Subgroups — #79898

Order 384.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5768	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	#5196	C₂	1	1	yes	maximal, characteristic, complemented
5	#5144	C₂	1	2	yes	maximal, complemented
6	#4895	C₂	1	1	yes	maximal, characteristic, complemented
7	#4893	C₂	1	1	yes	maximal, characteristic, complemented
8	#4823	C₂	1	2	yes	maximal, complemented
9	D₃×#104	C₂	1	2	yes	maximal, complemented
10	#4717	C₂	1	1	yes	maximal, characteristic, complemented
11	#4707	C₂	1	1	yes	maximal, characteristic, complemented
12	#2956	—	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	#918	C₄	1	2	yes	complemented
19	#915	C₄	1	2	yes	complemented
20	#914	C₄	1	2	yes	complemented
21	#911	C₄	1	2	yes	complemented
22	#907	—	2	2	no	—
23	#907	C₂²	1	2	yes	complemented
24	D₃×M₁₆	—	2	2	no	—
25	D₃×M₁₆	C₂²	1	2	yes	—
26	#904	C₂²	1	1	yes	characteristic
27	#901	—	2	2	no	—
28	#901	C₂²	1	2	yes	complemented
29	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
30	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
31	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
32	C₄²×D₃	C₂²	1	1	yes	characteristic
33	C₃×#108	C₂²	1	1	yes	characteristic, complemented
34	C₃×#104	C₂²	1	2	yes	complemented
35	C₃×#97	C₂²	1	1	yes	characteristic, complemented
36	#837	C₂²	1	2	yes	complemented
37	#825	C₂²	1	2	yes	complemented
38	#822	C₂²	1	2	yes	complemented
39	#819	C₂²	1	1	yes	characteristic, complemented
40	C₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
41	C₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
42	#805	C₂²	1	2	yes	complemented
43	#803	C₂²	1	1	yes	characteristic, complemented
44	#575	—	3	1	no	—
45	#568	—	3	2	no	—
46	#444	—	3	2	no	—
47	#444	—	3	1	no	—
48	#444	D₃	1	1	yes	characteristic, complemented
49	#443	—	3	1	no	—
50	#443	—	3	2	no	—
51	#443	—	3	1	no	—
52	#430	—	3	1	no	—
53	C₂×#104	—	3	2	no	—
54	C₄×M₁₆	—	3	1	no	—
55	C₃×#41	C₂³	1	2	yes	—
56	#239	C₂×C₄	1	2	yes	—
57	D₃×Q₈	C₂×C₄	1	2	yes	—
58	#237	—	2	2	no	—
59	#237	C₂×C₄	1	2	yes	—
60	D₃×D₄	—	2	2	no	—
61	D₃×D₄	C₂×C₄	1	2	yes	—
62	#235	—	2	2	no	—
63	#235	C₂³	1	2	yes	—
64	C₂×C₄×D₃	—	2	2	no	—
65	C₂×C₄×D₃	—	2	1	no	—
66	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
67	C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
68	C₃×SD₁₆	C₂×C₄	1	2	yes	complemented
69	C₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
70	C₃×M₁₆	—	2	2	no	—
71	C₃×M₁₆	C₂³	1	2	yes	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
75	C₃×C₄²	C₂³	1	1	yes	characteristic
76	#216	C₂×C₄	1	2	yes	complemented
77	#215	C₂×C₄	1	2	yes	complemented
78	#214	C₂×C₄	1	2	yes	complemented
79	#213	C₂×C₄	1	2	yes	complemented
80	C₄×Dic₃	—	2	1	no	—
81	C₄×Dic₃	C₂³	1	1	yes	characteristic
82	#208	—	2	2	no	—
83	#208	C₂³	1	2	yes	—
84	C₂×C₃⋊C₈	—	2	2	no	—
85	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
86	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
87	Dic₁₂	C₂×C₄	1	1	yes	characteristic, complemented
88	D₂₄	C₂×C₄	1	1	yes	characteristic, complemented
89	#204	C₂×C₄	1	2	yes	complemented
90	ΓL(1,𝔽₂₅)	—	2	2	no	—
91	ΓL(1,𝔽₂₅)	—	2	2	no	—
