Subgroups — #13154276

Order 1728.

1000 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊Dic₉	C₂	1	2	yes	maximal
3	#11721854	C₂	1	2	yes	maximal
4	#11721746	C₂	1	2	yes	maximal
5	#11721745	C₂	1	4	yes	maximal
6	#11721743	C₂	1	2	yes	maximal
7	C₄×#6226	C₂	1	1	yes	maximal, characteristic
8	C₄×#6226	C₂	1	1	yes	maximal, characteristic
9	#10596192	—	3	1	no	maximal
10	#10589823	—	3	3	no	maximal
11	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
12	C₂×#6226	C₂²	1	1	yes	characteristic
13	C₂×#6226	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₄⋊(C₃⋊Dic₉)	C₂²	1	4	yes	—
18	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
20	#88534	C₂²	1	4	yes	—
21	#88534	C₂²	1	2	yes	—
22	#88534	C₂²	1	2	yes	—
23	#88534	C₂²	1	4	yes	—
24	C₄×C₃⋊Dic₉	C₂²	1	4	yes	—
25	C₄×C₃⋊Dic₉	C₂²	1	2	yes	—
26	C₄×C₃⋊Dic₉	C₂²	1	2	yes	—
27	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
28	Q₈×C₃⋊Dic₃	—	3	2	no	—
29	#64153	—	3	2	no	—
30	#64099	—	3	2	no	—
31	#64098	—	3	4	no	—
32	#64096	—	3	2	no	—
33	C₄×#634	—	3	1	no	—
34	C₄×#634	—	3	1	no	—
35	C₉×C₄⋊Q₈	D₃	1	3	yes	—
36	Q₈×Dic₉	—	3	6	no	—
37	#63506	—	3	6	no	—
38	#63452	—	3	6	no	—
39	#63451	—	3	12	no	—
40	#63449	—	3	6	no	—
41	C₄×Dic₁₈	—	3	3	no	—
42	C₄×Dic₁₈	—	3	3	no	—
43	C₃×Q₈×C₉	—	2	4	no	—
44	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
45	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂×C₃⋊Dic₉	C₂³	1	2	yes	—
49	C₂×C₃⋊Dic₉	C₂³	1	2	yes	—
50	C₂×C₃⋊Dic₉	C₂³	1	2	yes	—
51	C₂×C₃⋊Dic₉	C₂³	1	2	yes	—
52	#6226	—	2	2	no	—
53	#6226	Q₈	1	4	yes	complemented
54	#5721	—	9	3	no	—
55	#5721	—	9	1	no	—
56	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
57	C₂×#634	—	3	1	no	—
58	C₂×#634	—	3	1	no	—
59	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
60	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
61	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
63	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
64	C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
65	#3750	—	3	2	no	—
66	#3750	—	3	2	no	—
67	#3750	—	3	4	no	—
68	#3750	—	3	4	no	—
69	C₄×C₃⋊Dic₃	—	3	2	no	—
70	C₄×C₃⋊Dic₃	—	3	2	no	—
71	C₄×C₃⋊Dic₃	—	3	4	no	—
72	C₂×Q₈×C₉	C₂×D₃	1	6	yes	—
73	C₂×Dic₁₈	—	3	3	no	—
74	C₂×Dic₁₈	—	3	3	no	—
75	C₉×C₄⋊C₄	C₂×D₃	1	6	yes	—
76	C₉×C₄⋊C₄	C₂×D₃	1	6	yes	—
77	C₄²×C₉	C₂×D₃	1	3	yes	—
78	C₄⋊Dic₉	—	3	3	no	—
79	C₄⋊Dic₉	—	3	12	no	—
80	C₄⋊Dic₉	—	3	3	no	—
81	Dic₉⋊C₄	—	3	12	no	—
82	Dic₉⋊C₄	—	3	12	no	—
83	Dic₉⋊C₄	—	3	6	no	—
84	Dic₉⋊C₄	—	3	6	no	—
85	C₄×Dic₉	—	3	6	no	—
86	C₄×Dic₉	—	3	12	no	—
87	C₄×Dic₉	—	3	6	no	—
88	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
89	C₃×C₄×C₉	—	2	4	no	—
90	C₃×C₄×C₉	—	2	1	no	—
91	C₃×C₄×C₉	C₂×Q₈	1	2	yes	—
92	C₃×C₄×C₉	#41	1	2	yes	—
93	C₃⋊Dic₉	—	2	2	no	—
94	C₃⋊Dic₉	—	2	2	no	—
95	C₃⋊Dic₉	—	2	2	no	—
96	C₃⋊Dic₉	C₂×Q₈	1	4	yes	—
97	C₃×C₄⋊Q₈	D₉	1	3	yes	—
98	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
99	Q₈×Dic₃	—	9	6	no	—
100	Q₈×Dic₃	—	9	2	no	—
101	#944	—	9	6	no	—
102	#944	—	9	2	no	—
103	#890	—	9	6	no	—
