Subgroups — C₂²×#88638 #13164694

Order 1728.

2826 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6226	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄×D₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#6194	C₂	1	2	yes	maximal, complemented
5	C₂²×D₃×Dic₉	C₂	1	2	yes	maximal, complemented
6	C₂×#88638	C₂	1	24	yes	maximal, direct product factor
7	C₂²×#3795	—	3	1	no	maximal
8	C₂²×#3699	—	3	1	no	maximal
9	C₂³×D₃×C₉	C₂²	1	2	yes	—
10	C₂³×C₃×D₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
13	C₂×#6226	C₂²	1	12	yes	complemented
14	C₂²×C₃×Dic₉	C₂²	1	1	yes	characteristic
15	C₂×C₉×D₁₂	C₂²	1	12	yes	complemented
16	C₂×C₃×C₄×D₉	C₂²	1	12	yes	complemented
17	C₂×#6194	C₂²	1	24	yes	complemented
18	C₂×D₃×Dic₉	C₂²	1	24	yes	complemented
19	#88638	C₂²	1	64	yes	direct product factor
20	C₂²×#634	—	3	1	no	—
21	C₂²×C₃×D₁₂	D₃	1	1	yes	characteristic
22	C₂²×C₃×C₄×D₃	—	3	1	no	—
23	C₂²×#625	—	3	2	no	—
24	C₂²×D₃×Dic₃	—	3	2	no	—
25	C₂×#3795	—	3	24	no	—
26	C₂²×D₄×C₉	—	3	1	no	—
27	C₂²×#611	—	3	2	no	—
28	C₂³×Dic₉	—	3	2	no	—
29	C₂×#3699	—	3	24	no	—
30	C₂²×C₄×D₉	D₃	1	1	yes	characteristic, complemented
31	C₂²×Dic₁₈	—	3	1	no	—
32	C₂²×Dic₁₈	—	6	1	no	—
33	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
34	C₂²×D₃×C₉	—	2	8	no	—
35	C₂²×D₃×C₉	C₂³	1	12	yes	—
36	C₂²×C₃×D₉	—	2	4	no	—
37	C₂²×C₃×D₉	C₂³	1	6	yes	—
38	C₂×C₃×C₄×C₉	C₂³	1	6	yes	—
39	C₂×C₃⋊Dic₉	C₂³	1	12	yes	—
40	#6226	C₂³	1	16	yes	complemented
41	C₂×C₃×Dic₉	C₂³	1	6	yes	—
42	C₉×D₁₂	C₂³	1	16	yes	complemented
43	C₃×C₄×D₉	C₂³	1	16	yes	complemented
44	#6194	C₂³	1	32	yes	complemented
45	D₃×Dic₉	C₂³	1	32	yes	complemented
46	C₂²×#237	—	9	1	no	—
47	C₂²×#235	—	9	1	no	—
48	C₂³×C₃×D₃	—	3	1	no	—
49	C₂³×C₃×D₃	C₂×D₃	1	2	yes	—
50	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃⋊Dic₃	—	3	2	no	—
52	C₂×#634	—	3	12	no	—
53	C₂²×C₃×Dic₃	—	3	1	no	—
54	C₂×C₃×D₁₂	C₂×D₃	1	12	yes	—
55	C₂×C₃×C₄×D₃	—	3	12	no	—
56	C₂×#625	—	3	24	no	—
57	C₂×D₃×Dic₃	—	3	24	no	—
58	#3795	—	3	64	no	—
59	C₂⁴×C₉	—	3	2	no	—
60	C₂³×D₉	C₂×D₃	1	1	yes	characteristic
61	C₂×D₄×C₉	—	3	12	no	—
62	C₂²×C₄×C₉	—	2	1	no	—
63	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
64	C₂×#611	—	3	24	no	—
65	C₂²×Dic₉	—	3	2	no	—
66	C₂²×Dic₉	—	3	24	no	—
67	C₂²×Dic₉	—	6	2	no	—
68	C₂²×Dic₉	C₂×D₃	1	1	yes	characteristic
69	#3699	—	3	64	no	—
70	C₂×C₄×D₉	C₂×D₃	1	12	yes	complemented
71	C₂×Dic₁₈	—	3	12	no	—
72	C₂×Dic₁₈	—	6	12	no	—
73	C₂²×C₃×C₉	#41	1	4	yes	complemented
74	C₂²×C₃×C₉	C₂⁴	1	3	yes	—
75	C₂×D₃×C₉	—	2	24	no	—
76	C₂×D₃×C₉	C₂⁴	1	8	yes	—
