Subgroups — #79899

Order 384.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×#135	C₂	1	1	yes	maximal, characteristic
2	#5750	C₂	1	2	yes	maximal, complemented
3	C₃×#444	C₂	1	1	yes	maximal, characteristic, complemented
4	#5197	C₂	1	1	yes	maximal, characteristic, complemented
5	#5144	C₂	1	2	yes	maximal, complemented
6	#4894	C₂	1	1	yes	maximal, characteristic, complemented
7	D₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
8	#4824	C₂	1	2	yes	maximal, complemented
9	#4821	C₂	1	2	yes	maximal, complemented
10	#4717	C₂	1	1	yes	maximal, characteristic, complemented
11	#4708	C₂	1	1	yes	maximal, characteristic, complemented
12	#2957	—	3	1	no	maximal, Sylow
13	D₃×#41	C₂²	1	2	yes	—
14	C₃×#135	C₂²	1	1	yes	characteristic
15	C₃×#131	C₂²	1	2	yes	complemented
16	#950	C₂²	1	2	yes	—
17	#948	C₂²	1	2	yes	complemented
18	#919	C₄	1	1	yes	characteristic, complemented
19	D₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
20	#916	C₄	1	2	yes	complemented
21	D₃×SD₁₆	C₄	1	2	yes	complemented
22	#912	C₄	1	1	yes	characteristic, complemented
23	D₃×D₈	C₄	1	1	yes	characteristic, complemented
24	#907	—	2	2	no	—
25	#907	C₂²	1	2	yes	complemented
26	D₃×M₁₆	—	2	2	no	—
27	D₃×M₁₆	C₂²	1	2	yes	—
28	#904	C₂²	1	1	yes	characteristic
29	#901	—	2	2	no	—
30	#901	C₂²	1	2	yes	complemented
31	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
32	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
33	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
34	#872	C₂²	1	1	yes	characteristic
35	C₃×#108	C₂²	1	1	yes	characteristic, complemented
36	C₃×#104	C₂²	1	2	yes	complemented
37	C₃×#97	C₂²	1	1	yes	characteristic, complemented
38	#837	C₂²	1	2	yes	complemented
39	#825	C₂²	1	2	yes	complemented
40	#822	C₂²	1	2	yes	complemented
41	#819	C₂²	1	1	yes	characteristic, complemented
42	#815	C₂²	1	1	yes	characteristic, complemented
43	C₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
44	#805	C₂²	1	2	yes	complemented
45	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	C₂×#135	—	3	1	no	—
47	#568	—	3	2	no	—
48	#444	—	3	1	no	—
49	#444	—	3	1	no	—
50	#444	—	3	1	no	—
51	#444	D₃	1	1	yes	characteristic, complemented
52	#443	—	3	2	no	—
53	#443	—	3	2	no	—
54	C₂×#108	—	3	1	no	—
55	#421	—	3	2	no	—
56	#405	—	3	1	no	—
57	C₃×#41	C₂³	1	2	yes	—
58	#239	C₂×C₄	1	2	yes	—
59	D₃×Q₈	C₂×C₄	1	2	yes	—
60	#237	—	2	2	no	—
61	#237	C₂×C₄	1	2	yes	—
62	D₃×D₄	—	2	2	no	—
63	D₃×D₄	C₂×C₄	1	2	yes	—
64	#235	—	2	2	no	—
65	#235	C₂³	1	2	yes	—
66	C₂×C₄×D₃	—	2	2	no	—
67	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
68	C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
69	C₃×SD₁₆	C₂×C₄	1	2	yes	—
70	C₃×D₈	C₂×C₄	1	1	yes	characteristic
71	C₃×M₁₆	—	2	2	no	—
72	C₃×M₁₆	C₂³	1	2	yes	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
76	C₃×C₄²	C₂³	1	1	yes	characteristic
77	#216	C₂×C₄	1	2	yes	—
78	#215	C₂×C₄	1	2	yes	—
79	#214	C₂×C₄	1	2	yes	—
80	#213	C₂×C₄	1	2	yes	—
81	(C₂×D₃)⋊C₄	—	2	1	no	—
82	Dic₃⋊C₄	—	2	1	no	—
83	C₄×Dic₃	C₂³	1	1	yes	characteristic
84	#208	—	2	2	no	—
85	#208	C₂³	1	2	yes	—
86	C₂×C₃⋊C₈	—	2	2	no	—
87	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
88	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
89	Dic₁₂	C₂×C₄	1	1	yes	characteristic
90	D₂₄	C₂×C₄	1	1	yes	characteristic
91	#204	C₂×C₄	1	2	yes	—
