Subgroups — C₂×#10588821 #11867963

Order 1152.

672 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	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#63511	C₂	1	2	yes	maximal
3	C₂²×Dic₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588821	C₂	1	8	yes	maximal, direct product factor
5	C₂×#63382	C₂	1	2	yes	maximal, complemented
6	C₂×#63372	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#5143	—	3	1	no	maximal
8	C₂×C₉×#41	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₈	C₂²	1	2	yes	—
10	C₉×#131	—	2	2	no	—
11	C₉×#131	C₂²	1	4	yes	—
12	C₂×C₉×M₁₆	—	2	1	no	—
13	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₉	—	2	1	no	—
15	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
16	#63511	—	2	4	no	—
17	#63511	C₂²	1	8	yes	—
18	C₂×#3674	—	2	2	no	—
19	C₂×#3672	—	2	2	no	—
20	C₂×#3667	C₂²	1	2	yes	—
21	C₂×Dic₃₆	—	2	1	no	—
22	C₂×Dic₃₆	—	2	1	no	—
23	C₂×Dic₃₆	—	2	2	no	—
24	C₂×Dic₃₆	C₂²	1	4	yes	complemented
25	#63382	C₂²	1	8	yes	complemented
26	#63372	C₂²	1	4	yes	complemented
27	C₂×C₃×#131	D₃	1	1	yes	characteristic
28	C₂×#951	—	3	2	no	—
29	C₂²×Dic₁₂	—	3	1	no	—
30	#5143	—	3	8	no	—
31	C₂×#824	—	3	2	no	—
32	C₂×#819	—	3	1	no	—
33	C₉×#41	—	2	1	no	—
34	C₉×#41	—	2	1	no	—
35	C₉×#41	D₄	1	2	yes	—
36	C₉×#41	C₂³	1	2	yes	—
37	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
38	C₂×D₄×C₉	—	2	1	no	—
39	C₂×D₄×C₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂²×Dic₉	—	2	2	no	—
43	C₂×Dic₁₈	—	2	2	no	—
44	C₂×Dic₁₈	—	2	2	no	—
45	C₂×Dic₁₈	—	2	2	no	—
46	C₂×Dic₁₈	—	2	2	no	—
47	C₂×Dic₁₈	—	2	2	no	—
48	C₂×Dic₁₈	C₂³	1	4	yes	—
49	C₉×M₁₆	—	2	1	no	—
50	C₉×M₁₆	—	2	4	no	—
51	C₉×M₁₆	C₂³	1	2	yes	—
52	C₂×C₈×C₉	—	2	1	no	—
53	C₂×C₈×C₉	—	2	4	no	—
54	C₂×C₈×C₉	D₄	1	1	yes	characteristic
55	C₂×C₈×C₉	D₄	1	1	yes	characteristic
56	C₂×C₈×C₉	D₄	1	2	yes	—
57	C₂×C₈×C₉	C₂³	1	2	yes	—
58	#3674	—	2	8	no	—
59	#3672	—	2	8	no	—
60	#3667	—	2	2	no	—
61	#3667	C₂³	1	4	yes	—
62	C₂×C₉⋊C₈	—	2	2	no	—
63	Dic₃₆	—	2	4	no	—
64	Dic₃₆	—	2	4	no	—
65	C₂×#473	—	9	1	no	Sylow
66	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
67	C₂²×Dic₆	—	3	2	no	—
68	C₃×#131	—	2	2	no	—
69	C₃×#131	C₂×D₃	1	4	yes	—
70	C₂×C₃×M₁₆	—	2	1	no	—
71	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
72	C₂²×C₃×C₈	—	2	1	no	—
73	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
74	#951	—	3	8	no	—
75	#951	—	6	4	no	—
76	C₂×#216	—	6	2	no	—
77	C₂×#214	—	6	2	no	—
78	C₂×#208	—	3	2	no	—
79	C₂×Dic₁₂	—	3	4	no	—
80	C₂×Dic₁₂	—	6	1	no	—
81	C₂×Dic₁₂	—	6	1	no	—
82	C₂×Dic₁₂	—	6	2	no	—
83	#824	—	3	8	no	—
84	#819	—	3	4	no	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	#41	1	1	yes	characteristic
87	Q₈×C₉	—	2	1	no	—
88	Q₈×C₉	C₂×D₄	1	2	yes	—
89	D₄×C₉	—	2	2	no	—
90	D₄×C₉	—	2	2	no	—
91	D₄×C₉	—	2	1	no	—
92	D₄×C₉	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₉	#41	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₉	—	2	4	no	—
103	C₂×Dic₉	—	4	2	no	—
104	C₂×Dic₉	—	4	2	no	—
105	Dic₁₈	—	2	4	no	—
106	Dic₁₈	—	2	4	no	—
107	Dic₁₈	—	4	2	no	—
108	Dic₁₈	—	4	2	no	—
109	C₈×C₉	—	2	2	no	—
110	C₈×C₉	C₂×D₄	1	2	yes	—
111	C₈×C₉	C₂×D₄	1	2	yes	—