77	C₂×C₃×D₉	—	2	12	no	—
78	C₂×C₃×D₉	C₂⁴	1	4	yes	—
79	C₃×C₄×C₉	C₂⁴	1	4	yes	—
80	C₃⋊Dic₉	C₂⁴	1	8	yes	—
81	C₃×Dic₉	C₂⁴	1	4	yes	—
82	C₂²×C₃×D₄	—	3	1	no	—
83	C₂³×C₃×C₄	—	9	1	no	—
84	C₂²×#67	—	9	2	no	—
85	C₂²×#67	—	9	2	no	—
86	C₂³×Dic₃	—	9	2	no	—
87	C₂×#237	—	9	24	no	—
88	C₂×#235	—	9	24	no	—
89	C₂²×D₁₂	D₉	1	1	yes	characteristic, complemented
90	C₂²×C₄×D₃	—	3	1	no	—
91	C₂²×C₄×D₃	—	9	2	no	—
92	C₂²×Dic₆	—	9	1	no	—
93	C₂²×Dic₆	—	9	1	no	—
94	C₂²×Dic₆	—	18	1	no	—
95	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
96	C₂²×C₃×D₃	—	2	8	no	—
97	C₂²×C₃×D₃	—	3	6	no	—
98	C₂²×C₃×D₃	—	6	4	no	—
99	C₂²×C₃×D₃	C₂²×D₃	1	12	yes	—
100	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
101	C₂×C₃⋊Dic₃	—	3	12	no	—
102	#634	—	3	16	no	—
103	C₂×C₃×Dic₃	—	3	6	no	—
104	C₃×D₁₂	C₂²×D₃	1	16	yes	—
105	C₃×C₄×D₃	—	3	16	no	—
106	#625	—	3	32	no	—
107	D₃×Dic₃	—	3	32	no	—
108	C₂³×C₉	—	2	1	no	—
109	C₂³×C₉	—	3	12	no	—
110	C₂³×C₉	—	6	8	no	—
111	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
112	C₂²×D₉	—	2	4	no	—
113	C₂²×D₉	C₂²×D₃	1	6	yes	—
114	D₄×C₉	—	3	16	no	—
115	C₂×C₄×C₉	—	2	6	no	—
116	C₂×C₄×C₉	C₂²×D₃	1	6	yes	—
117	#611	—	3	32	no	—
118	C₂×Dic₉	—	3	12	no	—
119	C₂×Dic₉	—	3	32	no	—
120	C₂×Dic₉	—	6	12	no	—
121	C₂×Dic₉	C₂²×D₃	1	6	yes	—
122	C₄×D₉	C₂²×D₃	1	16	yes	complemented
123	Dic₁₈	—	3	16	no	—
124	Dic₁₈	—	6	16	no	—
125	C₂²×#41	—	27	1	no	Sylow
126	C₂×C₃×C₉	C₂×#41	1	6	yes	complemented
127	C₂×C₃×C₉	C₂⁵	1	1	yes	characteristic
128	D₃×C₉	—	2	8	no	—
129	C₃×D₉	—	2	4	no	—
130	C₂⁴×C₃	—	3	2	no	—
131	C₂⁴×C₃	—	9	1	no	—
132	C₂³×D₃	—	3	1	no	—
133	C₂³×D₃	C₂×D₉	1	2	yes	—
134	C₂×C₃×D₄	—	3	12	no	—
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	—	9	1	no	—
137	C₂²×C₃×C₄	—	9	12	no	—
138	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
139	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
140	C₂×#67	—	9	24	no	—
141	C₂×#67	—	9	24	no	—
142	C₂²×Dic₃	—	3	1	no	—
143	C₂²×Dic₃	—	9	24	no	—
144	C₂²×Dic₃	—	9	2	no	—
145	C₂²×Dic₃	—	9	2	no	—
146	C₂²×Dic₃	—	18	2	no	—
147	#237	—	9	64	no	—
148	#235	—	9	64	no	—
149	C₂×D₁₂	C₂×D₉	1	12	yes	complemented
150	C₂×C₄×D₃	—	3	12	no	—
151	C₂×C₄×D₃	—	9	24	no	—
152	C₂×Dic₆	—	9	12	no	—
153	C₂×Dic₆	—	9	12	no	—
154	C₂×Dic₆	—	18	12	no	—
155	C₂²×C₃²	#237	1	4	yes	—
156	C₂²×C₃²	C₂³×D₃	1	3	yes	—
157	C₂×C₃×D₃	—	2	24	no	—
158	C₂×C₃×D₃	—	3	4	no	—
159	C₂×C₃×D₃	—	6	12	no	—
160	C₂×C₃×D₃	C₂³×D₃	1	8	yes	—
161	C₃²×C₄	C₂³×D₃	1	4	yes	—
162	C₃⋊Dic₃	—	3	8	no	—
163	C₃×Dic₃	—	3	4	no	—
164	C₂²×C₉	—	2	3	no	—
165	C₂²×C₉	—	2	4	no	—
166	C₂²×C₉	—	3	8	no	—
167	C₂²×C₉	—	6	24	no	—
