Subgroups — #10589742

Order 576.

324 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#63509	C₂	1	1	yes	maximal, characteristic, complemented
3	#63506	C₂	1	1	yes	maximal, characteristic, complemented
4	#63491	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#611	C₂	1	1	yes	maximal, characteristic, complemented
6	#63460	C₂	1	1	yes	maximal, characteristic, complemented
7	#63458	C₂	1	1	yes	maximal, characteristic, complemented
8	#63458	C₂	1	1	yes	maximal, characteristic, complemented
9	#63456	C₂	1	1	yes	maximal, characteristic, complemented
10	#63454	C₂	1	1	yes	maximal, characteristic, complemented
11	#63449	C₂	1	1	yes	maximal, characteristic, complemented
12	#63447	C₂	1	1	yes	maximal, characteristic, complemented
13	#63447	C₂	1	1	yes	maximal, characteristic, complemented
14	#63446	C₂	1	1	yes	maximal, characteristic
15	#63441	C₂	1	1	yes	maximal, characteristic, complemented
16	#5640	—	3	1	no	maximal
17	C₂×Q₈×C₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₉	C₂²	1	1	yes	characteristic
19	C₂×#611	C₂²	1	1	yes	characteristic
20	C₂×#611	C₂²	1	1	yes	characteristic
21	C₂×D₃₆	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₈	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₈	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	#3676	C₂²	1	1	yes	characteristic
32	#3676	C₂²	1	1	yes	characteristic
33	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
34	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
35	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
36	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
37	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
38	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
39	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
40	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
41	C₄⋊Dic₉	C₂²	1	1	yes	characteristic
42	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
43	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
44	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
45	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
46	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
47	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×Dic₉	C₂²	1	1	yes	characteristic
49	C₄×Dic₉	C₂²	1	1	yes	characteristic
50	C₄×Dic₉	C₂²	1	1	yes	characteristic
51	C₄×Dic₉	C₂²	1	1	yes	characteristic
52	C₃×C₂²⋊Q₈	D₃	1	1	yes	characteristic
53	#947	—	3	1	no	—
54	#944	—	3	1	no	—
55	#929	—	3	1	no	—
56	C₄×#67	—	3	1	no	—
57	#898	—	3	1	no	—
58	#896	—	3	1	no	—
59	#896	—	3	1	no	—
60	#894	—	3	1	no	—
61	#892	—	3	1	no	—
62	#887	—	3	1	no	—
63	#885	—	3	1	no	—
64	#885	—	3	1	no	—
65	#884	—	3	1	no	—
66	#879	—	3	1	no	—
67	C₂³×C₉	C₂³	1	1	yes	characteristic
68	C₂²×D₉	C₂³	1	1	yes	characteristic
69	C₂²×D₉	C₂³	1	1	yes	characteristic
70	Q₈×C₉	—	4	1	no	—
71	C₂×C₄×C₉	—	4	1	no	—
72	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
74	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
76	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
78	#611	—	2	2	no	—
79	#611	—	4	1	no	—
80	C₂×Dic₉	C₂³	1	1	yes	characteristic
