Subgroups — C₄×#88724 #13152734

Order 1728.

965 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88724	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#63455⋊C₃	C₂	1	3	yes	maximal, complemented
5	#63454⋊C₃	C₂	1	3	yes	maximal, complemented
6	C₄×C₄⋊AGL(1,ℤ/9)	C₂	1	3	yes	maximal, complemented
7	C₄²×AGL(1,ℤ/9)	C₂	1	3	yes	maximal, complemented
8	C₃×C₄×#239	—	3	1	no	maximal
9	C₄×#3701	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	#88724	C₄	1	8	yes	direct product factor
12	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	3	yes	—
13	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	3	yes	—
14	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	3	yes	—
15	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	3	yes	—
16	C₄⋊C₄×3₋¹⁺²	C₂²	1	3	yes	—
17	C₄²×3₋¹⁺²	C₂²	1	3	yes	—
18	(C₂×AGL(1,ℤ/9))⋊C₄	C₂²	1	6	yes	—
19	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	3	yes	—
20	C₄×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
21	C₄×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
22	C₄×3₋¹⁺²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×#239	—	3	1	no	—
24	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
25	C₃×Q₈×Dic₃	—	3	1	no	—
26	C₃×#893	—	3	3	no	—
27	C₃×#892	—	3	3	no	—
28	C₃×C₄×D₁₂	—	3	3	no	—
29	C₃×C₄²×D₃	—	3	3	no	—
30	C₂×#3701	C₂×C₃	1	1	yes	characteristic
31	C₄×Q₈×C₉	—	2	1	no	—
32	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
33	Q₈×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
34	#63455	C₂×C₃	1	3	yes	complemented
35	#63454	C₂×C₃	1	3	yes	complemented
36	C₄×D₃₆	C₂×C₃	1	3	yes	complemented
37	C₄²×D₉	C₂×C₃	1	3	yes	complemented
38	C₂²×AGL(1,ℤ/9)	C₂³	1	3	yes	—
39	Q₈×3₋¹⁺²	C₂×C₄	1	4	yes	complemented
40	C₂×C₄×3₋¹⁺²	C₂³	1	3	yes	—
41	C₂×C₄×3₋¹⁺²	C₂³	1	3	yes	—
42	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
43	C₂×3₋¹⁺²⋊C₄	C₂³	1	3	yes	—
44	C₂×3₋¹⁺²⋊C₄	C₂³	1	1	yes	characteristic
45	C₂×3₋¹⁺²⋊C₄	C₂³	1	1	yes	characteristic
46	C₄⋊AGL(1,ℤ/9)	C₂×C₄	1	12	yes	complemented
47	C₄×AGL(1,ℤ/9)	—	2	6	no	—
48	C₄×AGL(1,ℤ/9)	—	2	6	no	—
49	C₄×AGL(1,ℤ/9)	C₂×C₄	1	12	yes	complemented
50	C₃×C₄×#41	—	9	1	no	—
51	C₄×#239	—	3	1	no	—
52	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
53	C₃×#239	—	3	8	no	—
54	C₂×C₃×D₁₂	—	3	3	no	—
55	C₂×C₃×C₄×D₃	—	3	3	no	—
56	C₂×C₃×C₄×D₃	—	3	3	no	—
57	C₂×C₃×C₄×D₃	—	3	3	no	—
58	C₃²×C₄⋊C₄	C₂×D₃	1	3	yes	—
59	C₃²×C₄²	C₂×D₃	1	3	yes	—
60	C₃×(C₂×D₃)⋊C₄	—	3	6	no	—
61	C₃×C₄⋊Dic₃	—	3	3	no	—
62	C₃×C₄×Dic₃	—	3	3	no	—
63	C₃×C₄×Dic₃	—	3	3	no	—
64	C₃×C₄×Dic₃	—	3	1	no	—
65	C₂×Q₈×C₉	—	2	1	no	—
66	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
67	#3701	C₃×C₄	1	8	yes	complemented
68	C₂×D₃₆	C₂²×C₃	1	3	yes	—
69	C₂×C₄×D₉	C₂²×C₃	1	3	yes	—
70	C₂×C₄×D₉	C₂²×C₃	1	3	yes	—
71	C₂×C₄×D₉	C₂²×C₃	1	3	yes	—
72	C₉×C₄⋊C₄	—	2	3	no	—
