Subgroups — C₂×D₉×#108 #11858605

Order 1152.

720 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₉	C₂	1	2	yes	maximal
2	C₂²×C₈×D₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63381	C₂	1	2	yes	maximal, complemented
5	C₂×#63372	C₂	1	1	yes	maximal, characteristic, complemented
6	D₉×#108	C₂	1	8	yes	maximal, direct product factor
7	C₂×D₃×#108	—	3	1	no	maximal
8	C₂²×C₄×D₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×#3667	C₂²	1	2	yes	—
12	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
13	M₁₆×D₉	—	2	4	no	—
14	M₁₆×D₉	C₂²	1	8	yes	—
15	C₂×#3663	—	2	2	no	—
16	C₂×C₈×D₉	—	2	2	no	—
17	C₂×C₈×D₉	C₄	1	2	yes	—
18	C₂×C₈×D₉	C₄	1	4	yes	—
19	C₂×C₈×D₉	C₄	1	2	yes	—
20	C₂×C₈×D₉	C₂²	1	4	yes	—
21	C₉×#108	C₂²	1	4	yes	complemented
22	#63381	C₂²	1	8	yes	complemented
23	#63372	C₂²	1	4	yes	complemented
24	C₂×D₃×M₁₆	—	3	2	no	—
25	C₂²×D₃×C₈	—	3	1	no	—
26	C₂×C₃×#108	D₃	1	1	yes	characteristic
27	C₂×#822	—	3	2	no	—
28	C₂×#819	—	3	1	no	—
29	D₃×#108	—	3	8	no	—
30	C₂³×D₉	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₉	Q₈	1	1	yes	characteristic
33	C₂×C₄×D₉	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₉	D₄	1	1	yes	characteristic
37	C₂×C₄×D₉	D₄	1	1	yes	characteristic
38	C₂×C₄×D₉	D₄	1	2	yes	—
39	C₂×C₄×D₉	Q₈	1	2	yes	—
40	C₂×C₄×D₉	C₂³	1	2	yes	—
41	C₉×M₁₆	—	2	2	no	—
42	C₉×M₁₆	C₂³	1	4	yes	—
43	C₂×C₈×C₉	—	2	2	no	—
44	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
45	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
46	C₂×C₈×C₉	C₂³	1	2	yes	—
47	#3667	—	2	2	no	—
48	#3667	C₂³	1	4	yes	—
49	C₂×C₉⋊C₈	—	2	2	no	—
50	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
51	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
52	C₂×C₉⋊C₈	C₂³	1	2	yes	—
53	#3663	—	2	8	no	—
54	C₈×D₉	—	2	8	no	—
55	C₈×D₉	C₂×C₄	1	8	yes	—
56	C₈×D₉	C₂×C₄	1	8	yes	—
57	C₂²×#108	—	9	1	no	Sylow
58	C₂²×C₄×D₃	—	3	1	no	—
59	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂×#208	—	3	2	no	—
62	C₂²×C₃⋊C₈	—	3	1	no	—
63	D₃×M₁₆	—	3	8	no	—
64	D₃×M₁₆	—	6	4	no	—
65	C₂×ΓL(1,𝔽₂₅)	—	6	2	no	—
66	C₂×D₃×C₈	—	3	2	no	—
67	C₂×D₃×C₈	—	3	4	no	—
68	C₂×D₃×C₈	—	3	2	no	—
69	C₂×D₃×C₈	—	3	4	no	—
70	C₂×D₃×C₈	—	6	2	no	—
71	C₃×#108	C₂×D₃	1	4	yes	—
72	#822	—	3	8	no	—
73	#819	—	3	4	no	—
74	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
75	C₂²×D₉	—	2	4	no	—
76	C₂²×D₉	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×D₉	C₄⋊C₄	1	2	yes	—
78	C₂²×D₉	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×D₉	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
87	C₂×Dic₉	C₄⋊C₄	1	1	yes	characteristic
88	C₂×Dic₉	C₄⋊C₄	1	1	yes	characteristic
89	C₂×Dic₉	C₂×Q₈	1	2	yes	—
90	C₄×D₉	C₄⋊C₄	1	2	yes	—
91	C₄×D₉	C₄⋊C₄	1	2	yes	—
92	C₄×D₉	C₄⋊C₄	1	2	yes	—
93	C₄×D₉	C₄⋊C₄	1	2	yes	—
94	C₄×D₉	C₂²×C₄	1	2	yes	—
95	C₄×D₉	C₂²×C₄	1	2	yes	—
96	C₄×D₉	C₂×D₄	1	2	yes	—
97	C₄×D₉	C₂×D₄	1	2	yes	—
98	C₈×C₉	—	2	4	no	—
99	C₈×C₉	C₂²×C₄	1	4	yes	—
100	C₉⋊C₈	—	2	4	no	—
101	C₉⋊C₈	C₂²×C₄	1	4	yes	—
102	C₂²×M₁₆	—	9	2	no	—
103	C₂³×C₈	—	9	1	no	—
104	C₂×#108	—	9	2	no	—
105	C₂×#108	—	9	1	no	—
106	C₂×#108	—	9	8	no	—
107	C₂×#108	D₉	1	1	yes	characteristic, direct product factor
108	C₂³×D₃	—	3	1	no	—