168	C₂²×C₉	#235	1	4	yes	complemented
169	C₂²×C₉	C₂³×D₃	1	3	yes	—
170	C₂×D₉	—	2	12	no	—
171	C₂×D₉	C₂³×D₃	1	4	yes	—
172	C₄×C₉	—	2	4	no	—
173	C₄×C₉	C₂³×D₃	1	4	yes	—
174	Dic₉	—	3	8	no	—
175	Dic₉	—	6	8	no	—
176	Dic₉	C₂³×D₃	1	4	yes	—
177	C₂×#41	—	27	24	no	—
178	C₂²×Q₈	—	27	1	no	—
179	C₂²×D₄	—	3	1	no	—
180	C₂²×D₄	—	27	2	no	—
181	C₂³×C₄	—	9	1	no	—
182	C₂³×C₄	—	27	2	no	—
183	C₃×C₉	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₃	—	2	1	no	—
185	C₂³×C₃	—	3	12	no	—
186	C₂³×C₃	—	6	8	no	—
187	C₂³×C₃	—	9	6	no	—
188	C₂³×C₃	—	18	4	no	—
189	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
190	C₂³×C₃	C₂×D₃²	1	1	yes	characteristic
191	C₂²×D₃	—	2	8	no	—
192	C₂²×D₃	—	3	6	no	—
193	C₂²×D₃	—	6	4	no	—
194	C₂²×D₃	C₂²×D₉	1	12	yes	—
195	C₃×D₄	—	3	16	no	—
196	C₂×C₃×C₄	—	2	6	no	—
197	C₂×C₃×C₄	—	9	6	no	—
198	C₂×C₃×C₄	—	9	16	no	—
199	C₂×C₃×C₄	C₂²×D₉	1	6	yes	—
200	C₂×C₃×C₄	C₂×D₃²	1	6	yes	—
201	#67	—	9	32	no	—
202	#67	—	9	32	no	—
203	C₂×Dic₃	—	3	6	no	—
204	C₂×Dic₃	—	9	32	no	—
205	C₂×Dic₃	—	9	12	no	—
206	C₂×Dic₃	—	9	12	no	—
207	C₂×Dic₃	—	18	12	no	—
208	D₁₂	C₂²×D₉	1	16	yes	complemented
209	C₄×D₃	—	3	16	no	—
210	C₄×D₃	—	9	32	no	—
211	Dic₆	—	9	16	no	—
212	Dic₆	—	9	16	no	—
213	Dic₆	—	18	16	no	—
214	C₂×C₃²	C₂×#237	1	6	yes	—
215	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
216	C₃×D₃	—	2	8	no	—
217	C₃×D₃	—	6	4	no	—
218	C₂×C₉	—	2	6	no	—
219	C₂×C₉	—	2	1	no	—
220	C₂×C₉	—	6	8	no	—
221	C₂×C₉	C₂×#235	1	6	yes	complemented
222	C₂×C₉	C₂⁴×D₃	1	1	yes	characteristic
223	D₉	—	2	4	no	—
224	C₂⁴	—	3	2	no	—
225	C₂⁴	—	9	1	no	—
226	#41	—	27	64	no	—
227	C₂×Q₈	—	27	12	no	—
228	C₂×D₄	—	3	12	no	—
229	C₂×D₄	—	27	24	no	—
230	C₂²×C₄	—	9	1	no	—
231	C₂²×C₄	—	9	12	no	—
232	C₂²×C₄	—	27	24	no	—
233	C₂²×C₄	—	27	2	no	—
234	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
235	C₂²×C₃	—	2	4	no	—
236	C₂²×C₃	—	2	3	no	—
237	C₂²×C₃	—	3	8	no	—
238	C₂²×C₃	—	6	24	no	—
239	C₂²×C₃	—	9	4	no	—
240	C₂²×C₃	—	18	12	no	—
241	C₂²×C₃	#3699	1	4	yes	complemented
242	C₂²×C₃	C₂³×D₉	1	3	yes	—
243	C₂²×C₃	#3795	1	4	yes	—
244	C₂²×C₃	C₂²×D₃²	1	3	yes	—
245	C₂×D₃	—	2	24	no	—
246	C₂×D₃	—	3	4	no	—
247	C₂×D₃	—	6	12	no	—
248	C₂×D₃	C₂³×D₉	1	8	yes	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	9	4	no	—
251	C₃×C₄	C₂³×D₉	1	4	yes	—
252	C₃×C₄	C₂²×D₃²	1	4	yes	—
253	Dic₃	—	3	4	no	—
254	Dic₃	—	9	8	no	—
255	Dic₃	—	9	8	no	—
256	Dic₃	—	18	8	no	—
257	C₃²	C₂²×#237	1	1	yes	characteristic
258	C₉	—	2	1	no	—
259	C₉	C₂²×#235	1	1	yes	characteristic, complemented
260	C₂³	—	3	12	no	—
261	C₂³	—	6	8	no	—
