Subgroups — C₂×#10588877 #11861040

Order 1152.

696 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×#63510	C₂	1	2	yes	maximal, complemented
3	C₂×#63396	C₂	1	2	yes	maximal, complemented
4	#10588877	C₂	1	8	yes	maximal, direct product factor
5	C₂×#63375	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63372	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#5197	—	3	1	no	maximal
8	C₂×C₉×#41	C₂²	1	2	yes	—
9	C₉×#135	C₄	1	4	yes	complemented
10	C₉×#135	C₂²	1	4	yes	—
11	C₂×C₉×Q₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂×C₉×SD₁₆	C₄	1	2	yes	complemented
13	C₂×C₉×D₈	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
15	#63510	—	2	4	no	—
16	#63510	C₂²	1	8	yes	complemented
17	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
18	C₂×#3667	—	2	2	no	—
19	C₂×#3667	C₂²	1	2	yes	—
20	C₂²×C₉⋊C₈	—	2	2	no	—
21	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
22	#63396	C₂²	1	8	yes	complemented
23	#63375	C₂²	1	4	yes	complemented
24	#63372	C₂²	1	4	yes	complemented
25	C₂×C₃×#135	D₃	1	1	yes	characteristic
26	C₂×#948	—	3	2	no	—
27	C₂×#837	—	3	2	no	—
28	#5197	—	3	8	no	—
29	C₂×#819	—	3	1	no	—
30	C₂×#815	—	3	1	no	—
31	C₉×#41	—	2	2	no	—
32	C₉×#41	—	2	2	no	—
33	C₉×#41	C₂×C₄	1	4	yes	—
34	C₉×#41	C₂³	1	4	yes	—
35	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₉	—	2	2	no	—
37	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₉	—	2	2	no	—
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₉	—	2	1	no	—
41	C₉×Q₁₆	C₂×C₄	1	4	yes	complemented
42	C₉×SD₁₆	C₂×C₄	1	8	yes	complemented
43	C₉×D₈	C₂×C₄	1	4	yes	complemented
44	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₉	C₂³	1	2	yes	—
48	C₄×Dic₉	—	2	1	no	—
49	C₄×Dic₉	C₂³	1	2	yes	—
50	#3667	—	2	2	no	—
51	#3667	—	2	8	no	—
52	#3667	C₂³	1	4	yes	—
53	C₂×C₉⋊C₈	—	2	2	no	—
54	C₂×C₉⋊C₈	—	2	8	no	—
55	C₂×C₉⋊C₈	D₄	1	2	yes	—
56	C₂×C₉⋊C₈	D₄	1	2	yes	—
57	C₂×C₉⋊C₈	C₂³	1	2	yes	—
58	C₂×#444	—	9	1	no	Sylow
59	C₂×C₃×#41	C₂×D₃	1	2	yes	—
60	C₃×#135	Dic₃	1	4	yes	—
61	C₃×#135	C₂×D₃	1	4	yes	—
62	C₂×C₃×Q₁₆	Dic₃	1	1	yes	characteristic
63	C₂×C₃×SD₁₆	Dic₃	1	2	yes	—
64	C₂×C₃×D₈	Dic₃	1	1	yes	characteristic
65	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
66	#948	—	3	8	no	—
67	#948	—	6	4	no	—
68	C₂×C₄×Dic₃	—	3	1	no	—
69	C₂×#208	—	3	2	no	—
70	C₂×#208	—	6	2	no	—
71	C₂²×C₃⋊C₈	—	3	1	no	—
72	C₂²×C₃⋊C₈	—	6	2	no	—
73	#837	—	3	8	no	—
74	#819	—	3	4	no	—
75	#815	—	3	4	no	—
76	C₂³×C₉	—	2	2	no	—
77	C₂³×C₉	#41	1	1	yes	characteristic
78	Q₈×C₉	—	2	2	no	—
79	Q₈×C₉	C₂²×C₄	1	4	yes	—
80	D₄×C₉	—	2	4	no	—
81	D₄×C₉	—	2	4	no	—
82	D₄×C₉	—	2	2	no	—
83	D₄×C₉	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	4	no	—
86	C₂×C₄×C₉	—	2	4	no	—
87	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	#41	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
93	C₂×Dic₉	—	2	2	no	—
94	C₂×Dic₉	—	4	2	no	—
95	C₈×C₉	C₂²×C₄	1	2	yes	—
96	C₈×C₉	C₂²×C₄	1	2	yes	—
97	C₉⋊C₈	—	2	4	no	—
98	C₉⋊C₈	C₂×D₄	1	4	yes	—
99	C₂×#135	D₉	1	1	yes	characteristic
100	C₂×#131	—	9	2	no	—
101	#444	—	9	8	no	—
102	C₂×#108	—	9	1	no	—
103	C₂×#104	—	9	2	no	—
104	C₂×#97	—	9	1	no	—
105	C₃×#41	—	2	2	no	—
