Subgroups — #13179767

Order 1728.

2996 conjugacy classes of subgroups.

The subgroup lattice has 292 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	#11723987	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6312	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	#11723933	C₂	1	1	yes	maximal, characteristic, complemented
8	#11723931	C₂	1	2	yes	maximal, complemented
9	#11723931	C₂	1	1	yes	maximal, characteristic, complemented
10	#11723927	C₂	1	2	yes	maximal
11	#11723918	C₂	1	2	yes	maximal, complemented
12	#10596113	—	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	1	yes	characteristic
16	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
17	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
18	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
21	#88864	C₂²	1	1	yes	characteristic
22	#88859	C₂²	1	2	yes	—
23	#88859	C₂²	1	2	yes	—
24	#88859	C₂²	1	1	yes	characteristic
25	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
26	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
27	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
28	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
29	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
30	#88857	C₂²	1	2	yes	—
31	#88857	C₂²	1	2	yes	—
32	#88857	C₂²	1	1	yes	characteristic
33	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
34	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
36	C₃²×C₂²⋊Q₈	D₃	1	13	yes	complemented
37	#64155	—	3	13	no	—
38	Q₈×C₃⋊Dic₃	—	3	13	no	—
39	C₄×#638	—	3	13	no	—
40	C₂×C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
41	#64107	—	3	26	no	—
42	#64101	—	3	13	no	—
43	#64099	—	3	26	no	—
44	#64099	—	3	13	no	—
45	#64095	—	3	26	no	—
46	#64086	—	3	26	no	—
47	C₂³×C₃³	C₂³	1	1	yes	characteristic
48	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
49	C₃³×Q₈	—	2	2	no	—
50	C₂×C₃³×C₄	—	2	2	no	—
51	C₂×C₃³×C₄	C₂³	1	2	yes	—
52	C₂×C₃³×C₄	C₂³	1	2	yes	—
53	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
54	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
55	#6312	—	2	2	no	—
56	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
57	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
58	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
59	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
60	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
61	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
62	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
63	#5642	—	9	13	no	—
64	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
65	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
66	C₂×#638	—	3	13	no	—
67	C₂²×C₃⋊Dic₃	—	3	26	no	—
68	C₂×C₄×C₃⋊D₃	—	3	13	no	—
69	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
70	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
71	C₃²×C₂²⋊C₄	C₂×D₃	1	26	yes	—
72	#3757	—	3	13	no	—
73	#3752	—	3	26	no	—
74	#3752	—	3	26	no	—
75	#3752	—	3	13	no	—
76	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
77	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
78	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
79	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
80	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
81	#3750	—	3	26	no	—
82	#3750	—	3	26	no	—
83	#3750	—	3	13	no	—
84	C₄×C₃⋊Dic₃	—	3	26	no	—
85	C₄×C₃⋊Dic₃	—	3	26	no	—
86	C₄×C₃⋊Dic₃	—	3	13	no	—
87	C₂²×C₃³	—	2	1	no	—
88	C₂²×C₃³	—	2	1	no	—
89	C₂²×C₃³	#41	1	2	yes	—