104	#890	—	9	2	no	—
105	#889	—	9	12	no	—
106	#889	—	9	4	no	—
107	#887	—	9	6	no	—
108	#887	—	9	2	no	—
109	C₄×Dic₆	—	9	3	no	—
110	C₄×Dic₆	—	9	3	no	—
111	C₄×Dic₆	—	9	1	no	—
112	C₄×Dic₆	—	9	1	no	—
113	C₃²×Q₈	—	2	4	no	—
114	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
115	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₃⋊Dic₃	—	3	2	no	—
119	C₂×C₃⋊Dic₃	—	3	2	no	—
120	C₂×C₃⋊Dic₃	—	3	2	no	—
121	C₂×C₃⋊Dic₃	—	3	2	no	—
122	#634	—	3	4	no	—
123	#634	—	6	2	no	—
124	Q₈×C₉	—	2	12	no	—
125	C₂×C₄×C₉	C₂²×D₃	1	12	yes	—
126	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
127	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
128	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
129	C₂×Dic₉	—	3	6	no	—
130	C₂×Dic₉	—	3	6	no	—
131	C₂×Dic₉	—	3	6	no	—
132	C₂×Dic₉	—	3	6	no	—
133	Dic₁₈	—	3	12	no	—
134	Dic₁₈	—	6	6	no	—
135	#557	—	27	1	no	Sylow
136	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
138	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
139	C₂×C₃×Q₈	C₂×D₉	1	6	yes	—
140	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
141	C₂×Dic₆	—	9	3	no	—
142	C₂×Dic₆	—	9	1	no	—
143	C₂×Dic₆	—	9	3	no	—
144	C₂×Dic₆	—	9	1	no	—
145	C₃×C₄⋊C₄	C₂×D₉	1	6	yes	—
146	C₃×C₄⋊C₄	C₂×D₉	1	6	yes	—
147	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
148	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
149	C₃×C₄²	C₂×D₉	1	3	yes	—
150	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
151	C₄⋊Dic₃	—	9	3	no	—
152	C₄⋊Dic₃	—	9	3	no	—
153	C₄⋊Dic₃	—	9	12	no	—
154	C₄⋊Dic₃	—	9	1	no	—
155	C₄⋊Dic₃	—	9	1	no	—
156	C₄⋊Dic₃	—	9	4	no	—
157	Dic₃⋊C₄	—	9	6	no	—
158	Dic₃⋊C₄	—	9	6	no	—
159	Dic₃⋊C₄	—	9	12	no	—
160	Dic₃⋊C₄	—	9	12	no	—
161	Dic₃⋊C₄	—	9	2	no	—
162	Dic₃⋊C₄	—	9	2	no	—
163	Dic₃⋊C₄	—	9	4	no	—
164	Dic₃⋊C₄	—	9	4	no	—
165	C₄×Dic₃	—	9	6	no	—
166	C₄×Dic₃	—	9	6	no	—
167	C₄×Dic₃	—	9	12	no	—
168	C₄×Dic₃	—	9	2	no	—
169	C₄×Dic₃	—	9	2	no	—
170	C₄×Dic₃	—	9	4	no	—
171	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
172	C₃²×C₄	—	2	4	no	—
173	C₃²×C₄	—	2	1	no	—
174	C₃²×C₄	#237	1	2	yes	—
175	C₃²×C₄	D₃×Q₈	1	2	yes	—
176	C₃⋊Dic₃	—	3	4	no	—
177	C₃⋊Dic₃	—	6	2	no	—
178	C₃⋊Dic₃	—	6	2	no	—
179	C₃⋊Dic₃	—	6	2	no	—
180	C₂²×C₉	C₂³×D₃	1	3	yes	—
181	C₄×C₉	—	2	12	no	—
182	C₄×C₉	—	2	3	no	—
183	C₄×C₉	#237	1	6	yes	—
184	C₄×C₉	D₃×Q₈	1	6	yes	—
185	Dic₉	—	3	12	no	—
186	Dic₉	—	6	6	no	—
187	Dic₉	—	6	6	no	—
188	Dic₉	—	6	6	no	—
189	C₄⋊Q₈	—	27	2	no	—
190	C₄⋊Q₈	C₃⋊D₉	1	1	yes	characteristic
191	#125	—	27	4	no	—
192	#125	—	27	2	no	—
193	C₄×Q₈	—	27	2	no	—
194	C₄×Q₈	—	27	2	no	—
195	C₄×Q₈	—	27	1	no	—
196	C₄×Q₈	—	27	1	no	—
197	C₃×C₉	#557	1	1	yes	characteristic, complemented, Sylow
198	C₃×Q₈	—	2	12	no	—
199	C₃×Q₈	—	2	4	no	—
200	C₂×C₃×C₄	C₂²×D₉	1	12	yes	—
201	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
202	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
203	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
204	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	4	yes	—