106	C₃×#41	—	2	2	no	—
107	C₃×#41	C₂×Dic₃	1	4	yes	—
108	C₃×#41	C₂²×D₃	1	4	yes	—
109	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
110	C₂×C₃×D₄	—	2	2	no	—
111	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
114	C₂²×Dic₃	—	6	1	no	—
115	C₃×Q₁₆	C₂×Dic₃	1	4	yes	—
116	C₃×SD₁₆	C₂×Dic₃	1	8	yes	—
117	C₃×D₈	C₂×Dic₃	1	4	yes	—
118	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
121	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
122	C₄×Dic₃	—	3	2	no	—
123	C₄×Dic₃	—	6	1	no	—
124	#208	—	3	4	no	—
125	#208	—	6	8	no	—
126	#208	—	6	2	no	—
127	C₂×C₃⋊C₈	—	3	2	no	—
128	C₂×C₃⋊C₈	—	3	2	no	—
129	C₂×C₃⋊C₈	—	3	2	no	—
130	C₂×C₃⋊C₈	—	6	8	no	—
131	C₂×C₃⋊C₈	—	6	2	no	—
132	C₂²×C₉	—	2	2	no	—
133	C₂²×C₉	—	2	4	no	—
134	C₂²×C₉	—	4	4	no	—
135	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
136	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
137	C₂²×C₉	C₂×#41	1	1	yes	characteristic
138	C₄×C₉	—	2	4	no	—
139	C₄×C₉	C₄×D₄	1	1	yes	characteristic
140	C₄×C₉	C₄×D₄	1	1	yes	characteristic
141	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
143	Dic₉	—	4	2	no	—
144	C₂×#41	C₂×D₉	1	2	yes	—
145	#135	Dic₉	1	4	yes	complemented
146	#135	C₂×D₉	1	4	yes	—
147	C₂×Q₁₆	Dic₉	1	1	yes	characteristic, complemented
148	C₂×SD₁₆	Dic₉	1	2	yes	complemented
149	C₂×D₈	Dic₉	1	1	yes	characteristic, complemented
150	#131	—	9	8	no	—
151	#131	—	18	4	no	—
152	C₂×M₁₆	—	9	2	no	—
153	C₂×M₁₆	—	18	2	no	—
154	C₂²×C₈	—	9	1	no	—
155	C₂²×C₈	—	18	2	no	—
156	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
157	C₂×C₄²	—	9	1	no	—
158	#108	—	9	4	no	—
159	#104	—	9	8	no	—
160	#97	—	9	4	no	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	#237	1	1	yes	characteristic
163	C₃×Q₈	—	2	2	no	—
164	C₃×Q₈	C₂²×Dic₃	1	4	yes	—
165	C₃×D₄	—	2	4	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	2	no	—
168	C₃×D₄	C₂²×Dic₃	1	4	yes	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	#237	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
178	C₂×Dic₃	—	6	2	no	—
179	C₂×Dic₃	—	12	2	no	—
180	C₃×C₈	C₂²×Dic₃	1	2	yes	—
181	C₃×C₈	C₂²×Dic₃	1	2	yes	—
182	C₃⋊C₈	—	3	4	no	—
183	C₃⋊C₈	—	6	4	no	—
184	C₂×C₉	—	2	1	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	4	4	no	—
187	C₂×C₉	#444	1	2	yes	complemented
188	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	C₂×Dic₉	1	4	yes	—
192	#41	C₂²×D₉	1	4	yes	—
193	C₂×Q₈	C₂×Dic₉	1	2	yes	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	C₂×Dic₉	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	18	1	no	—
198	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
199	Q₁₆	C₂×Dic₉	1	4	yes	complemented
200	SD₁₆	C₂×Dic₉	1	8	yes	complemented
201	D₈	C₂×Dic₉	1	4	yes	complemented
202	M₁₆	—	9	4	no	—
203	M₁₆	—	18	8	no	—
204	M₁₆	—	18	2	no	—
205	C₂×C₈	—	9	2	no	—
206	C₂×C₈	—	9	2	no	—
207	C₂×C₈	—	9	2	no	—
208	C₂×C₈	—	18	8	no	—
209	C₂×C₈	—	18	2	no	—
210	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
211	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
212	C₂×C₈	C₂×Dic₉	1	2	yes	—
213	C₂×C₈	C₂²×D₉	1	2	yes	—
214	C₄²	—	9	2	no	—
215	C₄²	—	18	1	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	4	no	—