81	C₂×Dic₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₉	C₂³	1	1	yes	characteristic
83	C₂×Dic₉	C₂³	1	1	yes	characteristic
84	C₂×Dic₉	C₂³	1	1	yes	characteristic
85	C₂×Dic₉	C₂³	1	1	yes	characteristic
86	D₃₆	—	4	1	no	—
87	C₄×D₉	—	2	2	no	—
88	C₄×D₉	—	4	1	no	—
89	Dic₁₈	—	2	2	no	—
90	Dic₁₈	—	4	1	no	—
91	#542	—	9	1	no	Sylow
92	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
94	C₂×#67	—	3	1	no	—
95	C₂×#67	—	3	1	no	—
96	C₂×D₁₂	—	3	1	no	—
97	C₂×C₄×D₃	—	3	1	no	—
98	C₂×C₄×D₃	—	3	1	no	—
99	C₂×Dic₆	—	3	1	no	—
100	C₂×Dic₆	—	3	1	no	—
101	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
102	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
103	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
104	C₃×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
105	C₃×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
106	#217	—	3	1	no	—
107	#217	—	3	1	no	—
108	(C₂×D₃)⋊C₄	—	3	1	no	—
109	(C₂×D₃)⋊C₄	—	3	1	no	—
110	(C₂×D₃)⋊C₄	—	3	1	no	—
111	(C₂×D₃)⋊C₄	—	3	1	no	—
112	(C₂×D₃)⋊C₄	—	3	1	no	—
113	(C₂×D₃)⋊C₄	—	3	1	no	—
114	(C₂×D₃)⋊C₄	—	3	1	no	—
115	(C₂×D₃)⋊C₄	—	3	1	no	—
116	C₄⋊Dic₃	—	3	1	no	—
117	Dic₃⋊C₄	—	3	1	no	—
118	Dic₃⋊C₄	—	3	1	no	—
119	Dic₃⋊C₄	—	3	1	no	—
120	Dic₃⋊C₄	—	3	1	no	—
121	Dic₃⋊C₄	—	3	1	no	—
122	Dic₃⋊C₄	—	3	1	no	—
123	C₄×Dic₃	—	3	1	no	—
124	C₄×Dic₃	—	3	1	no	—
125	C₄×Dic₃	—	3	1	no	—
126	C₄×Dic₃	—	3	1	no	—
127	C₂²×C₉	—	2	1	no	—
128	C₂²×C₉	—	2	1	no	—
129	C₂²×C₉	—	2	1	no	—
130	C₂²×C₉	C₂⁴	1	1	yes	characteristic
131	C₂×D₉	—	2	1	no	—
132	C₂×D₉	—	2	1	no	—
133	C₂×D₉	—	2	1	no	—
134	C₂×D₉	—	2	1	no	—
135	C₂×D₉	—	2	1	no	—
136	C₂×D₉	—	2	1	no	—
137	C₄×C₉	—	2	1	no	—
138	C₄×C₉	—	2	1	no	—
139	C₄×C₉	—	2	1	no	—
140	C₄×C₉	—	2	1	no	—
141	C₄×C₉	—	2	1	no	—
142	C₄×C₉	—	2	1	no	—
143	Dic₉	—	2	1	no	—
144	Dic₉	—	2	1	no	—
145	Dic₉	—	2	1	no	—
146	Dic₉	—	2	1	no	—
147	Dic₉	—	2	1	no	—
148	Dic₉	#41	1	2	yes	—
149	C₄⋊Q₈	—	9	1	no	—
150	#126	—	9	1	no	—
151	#126	—	9	1	no	—
152	#124	—	9	1	no	—
153	#124	—	9	1	no	—
154	#124	—	9	1	no	—
155	#123	—	9	1	no	—
156	#123	—	9	1	no	—
157	C₂²⋊Q₈	—	9	1	no	—
158	C₂²⋊Q₈	—	9	1	no	—
159	C₂²⋊Q₈	D₉	1	1	yes	characteristic, complemented
160	#121	—	9	1	no	—
161	C₄×Q₈	—	9	1	no	—
162	C₄×D₄	—	9	1	no	—
163	#117	—	9	1	no	—
164	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
165	C₂²×D₃	—	3	1	no	—
166	C₂²×D₃	—	3	1	no	—
167	C₃×Q₈	—	4	1	no	—
168	C₂×C₃×C₄	—	4	1	no	—
169	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
175	#67	—	6	2	no	—
176	#67	—	12	1	no	—
177	C₂×Dic₃	—	3	1	no	—
178	C₂×Dic₃	—	3	1	no	—
179	C₂×Dic₃	—	3	1	no	—
180	C₂×Dic₃	—	3	1	no	—
181	C₂×Dic₃	—	3	1	no	—
182	C₂×Dic₃	—	3	1	no	—
183	D₁₂	—	12	1	no	—
184	C₄×D₃	—	6	2	no	—
185	C₄×D₃	—	12	1	no	—
186	Dic₆	—	6	2	no	—
187	Dic₆	—	12	1	no	—
188	C₂×C₉	—	4	1	no	—
189	C₂×C₉	C₂×#41	1	1	yes	characteristic
190	C₂×C₉	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×C₉	2₋¹⁺⁴	1	1	yes	characteristic
192	D₉	—	4	1	no	—
193	D₉	—	4	1	no	—
194	C₂×Q₈	—	9	1	no	—
195	C₂×Q₈	—	9	1	no	—
196	C₂×Q₈	C₂×D₉	1	1	yes	characteristic
197	C₂×D₄	—	9	1	no	—
198	C₂×D₄	—	9	1	no	—
199	C₂×D₄	—	9	1	no	—
200	C₂²×C₄	—	9	1	no	—
201	C₂²×C₄	—	9	1	no	—
202	C₂²×C₄	C₂×D₉	1	1	yes	characteristic
203	C₄⋊C₄	—	9	1	no	—
