
258	D₄	—	54	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	27	1	no	—
261	C₂×C₄	—	27	1	no	—
262	C₂×C₄	—	27	1	no	—
263	C₂×C₄	—	27	1	no	—
264	C₂×C₄	—	27	2	no	—
265	C₂×C₄	—	54	4	no	—
266	C₂×C₄	—	54	2	no	—
267	C₂×C₄	—	54	2	no	—
268	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
270	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
271	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
272	C₂×C₃	—	2	26	no	—
273	C₂×C₃	C₂×#3830	1	13	yes	—
274	C₂×C₃	C₂×#3832	1	13	yes	—
275	C₂×C₃	#41×C₃⋊D₃	1	13	yes	—
276	D₃	—	36	26	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	54	2	no	—
280	C₂²	—	54	2	no	—
281	C₂²	—	54	2	no	—
282	C₂²	#89080	1	2	yes	—
283	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	27	2	no	—
288	C₄	—	54	1	no	—
289	C₄	—	54	1	no	—
290	C₄	—	54	2	no	—
291	C₄	—	54	1	no	—
292	C₄	#89078	1	2	yes	—
293	C₃	#10596095	1	13	yes	complemented
294	C₂	—	2	2	no	—
295	C₂	—	108	2	no	—
296	C₂	C₂×#89078	1	1	yes	characteristic
297	C₂	C₂×#89080	1	1	yes	characteristic
298	C₂	#41×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Fitting	C₃³×C₂²⋊Q₈	C₂
prime radical	#11723971	C₂
hypercenter	C₂²⋊Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#529
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₂²⋊Q₈	C₂	semidirect
2. #11723988	C₂	semidirect
3. Q₈×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
5. C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
6. #11723939	C₂	semidirect
7. #11723934	C₂	semidirect
8. #11723933	C₂	semidirect
9. #11723931	C₂	semidirect
10. #11723927	C₂	semidirect
11. #11723922	C₂	semidirect
36. C₃²×C₂²⋊Q₈	D₃	semidirect
105. C₃×C₂²⋊Q₈	C₃⋊D₃	semidirect
183. C₂²⋊Q₈	C₃⋊(C₃⋊D₃)	semidirect
190. C₃³	#529	semidirect
253. C₃²	#5624	semidirect
293. C₃	#10596095	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₂²⋊Q₈	2
#11723988	2
Q₈×C₃⋊(C₃⋊Dic₃)	2
#11723971	2
C₂×C₄×C₃⋊(C₃⋊Dic₃)	2
#11723939	2
#11723934	2
#11723933	2
#11723931	2
#11723927	2
#11723922	2
#10596095	3