205	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
208	C₂×Dic₃	—	9	2	no	—
209	C₂×Dic₃	—	9	6	no	—
210	C₂×Dic₃	—	9	2	no	—
211	C₂×Dic₃	—	9	2	no	—
212	C₂×Dic₃	—	9	2	no	—
213	C₂×Dic₃	—	9	6	no	—
214	C₂×Dic₃	—	9	6	no	—
215	C₂×Dic₃	—	9	6	no	—
216	Dic₆	—	9	12	no	—
217	Dic₆	—	9	4	no	—
218	Dic₆	—	18	6	no	—
219	Dic₆	—	18	2	no	—
220	C₂×C₃²	C₂×#237	1	1	yes	characteristic
221	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
222	C₂×C₃²	#1007	1	1	yes	characteristic
223	C₂×C₉	C₂×#237	1	3	yes	—
224	C₂×C₉	C₂×D₃×Q₈	1	3	yes	—
225	C₂×C₉	#1007	1	3	yes	—
226	C₂×Q₈	—	27	1	no	—
227	C₂×Q₈	—	27	1	no	—
228	C₂×Q₈	C₂×C₃⋊D₉	1	2	yes	—
229	C₄⋊C₄	—	27	1	no	—
230	C₄⋊C₄	—	27	2	no	—
231	C₄⋊C₄	—	27	2	no	—
232	C₄⋊C₄	—	27	4	no	—
233	C₄⋊C₄	—	27	1	no	—
234	C₄⋊C₄	—	27	4	no	—
235	C₄⋊C₄	—	27	4	no	—
236	C₄⋊C₄	C₂×C₃⋊D₉	1	2	yes	—
237	C₄⋊C₄	C₂×C₃⋊D₉	1	2	yes	—
238	C₄²	—	27	4	no	—
239	C₄²	—	27	2	no	—
240	C₄²	—	27	2	no	—
241	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
242	C₂²×C₃	C₂³×D₉	1	3	yes	—
243	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
244	C₃×C₄	—	2	4	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	12	no	—
247	C₃×C₄	—	2	3	no	—
248	C₃×C₄	#3699	1	6	yes	—
249	C₃×C₄	Q₈×D₉	1	6	yes	—
250	C₃×C₄	#3830	1	2	yes	—
251	C₃×C₄	Q₈×C₃⋊D₃	1	2	yes	—
252	Dic₃	—	9	4	no	—
253	Dic₃	—	9	12	no	—
254	Dic₃	—	18	2	no	—
255	Dic₃	—	18	2	no	—
256	Dic₃	—	18	2	no	—
257	Dic₃	—	18	6	no	—
258	Dic₃	—	18	6	no	—
259	Dic₃	—	18	6	no	—
260	C₃²	#5721	1	1	yes	characteristic
261	C₉	#5721	1	3	yes	complemented
262	Q₈	—	2	4	no	—
263	Q₈	—	27	4	no	—
264	Q₈	—	54	2	no	—
265	C₂×C₄	—	27	2	no	—
266	C₂×C₄	—	27	2	no	—
267	C₂×C₄	—	27	2	no	—
268	C₂×C₄	—	27	2	no	—
269	C₂×C₄	C₂²×C₃⋊D₉	1	4	yes	—
270	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
273	C₂×C₃	C₂×#3699	1	3	yes	—
274	C₂×C₃	C₂×Q₈×D₉	1	3	yes	—
275	C₂×C₃	#63713	1	3	yes	—
276	C₂×C₃	C₂×#3830	1	1	yes	characteristic
277	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
278	C₂×C₃	#64364	1	1	yes	characteristic
279	C₂²	C₂³×C₃⋊D₉	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	1	no	—
282	C₄	—	27	4	no	—
283	C₄	—	54	2	no	—
284	C₄	—	54	2	no	—
285	C₄	—	54	2	no	—
286	C₄	#88742	1	2	yes	—
287	C₄	Q₈×C₃⋊D₉	1	2	yes	—
288	C₃	#10589823	1	3	yes	complemented
289	C₃	#10596192	1	1	yes	characteristic
290	C₂	C₂×#88742	1	1	yes	characteristic
291	C₂	C₂×Q₈×C₃⋊D₉	1	1	yes	characteristic
292	C₂	#11722667	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×C₃⋊D₉
commutator, cosocle	C₂²×C₃×C₉	C₂⁴
Frattini	C₂²×C₃	C₂³×C₃⋊D₃
Fitting	C₃×C₉×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃²	C₂³×D₃
hypercenter	C₄⋊Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#557
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
53. #6226	Q₈	semidirect
197. C₃×C₉	#557	semidirect
261. C₉	#5721	semidirect
288. C₃	#10589823	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×C₄⋊Q₈	2
Q₈×C₃⋊Dic₉	2
#11721854	2
#11721746	2
#11721745	2
#11721743	2
C₄×#6226	2
#10596192	3
#10589823	3