Subgroups — #11861200

Order 1152.

1356 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#135	C₂	1	1	yes	maximal, characteristic
2	#10596290	C₂	1	2	yes	maximal, complemented
3	#10592935	C₂	1	2	yes	maximal, complemented
4	#10592896	C₂	1	4	yes	maximal, complemented
5	#10592895	C₂	1	4	yes	maximal, complemented
6	C₂×#63646	C₂	1	1	yes	maximal, characteristic, complemented
7	#10592800	C₂	1	1	yes	maximal, characteristic, complemented
8	#83910	—	3	4	no	maximal
9	C₂×C₃²×#41	C₂²	1	2	yes	—
10	C₃²×#135	C₄	1	2	yes	complemented
11	C₃²×#135	C₄	1	2	yes	complemented
12	C₃²×#135	C₂²	1	2	yes	—
13	C₃²×#135	C₂²	1	2	yes	—
14	C₂×C₃²×Q₁₆	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃²×SD₁₆	C₄	1	2	yes	complemented
16	C₂×C₃²×D₈	C₄	1	1	yes	characteristic, complemented
17	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
18	#64157	—	2	4	no	—
19	#64157	C₂²	1	8	yes	complemented
20	#64135	C₂²	1	1	yes	characteristic
21	C₂×#3748	—	2	4	no	—
22	C₂×#3748	C₂²	1	2	yes	—
23	C₂×#3748	C₂²	1	1	yes	characteristic
24	#63664	C₂²	1	8	yes	complemented
25	#63646	C₂²	1	2	yes	complemented
26	#63646	C₂²	1	2	yes	complemented
27	#63642	C₂²	1	2	yes	complemented
28	C₈×C₃⋊Dic₃	C₂²	1	2	yes	complemented
29	C₂×C₃×#135	D₃	1	4	yes	—
30	#5819	—	3	8	no	—
31	#5236	—	3	8	no	—
32	#5197	—	3	16	no	—
33	#5196	—	3	16	no	—
34	C₂×#819	—	3	4	no	—
35	#5101	—	3	4	no	—
36	C₃²×#41	—	2	2	no	—
37	C₃²×#41	—	2	2	no	—
38	C₃²×#41	C₂×C₄	1	4	yes	—
39	C₃²×#41	C₂³	1	4	yes	—
40	C₂×C₃²×Q₈	C₂×C₄	1	2	yes	—
41	C₂×C₃²×D₄	—	2	2	no	—
42	C₂×C₃²×D₄	C₂×C₄	1	2	yes	—
43	C₂²×C₃²×C₄	—	2	2	no	—
44	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
45	C₃²×Q₁₆	C₂×C₄	1	2	yes	—
46	C₃²×Q₁₆	C₂×C₄	1	2	yes	—
47	C₃²×SD₁₆	C₂×C₄	1	4	yes	—
48	C₃²×SD₁₆	C₂×C₄	1	4	yes	—
49	C₃²×D₈	C₂×C₄	1	2	yes	—
50	C₃²×D₈	C₂×C₄	1	2	yes	—
51	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
57	#3757	—	2	1	no	—
58	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
59	C₄×C₃⋊Dic₃	C₂³	1	2	yes	—
60	#3748	—	2	2	no	—
61	#3748	—	2	8	no	—
62	#3748	D₄	1	2	yes	—
63	#3748	D₄	1	2	yes	—
64	#3748	C₂³	1	4	yes	—
65	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
66	C₂×C₃⋊(C₃⋊C₈)	—	2	8	no	—
67	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
68	#2957	—	9	1	no	Sylow
69	C₂×C₃×#41	C₂×D₃	1	8	yes	—
70	C₃×#135	Dic₃	1	8	yes	complemented
71	C₃×#135	Dic₃	1	8	yes	complemented
72	C₃×#135	C₂×D₃	1	8	yes	—
73	C₃×#135	C₂×D₃	1	8	yes	—
74	C₂×C₃×Q₁₆	Dic₃	1	4	yes	complemented
75	C₂×C₃×SD₁₆	Dic₃	1	8	yes	complemented
76	C₂×C₃×D₈	Dic₃	1	4	yes	complemented
77	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
78	#948	—	3	32	no	—
79	#948	—	6	16	no	—
80	#926	—	3	4	no	—
