Subgroups — C₂²×D₃²×C₈ #11918426

Order 1152.

4121 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×C₄×D₃²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×D₃×C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×#3710	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×D₃²×C₈	C₂	1	24	yes	maximal, direct product factor
7	C₂³×D₃×C₈	—	3	2	no	maximal
8	C₂³×D₃²	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×D₃	C₂²	1	2	yes	—
11	C₂²×#624	C₄	1	1	yes	characteristic
12	C₂²×D₃×Dic₃	C₄	1	2	yes	—
13	C₂×C₄×D₃²	C₄	1	12	yes	—
14	C₂×C₄×D₃²	C₂²	1	12	yes	—
15	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₈×C₃⋊D₃	C₂²	1	12	yes	complemented
18	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	complemented
19	C₂×C₃×D₃×C₈	C₂²	1	24	yes	complemented
20	C₂×#3710	C₂²	1	12	yes	complemented
21	C₂×D₃×C₃⋊C₈	C₂²	1	24	yes	complemented
22	D₃²×C₈	C₂²	1	64	yes	direct product factor
23	C₂³×C₄×D₃	—	3	2	no	—
24	C₂³×C₃×C₈	—	3	2	no	—
25	C₂³×C₃⋊C₈	—	3	2	no	—
26	C₂²×D₃×C₈	—	3	2	no	—
27	C₂²×D₃×C₈	—	3	2	no	—
28	C₂²×D₃×C₈	—	3	2	no	—
29	C₂²×D₃×C₈	—	3	48	no	—
30	C₂²×D₃×C₈	—	6	1	no	—
31	C₂²×D₃×C₈	D₃	1	2	yes	direct product factor
32	C₂³×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₃×D₃	C₂×C₄	1	2	yes	—
34	C₂²×D₃²	C₈	1	16	yes	direct product factor
35	C₂²×D₃²	C₂×C₄	1	12	yes	—
36	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
37	C₂²×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃⋊D₃	C₂×C₄	1	6	yes	—
39	C₂×C₄×C₃⋊D₃	C₂³	1	6	yes	—
40	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₄×D₃	C₂×C₄	1	12	yes	—
42	C₂×C₃×C₄×D₃	C₂³	1	12	yes	—
43	C₂×#624	C₂×C₄	1	12	yes	—
44	C₂×D₃×Dic₃	C₂×C₄	1	24	yes	—
45	C₄×D₃²	C₂×C₄	1	48	yes	—
46	C₄×D₃²	C₂³	1	16	yes	—
47	C₂×C₃²×C₈	C₂³	1	6	yes	complemented
48	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	6	yes	complemented
49	C₈×C₃⋊D₃	C₂³	1	16	yes	complemented
50	C₂×C₃×C₃⋊C₈	C₂³	1	12	yes	complemented
51	C₃×D₃×C₈	C₂³	1	32	yes	complemented
52	#3710	C₂³	1	16	yes	complemented
53	D₃×C₃⋊C₈	C₂³	1	32	yes	complemented
54	C₂⁴×C₈	—	9	1	no	Sylow
55	C₂⁴×D₃	—	3	2	no	—
56	C₂³×C₃×C₄	—	3	2	no	—
57	C₂³×Dic₃	—	3	2	no	—
58	C₂²×C₄×D₃	—	3	2	no	—
59	C₂²×C₄×D₃	—	3	2	no	—
60	C₂²×C₄×D₃	—	3	2	no	—
61	C₂²×C₄×D₃	—	3	24	no	—
