Subgroups — C₂×#10588799 #11858634

Order 1152.

690 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3667	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3667	C₂	1	1	yes	maximal, characteristic
4	#10588799	C₂	1	8	yes	maximal, direct product factor
5	C₂×#63381	C₂	1	4	yes	maximal, complemented
6	C₂×#5124	—	3	1	no	maximal
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×M₁₆	—	2	1	no	—
9	C₂×C₉×M₁₆	—	2	2	no	—
10	C₂×C₉×M₁₆	C₂²	1	2	yes	—
11	C₂×C₉×M₁₆	C₂²	1	4	yes	—
12	C₂²×C₈×C₉	—	2	1	no	—
13	C₂×#3667	—	2	1	no	—
14	C₂×#3667	—	2	2	no	—
15	C₂×#3667	C₄	1	2	yes	—
16	C₂×#3667	C₄	1	2	yes	—
17	C₂×#3667	C₄	1	4	yes	—
18	C₂×#3667	C₂²	1	2	yes	—
19	C₂×#3667	C₂²	1	4	yes	—
20	C₂×#3667	C₂²	1	4	yes	—
21	C₂²×C₉⋊C₈	—	2	1	no	—
22	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
23	#63381	C₂²	1	16	yes	complemented
24	C₂²×C₃×M₁₆	D₃	1	1	yes	characteristic
25	C₂²×#208	—	3	1	no	—
26	C₂²×#208	—	3	1	no	—
27	#5124	—	3	8	no	—
28	C₂×#822	—	3	4	no	—
29	C₂⁴×C₉	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	D₄	1	2	yes	—
36	C₂²×C₄×C₉	Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₉	C₂³	1	2	yes	—
40	C₉×M₁₆	—	2	4	no	—
41	C₉×M₁₆	—	2	2	no	—
42	C₉×M₁₆	C₂³	1	4	yes	—
43	C₂×C₈×C₉	—	2	2	no	—
44	C₂×C₈×C₉	—	2	4	no	—
45	#3667	—	2	4	no	—
46	#3667	—	2	2	no	—
47	#3667	C₂×C₄	1	8	yes	—
48	#3667	C₂×C₄	1	8	yes	—
49	#3667	C₂³	1	4	yes	—
50	C₂×C₉⋊C₈	—	2	2	no	—
51	C₂×C₉⋊C₈	—	2	4	no	—
52	C₂×C₉⋊C₈	C₂×C₄	1	4	yes	—
53	C₂×C₉⋊C₈	C₂×C₄	1	4	yes	—
54	C₂×C₉⋊C₈	C₂³	1	4	yes	—
55	C₂×#430	—	9	1	no	Sylow
56	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×M₁₆	—	2	1	no	—
58	C₂×C₃×M₁₆	—	2	2	no	—
59	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
60	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
61	C₂²×C₃×C₈	—	2	1	no	—
62	C₂×#208	—	3	2	no	—
63	C₂×#208	—	3	2	no	—
64	C₂×#208	—	3	4	no	—
65	C₂×#208	—	3	4	no	—
66	C₂×#208	—	3	2	no	—
67	C₂×#208	—	3	4	no	—
68	C₂×#208	—	6	1	no	—
69	C₂×#208	—	6	2	no	—
70	C₂²×C₃⋊C₈	—	3	2	no	—
71	C₂²×C₃⋊C₈	—	6	1	no	—
72	#822	—	3	16	no	—
73	C₂³×C₉	—	4	2	no	—
74	C₂³×C₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂³×C₉	C₄⋊C₄	1	1	yes	characteristic
76	C₂³×C₉	C₄⋊C₄	1	2	yes	—
77	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
78	C₂³×C₉	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
97	C₈×C₉	—	2	4	no	—
98	C₉⋊C₈	—	2	4	no	—
99	C₉⋊C₈	C₂²×C₄	1	8	yes	—
100	C₂²×M₁₆	—	9	1	no	—
101	C₂²×M₁₆	—	9	1	no	—
102	C₂²×M₁₆	D₉	1	1	yes	characteristic
103	#430	—	9	8	no	—
104	C₂×#108	—	9	4	no	—
105	C₂⁴×C₃	Dic₆	1	1	yes	characteristic
106	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
107	C₂²×C₃×C₄	Dic₆	1	2	yes	—
108	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
111	C₂²×C₃×C₄	#67	1	1	yes	characteristic
112	C₂²×C₃×C₄	#67	1	1	yes	characteristic
113	C₂²×C₃×C₄	#67	1	2	yes	—
114	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
116	C₃×M₁₆	—	2	4	no	—
117	C₃×M₁₆	—	2	2	no	—
118	C₃×M₁₆	C₂²×D₃	1	4	yes	—
119	C₂×C₃×C₈	—	2	4	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	#208	—	3	8	no	—
122	#208	—	3	8	no	—
123	#208	—	3	4	no	—
124	#208	—	6	4	no	—
125	#208	—	6	2	no	—
126	C₂×C₃⋊C₈	—	3	4	no	—
127	C₂×C₃⋊C₈	—	3	4	no	—
128	C₂×C₃⋊C₈	—	3	4	no	—
129	C₂×C₃⋊C₈	—	6	4	no	—
130	C₂×C₃⋊C₈	—	6	2	no	—
131	C₂²×C₉	—	2	2	no	—
132	C₂²×C₉	—	2	2	no	—
133	C₂²×C₉	—	2	2	no	—
134	C₂²×C₉	—	4	2	no	—
135	C₂²×C₉	—	4	2	no	—
136	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₉	C₂×C₄⋊C₄	1	2	yes	—
138	C₂²×C₉	C₂×C₄⋊C₄	1	2	yes	—
139	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
141	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₉	C₂×C₄⋊C₄	1	2	yes	—
144	C₄×C₉	C₂×C₄⋊C₄	1	2	yes	—
145	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
146	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
147	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	1	no	—
150	C₂×M₁₆	—	9	2	no	—
151	C₂×M₁₆	—	9	2	no	—
152	C₂×M₁₆	—	9	4	no	—
153	C₂×M₁₆	—	9	4	no	—
154	C₂×M₁₆	—	9	2	no	—
155	C₂×M₁₆	—	9	4	no	—
156	C₂×M₁₆	—	18	2	no	—
157	C₂×M₁₆	—	18	1	no	—
158	C₂×M₁₆	C₂×D₉	1	2	yes	—
159	C₂×M₁₆	C₂×D₉	1	4	yes	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	9	2	no	—
162	C₂²×C₈	—	18	1	no	—
163	#108	—	9	16	no	—
164	C₂³×C₃	—	4	2	no	—
165	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
166	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
167	C₂³×C₃	Dic₃⋊C₄	1	2	yes	—
168	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
169	C₂³×C₃	C₂×Dic₆	1	2	yes	—
170	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
171	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
172	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