204	C₄⋊C₄	—	9	1	no	—
205	C₄⋊C₄	—	9	1	no	—
206	C₄⋊C₄	—	9	1	no	—
207	C₄⋊C₄	—	9	1	no	—
208	C₄⋊C₄	—	9	1	no	—
209	C₄⋊C₄	—	9	1	no	—
210	C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
211	C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
212	C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
213	C₂²⋊C₄	—	9	1	no	—
214	C₂²⋊C₄	—	9	1	no	—
215	C₂²⋊C₄	—	9	1	no	—
216	C₂²⋊C₄	—	9	1	no	—
217	C₂²⋊C₄	—	9	1	no	—
218	C₂²⋊C₄	—	9	1	no	—
219	C₂²⋊C₄	—	9	1	no	—
220	C₂²⋊C₄	—	9	1	no	—
221	C₂²⋊C₄	—	9	1	no	—
222	C₂²⋊C₄	—	9	1	no	—
223	C₂²⋊C₄	C₂×D₉	1	1	yes	characteristic
224	C₂²⋊C₄	C₂×D₉	1	1	yes	characteristic
225	C₄²	—	9	1	no	—
226	C₄²	—	9	1	no	—
227	C₄²	—	9	1	no	—
228	C₄²	—	9	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
233	C₂×D₃	—	6	1	no	—
234	C₂×D₃	—	6	1	no	—
235	C₂×D₃	—	6	1	no	—
236	C₂×D₃	—	6	1	no	—
237	C₂×D₃	—	6	1	no	—
238	C₂×D₃	—	6	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	Dic₃	—	3	2	no	—
246	Dic₃	—	6	1	no	—
247	Dic₃	—	6	1	no	—
248	Dic₃	—	6	1	no	—
249	Dic₃	—	6	1	no	—
250	Dic₃	—	6	1	no	—
251	C₉	#542	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	9	1	no	—
253	C₂³	—	9	1	no	—
254	C₂³	C₂²×D₉	1	1	yes	characteristic
255	Q₈	—	4	1	no	—
256	Q₈	—	18	2	no	—
257	Q₈	—	36	1	no	—
258	D₄	—	18	2	no	—
259	D₄	—	36	1	no	—
260	D₄	—	36	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	9	1	no	—
263	C₂×C₄	—	9	1	no	—
264	C₂×C₄	—	9	1	no	—
265	C₂×C₄	—	9	1	no	—
266	C₂×C₄	—	9	1	no	—
267	C₂×C₄	—	9	1	no	—
268	C₂×C₄	—	18	2	no	—
269	C₂×C₄	—	36	1	no	—
270	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
276	C₂×C₃	—	4	1	no	—
277	C₂×C₃	#1004	1	1	yes	characteristic
278	C₂×C₃	#1007	1	1	yes	characteristic
279	C₂×C₃	D₃×#41	1	1	yes	characteristic
280	D₃	—	12	1	no	—
281	D₃	—	12	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	18	1	no	—
286	C₂²	—	18	1	no	—
287	C₂²	—	18	1	no	—
288	C₂²	—	18	1	no	—
289	C₂²	—	18	1	no	—
290	C₂²	—	18	1	no	—
291	C₂²	C₂³×D₉	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	9	2	no	—
299	C₄	—	18	1	no	—
300	C₄	—	18	1	no	—
301	C₄	—	18	1	no	—
302	C₄	—	18	1	no	—
303	C₄	—	18	1	no	—
304	C₃	#5640	1	1	yes	characteristic
305	C₂	—	4	1	no	—
306	C₂	—	36	1	no	—
307	C₂	—	36	1	no	—
308	C₂	#63710	1	1	yes	characteristic
309	C₂	#63713	1	1	yes	characteristic
310	C₂	#41×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴
Frattini, socle	C₂²×C₃	C₂³×D₃
Fitting	C₉×C₂²⋊Q₈	C₂
prime radical	#63446	C₂
hypercenter	C₂²⋊Q₈	D₉
hypocenter	C₉	#542
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₉×C₂²⋊Q₈	C₂	semidirect
2. #63509	C₂	semidirect
3. #63506	C₂	semidirect
4. #63491	C₂	semidirect
5. C₄×#611	C₂	semidirect
6. #63460	C₂	semidirect
7. #63458	C₂	semidirect
8. #63458	C₂	semidirect
9. #63456	C₂	semidirect
10. #63454	C₂	semidirect
11. #63449	C₂	semidirect
12. #63447	C₂	semidirect
13. #63447	C₂	semidirect
15. #63441	C₂	semidirect
159. C₂²⋊Q₈	D₉	semidirect
251. C₉	#542	semidirect

Maximal subgroups

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

subgroup	index
C₉×C₂²⋊Q₈	2
#63509	2
#63506	2
#63491	2
C₄×#611	2
#63460	2
#63458	2
#63456	2
#63454	2
#63449	2
#63447	2
#63446	2
#63441	2
#5640	3