92	ΓL(1,𝔽₂₅)	—	2	2	no	—
93	ΓL(1,𝔽₂₅)	—	2	2	no	—
94	ΓL(1,𝔽₂₅)	—	2	2	no	—
95	ΓL(1,𝔽₂₅)	D₄	1	2	yes	—
96	ΓL(1,𝔽₂₅)	D₄	1	2	yes	—
97	D₃×C₈	—	2	2	no	—
98	D₃×C₈	—	2	2	no	—
99	D₃×C₈	—	2	2	no	—
100	D₃×C₈	C₂×C₄	1	1	yes	characteristic
101	D₃×C₈	C₂×C₄	1	1	yes	characteristic
102	D₃×C₈	C₂×C₄	1	1	yes	characteristic
103	D₃×C₈	C₂×C₄	1	1	yes	characteristic
104	C₂×#41	—	3	2	no	—
105	#135	—	3	2	no	—
106	#135	—	3	1	no	—
107	#135	—	3	2	no	—
108	#135	—	3	1	no	—
109	#135	—	3	1	no	—
110	#135	C₂×D₃	1	1	yes	characteristic
111	C₂×Q₁₆	—	3	1	no	—
112	C₂×SD₁₆	—	3	2	no	—
113	C₂×D₈	—	3	1	no	—
114	#131	—	3	2	no	—
115	#131	—	3	2	no	—
116	#131	—	3	2	no	—
117	#131	—	6	2	no	—
118	#131	—	6	2	no	—
119	#131	C₂×D₃	1	2	yes	complemented
120	C₂×M₁₆	—	3	1	no	—
121	C₂×M₁₆	—	3	2	no	—
122	C₂×M₁₆	—	6	2	no	—
123	C₂×M₁₆	—	6	2	no	—
124	C₂²×C₈	—	3	1	no	—
125	#117	—	3	1	no	—
126	#108	—	3	2	no	—
127	#108	—	3	1	no	—
128	#108	C₂×D₃	1	1	yes	characteristic, complemented
129	#104	—	3	2	no	—
130	#104	—	3	2	no	—
131	#104	—	3	2	no	—
132	#104	C₂×D₃	1	2	yes	complemented
133	#97	—	3	1	no	—
134	#97	C₂×D₃	1	1	yes	characteristic, complemented
135	C₄×C₈	—	3	1	no	—
136	C₄×C₈	—	3	1	no	—
137	C₂²×D₃	—	2	2	no	—
138	C₂²×D₃	#41	1	1	yes	characteristic
139	C₃×Q₈	C₂²×C₄	1	2	yes	—
140	C₃×D₄	—	2	2	no	—
141	C₃×D₄	C₂²×C₄	1	2	yes	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
145	#67	—	2	2	no	—
146	#67	—	2	2	no	—
147	#67	—	2	2	no	—
148	C₂×Dic₃	—	2	2	no	—
149	C₂×Dic₃	—	2	1	no	—
150	C₂×Dic₃	#41	1	1	yes	characteristic
151	D₁₂	—	2	2	no	—
152	D₁₂	C₂²×C₄	1	2	yes	—
153	C₄×D₃	—	2	2	no	—
154	C₄×D₃	—	2	2	no	—
155	C₄×D₃	—	2	2	no	—
156	C₄×D₃	C₂²×C₄	1	1	yes	characteristic
157	C₄×D₃	C₂²×C₄	1	1	yes	characteristic
158	C₄×D₃	C₂×D₄	1	1	yes	characteristic
159	C₄×D₃	C₂×D₄	1	1	yes	characteristic
160	Dic₆	—	2	2	no	—
161	Dic₆	C₂²×C₄	1	2	yes	—
162	C₃×C₈	—	2	2	no	—
163	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
164	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
165	C₃×C₈	C₂×D₄	1	2	yes	—
166	C₃⋊C₈	—	2	2	no	—
167	C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
168	C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
169	C₃⋊C₈	C₂×D₄	1	2	yes	—
170	#41	—	3	2	no	—
171	#41	—	3	2	no	—
172	#41	—	3	2	no	—
173	#41	—	6	2	no	—
174	#41	—	6	2	no	—
175	#41	C₂²×D₃	1	2	yes	—
176	C₂×Q₈	—	3	2	no	—
177	C₂×D₄	—	3	2	no	—
178	C₂×D₄	—	6	2	no	—
179	C₂²×C₄	—	3	1	no	—
180	C₂²×C₄	—	6	2	no	—
181	Q₁₆	—	3	2	no	—
182	Q₁₆	—	3	1	no	—
183	Q₁₆	C₄×D₃	1	1	yes	characteristic
184	SD₁₆	—	3	2	no	—
185	SD₁₆	—	3	2	no	—
186	SD₁₆	—	3	2	no	—
187	SD₁₆	C₄×D₃	1	2	yes	—
188	D₈	—	3	2	no	—
189	D₈	—	3	1	no	—
190	D₈	C₄×D₃	1	1	yes	characteristic
191	M₁₆	—	2	2	no	—
192	M₁₆	—	3	2	no	—
193	M₁₆	—	3	2	no	—
194	M₁₆	—	3	2	no	—
195	M₁₆	—	6	2	no	—
196	M₁₆	—	6	2	no	—
197	M₁₆	—	6	2	no	—
198	M₁₆	—	6	2	no	—
199	M₁₆	C₂²×D₃	1	2	yes	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	3	1	no	—
202	C₂×C₈	—	3	1	no	—
203	C₂×C₈	—	3	1	no	—
204	C₂×C₈	—	3	1	no	—
205	C₂×C₈	—	3	1	no	—
206	C₂×C₈	—	3	1	no	—
207	C₂×C₈	—	6	2	no	—
208	C₂×C₈	—	6	2	no	—
209	C₂×C₈	—	6	2	no	—
210	C₂×C₈	—	6	2	no	—
211	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
213	C₄⋊C₄	—	6	1	no	—
214	C₂²⋊C₄	—	6	1	no	—
215	C₄²	—	3	1	no	—
216	C₄²	C₂²×D₃	1	1	yes	characteristic
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	C₂×#41	1	1	yes	characteristic