81	C₂×#208	—	3	4	no	—
82	C₂×#208	—	3	8	no	—
83	C₂×#208	—	6	16	no	—
84	#837	—	3	32	no	—
85	#819	—	3	8	no	—
86	#819	—	3	8	no	—
87	#815	—	3	8	no	—
88	C₈×Dic₃	—	3	8	no	—
89	C₂³×C₃²	—	2	2	no	—
90	C₂³×C₃²	#41	1	1	yes	characteristic
91	C₃²×Q₈	—	2	2	no	—
92	C₃²×Q₈	C₂²×C₄	1	4	yes	—
93	C₃²×D₄	—	2	4	no	—
94	C₃²×D₄	—	2	4	no	—
95	C₃²×D₄	—	2	2	no	—
96	C₃²×D₄	C₂²×C₄	1	4	yes	—
97	C₂×C₃²×C₄	—	2	2	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	—	2	4	no	—
100	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
104	C₂×C₃²×C₄	#41	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
106	C₂×C₃⋊Dic₃	—	2	2	no	—
107	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₃⋊(C₃⋊C₈)	—	2	4	no	—
112	C₃⋊(C₃⋊C₈)	C₂×D₄	1	4	yes	—
113	C₂×#135	C₃⋊D₃	1	1	yes	characteristic
114	#568	—	9	2	no	—
115	#444	—	9	4	no	—
116	#443	—	9	4	no	—
117	C₂×#108	—	9	1	no	—
118	#421	—	9	2	no	—
119	#405	—	9	1	no	—
120	C₃×#41	—	2	8	no	—
121	C₃×#41	—	2	8	no	—
122	C₃×#41	C₂×Dic₃	1	16	yes	—
123	C₃×#41	C₂²×D₃	1	16	yes	—
124	C₂×C₃×Q₈	C₂×Dic₃	1	8	yes	—
125	C₂×C₃×D₄	—	2	8	no	—
126	C₂×C₃×D₄	C₂×Dic₃	1	8	yes	—
127	C₂²×C₃×C₄	—	2	8	no	—
128	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
129	C₃×Q₁₆	C₂×Dic₃	1	8	yes	—
130	C₃×Q₁₆	C₂×Dic₃	1	8	yes	—
131	C₃×SD₁₆	C₂×Dic₃	1	16	yes	—
132	C₃×SD₁₆	C₂×Dic₃	1	16	yes	—
133	C₃×D₈	C₂×Dic₃	1	8	yes	—
134	C₃×D₈	C₂×Dic₃	1	8	yes	—
135	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
136	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
137	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
138	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
139	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
140	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
141	#217	—	6	4	no	—
142	C₄⋊Dic₃	—	6	4	no	—
143	C₄×Dic₃	—	3	8	no	—
144	#208	—	3	8	no	—
145	#208	—	3	8	no	—
146	#208	—	3	16	no	—
147	#208	—	6	32	no	—
148	#208	—	6	8	no	—
149	C₂×C₃⋊C₈	—	3	8	no	—
150	C₂×C₃⋊C₈	—	6	32	no	—
151	C₂×C₃⋊C₈	—	6	8	no	—
152	C₂²×C₃²	—	2	4	no	—
153	C₂²×C₃²	—	4	1	no	—
154	C₂²×C₃²	—	8	2	no	—
155	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
156	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
157	C₂²×C₃²	C₂×#41	1	1	yes	characteristic
158	C₃²×C₄	—	2	4	no	—
159	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
160	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
161	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
162	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
163	C₃⋊Dic₃	—	4	2	no	—
164	C₂×#41	C₂×C₃⋊D₃	1	2	yes	—
165	#135	C₃⋊Dic₃	1	2	yes	complemented
166	#135	C₃⋊Dic₃	1	2	yes	complemented
167	#135	C₂×C₃⋊D₃	1	2	yes	—
168	#135	C₂×C₃⋊D₃	1	2	yes	—