73	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
74	C₄²×C₉	—	2	3	no	—
75	C₄²×C₉	C₂²×C₃	1	3	yes	—
76	(C₂×D₉)⋊C₄	C₂²×C₃	1	6	yes	—
77	C₄⋊Dic₉	C₂²×C₃	1	3	yes	—
78	C₄×Dic₉	C₂²×C₃	1	3	yes	—
79	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
80	C₄×Dic₉	C₂²×C₃	1	3	yes	—
81	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
82	C₂×AGL(1,ℤ/9)	—	2	3	no	—
83	C₂×AGL(1,ℤ/9)	—	2	3	no	—
84	C₂×AGL(1,ℤ/9)	C₂²×C₄	1	6	yes	—
85	C₄×3₋¹⁺²	—	2	3	no	—
86	C₄×3₋¹⁺²	C₂²×C₄	1	6	yes	—
87	C₄×3₋¹⁺²	#41	1	2	yes	complemented
88	3₋¹⁺²⋊C₄	—	2	3	no	—
89	3₋¹⁺²⋊C₄	C₂²×C₄	1	2	yes	—
90	3₋¹⁺²⋊C₄	#41	1	2	yes	complemented
91	C₂×C₃×#41	—	9	1	no	—
92	C₂×#239	—	3	1	no	—
93	C₃×C₄×Q₈	—	3	1	no	—
94	C₃×C₄×Q₈	—	9	1	no	—
95	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
96	C₃×C₄×D₄	—	9	3	no	—
97	C₃×C₄×D₄	—	9	3	no	—
98	C₃×#117	—	9	3	no	—
99	C₂×C₃×C₄²	—	9	3	no	—
100	Q₈×Dic₃	—	3	1	no	—
101	#893	—	3	3	no	—
102	#892	—	3	3	no	—
103	C₄×D₁₂	—	3	3	no	—
104	C₄²×D₃	—	3	3	no	—
105	C₂²×C₃×D₃	—	3	3	no	—
106	C₃²×Q₈	C₄×D₃	1	4	yes	—
107	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
108	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
109	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₃×Dic₃	—	3	1	no	—
111	C₂×C₃×Dic₃	—	3	1	no	—
112	C₂×C₃×Dic₃	—	3	3	no	—
113	C₃×D₁₂	—	3	12	no	—
114	C₃×C₄×D₃	—	3	12	no	—
115	C₃×C₄×D₃	—	6	6	no	—
116	C₃×C₄×D₃	—	6	6	no	—
117	C₂²×D₉	C₂³×C₃	1	3	yes	—
118	Q₈×C₉	—	2	4	no	—
119	Q₈×C₉	C₂×C₃×C₄	1	4	yes	complemented
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	2	3	no	—
122	C₂×C₄×C₉	—	2	3	no	—
123	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
124	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
125	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₉	C₂³×C₃	1	3	yes	—
127	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
129	D₃₆	C₂×C₃×C₄	1	12	yes	complemented
130	C₄×D₉	—	2	6	no	—
131	C₄×D₉	—	2	6	no	—
132	C₄×D₉	C₂×C₃×C₄	1	12	yes	complemented
133	C₄×#41	—	9	1	no	Sylow
134	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
135	C₂×3₋¹⁺²	C₂×#41	1	1	yes	characteristic
136	C₂×3₋¹⁺²	C₂×#41	1	1	yes	characteristic
137	AGL(1,ℤ/9)	—	2	6	no	—
138	C₃×#41	—	9	8	no	—
139	C₂×C₃×Q₈	—	3	1	no	—
140	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
141	C₂×C₃×D₄	—	9	3	no	—
142	C₂²×C₃×C₄	—	9	3	no	—
143	C₂²×C₃×C₄	—	9	3	no	—
144	C₂²×C₃×C₄	—	9	3	no	—
145	#239	—	3	8	no	—
146	C₂×D₁₂	—	3	3	no	—
147	C₂×C₄×D₃	—	3	3	no	—
148	C₂×C₄×D₃	—	3	3	no	—
149	C₂×C₄×D₃	—	3	3	no	—
150	C₃×C₄⋊C₄	—	3	3	no	—
151	C₃×C₄⋊C₄	—	9	3	no	—
152	C₃×C₄⋊C₄	C₂×C₃×D₃	1	3	yes	—
153	C₃×C₂²⋊C₄	—	9	6	no	—
154	C₃×C₄²	—	3	3	no	—
155	C₃×C₄²	—	9	1	no	—
156	C₃×C₄²	—	9	3	no	—
157	C₃×C₄²	—	9	3	no	—
158	C₃×C₄²	C₂×C₃×D₃	1	3	yes	—
159	(C₂×D₃)⋊C₄	—	3	6	no	—
160	C₄⋊Dic₃	—	3	3	no	—
161	C₄×Dic₃	—	3	1	no	—
162	C₄×Dic₃	—	3	3	no	—
163	C₄×Dic₃	—	3	3	no	—