90	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
91	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
92	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
93	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
94	C₃³×C₄	—	2	1	no	—
95	C₃³×C₄	—	2	2	no	—
96	C₃³×C₄	—	2	2	no	—
97	C₃³×C₄	#41	1	2	yes	—
98	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
99	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
100	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
101	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
102	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
103	C₃×C₂²⋊Q₈	C₃⋊D₃	1	13	yes	complemented
104	#946	—	9	13	no	—
105	Q₈×Dic₃	—	9	13	no	—
106	C₄×#67	—	9	13	no	—
107	C₂×C₄⋊Dic₃	—	9	13	no	—
108	#898	—	9	26	no	—
109	#892	—	9	13	no	—
110	#890	—	9	26	no	—
111	#890	—	9	13	no	—
112	#886	—	9	26	no	—
113	#877	—	9	26	no	—
114	C₂³×C₃²	C₂²×D₃	1	13	yes	—
115	C₂²×C₃⋊D₃	—	3	13	no	—
116	C₃²×Q₈	—	2	26	no	—
117	C₂×C₃²×C₄	—	2	26	no	—
118	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
119	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
120	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
121	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
122	#638	—	6	26	no	—
123	C₂×C₃⋊Dic₃	—	3	13	no	—
124	C₂×C₃⋊Dic₃	—	3	13	no	—
125	C₂×C₃⋊Dic₃	—	3	26	no	—
126	C₂×C₃⋊Dic₃	—	3	13	no	—
127	C₂×C₃⋊Dic₃	—	3	26	no	—
128	C₂×C₃⋊Dic₃	—	6	52	no	—
129	C₄×C₃⋊D₃	—	6	26	no	—
130	#552	—	27	1	no	Sylow
131	C₂×C₃³	—	2	2	no	—
132	C₂×C₃³	C₂×#41	1	1	yes	characteristic
133	C₂×C₃³	C₂×#41	1	1	yes	characteristic
134	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
135	C₃⋊(C₃⋊D₃)	—	4	1	no	—
136	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
137	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
138	C₂×#67	—	9	13	no	—
139	C₂²×Dic₃	—	9	26	no	—
140	C₂×C₄×D₃	—	9	13	no	—
141	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
142	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
143	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
144	#217	—	9	13	no	—
145	(C₂×D₃)⋊C₄	—	9	26	no	—
146	(C₂×D₃)⋊C₄	—	9	26	no	—
147	(C₂×D₃)⋊C₄	—	9	13	no	—
148	C₄⋊Dic₃	—	9	26	no	—
149	C₄⋊Dic₃	—	9	26	no	—
150	C₄⋊Dic₃	—	9	26	no	—
151	C₄⋊Dic₃	—	9	13	no	—
152	C₄⋊Dic₃	—	9	13	no	—
153	Dic₃⋊C₄	—	9	26	no	—
154	Dic₃⋊C₄	—	9	26	no	—
155	Dic₃⋊C₄	—	9	13	no	—
156	C₄×Dic₃	—	9	26	no	—
157	C₄×Dic₃	—	9	26	no	—
158	C₄×Dic₃	—	9	13	no	—
159	C₂²×C₃²	—	2	13	no	—
160	C₂²×C₃²	—	2	13	no	—
161	C₂²×C₃²	#239	1	26	yes	—
162	C₂²×C₃²	C₂³×D₃	1	13	yes	—
163	C₂×C₃⋊D₃	—	6	13	no	—
164	C₂×C₃⋊D₃	—	6	13	no	—
165	C₂×C₃⋊D₃	—	6	13	no	—
166	C₃²×C₄	—	2	13	no	—
167	C₃²×C₄	—	2	26	no	—
168	C₃²×C₄	—	2	26	no	—
169	C₃²×C₄	#237	1	26	yes	—
170	C₃⋊Dic₃	—	6	13	no	—
171	C₃⋊Dic₃	—	6	26	no	—
172	C₃⋊Dic₃	—	6	26	no	—
173	C₃⋊Dic₃	—	6	13	no	—
174	C₃⋊Dic₃	—	6	13	no	—
175	#126	—	27	2	no	—
176	#125	—	27	2	no	—
177	#125	—	27	1	no	—
178	#123	—	27	2	no	—
179	C₂²⋊Q₈	—	27	1	no	—
180	C₂²⋊Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
181	C₄×Q₈	—	27	1	no	—
182	C₄×D₄	—	27	1	no	—
183	#117	—	27	1	no	—
184	#117	—	27	2	no	—
185	C₂×C₄⋊C₄	—	27	1	no	—
186	C₃³	#552	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
188	C₂²×D₃	—	9	13	no	—
189	C₃×Q₈	—	2	26	no	—
190	C₂×C₃×C₄	—	2	26	no	—
191	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
192	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
193	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
194	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