109	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂²×Dic₃	—	3	1	no	—
111	C₂×C₄×D₃	—	3	1	no	—
112	C₂×C₄×D₃	—	3	2	no	—
113	C₂×C₄×D₃	—	3	1	no	—
114	C₂×C₄×D₃	—	3	1	no	—
115	C₂×C₄×D₃	—	3	1	no	—
116	C₂×C₄×D₃	—	3	2	no	—
117	C₂×C₄×D₃	—	3	2	no	—
118	C₂×C₄×D₃	—	3	2	no	—
119	C₃×M₁₆	—	2	2	no	—
120	C₃×M₁₆	C₂²×D₃	1	4	yes	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
123	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
124	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
125	#208	—	3	4	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	2	no	—
131	ΓL(1,𝔽₂₅)	—	6	8	no	—
132	D₃×C₈	—	3	8	no	—
133	D₃×C₈	—	3	8	no	—
134	D₃×C₈	—	6	8	no	—
135	C₂²×C₉	—	2	2	no	—
136	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
139	C₂×D₉	—	2	2	no	—
140	C₂×D₉	—	2	4	no	—
141	C₂×D₉	—	2	4	no	—
142	C₂×D₉	#108	1	4	yes	direct product factor
143	C₂×D₉	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×D₉	C₂×C₄⋊C₄	1	2	yes	—
145	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
148	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
149	Dic₉	C₂×C₄⋊C₄	1	2	yes	—
150	Dic₉	C₂×C₄⋊C₄	1	2	yes	—
151	C₂³×C₄	—	9	1	no	—
152	C₂×M₁₆	—	9	2	no	—
153	C₂×M₁₆	—	9	8	no	—
154	C₂×M₁₆	—	18	2	no	—
155	C₂×M₁₆	—	18	4	no	—
156	C₂×M₁₆	C₂×D₉	1	2	yes	—
157	C₂²×C₈	—	9	1	no	—
158	C₂²×C₈	—	9	2	no	—
159	C₂²×C₈	—	9	2	no	—
160	C₂²×C₈	—	9	4	no	—
161	C₂²×C₈	—	9	4	no	—
162	C₂²×C₈	—	18	2	no	—
163	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
164	#108	—	9	8	no	—
165	#108	—	9	4	no	—
166	#108	C₂×D₉	1	4	yes	direct product factor
167	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
168	C₂²×D₃	—	3	1	no	—
169	C₂²×D₃	—	3	2	no	—
170	C₂²×D₃	—	3	1	no	—
171	C₂²×D₃	—	3	2	no	—
172	C₂²×D₃	—	6	4	no	—
173	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
179	C₂×Dic₃	—	3	2	no	—
180	C₂×Dic₃	—	3	1	no	—
181	C₂×Dic₃	—	3	1	no	—
182	C₂×Dic₃	—	3	2	no	—
183	C₄×D₃	—	3	2	no	—
184	C₄×D₃	—	3	2	no	—
185	C₄×D₃	—	3	2	no	—
186	C₄×D₃	—	3	2	no	—
187	C₄×D₃	—	3	2	no	—
188	C₄×D₃	—	3	2	no	—
189	C₄×D₃	—	3	2	no	—
190	C₄×D₃	—	3	2	no	—
191	C₃×C₈	—	2	4	no	—
192	C₃×C₈	C₂×C₄×D₃	1	4	yes	—
193	C₃⋊C₈	—	3	4	no	—
194	C₃⋊C₈	—	6	4	no	—
195	C₂×C₉	—	2	1	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	C₂×#108	1	2	yes	complemented
198	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
199	D₉	—	2	2	no	—
200	D₉	C₂×#108	1	4	yes	direct product factor
201	C₂⁴	—	9	1	no	—
202	C₂²×C₄	—	9	1	no	—
203	C₂²×C₄	—	9	1	no	—
204	C₂²×C₄	—	9	2	no	—
205	C₂²×C₄	—	9	1	no	—
206	C₂²×C₄	—	9	2	no	—
207	C₂²×C₄	—	9	1	no	—
208	C₂²×C₄	—	9	1	no	—
209	C₂²×C₄	—	9	2	no	—
210	C₂²×C₄	—	9	2	no	—
211	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
212	M₁₆	—	2	2	no	—
213	M₁₆	—	9	4	no	—
214	M₁₆	—	18	8	no	—
215	M₁₆	—	18	2	no	—
216	M₁₆	C₂²×D₉	1	4	yes	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	9	2	no	—
219	C₂×C₈	—	9	2	no	—
220	C₂×C₈	—	9	2	no	—
221	C₂×C₈	—	9	8	no	—
222	C₂×C₈	—	9	8	no	—
223	C₂×C₈	—	18	8	no	—
224	C₂×C₈	—	18	2	no	—
225	C₂×C₈	C₄×D₉	1	2	yes	—
226	C₂×C₈	C₄×D₉	1	2	yes	—
227	C₂×C₈	C₂²×D₉	1	2	yes	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