92	ΓL(1,𝔽₂₅)	—	2	2	no	—
93	ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
94	ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
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	C₂×#41	—	3	2	no	—
101	#137	—	3	2	no	—
102	#137	—	3	2	no	—
103	AGL(1,ℤ/8)	—	3	2	no	—
104	AGL(1,ℤ/8)	—	3	2	no	—
105	#135	—	3	2	no	—
106	#135	—	3	1	no	—
107	#135	C₂×D₃	1	1	yes	characteristic
108	#131	—	3	2	no	—
109	#131	—	3	2	no	—
110	#131	—	3	2	no	—
111	#131	—	6	2	no	—
112	#131	—	6	2	no	—
113	#131	C₂×D₃	1	2	yes	complemented
114	C₂×M₁₆	—	3	1	no	—
115	C₂×M₁₆	—	3	1	no	—
116	C₂×M₁₆	—	3	2	no	—
117	C₂×M₁₆	—	6	2	no	—
118	C₂×M₁₆	—	6	2	no	—
119	C₂×C₄²	—	3	1	no	—
120	#108	—	3	2	no	—
121	#108	—	3	1	no	—
122	#108	C₂×D₃	1	1	yes	characteristic, complemented
123	#104	—	3	2	no	—
124	#104	—	3	2	no	—
125	#104	—	3	2	no	—
126	#104	C₂×D₃	1	2	yes	complemented
127	#97	—	3	1	no	—
128	#97	C₂×D₃	1	1	yes	characteristic, complemented
129	C₄×C₈	—	3	1	no	—
130	C₄×C₈	—	3	1	no	—
131	C₂²×D₃	—	2	2	no	—
132	C₂²×D₃	#41	1	1	yes	characteristic
133	C₃×Q₈	C₂²×C₄	1	2	yes	—
134	C₃×D₄	—	2	2	no	—
135	C₃×D₄	C₂²×C₄	1	2	yes	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
139	#67	—	2	2	no	—
140	#67	—	2	2	no	—
141	#67	—	2	2	no	—
142	C₂×Dic₃	—	2	1	no	—
143	C₂×Dic₃	—	2	2	no	—
144	C₂×Dic₃	#41	1	1	yes	characteristic
145	D₁₂	—	2	2	no	—
146	D₁₂	C₂²×C₄	1	2	yes	—
147	C₄×D₃	—	2	2	no	—
148	C₄×D₃	—	2	2	no	—
149	C₄×D₃	—	2	2	no	—
150	C₄×D₃	—	4	2	no	—
151	C₄×D₃	C₂²×C₄	1	2	yes	—
152	C₄×D₃	C₂×D₄	1	2	yes	—
153	Dic₆	—	2	2	no	—
154	Dic₆	C₂²×C₄	1	2	yes	—
155	C₃×C₈	—	2	2	no	—
156	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₃×C₈	C₂×D₄	1	2	yes	—
159	C₃⋊C₈	—	2	2	no	—
160	C₃⋊C₈	C₂²×C₄	1	2	yes	—
161	C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
162	C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
163	#41	—	3	2	no	—
164	#41	—	3	2	no	—
165	#41	—	3	2	no	—
166	#41	—	6	2	no	—
167	#41	—	6	2	no	—
168	#41	C₂²×D₃	1	2	yes	—
169	C₂×Q₈	—	3	2	no	—
170	C₂×D₄	—	3	2	no	—
171	C₂×D₄	—	6	2	no	—
172	C₂²×C₄	—	3	1	no	—
173	C₂²×C₄	—	6	2	no	—
174	C₂²×C₄	—	6	1	no	—
175	Q₁₆	—	3	2	no	—
176	Q₁₆	—	3	1	no	—
177	Q₁₆	C₄×D₃	1	1	yes	characteristic, complemented
178	SD₁₆	—	3	2	no	—
179	SD₁₆	—	3	2	no	—
180	SD₁₆	—	3	2	no	—
181	SD₁₆	C₄×D₃	1	2	yes	complemented
182	D₈	—	3	2	no	—
183	D₈	—	3	1	no	—
184	D₈	C₄×D₃	1	1	yes	characteristic, complemented
185	M₁₆	—	2	2	no	—
186	M₁₆	—	3	2	no	—
187	M₁₆	—	3	2	no	—
188	M₁₆	—	3	2	no	—
189	M₁₆	—	3	2	no	—
190	M₁₆	—	3	2	no	—
191	M₁₆	—	6	2	no	—
192	M₁₆	—	6	2	no	—
193	M₁₆	—	6	2	no	—
194	M₁₆	—	6	2	no	—
195	M₁₆	C₂²×D₃	1	2	yes	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	3	1	no	—
198	C₂×C₈	—	3	1	no	—
199	C₂×C₈	—	6	2	no	—
200	C₂×C₈	—	6	2	no	—
201	C₂×C₈	—	6	2	no	—
202	C₂×C₈	—	6	2	no	—
203	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
205	C₄²	—	3	1	no	—
206	C₄²	—	6	1	no	—
207	C₄²	C₂²×D₃	1	1	yes	characteristic
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	C₂×#41	1	1	yes	characteristic
210	C₂×D₃	—	2	2	no	—
211	C₂×D₃	—	4	1	no	—
212	C₂×D₃	—	8	2	no	—
213	C₂×D₃	C₄×D₄	1	2	yes	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