62	C₂²×C₄×D₃	—	3	24	no	—
63	C₂²×C₄×D₃	—	6	1	no	—
64	C₂²×C₄×D₃	C₂×D₃	1	2	yes	—
65	C₂²×C₃×C₈	—	2	1	no	—
66	C₂²×C₃×C₈	—	3	2	no	—
67	C₂²×C₃×C₈	—	3	24	no	—
68	C₂²×C₃×C₈	C₂×D₃	1	2	yes	complemented
69	C₂²×C₃⋊C₈	—	3	2	no	—
70	C₂²×C₃⋊C₈	—	3	24	no	—
71	C₂²×C₃⋊C₈	—	6	1	no	—
72	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	complemented
73	C₂×D₃×C₈	—	3	24	no	—
74	C₂×D₃×C₈	—	3	24	no	—
75	C₂×D₃×C₈	—	3	24	no	—
76	C₂×D₃×C₈	—	3	128	no	—
77	C₂×D₃×C₈	—	6	12	no	—
78	C₂×D₃×C₈	C₂×D₃	1	24	yes	direct product factor
79	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₃⋊D₃	C₂×C₈	1	8	yes	complemented
81	C₂²×C₃⋊D₃	C₂²×C₄	1	6	yes	—
82	C₂²×C₃×D₃	C₂×C₈	1	16	yes	complemented
83	C₂²×C₃×D₃	C₂²×C₄	1	12	yes	—
84	C₂×D₃²	C₂×C₈	1	96	yes	direct product factor
85	C₂×D₃²	C₂²×C₄	1	16	yes	—
86	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
87	C₂×C₃²×C₄	C₂⁴	1	3	yes	—
88	C₂×C₃⋊Dic₃	C₂²×C₄	1	6	yes	—
89	C₄×C₃⋊D₃	C₂²×C₄	1	12	yes	—
90	C₄×C₃⋊D₃	C₂⁴	1	4	yes	—
91	C₂×C₃×Dic₃	C₂²×C₄	1	12	yes	—
92	C₃×C₄×D₃	C₂²×C₄	1	24	yes	—
93	C₃×C₄×D₃	C₂⁴	1	8	yes	—
94	#624	C₂²×C₄	1	16	yes	—
95	D₃×Dic₃	C₂²×C₄	1	32	yes	—
96	C₃²×C₈	C₂⁴	1	4	yes	complemented
97	C₃⋊(C₃⋊C₈)	C₂⁴	1	4	yes	complemented
98	C₃×C₃⋊C₈	C₂⁴	1	8	yes	complemented
99	C₂⁴×C₄	—	9	1	no	—
100	C₂³×C₈	—	3	2	no	—
101	C₂³×C₈	—	9	1	no	—
102	C₂³×C₈	—	9	24	no	—
103	C₂³×C₈	—	9	2	no	—
104	C₂³×C₈	—	9	1	no	—
105	C₂⁴×C₃	—	3	2	no	—
106	C₂³×D₃	—	3	2	no	—
107	C₂³×D₃	—	3	32	no	—
108	C₂³×D₃	—	3	24	no	—
109	C₂³×D₃	—	6	1	no	—
110	C₂³×D₃	C₄×D₃	1	2	yes	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	—	3	2	no	—
113	C₂²×C₃×C₄	—	3	12	no	—
114	C₂²×C₃×C₄	—	3	12	no	—
115	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
116	C₂²×Dic₃	—	3	2	no	—
117	C₂²×Dic₃	—	3	24	no	—
118	C₂²×Dic₃	—	6	1	no	—
119	C₂²×Dic₃	C₄×D₃	1	2	yes	—
120	C₂×C₄×D₃	—	3	24	no	—
121	C₂×C₄×D₃	—	3	12	no	—
122	C₂×C₄×D₃	—	3	12	no	—
123	C₂×C₄×D₃	—	3	24	no	—
124	C₂×C₄×D₃	—	3	96	no	—
125	C₂×C₄×D₃	—	3	32	no	—
126	C₂×C₄×D₃	—	6	6	no	—
127	C₂×C₄×D₃	—	6	6	no	—
128	C₂×C₄×D₃	C₄×D₃	1	12	yes	—
129	C₂×C₄×D₃	C₂²×D₃	1	12	yes	—
130	C₂×C₃×C₈	—	2	6	no	—
131	C₂×C₃×C₈	—	3	12	no	—
132	C₂×C₃×C₈	—	3	32	no	—
133	C₂×C₃×C₈	C₂²×D₃	1	12	yes	complemented
134	C₂×C₃⋊C₈	—	3	12	no	—
135	C₂×C₃⋊C₈	—	3	32	no	—