230	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
231	C₂²×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
232	C₂×D₃	—	3	4	no	—
233	C₂×D₃	—	3	2	no	—
234	C₂×D₃	—	3	2	no	—
235	C₂×D₃	—	6	4	no	—
236	C₂×D₃	—	6	4	no	—
237	C₂×D₃	—	6	2	no	—
238	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
239	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
240	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
241	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
242	Dic₃	—	3	2	no	—
243	Dic₃	—	3	2	no	—
244	C₉	C₂²×#108	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	9	1	no	—
246	C₂³	—	9	2	no	—
247	C₂³	—	9	1	no	—
248	C₂³	—	9	2	no	—
249	C₂³	—	18	4	no	—
250	C₂³	Q₈×D₉	1	1	yes	characteristic
251	C₂×C₄	—	9	2	no	—
252	C₂×C₄	—	9	1	no	—
253	C₂×C₄	—	9	1	no	—
254	C₂×C₄	—	9	2	no	—
255	C₂×C₄	—	9	2	no	—
256	C₂×C₄	—	9	2	no	—
257	C₂×C₄	—	9	2	no	—
258	C₂×C₄	—	9	2	no	—
259	C₂×C₄	—	9	2	no	—
260	C₂×C₄	—	9	2	no	—
261	C₂×C₄	—	9	2	no	—
262	C₂×C₄	—	9	2	no	—
263	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
265	C₂×C₄	D₄×D₉	1	1	yes	characteristic
266	C₂×C₄	D₄×D₉	1	1	yes	characteristic
267	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
268	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
269	C₈	—	2	4	no	—
270	C₈	—	9	4	no	—
271	C₈	—	18	4	no	—
272	C₈	C₂×C₄×D₉	1	4	yes	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	D₃×#108	1	2	yes	—
276	C₂×C₃	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
277	D₃	—	3	4	no	—
278	D₃	—	6	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	9	4	no	—
281	C₂²	—	9	2	no	—
282	C₂²	—	9	2	no	—
283	C₂²	—	18	4	no	—
284	C₂²	—	18	4	no	—
285	C₂²	—	18	2	no	—
286	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
287	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
288	C₂²	C₂×Q₈×D₉	1	1	yes	characteristic
289	C₄	—	9	2	no	—
290	C₄	—	9	2	no	—
291	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
292	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
293	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
294	C₄	C₂×D₄×D₉	1	1	yes	characteristic
295	C₃	C₂×D₃×#108	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	9	4	no	—
299	C₂	—	18	2	no	—
300	C₂	D₉×#108	1	2	yes	direct product factor
301	C₂	C₂×C₄⋊C₄×D₉	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×#108	C₂
socle	C₂²×C₃	D₃×C₄⋊C₄
prime radical	C₂³×D₉	Q₈
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×#108	D₉
hypocenter	C₉	C₂²×#108
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. C₂×C₉×#108	C₂	semidirect
4. C₂×#63381	C₂	semidirect
5. C₂×#63372	C₂	semidirect
6. D₉×#108	C₂	direct (N × H)
21. C₉×#108	C₂²	semidirect
22. #63381	C₂²	semidirect
23. #63372	C₂²	semidirect
107. C₂×#108	D₉	direct (N × H)
142. C₂×D₉	#108	direct (N × H)
166. #108	C₂×D₉	direct (N × H)
197. C₂×C₉	C₂×#108	semidirect
200. D₉	C₂×#108	direct (N × H)
244. C₉	C₂²×#108	semidirect
300. C₂	D₉×#108	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×M₁₆×D₉	2
C₂²×C₈×D₉	2
C₂×C₉×#108	2
C₂×#63381	2
C₂×#63372	2
D₉×#108	2
C₂×D₃×#108	3

Subgroups — C2×D9×#108 #11858605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₉×#108 #11858605