217	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
218	Dic₃	—	2	2	no	—
219	Dic₃	—	2	2	no	—
220	Dic₃	C₄×D₄	1	2	yes	—
221	C₂³	—	3	1	no	—
222	C₂³	—	6	2	no	—
223	Q₈	—	3	2	no	—
224	Q₈	—	6	2	no	—
225	Q₈	C₂×C₄×D₃	1	2	yes	—
226	D₄	—	2	2	no	—
227	D₄	—	3	2	no	—
228	D₄	—	6	2	no	—
229	D₄	—	6	2	no	—
230	D₄	—	6	2	no	—
231	D₄	—	6	2	no	—
232	D₄	C₂×C₄×D₃	1	2	yes	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	3	1	no	—
236	C₂×C₄	—	3	2	no	—
237	C₂×C₄	—	3	2	no	—
238	C₂×C₄	—	6	2	no	—
239	C₂×C₄	—	6	2	no	—
240	C₂×C₄	—	6	2	no	—
241	C₂×C₄	—	6	1	no	—
242	C₂×C₄	—	6	2	no	—
243	C₂×C₄	—	12	2	no	—
244	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	3	2	no	—
247	C₈	—	3	1	no	—
248	C₈	—	3	1	no	—
249	C₈	—	6	2	no	—
250	C₈	C₂×C₄×D₃	1	1	yes	characteristic
251	C₈	C₂×C₄×D₃	1	1	yes	characteristic
252	C₈	D₃×D₄	1	2	yes	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	4	2	no	—
255	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
256	D₃	—	2	2	no	—
257	D₃	—	4	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	3	2	no	—
260	C₂²	—	6	2	no	—
261	C₂²	—	12	1	no	—
262	C₂²	—	24	2	no	—
263	C₂²	D₃×#41	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	3	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	6	2	no	—
268	C₄	—	6	2	no	—
269	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
270	C₄	C₂×D₃×D₄	1	1	yes	characteristic
271	C₃	#2956	1	1	yes	characteristic, complemented, Sylow
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	6	2	no	—
275	C₂	—	12	2	no	—
276	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	#5768	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#444	D₃
hypocenter	C₃	#2956
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. #5196	C₂	semidirect
5. #5144	C₂	semidirect
6. #4895	C₂	semidirect
7. #4893	C₂	semidirect
8. #4823	C₂	semidirect
9. D₃×#104	C₂	semidirect
10. #4717	C₂	semidirect
11. #4707	C₂	semidirect
15. C₃×#131	C₂²	semidirect
17. #948	C₂²	semidirect
18. #918	C₄	semidirect
19. #915	C₄	semidirect
20. #914	C₄	semidirect
21. #911	C₄	semidirect
23. #907	C₂²	semidirect
28. #901	C₂²	semidirect
33. C₃×#108	C₂²	semidirect
34. C₃×#104	C₂²	semidirect
35. C₃×#97	C₂²	semidirect
36. #837	C₂²	semidirect
37. #825	C₂²	semidirect
38. #822	C₂²	semidirect
39. #819	C₂²	semidirect
40. C₈×Dic₃	C₂²	semidirect
41. C₈×Dic₃	C₂²	semidirect
42. #805	C₂²	semidirect
43. #803	C₂²	semidirect
48. #444	D₃	semidirect
67. C₃×Q₁₆	C₂×C₄	semidirect
68. C₃×SD₁₆	C₂×C₄	semidirect
69. C₃×D₈	C₂×C₄	semidirect
76. #216	C₂×C₄	semidirect
77. #215	C₂×C₄	semidirect
78. #214	C₂×C₄	semidirect
79. #213	C₂×C₄	semidirect
87. Dic₁₂	C₂×C₄	semidirect
88. D₂₄	C₂×C₄	semidirect
89. #204	C₂×C₄	semidirect
113. #131	C₂×D₃	semidirect
122. #108	C₂×D₃	semidirect
126. #104	C₂×D₃	semidirect
128. #97	C₂×D₃	semidirect
177. Q₁₆	C₄×D₃	semidirect
181. SD₁₆	C₄×D₃	semidirect
184. D₈	C₄×D₃	semidirect
271. C₃	#2956	semidirect

Maximal subgroups

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

subgroup	index
#5768	2
#5750	2
C₃×#444	2
#5196	2
#5144	2
#4895	2
#4893	2
#4823	2
D₃×#104	2
#4717	2
#4707	2
#2956	3