112	C₉⋊C₈	—	2	4	no	—
113	C₂×#137	—	9	2	no	—
114	C₂²×Q₁₆	—	9	1	no	—
115	C₂×#131	D₉	1	1	yes	characteristic
116	#473	—	9	8	no	—
117	C₂×#108	—	9	1	no	—
118	C₂×#101	—	9	2	no	—
119	C₃×#41	—	2	1	no	—
120	C₃×#41	—	2	1	no	—
121	C₃×#41	D₁₂	1	2	yes	—
122	C₃×#41	C₂²×D₃	1	2	yes	—
123	C₂×C₃×Q₈	D₁₂	1	1	yes	characteristic
124	C₂×C₃×D₄	—	2	1	no	—
125	C₂×C₃×D₄	D₁₂	1	1	yes	characteristic
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
128	C₂²×Dic₃	—	6	2	no	—
129	C₂×Dic₆	—	3	4	no	—
130	C₂×Dic₆	—	6	2	no	—
131	C₂×Dic₆	—	6	2	no	—
132	C₂×Dic₆	—	6	2	no	—
133	C₂×Dic₆	—	6	2	no	—
134	C₂×Dic₆	—	6	2	no	—
135	C₃×M₁₆	—	2	1	no	—
136	C₃×M₁₆	—	2	4	no	—
137	C₃×M₁₆	C₂²×D₃	1	2	yes	—
138	C₂×C₃×C₈	—	2	1	no	—
139	C₂×C₃×C₈	—	2	4	no	—
140	C₂×C₃×C₈	#67	1	1	yes	characteristic
141	C₂×C₃×C₈	#67	1	1	yes	characteristic
142	C₂×C₃×C₈	#67	1	2	yes	—
143	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
144	#216	—	6	8	no	—
145	#214	—	6	8	no	—
146	#208	—	3	4	no	—
147	#208	—	6	2	no	—
148	C₂×C₃⋊C₈	—	6	2	no	—
149	Dic₁₂	—	6	4	no	—
150	Dic₁₂	—	6	4	no	—
151	C₂²×C₉	—	2	2	no	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	4	2	no	—
154	C₂²×C₉	#121	1	1	yes	characteristic
155	C₂²×C₉	#121	1	1	yes	characteristic
156	C₂²×C₉	C₂×#41	1	1	yes	characteristic
157	C₄×C₉	—	2	2	no	—
158	C₄×C₉	#121	1	1	yes	characteristic
159	C₄×C₉	#121	1	1	yes	characteristic
160	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
161	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
162	Dic₉	—	4	4	no	—
163	C₂×#41	C₂×D₉	1	1	yes	characteristic
164	C₂²×Q₈	—	9	2	no	—
165	#137	—	9	8	no	—
166	#137	—	18	4	no	—
167	C₂×Q₁₆	—	9	4	no	—
168	C₂×Q₁₆	—	18	2	no	—
169	C₂×Q₁₆	—	18	1	no	—
170	C₂×Q₁₆	—	18	1	no	—
171	C₂×Q₁₆	—	18	2	no	—
172	C₂×SD₁₆	—	18	2	no	—
173	#131	—	2	2	no	—
174	#131	C₂×D₉	1	4	yes	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	9	2	no	—
177	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
180	#108	—	9	4	no	—
181	#101	—	9	8	no	—
182	C₂³×C₃	—	2	1	no	—
183	C₂³×C₃	#235	1	1	yes	characteristic
184	C₃×Q₈	—	2	1	no	—
185	C₃×Q₈	C₂×D₁₂	1	2	yes	—
186	C₃×D₄	—	2	2	no	—
187	C₃×D₄	—	2	2	no	—
188	C₃×D₄	—	2	1	no	—
189	C₃×D₄	C₂×D₁₂	1	2	yes	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
195	C₂×C₃×C₄	#235	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
199	C₂×Dic₃	—	6	4	no	—
200	C₂×Dic₃	—	12	2	no	—
201	C₂×Dic₃	—	12	2	no	—
202	Dic₆	—	6	4	no	—
203	Dic₆	—	6	4	no	—
204	Dic₆	—	12	2	no	—
205	Dic₆	—	12	2	no	—
206	C₃×C₈	—	2	2	no	—
207	C₃×C₈	C₂×#67	1	2	yes	—
208	C₃×C₈	C₂×#67	1	2	yes	—
209	C₃⋊C₈	—	6	4	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	—	4	2	no	—
213	C₂×C₉	#473	1	2	yes	complemented
214	C₂×C₉	C₂×#121	1	1	yes	characteristic
215	#41	—	2	1	no	—
216	#41	—	2	1	no	—
217	#41	D₃₆	1	2	yes	—
218	#41	C₂²×D₉	1	2	yes	—
219	C₂×Q₈	—	9	4	no	—
220	C₂×Q₈	—	18	2	no	—
221	C₂×Q₈	—	18	2	no	—
222	C₂×Q₈	—	18	2	no	—
223	C₂×Q₈	—	18	2	no	—
224	C₂×Q₈	—	18	2	no	—
225	C₂×Q₈	D₃₆	1	1	yes	characteristic
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	D₃₆	1	1	yes	characteristic
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	18	2	no	—