136	C₂×C₃⋊C₈	—	6	6	no	—
137	C₂×C₃⋊C₈	C₂²×D₃	1	12	yes	complemented
138	D₃×C₈	—	3	32	no	—
139	D₃×C₈	—	3	32	no	—
140	D₃×C₈	—	3	32	no	—
141	D₃×C₈	—	6	16	no	—
142	D₃×C₈	C₂²×D₃	1	32	yes	direct product factor
143	C₂²×C₃²	C₂²×C₈	1	4	yes	complemented
144	C₂²×C₃²	C₂³×C₄	1	3	yes	—
145	C₂×C₃⋊D₃	C₂²×C₈	1	24	yes	complemented
146	C₂×C₃⋊D₃	C₂³×C₄	1	4	yes	—
147	C₂×C₃×D₃	C₂²×C₈	1	48	yes	complemented
148	C₂×C₃×D₃	C₂³×C₄	1	8	yes	—
149	D₃²	C₂²×C₈	1	64	yes	direct product factor
150	C₃²×C₄	C₂³×C₄	1	3	yes	—
151	C₃²×C₄	C₂⁵	1	1	yes	characteristic
152	C₃⋊Dic₃	C₂³×C₄	1	4	yes	—
153	C₃×Dic₃	C₂³×C₄	1	8	yes	—
154	C₂⁵	—	9	1	no	—
155	C₂³×C₄	—	3	2	no	—
156	C₂³×C₄	—	9	12	no	—
157	C₂³×C₄	—	9	1	no	—
158	C₂³×C₄	—	9	12	no	—
159	C₂³×C₄	—	9	2	no	—
160	C₂³×C₄	—	9	1	no	—
161	C₂²×C₈	—	3	2	no	—
162	C₂²×C₈	—	3	24	no	—
163	C₂²×C₈	—	9	12	no	—
164	C₂²×C₈	—	9	64	no	—
165	C₂²×C₈	—	9	24	no	—
166	C₂²×C₈	—	9	1	no	—
167	C₂²×C₈	—	9	12	no	—
168	C₂²×C₈	D₃²	1	1	yes	characteristic, direct product factor
169	C₂³×C₃	—	2	1	no	—
170	C₂³×C₃	—	3	16	no	—
171	C₂³×C₃	—	3	12	no	—
172	C₂³×C₃	C₂×C₄×D₃	1	2	yes	—
173	C₂²×D₃	—	3	16	no	—
174	C₂²×D₃	—	3	12	no	—
175	C₂²×D₃	—	3	192	no	—
176	C₂²×D₃	—	3	32	no	—
177	C₂²×D₃	—	6	8	no	—
178	C₂²×D₃	—	6	6	no	—
179	C₂²×D₃	D₃×C₈	1	16	yes	direct product factor
180	C₂²×D₃	C₂×C₄×D₃	1	12	yes	—
181	C₂×C₃×C₄	—	2	3	no	—
182	C₂×C₃×C₄	—	2	3	no	—
183	C₂×C₃×C₄	—	3	12	no	—
184	C₂×C₃×C₄	—	3	24	no	—
185	C₂×C₃×C₄	—	3	8	no	—
186	C₂×C₃×C₄	C₂×C₄×D₃	1	6	yes	—
187	C₂×C₃×C₄	C₂³×D₃	1	6	yes	—
188	C₂×Dic₃	—	3	12	no	—
189	C₂×Dic₃	—	3	32	no	—
190	C₂×Dic₃	—	6	6	no	—
191	C₂×Dic₃	C₂×C₄×D₃	1	12	yes	—
192	C₄×D₃	—	3	32	no	—
193	C₄×D₃	—	3	24	no	—
194	C₄×D₃	—	3	8	no	—
195	C₄×D₃	—	3	32	no	—
196	C₄×D₃	—	6	12	no	—
197	C₄×D₃	—	6	4	no	—
198	C₄×D₃	C₂×C₄×D₃	1	24	yes	—
199	C₄×D₃	C₂³×D₃	1	8	yes	—
200	C₃×C₈	—	2	4	no	—
201	C₃×C₈	—	3	8	no	—
202	C₃×C₈	C₂³×D₃	1	8	yes	complemented
203	C₃⋊C₈	—	3	8	no	—
204	C₃⋊C₈	—	6	4	no	—
205	C₃⋊C₈	C₂³×D₃	1	8	yes	complemented
206	C₂×C₃²	C₂³×C₈	1	6	yes	complemented
207	C₂×C₃²	C₂⁴×C₄	1	1	yes	characteristic
208	C₃⋊D₃	C₂³×C₈	1	8	yes	complemented
209	C₃×D₃	C₂³×C₈	1	16	yes	complemented
210	C₂⁴	—	3	2	no	—
211	C₂⁴	—	9	12	no	—
212	C₂⁴	—	9	16	no	—
213	C₂⁴	—	9	1	no	—
214	C₂²×C₄	—	3	2	no	—
215	C₂²×C₄	—	3	12	no	—