262	C₂³	—	9	6	no	—
263	C₂³	—	18	4	no	—
264	C₂³	C₂×D₃×D₉	1	1	yes	characteristic
265	Q₈	—	27	16	no	—
266	D₄	—	3	16	no	—
267	D₄	—	27	32	no	—
268	C₂×C₄	—	9	6	no	—
269	C₂×C₄	—	9	16	no	—
270	C₂×C₄	—	27	32	no	—
271	C₂×C₄	—	27	12	no	—
272	C₂×C₄	C₂×D₃×D₉	1	6	yes	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	6	no	—
275	C₂×C₃	—	6	8	no	—
276	C₂×C₃	—	18	4	no	—
277	C₂×C₃	C₂×#3699	1	6	yes	complemented
278	C₂×C₃	C₂⁴×D₉	1	1	yes	characteristic
279	C₂×C₃	C₂×#3795	1	6	yes	—
280	C₂×C₃	C₂³×D₃²	1	1	yes	characteristic
281	D₃	—	2	8	no	—
282	D₃	—	6	4	no	—
283	C₂²	—	3	8	no	—
284	C₂²	—	6	24	no	—
285	C₂²	—	9	4	no	—
286	C₂²	—	18	12	no	—
287	C₂²	#88638	1	4	yes	direct product factor
288	C₂²	C₂²×D₃×D₉	1	3	yes	—
289	C₄	—	9	4	no	—
290	C₄	—	27	8	no	—
291	C₄	C₂²×D₃×D₉	1	4	yes	—
292	C₃	—	2	1	no	—
293	C₃	C₂²×#3699	1	1	yes	characteristic, complemented
294	C₃	C₂²×#3795	1	1	yes	characteristic
295	C₂	—	6	8	no	—
296	C₂	—	18	4	no	—
297	C₂	C₂×#88638	1	6	yes	direct product factor
298	C₂	C₂³×D₃×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃×D₉
commutator, cosocle	C₂×C₃×C₉	C₂⁵
Frattini	C₂×C₃	C₂³×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂³×C₃²	C₂²×D₃
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₂²×#41
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#6226	C₂	semidirect
2. C₂²×C₉×D₁₂	C₂	semidirect
3. C₂²×C₃×C₄×D₉	C₂	semidirect
4. C₂²×#6194	C₂	semidirect
5. C₂²×D₃×Dic₉	C₂	semidirect
6. C₂×#88638	C₂	direct (N × H)
13. C₂×#6226	C₂²	semidirect
15. C₂×C₉×D₁₂	C₂²	semidirect
16. C₂×C₃×C₄×D₉	C₂²	semidirect
17. C₂×#6194	C₂²	semidirect
18. C₂×D₃×Dic₉	C₂²	semidirect
19. #88638	C₂²	direct (N × H)
30. C₂²×C₄×D₉	D₃	semidirect
40. #6226	C₂³	semidirect
42. C₉×D₁₂	C₂³	semidirect
43. C₃×C₄×D₉	C₂³	semidirect
44. #6194	C₂³	semidirect
45. D₃×Dic₉	C₂³	semidirect
70. C₂×C₄×D₉	C₂×D₃	semidirect
73. C₂²×C₃×C₉	#41	semidirect
89. C₂²×D₁₂	D₉	semidirect
122. C₄×D₉	C₂²×D₃	semidirect
126. C₂×C₃×C₉	C₂×#41	semidirect
149. C₂×D₁₂	C₂×D₉	semidirect
168. C₂²×C₉	#235	semidirect
183. C₃×C₉	C₂²×#41	semidirect
208. D₁₂	C₂²×D₉	semidirect
221. C₂×C₉	C₂×#235	semidirect
241. C₂²×C₃	#3699	semidirect
259. C₉	C₂²×#235	semidirect
277. C₂×C₃	C₂×#3699	semidirect
287. C₂²	#88638	direct (N × H)
293. C₃	C₂²×#3699	semidirect
297. C₂	C₂×#88638	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×#6226	2
C₂²×C₉×D₁₂	2
C₂²×C₃×C₄×D₉	2
C₂²×#6194	2
C₂²×D₃×Dic₉	2
C₂×#88638	2
C₂²×#3795	3
C₂²×#3699	3

Subgroups — C22×#88638 #13164694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#88638 #13164694