230	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
231	Q₁₆	—	18	8	no	—
232	Q₁₆	—	18	4	no	—
233	Q₁₆	—	18	4	no	—
234	SD₁₆	—	18	8	no	—
235	M₁₆	—	2	1	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	9	4	no	—
238	M₁₆	—	18	2	no	—
239	M₁₆	C₂²×D₉	1	2	yes	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	18	2	no	—
243	C₂×C₈	#611	1	1	yes	characteristic
244	C₂×C₈	#611	1	1	yes	characteristic
245	C₂×C₈	#611	1	2	yes	—
246	C₂×C₈	C₂²×D₉	1	2	yes	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	4	2	no	—
250	C₂²×C₃	#930	1	1	yes	characteristic
251	C₂²×C₃	#930	1	1	yes	characteristic
252	C₂²×C₃	C₂×#235	1	1	yes	characteristic
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	#930	1	1	yes	characteristic
255	C₃×C₄	#930	1	1	yes	characteristic
256	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
257	C₃×C₄	C₂²×#67	1	1	yes	characteristic
258	Dic₃	—	12	4	no	—
259	C₉	C₂×#473	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	2	1	no	—
261	C₂³	#3697	1	1	yes	characteristic
262	Q₈	—	2	1	no	—
263	Q₈	—	18	4	no	—
264	Q₈	—	18	4	no	—
265	Q₈	—	36	2	no	—
266	Q₈	—	36	2	no	—
267	Q₈	C₂×D₃₆	1	2	yes	—
268	D₄	—	2	2	no	—
269	D₄	—	2	2	no	—
270	D₄	—	2	1	no	—
271	D₄	C₂×D₃₆	1	2	yes	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	18	4	no	—
276	C₂×C₄	—	36	2	no	—
277	C₂×C₄	—	36	2	no	—
278	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
279	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
280	C₂×C₄	#3697	1	1	yes	characteristic
281	C₂×C₄	C₂×#611	1	1	yes	characteristic
282	C₂×C₄	C₂×#611	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	18	4	no	—
286	C₈	C₂×#611	1	2	yes	—
287	C₈	C₂×#611	1	2	yes	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	4	2	no	—
291	C₂×C₃	#5143	1	2	yes	—
292	C₂×C₃	C₂×#930	1	1	yes	characteristic
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	#63492	1	1	yes	characteristic
297	C₂²	#63492	1	1	yes	characteristic
298	C₂²	C₂×#3697	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	36	4	no	—
301	C₄	#63492	1	1	yes	characteristic
302	C₄	#63492	1	1	yes	characteristic
303	C₄	C₂²×D₃₆	1	1	yes	characteristic
304	C₄	C₂²×#611	1	1	yes	characteristic
305	C₃	C₂×#5143	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	4	2	no	—
309	C₂	#10588821	1	2	yes	direct product factor
310	C₂	C₂×#63492	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63492
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#131	C₂
socle	C₂²×C₃	#930
prime radical	C₂×C₃×#41	C₂×D₃
hypercenter	C₂×#131	D₉
hypocenter	C₉	C₂×#473
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
3. C₂²×Dic₃₆	C₂	semidirect
4. #10588821	C₂	direct (N × H)
5. C₂×#63382	C₂	semidirect
6. C₂×#63372	C₂	semidirect
24. C₂×Dic₃₆	C₂²	semidirect
25. #63382	C₂²	semidirect
26. #63372	C₂²	semidirect
213. C₂×C₉	#473	semidirect
259. C₉	C₂×#473	semidirect
309. C₂	#10588821	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×C₉×#131	2
C₂×#63511	2
C₂²×Dic₃₆	2
#10588821	2
C₂×#63382	2
C₂×#63372	2
C₂×#5143	3

Subgroups — C2×#10588821 #11867963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588821 #11867963