216	C₂²×C₄	—	3	12	no	—
217	C₂²×C₄	—	9	16	no	—
218	C₂²×C₄	—	9	12	no	—
219	C₂²×C₄	—	9	48	no	—
220	C₂²×C₄	—	9	24	no	—
221	C₂²×C₄	—	9	1	no	—
222	C₂²×C₄	—	9	6	no	—
223	C₂²×C₄	—	9	6	no	—
224	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
225	C₂×C₈	—	3	12	no	—
226	C₂×C₈	—	3	32	no	—
227	C₂×C₈	—	9	16	no	—
228	C₂×C₈	—	9	32	no	—
229	C₂×C₈	—	9	6	no	—
230	C₂×C₈	—	9	16	no	—
231	C₂×C₈	C₂×D₃²	1	6	yes	direct product factor
232	C₂²×C₃	—	2	4	no	—
233	C₂²×C₃	—	2	3	no	—
234	C₂²×C₃	—	3	8	no	—
235	C₂²×C₃	—	3	48	no	—
236	C₂²×C₃	C₂×D₃×C₈	1	8	yes	complemented
237	C₂²×C₃	C₂²×C₄×D₃	1	6	yes	—
238	C₂×D₃	—	3	8	no	—
239	C₂×D₃	—	3	48	no	—
240	C₂×D₃	—	3	128	no	—
241	C₂×D₃	—	6	4	no	—
242	C₂×D₃	—	6	24	no	—
243	C₂×D₃	C₂×D₃×C₈	1	48	yes	direct product factor
244	C₂×D₃	C₂²×C₄×D₃	1	8	yes	—
245	C₃×C₄	—	2	3	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	3	8	no	—
248	C₃×C₄	C₂²×C₄×D₃	1	6	yes	—
249	C₃×C₄	C₂⁴×D₃	1	2	yes	—
250	Dic₃	—	3	8	no	—
251	Dic₃	—	6	4	no	—
252	Dic₃	C₂²×C₄×D₃	1	8	yes	—
253	C₃²	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	3	16	no	—
255	C₂³	—	3	12	no	—
256	C₂³	—	9	16	no	—
257	C₂³	—	9	96	no	—
258	C₂³	—	9	8	no	—
259	C₂³	—	9	6	no	—
260	C₂³	C₄×D₃²	1	1	yes	characteristic
261	C₂×C₄	—	3	12	no	—
262	C₂×C₄	—	3	24	no	—
263	C₂×C₄	—	3	8	no	—
264	C₂×C₄	—	9	16	no	—
265	C₂×C₄	—	9	32	no	—
266	C₂×C₄	—	9	6	no	—
267	C₂×C₄	—	9	12	no	—
268	C₂×C₄	—	9	4	no	—
269	C₂×C₄	C₄×D₃²	1	3	yes	—
270	C₂×C₄	C₂²×D₃²	1	3	yes	—
271	C₈	—	3	8	no	—
272	C₈	—	9	4	no	—
273	C₈	C₂²×D₃²	1	4	yes	direct product factor
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	6	no	—
276	C₂×C₃	—	3	16	no	—
277	C₂×C₃	C₂²×D₃×C₈	1	12	yes	complemented
278	C₂×C₃	C₂³×C₄×D₃	1	2	yes	—
279	D₃	—	3	16	no	—
280	D₃	—	6	8	no	—
281	D₃	C₂²×D₃×C₈	1	16	yes	direct product factor
282	C₂²	—	3	8	no	—
283	C₂²	—	3	48	no	—
284	C₂²	—	9	64	no	—
285	C₂²	—	9	4	no	—
286	C₂²	—	9	24	no	—
287	C₂²	D₃²×C₈	1	4	yes	direct product factor
288	C₂²	C₂×C₄×D₃²	1	3	yes	—
289	C₄	—	3	8	no	—
290	C₄	—	9	4	no	—
291	C₄	C₂×C₄×D₃²	1	3	yes	—
292	C₄	C₂³×D₃²	1	1	yes	characteristic
293	C₃	—	2	1	no	—
294	C₃	C₂³×D₃×C₈	1	2	yes	complemented
295	C₂	—	3	16	no	—
296	C₂	—	9	8	no	—
297	C₂	C₂×D₃²×C₈	1	6	yes	direct product factor