195	#67	—	18	26	no	—
196	C₂×Dic₃	—	9	13	no	—
197	C₂×Dic₃	—	9	13	no	—
198	C₂×Dic₃	—	9	26	no	—
199	C₂×Dic₃	—	9	13	no	—
200	C₂×Dic₃	—	9	26	no	—
201	C₂×Dic₃	—	18	52	no	—
202	C₄×D₃	—	18	26	no	—
203	C₂×C₃²	—	2	26	no	—
204	C₂×C₃²	C₂×#237	1	13	yes	—
205	C₂×C₃²	C₂×#239	1	13	yes	—
206	C₂×C₃²	#1010	1	13	yes	—
207	C₃⋊D₃	—	12	13	no	—
208	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
209	C₂×D₄	—	27	1	no	—
210	C₂²×C₄	—	27	2	no	—
211	C₂²×C₄	—	27	1	no	—
212	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
213	C₄⋊C₄	—	27	1	no	—
214	C₄⋊C₄	—	27	1	no	—
215	C₄⋊C₄	—	27	2	no	—
216	C₄⋊C₄	—	27	2	no	—
217	C₄⋊C₄	—	27	2	no	—
218	C₄⋊C₄	—	27	1	no	—
219	C₄⋊C₄	—	27	2	no	—
220	C₄⋊C₄	—	27	2	no	—
221	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
222	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
223	C₂²⋊C₄	—	27	1	no	—
224	C₂²⋊C₄	—	27	2	no	—
225	C₂²⋊C₄	—	27	1	no	—
226	C₂²⋊C₄	—	27	2	no	—
227	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
228	C₄²	—	27	2	no	—
229	C₄²	—	27	2	no	—
230	C₄²	—	27	1	no	—
231	C₂²×C₃	—	2	13	no	—
232	C₂²×C₃	—	2	13	no	—
233	C₂²×C₃	#3832	1	26	yes	—
234	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
235	C₂×D₃	—	18	13	no	—
236	C₂×D₃	—	18	13	no	—
237	C₂×D₃	—	18	13	no	—
238	C₃×C₄	—	2	13	no	—
239	C₃×C₄	—	2	26	no	—
240	C₃×C₄	—	2	26	no	—
241	C₃×C₄	#3830	1	26	yes	—
242	Dic₃	—	18	13	no	—
243	Dic₃	—	18	26	no	—
244	Dic₃	—	18	26	no	—
245	Dic₃	—	18	13	no	—
246	Dic₃	—	18	13	no	—
247	C₃²	#5642	1	13	yes	complemented
248	C₂³	—	27	1	no	—
249	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	D₄	—	54	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	27	1	no	—
254	C₂×C₄	—	27	1	no	—
255	C₂×C₄	—	27	2	no	—
256	C₂×C₄	—	27	1	no	—
257	C₂×C₄	—	27	2	no	—
258	C₂×C₄	—	54	4	no	—
259	C₂×C₄	—	54	2	no	—
260	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
261	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
262	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
264	C₂×C₃	—	2	26	no	—
265	C₂×C₃	C₂×#3830	1	13	yes	—
266	C₂×C₃	C₂×#3832	1	13	yes	—
267	C₂×C₃	#64367	1	13	yes	—
268	D₃	—	36	13	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	54	1	no	—
272	C₂²	—	54	1	no	—
273	C₂²	—	54	1	no	—
274	C₂²	#89080	1	2	yes	—
275	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	54	1	no	—
280	C₄	—	54	2	no	—
281	C₄	—	54	2	no	—
282	C₄	—	54	1	no	—
283	C₄	—	54	1	no	—
284	C₄	#89078	1	2	yes	—
285	C₃	#10596113	1	13	yes	complemented
286	C₂	—	2	2	no	—
287	C₂	—	108	1	no	—
288	C₂	C₂×#89078	1	1	yes	characteristic
289	C₂	C₂×#89080	1	1	yes	characteristic
290	C₂	#11724685	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	C₂×#6312	C₂²
hypercenter	C₂²⋊Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#552
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. #11723987	C₂	semidirect
3. Q₈×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
5. C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	semidirect
6. #11723939	C₂	semidirect
7. #11723933	C₂	semidirect
8. #11723931	C₂	semidirect
9. #11723931	C₂	semidirect
11. #11723918	C₂	semidirect
36. C₃²×C₂²⋊Q₈	D₃	semidirect
103. C₃×C₂²⋊Q₈	C₃⋊D₃	semidirect
180. C₂²⋊Q₈	C₃⋊(C₃⋊D₃)	semidirect
186. C₃³	#552	semidirect
247. C₃²	#5642	semidirect
285. C₃	#10596113	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
#11723987	2
Q₈×C₃⋊(C₃⋊Dic₃)	2
C₄×#6312	2
C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	2
#11723939	2
#11723933	2
#11723931	2
#11723927	2
#11723918	2
#10596113	3