164	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
165	C₂×C₃×D₃	—	3	6	no	—
166	C₂×C₃×D₃	—	6	3	no	—
167	C₂×C₃×D₃	—	6	3	no	—
168	C₃²×C₄	—	2	3	no	—
169	C₃²×C₄	C₂×C₄×D₃	1	6	yes	—
170	C₃²×C₄	#239	1	2	yes	—
171	C₃×Dic₃	—	3	2	no	—
172	C₃×Dic₃	—	3	2	no	—
173	C₃×Dic₃	—	6	3	no	—
174	C₂²×C₉	—	2	1	no	—
175	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
176	C₂×D₉	—	2	3	no	—
177	C₂×D₉	—	2	3	no	—
178	C₂×D₉	C₂²×C₃×C₄	1	6	yes	—
179	C₄×C₉	—	2	3	no	—
180	C₄×C₉	—	2	6	no	—
181	C₄×C₉	—	2	2	no	—
182	C₄×C₉	—	4	3	no	—
183	C₄×C₉	C₂²×C₃×C₄	1	6	yes	—
184	C₄×C₉	C₃×#41	1	2	yes	complemented
185	Dic₉	—	2	3	no	—
186	Dic₉	C₂²×C₃×C₄	1	2	yes	—
187	Dic₉	C₃×#41	1	2	yes	complemented
188	C₂×#41	—	9	1	no	—
189	C₄×Q₈	—	9	1	no	—
190	C₄×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
191	C₄×D₄	—	9	3	no	—
192	C₄×D₄	—	9	3	no	—
193	#117	—	9	3	no	—
194	C₂×C₄²	—	9	3	no	—
195	3₋¹⁺²	C₄×#41	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₃	—	9	3	no	—
197	C₂²×D₃	—	3	3	no	—
198	C₃×Q₈	—	3	4	no	—
199	C₃×Q₈	C₃×C₄×D₃	1	4	yes	—
200	C₃×D₄	—	9	12	no	—
201	C₂×C₃×C₄	—	3	3	no	—
202	C₂×C₃×C₄	—	3	3	no	—
203	C₂×C₃×C₄	—	3	1	no	—
204	C₂×C₃×C₄	—	9	12	no	—
205	C₂×C₃×C₄	—	9	1	no	—
206	C₂×C₃×C₄	—	9	1	no	—
207	C₂×C₃×C₄	—	9	3	no	—
208	C₂×C₃×C₄	—	18	6	no	—
209	C₂×C₃×C₄	—	18	6	no	—
210	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
211	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
212	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
213	C₂×Dic₃	—	3	1	no	—
214	C₂×Dic₃	—	3	1	no	—
215	C₂×Dic₃	—	3	3	no	—
216	D₁₂	—	3	12	no	—
217	C₄×D₃	—	3	12	no	—
218	C₄×D₃	—	6	6	no	—
219	C₄×D₃	—	6	6	no	—
220	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
221	C₂×C₃²	C₂×#239	1	1	yes	characteristic
222	C₂×C₃²	D₃×#41	1	1	yes	characteristic
223	C₃×D₃	—	6	6	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	—	2	1	no	—
227	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
228	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
229	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
230	D₉	—	2	6	no	—
231	#41	—	9	8	no	—
232	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
233	C₂×D₄	—	9	3	no	—
234	C₂²×C₄	—	9	3	no	—
235	C₂²×C₄	—	9	3	no	—
236	C₂²×C₄	—	9	3	no	—
237	C₄⋊C₄	—	9	3	no	—
238	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
239	C₂²⋊C₄	—	9	6	no	—
240	C₄²	—	9	3	no	—
241	C₄²	—	9	3	no	—
242	C₄²	—	9	1	no	—
243	C₄²	C₂×AGL(1,ℤ/9)	1	3	yes	—
244	C₂²×C₃	—	3	1	no	—
245	C₂²×C₃	—	9	6	no	—
246	C₂²×C₃	—	18	3	no	—
247	C₂²×C₃	—	18	3	no	—
248	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
249	C₂×D₃	—	3	6	no	—
250	C₂×D₃	—	6	3	no	—
251	C₂×D₃	—	6	3	no	—
252	C₃×C₄	—	2	3	no	—
253	C₃×C₄	—	3	6	no	—
254	C₃×C₄	—	3	2	no	—
255	C₃×C₄	—	6	3	no	—
256	C₃×C₄	—	9	2	no	—
257	C₃×C₄	—	9	2	no	—
258	C₃×C₄	—	18	3	no	—
259	C₃×C₄	C₂×C₃×C₄×D₃	1	6	yes	—
260	C₃×C₄	C₃×#239	1	2	yes	—