219	C₂×D₃	—	2	2	no	—
220	C₂×D₃	—	4	1	no	—
221	C₂×D₃	—	8	2	no	—
222	C₂×D₃	C₄×D₄	1	1	yes	characteristic
223	C₂×D₃	C₄×D₄	1	1	yes	characteristic
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	4	1	no	—
226	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
227	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
228	Dic₃	—	2	2	no	—
229	Dic₃	—	4	1	no	—
230	Dic₃	C₄×D₄	1	1	yes	characteristic
231	Dic₃	C₄×D₄	1	1	yes	characteristic
232	C₂³	—	3	1	no	—
233	C₂³	—	6	2	no	—
234	Q₈	—	3	2	no	—
235	Q₈	—	6	2	no	—
236	Q₈	C₂×C₄×D₃	1	2	yes	—
237	D₄	—	2	2	no	—
238	D₄	—	3	2	no	—
239	D₄	—	6	2	no	—
240	D₄	—	6	2	no	—
241	D₄	—	6	2	no	—
242	D₄	—	6	2	no	—
243	D₄	C₂×C₄×D₃	1	2	yes	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	3	1	no	—
247	C₂×C₄	—	3	1	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	1	no	—
251	C₂×C₄	—	6	2	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	—	6	2	no	—
254	C₂×C₄	—	6	2	no	—
255	C₂×C₄	—	6	1	no	—
256	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	3	1	no	—
259	C₈	—	3	1	no	—
260	C₈	—	3	2	no	—
261	C₈	—	6	2	no	—
262	C₈	C₂×C₄×D₃	1	1	yes	characteristic
263	C₈	C₂×C₄×D₃	1	1	yes	characteristic
264	C₈	D₃×D₄	1	2	yes	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	4	2	no	—
267	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
268	D₃	—	2	1	no	—
269	D₃	—	2	1	no	—
270	D₃	—	4	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	3	1	no	—
273	C₂²	—	3	1	no	—
274	C₂²	—	6	2	no	—
275	C₂²	—	12	1	no	—
276	C₂²	—	24	2	no	—
277	C₂²	D₃×#41	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	3	1	no	—
280	C₄	—	3	1	no	—
281	C₄	—	4	1	no	—
282	C₄	—	6	2	no	—
283	C₄	—	12	1	no	—
284	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
285	C₄	C₂×D₃×D₄	1	1	yes	characteristic
286	C₃	#2957	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	6	1	no	—
290	C₂	—	6	1	no	—
291	C₂	—	12	2	no	—
292	C₂	C₄×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#444	C₂
socle	C₂×C₃	C₂×C₄×D₄
prime radical	D₃×#135	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#444	D₃
hypocenter	C₃	#2957
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #5750	C₂	semidirect
3. C₃×#444	C₂	semidirect
4. #5197	C₂	semidirect
5. #5144	C₂	semidirect
6. #4894	C₂	semidirect
7. D₃×#108	C₂	semidirect
8. #4824	C₂	semidirect
9. #4821	C₂	semidirect
10. #4717	C₂	semidirect
11. #4708	C₂	semidirect
15. C₃×#131	C₂²	semidirect
17. #948	C₂²	semidirect
18. #919	C₄	semidirect
19. D₃×Q₁₆	C₄	semidirect
20. #916	C₄	semidirect
21. D₃×SD₁₆	C₄	semidirect
22. #912	C₄	semidirect
23. D₃×D₈	C₄	semidirect
25. #907	C₂²	semidirect
30. #901	C₂²	semidirect
35. C₃×#108	C₂²	semidirect
36. C₃×#104	C₂²	semidirect
37. C₃×#97	C₂²	semidirect
38. #837	C₂²	semidirect
39. #825	C₂²	semidirect
40. #822	C₂²	semidirect
41. #819	C₂²	semidirect
42. #815	C₂²	semidirect
43. C₈×Dic₃	C₂²	semidirect
44. #805	C₂²	semidirect
45. C₄×C₃⋊C₈	C₂²	semidirect
51. #444	D₃	semidirect
119. #131	C₂×D₃	semidirect
128. #108	C₂×D₃	semidirect
132. #104	C₂×D₃	semidirect
134. #97	C₂×D₃	semidirect
286. C₃	#2957	semidirect

Maximal subgroups

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

subgroup	index
D₃×#135	2
#5750	2
C₃×#444	2
#5197	2
#5144	2
#4894	2
D₃×#108	2
#4824	2
#4821	2
#4717	2
#4708	2
#2957	3