173	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
174	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
175	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
179	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
182	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
183	C₂×C₃×C₄	C₂×#67	1	2	yes	—
184	C₂×C₃×C₄	C₂×#67	1	2	yes	—
185	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
188	C₃×C₈	—	2	4	no	—
189	C₃⋊C₈	—	3	8	no	—
190	C₃⋊C₈	—	6	4	no	—
191	C₂×C₉	—	2	2	no	—
192	C₂×C₉	—	2	2	no	—
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	—	2	1	no	—
195	C₂×C₉	#430	1	2	yes	complemented
196	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂⁴	Dic₁₈	1	1	yes	characteristic
198	C₂²×C₄	Dic₁₈	1	1	yes	characteristic
199	C₂²×C₄	Dic₁₈	1	2	yes	—
200	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
201	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
202	C₂²×C₄	C₄×D₉	1	2	yes	—
203	C₂²×C₄	#611	1	2	yes	—
204	C₂²×C₄	#611	1	1	yes	characteristic
205	C₂²×C₄	#611	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₉	1	2	yes	—
207	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	9	8	no	—
211	M₁₆	—	9	8	no	—
212	M₁₆	—	9	4	no	—
213	M₁₆	—	18	4	no	—
214	M₁₆	—	18	2	no	—
215	M₁₆	C₂²×D₉	1	4	yes	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	9	4	no	—
219	C₂×C₈	—	9	4	no	—
220	C₂×C₈	—	9	4	no	—
221	C₂×C₈	—	18	4	no	—
222	C₂×C₈	—	18	2	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	4	2	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
229	C₂²×C₃	C₂×Dic₃⋊C₄	1	2	yes	—
230	C₂²×C₃	C₂×Dic₃⋊C₄	1	2	yes	—
231	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
232	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
233	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
234	C₃×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
235	C₃×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
236	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
237	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
238	C₃×C₄	C₂²×#67	1	1	yes	characteristic
239	C₉	C₂×#430	1	1	yes	characteristic, complemented, Sylow
240	C₂³	—	4	2	no	—
241	C₂³	Dic₉⋊C₄	1	1	yes	characteristic
242	C₂³	Dic₉⋊C₄	1	1	yes	characteristic
243	C₂³	Dic₉⋊C₄	1	2	yes	—
244	C₂³	C₂×Dic₁₈	1	1	yes	characteristic
245	C₂³	C₂×Dic₁₈	1	2	yes	—
246	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
247	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
248	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
249	C₂×C₄	Dic₉⋊C₄	1	1	yes	characteristic
250	C₂×C₄	Dic₉⋊C₄	1	1	yes	characteristic
251	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
252	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
253	C₂×C₄	C₂×Dic₁₈	1	2	yes	—
254	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
258	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
259	C₂×C₄	C₂×#611	1	2	yes	—
260	C₂×C₄	C₂×#611	1	2	yes	—
261	C₂×C₄	C₂×#611	1	1	yes	characteristic
262	C₂×C₄	C₂×#611	1	1	yes	characteristic
263	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
264	C₈	—	2	4	no	—
265	C₈	—	9	8	no	—
266	C₈	—	18	4	no	—
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	2	2	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	#5124	1	2	yes	—
272	C₂×C₃	C₂²×Dic₃⋊C₄	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
279	C₂²	C₂×Dic₉⋊C₄	1	2	yes	—
280	C₂²	C₂×Dic₉⋊C₄	1	2	yes	—
281	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
282	C₂²	C₂²×Dic₁₈	1	1	yes	characteristic
283	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
284	C₄	C₂×Dic₉⋊C₄	1	2	yes	—
285	C₄	C₂×Dic₉⋊C₄	1	2	yes	—
286	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
287	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
288	C₄	C₂²×#611	1	1	yes	characteristic
289	C₃	C₂×#5124	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	1	no	—
294	C₂	#10588799	1	2	yes	direct product factor
295	C₂	C₂²×Dic₉⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂²×C₉×M₁₆	C₂
socle	C₂²×C₃	C₂×Dic₃⋊C₄
prime radical	C₂⁴×C₃	Dic₆
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂²×M₁₆	D₉
hypocenter	C₉	C₂×#430
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10588799	C₂	direct (N × H)
5. C₂×#63381	C₂	semidirect
23. #63381	C₂²	semidirect
195. C₂×C₉	#430	semidirect
239. C₉	C₂×#430	semidirect
294. C₂	#10588799	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×M₁₆	2
C₂²×#3667	2
#10588799	2
C₂×#63381	2
C₂×#5124	3

Subgroups — C2×#10588799 #11858634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588799 #11858634