261	Dic₃	—	3	2	no	—
262	Dic₃	—	3	2	no	—
263	Dic₃	—	6	3	no	—
264	C₃²	C₄×#239	1	1	yes	characteristic
265	C₉	—	2	1	no	—
266	C₉	C₃×C₄×#41	1	1	yes	characteristic, complemented
267	C₂³	—	9	3	no	—
268	Q₈	C₄×AGL(1,ℤ/9)	1	4	yes	complemented
269	D₄	—	9	12	no	—
270	C₂×C₄	—	9	1	no	—
271	C₂×C₄	—	9	1	no	—
272	C₂×C₄	—	9	12	no	—
273	C₂×C₄	—	9	3	no	—
274	C₂×C₄	—	18	6	no	—
275	C₂×C₄	—	18	6	no	—
276	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
277	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
278	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
279	C₂×C₃	—	3	1	no	—
280	C₂×C₃	—	3	1	no	—
281	C₂×C₃	—	3	1	no	—
282	C₂×C₃	—	18	6	no	—
283	C₂×C₃	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
284	C₂×C₃	C₂×C₃×#239	1	1	yes	characteristic
285	C₂×C₃	C₃×D₃×#41	1	1	yes	characteristic
286	D₃	—	6	6	no	—
287	C₂²	—	9	6	no	—
288	C₂²	—	18	3	no	—
289	C₂²	—	18	3	no	—
290	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
291	C₄	—	2	3	no	—
292	C₄	—	9	2	no	—
293	C₄	—	9	2	no	—
294	C₄	—	18	3	no	—
295	C₄	C₂×C₄×AGL(1,ℤ/9)	1	6	yes	—
296	C₄	#88724	1	2	yes	direct product factor
297	C₃	—	3	1	no	—
298	C₃	C₃×C₄×#239	1	1	yes	characteristic
299	C₂	—	18	6	no	—
300	C₂	C₂²×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
301	C₂	C₂×#88724	1	1	yes	characteristic
302	C₂	#41×AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂³×C₃×C₄
Frattini, socle	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₄×Q₈×3₋¹⁺²	C₂
prime radical	C₂×#88724	C₂
cosocle	C₂²×C₉	C₂⁴×C₃
hypercenter	C₄×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×C₄×#41
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×3₋¹⁺²	C₂	semidirect
3. Q₈×3₋¹⁺²⋊C₄	C₂	semidirect
4. #63455⋊C₃	C₂	semidirect
5. #63454⋊C₃	C₂	semidirect
6. C₄×C₄⋊AGL(1,ℤ/9)	C₂	semidirect
7. C₄²×AGL(1,ℤ/9)	C₂	semidirect
9. C₄×#3701	C₃	semidirect
11. #88724	C₄	direct (N × H)
32. C₄×Q₈×C₉	C₂×C₃	semidirect
33. Q₈×Dic₉	C₂×C₃	semidirect
34. #63455	C₂×C₃	semidirect
35. #63454	C₂×C₃	semidirect
36. C₄×D₃₆	C₂×C₃	semidirect
37. C₄²×D₉	C₂×C₃	semidirect
39. Q₈×3₋¹⁺²	C₂×C₄	semidirect
46. C₄⋊AGL(1,ℤ/9)	C₂×C₄	semidirect
49. C₄×AGL(1,ℤ/9)	C₂×C₄	semidirect
67. #3701	C₃×C₄	semidirect
87. C₄×3₋¹⁺²	#41	semidirect
90. 3₋¹⁺²⋊C₄	#41	semidirect
119. Q₈×C₉	C₂×C₃×C₄	semidirect
129. D₃₆	C₂×C₃×C₄	semidirect
132. C₄×D₉	C₂×C₃×C₄	semidirect
184. C₄×C₉	C₃×#41	semidirect
187. Dic₉	C₃×#41	semidirect
190. C₄×Q₈	AGL(1,ℤ/9)	semidirect
195. 3₋¹⁺²	C₄×#41	semidirect
266. C₉	C₃×C₄×#41	semidirect
268. Q₈	C₄×AGL(1,ℤ/9)	semidirect
296. C₄	#88724	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#88724	2
C₄×Q₈×3₋¹⁺²	2
Q₈×3₋¹⁺²⋊C₄	2
#63455⋊C₃	2
#63454⋊C₃	2
C₄×C₄⋊AGL(1,ℤ/9)	2
C₄²×AGL(1,ℤ/9)	2
C₃×C₄×#239	3
C₄×#3701	3

Subgroups — C4×#88724 #13152734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88724 #13152734