218	C₂²×C₃	—	4	4	no	—
219	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
220	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
221	C₂²×C₃	C₂×#237	1	1	yes	characteristic
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
224	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
225	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
226	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
227	Dic₃	—	12	2	no	—
228	C₉	C₂×#444	1	1	yes	characteristic, complemented, Sylow
229	C₂³	—	2	2	no	—
230	C₂³	#3699	1	1	yes	characteristic
231	Q₈	—	2	2	no	—
232	Q₈	C₂²×Dic₉	1	4	yes	—
233	D₄	—	2	4	no	—
234	D₄	—	2	4	no	—
235	D₄	—	2	2	no	—
236	D₄	C₂²×Dic₉	1	4	yes	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	18	2	no	—
241	C₂×C₄	—	36	2	no	—
242	C₂×C₄	D₄×D₉	1	1	yes	characteristic
243	C₂×C₄	D₄×D₉	1	1	yes	characteristic
244	C₂×C₄	#3699	1	1	yes	characteristic
245	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
246	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
247	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
248	C₈	—	9	4	no	—
249	C₈	—	18	4	no	—
250	C₈	C₂²×Dic₉	1	2	yes	—
251	C₈	C₂²×Dic₉	1	2	yes	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	4	4	no	—
255	C₂×C₃	#5197	1	2	yes	—
256	C₂×C₃	C₂×D₄×Dic₃	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	4	4	no	—
260	C₂²	D₄×Dic₉	1	1	yes	characteristic
261	C₂²	D₄×Dic₉	1	1	yes	characteristic
262	C₂²	C₂×#3699	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	36	2	no	—
265	C₄	D₄×Dic₉	1	1	yes	characteristic
266	C₄	D₄×Dic₉	1	1	yes	characteristic
267	C₄	C₂×D₄×D₉	1	1	yes	characteristic
268	C₄	C₂³×Dic₉	1	1	yes	characteristic
269	C₃	C₂×#5197	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	4	4	no	—
273	C₂	#10588877	1	2	yes	direct product factor
274	C₂	C₂×D₄×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₉
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#135	C₂
socle	C₂²×C₃	D₄×Dic₃
prime radical	C₂×C₃×#135	D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×#135	D₉
hypocenter	C₉	C₂×#444
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#63510	C₂	semidirect
3. C₂×#63396	C₂	semidirect
4. #10588877	C₂	direct (N × H)
5. C₂×#63375	C₂	semidirect
6. C₂×#63372	C₂	semidirect
9. C₉×#135	C₄	semidirect
11. C₂×C₉×Q₁₆	C₄	semidirect
12. C₂×C₉×SD₁₆	C₄	semidirect
13. C₂×C₉×D₈	C₄	semidirect
16. #63510	C₂²	semidirect
22. #63396	C₂²	semidirect
23. #63375	C₂²	semidirect
24. #63372	C₂²	semidirect
41. C₉×Q₁₆	C₂×C₄	semidirect
42. C₉×SD₁₆	C₂×C₄	semidirect
43. C₉×D₈	C₂×C₄	semidirect
145. #135	Dic₉	semidirect
147. C₂×Q₁₆	Dic₉	semidirect
148. C₂×SD₁₆	Dic₉	semidirect
149. C₂×D₈	Dic₉	semidirect
187. C₂×C₉	#444	semidirect
199. Q₁₆	C₂×Dic₉	semidirect
200. SD₁₆	C₂×Dic₉	semidirect
201. D₈	C₂×Dic₉	semidirect
228. C₉	C₂×#444	semidirect
273. C₂	#10588877	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×#135	2
C₂×#63510	2
C₂×#63396	2
#10588877	2
C₂×#63375	2
C₂×#63372	2
C₂×#5197	3

Subgroups — C2×#10588877 #11861040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588877 #11861040