169	C₂×Q₁₆	C₃⋊Dic₃	1	1	yes	characteristic, complemented
170	C₂×SD₁₆	C₃⋊Dic₃	1	2	yes	complemented
171	C₂×D₈	C₃⋊Dic₃	1	1	yes	characteristic, complemented
172	#131	—	9	8	no	—
173	#131	—	18	4	no	—
174	C₂×M₁₆	—	9	1	no	—
175	C₂×M₁₆	—	9	2	no	—
176	C₂×M₁₆	—	18	4	no	—
177	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
178	#117	—	9	1	no	—
179	#108	—	9	2	no	—
180	#108	—	9	2	no	—
181	#104	—	9	8	no	—
182	#97	—	9	2	no	—
183	C₄×C₈	—	9	2	no	—
184	C₂³×C₃	—	2	8	no	—
185	C₂³×C₃	#237	1	4	yes	—
186	C₃×Q₈	—	2	8	no	—
187	C₃×Q₈	C₂²×Dic₃	1	16	yes	—
188	C₃×D₄	—	2	16	no	—
189	C₃×D₄	—	2	16	no	—
190	C₃×D₄	—	2	8	no	—
191	C₃×D₄	C₂²×Dic₃	1	16	yes	—
192	C₂×C₃×C₄	—	2	8	no	—
193	C₂×C₃×C₄	—	2	16	no	—
194	C₂×C₃×C₄	—	2	16	no	—
195	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
196	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
197	C₂×C₃×C₄	#237	1	4	yes	—
198	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
199	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
200	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
201	C₂×Dic₃	—	6	8	no	—
202	C₃×C₈	C₂²×Dic₃	1	4	yes	—
203	C₃×C₈	C₂²×Dic₃	1	4	yes	—
204	C₃×C₈	C₂²×Dic₃	1	4	yes	—
205	C₃×C₈	C₂²×Dic₃	1	4	yes	—
206	C₃⋊C₈	—	3	16	no	—
207	C₃⋊C₈	—	6	16	no	—
208	C₂×C₃²	—	2	1	no	—
209	C₂×C₃²	—	2	1	no	—
210	C₂×C₃²	—	2	1	no	—
211	C₂×C₃²	—	4	4	no	—
212	C₂×C₃²	C₂×C₄×D₄	1	1	yes	characteristic
213	#41	—	2	2	no	—
214	#41	—	2	2	no	—
215	#41	C₂×C₃⋊Dic₃	1	4	yes	—
216	#41	C₂²×C₃⋊D₃	1	4	yes	—
217	C₂×Q₈	C₂×C₃⋊Dic₃	1	2	yes	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	C₂×C₃⋊Dic₃	1	2	yes	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
222	Q₁₆	C₂×C₃⋊Dic₃	1	2	yes	—
223	Q₁₆	C₂×C₃⋊Dic₃	1	2	yes	—
224	SD₁₆	C₂×C₃⋊Dic₃	1	4	yes	—
225	SD₁₆	C₂×C₃⋊Dic₃	1	4	yes	—
226	D₈	C₂×C₃⋊Dic₃	1	2	yes	—
227	D₈	C₂×C₃⋊Dic₃	1	2	yes	—
228	M₁₆	—	9	2	no	—
229	M₁₆	—	9	2	no	—
230	M₁₆	—	9	4	no	—
231	M₁₆	—	18	8	no	—
232	M₁₆	—	18	2	no	—
233	C₂×C₈	—	9	2	no	—
234	C₂×C₈	—	18	8	no	—
235	C₂×C₈	—	18	2	no	—
236	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
239	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
240	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
242	C₄⋊C₄	—	18	1	no	—
243	C₂²⋊C₄	—	18	1	no	—
244	C₄²	—	9	2	no	—
245	C₂²×C₃	—	2	16	no	—
246	C₂²×C₃	—	4	4	no	—
247	C₂²×C₃	—	8	8	no	—
248	C₂²×C₃	D₄×Dic₃	1	4	yes	—
249	C₂²×C₃	D₄×Dic₃	1	4	yes	—
250	C₂²×C₃	C₂×#237	1	4	yes	—
251	C₃×C₄	—	2	16	no	—
252	C₃×C₄	D₄×Dic₃	1	4	yes	—
253	C₃×C₄	D₄×Dic₃	1	4	yes	—
254	C₃×C₄	C₂×D₃×D₄	1	4	yes	—
255	C₃×C₄	C₂³×Dic₃	1	4	yes	—
256	Dic₃	—	12	8	no	—
257	C₃²	#2957	1	1	yes	characteristic, complemented, Sylow
258	C₂³	—	2	2	no	—
259	C₂³	#3830	1	1	yes	characteristic