298	C₂	C₂²×C₄×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₃²
commutator, hypocenter	C₃²	C₂⁴×C₈
Frattini	C₄	C₂³×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂³×C₃²	C₂²×C₄
prime radical	C₂³×D₃²	C₄
cosocle	C₃²×C₄	C₂⁵
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×C₃⋊D₃	C₂	semidirect
3. C₂²×C₃×D₃×C₈	C₂	semidirect
4. C₂²×#3710	C₂	semidirect
5. C₂²×D₃×C₃⋊C₈	C₂	semidirect
6. C₂×D₃²×C₈	C₂	direct (N × H)
15. C₂²×C₃²×C₈	C₂²	semidirect
16. C₂²×C₃⋊(C₃⋊C₈)	C₂²	semidirect
17. C₂×C₈×C₃⋊D₃	C₂²	semidirect
18. C₂²×C₃×C₃⋊C₈	C₂²	semidirect
19. C₂×C₃×D₃×C₈	C₂²	semidirect
20. C₂×#3710	C₂²	semidirect
21. C₂×D₃×C₃⋊C₈	C₂²	semidirect
22. D₃²×C₈	C₂²	direct (N × H)
31. C₂²×D₃×C₈	D₃	direct (N × H)
34. C₂²×D₃²	C₈	direct (N × H)
47. C₂×C₃²×C₈	C₂³	semidirect
48. C₂×C₃⋊(C₃⋊C₈)	C₂³	semidirect
49. C₈×C₃⋊D₃	C₂³	semidirect
50. C₂×C₃×C₃⋊C₈	C₂³	semidirect
51. C₃×D₃×C₈	C₂³	semidirect
52. #3710	C₂³	semidirect
53. D₃×C₃⋊C₈	C₂³	semidirect
68. C₂²×C₃×C₈	C₂×D₃	semidirect
72. C₂²×C₃⋊C₈	C₂×D₃	semidirect
78. C₂×D₃×C₈	C₂×D₃	direct (N × H)
80. C₂²×C₃⋊D₃	C₂×C₈	semidirect
82. C₂²×C₃×D₃	C₂×C₈	semidirect
84. C₂×D₃²	C₂×C₈	direct (N × H)
96. C₃²×C₈	C₂⁴	semidirect
97. C₃⋊(C₃⋊C₈)	C₂⁴	semidirect
98. C₃×C₃⋊C₈	C₂⁴	semidirect
133. C₂×C₃×C₈	C₂²×D₃	semidirect
137. C₂×C₃⋊C₈	C₂²×D₃	semidirect
142. D₃×C₈	C₂²×D₃	direct (N × H)
143. C₂²×C₃²	C₂²×C₈	semidirect
145. C₂×C₃⋊D₃	C₂²×C₈	semidirect
147. C₂×C₃×D₃	C₂²×C₈	semidirect
149. D₃²	C₂²×C₈	direct (N × H)
168. C₂²×C₈	D₃²	direct (N × H)
179. C₂²×D₃	D₃×C₈	direct (N × H)
202. C₃×C₈	C₂³×D₃	semidirect
205. C₃⋊C₈	C₂³×D₃	semidirect
206. C₂×C₃²	C₂³×C₈	semidirect
208. C₃⋊D₃	C₂³×C₈	semidirect
209. C₃×D₃	C₂³×C₈	semidirect
231. C₂×C₈	C₂×D₃²	direct (N × H)
236. C₂²×C₃	C₂×D₃×C₈	semidirect
243. C₂×D₃	C₂×D₃×C₈	direct (N × H)
253. C₃²	C₂⁴×C₈	semidirect
273. C₈	C₂²×D₃²	direct (N × H)
277. C₂×C₃	C₂²×D₃×C₈	semidirect
281. D₃	C₂²×D₃×C₈	direct (N × H)
287. C₂²	D₃²×C₈	direct (N × H)
294. C₃	C₂³×D₃×C₈	semidirect
297. C₂	C₂×D₃²×C₈	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×C₄×D₃²	2
C₂²×C₈×C₃⋊D₃	2
C₂²×C₃×D₃×C₈	2
C₂²×#3710	2
C₂²×D₃×C₃⋊C₈	2
C₂×D₃²×C₈	2
C₂³×D₃×C₈	3

Subgroups — C22×D32×C8 #11918426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₃²×C₈ #11918426