260	Q₈	—	2	2	no	—
261	Q₈	C₂²×C₃⋊Dic₃	1	4	yes	—
262	D₄	—	2	4	no	—
263	D₄	—	2	4	no	—
264	D₄	—	2	2	no	—
265	D₄	C₂²×C₃⋊Dic₃	1	4	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	18	2	no	—
270	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
271	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
272	C₂×C₄	#3830	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
275	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
276	C₈	—	9	4	no	—
277	C₈	—	18	4	no	—
278	C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
279	C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
280	C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
281	C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
282	C₂×C₃	—	2	4	no	—
283	C₂×C₃	—	2	4	no	—
284	C₂×C₃	—	2	4	no	—
285	C₂×C₃	—	4	16	no	—
286	C₂×C₃	C₂×D₄×Dic₃	1	4	yes	—
287	C₂²	—	2	4	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	8	2	no	—
290	C₂²	D₄×C₃⋊Dic₃	1	1	yes	characteristic
291	C₂²	D₄×C₃⋊Dic₃	1	1	yes	characteristic
292	C₂²	C₂×#3830	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	36	2	no	—
295	C₄	D₄×C₃⋊Dic₃	1	1	yes	characteristic
296	C₄	D₄×C₃⋊Dic₃	1	1	yes	characteristic
297	C₄	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
298	C₄	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
299	C₃	#83910	1	4	yes	complemented
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	4	no	—
304	C₂	C₂×D₄×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting, prime radical	C₂×C₃²×#135	C₂
socle	C₂×C₃²	C₂×C₄×D₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂×#135	C₃⋊D₃
hypocenter	C₃²	#2957
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. #10596290	C₂	semidirect
3. #10592935	C₂	semidirect
4. #10592896	C₂	semidirect
5. #10592895	C₂	semidirect
6. C₂×#63646	C₂	semidirect
7. #10592800	C₂	semidirect
10. C₃²×#135	C₄	semidirect
11. C₃²×#135	C₄	semidirect
14. C₂×C₃²×Q₁₆	C₄	semidirect
15. C₂×C₃²×SD₁₆	C₄	semidirect
16. C₂×C₃²×D₈	C₄	semidirect
19. #64157	C₂²	semidirect
24. #63664	C₂²	semidirect
25. #63646	C₂²	semidirect
26. #63646	C₂²	semidirect
27. #63642	C₂²	semidirect
28. C₈×C₃⋊Dic₃	C₂²	semidirect
70. C₃×#135	Dic₃	semidirect
71. C₃×#135	Dic₃	semidirect
74. C₂×C₃×Q₁₆	Dic₃	semidirect
75. C₂×C₃×SD₁₆	Dic₃	semidirect
76. C₂×C₃×D₈	Dic₃	semidirect
165. #135	C₃⋊Dic₃	semidirect
166. #135	C₃⋊Dic₃	semidirect
169. C₂×Q₁₆	C₃⋊Dic₃	semidirect
170. C₂×SD₁₆	C₃⋊Dic₃	semidirect
171. C₂×D₈	C₃⋊Dic₃	semidirect
257. C₃²	#2957	semidirect
299. C₃	#83910	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×#135	2
#10596290	2
#10592935	2
#10592896	2
#10592895	2
C₂×#63646	2
#10592800	2
#83910	3