Subgroups — #78523

Order 384.

412 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	C₄×#235	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#5128	C₂	1	2	yes	maximal, complemented
4	#5127	C₂	1	2	yes	maximal, complemented
5	#4925	C₂	1	1	yes	maximal, characteristic, complemented
6	#4706	C₂	1	2	yes	maximal, complemented
7	#4701	C₂	1	2	yes	maximal, complemented
8	C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
9	#4685	C₂	1	1	yes	maximal, characteristic, complemented
10	#4685	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#2965	—	3	1	no	maximal, Sylow
13	C₂×#235	C₄	1	1	yes	characteristic
14	C₂×C₃×M₁₆	C₂²	1	2	yes	—
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₄×#67	C₄	1	1	yes	characteristic
17	C₄×#67	C₄	1	1	yes	characteristic
18	C₄×#67	C₂²	1	1	yes	characteristic
19	C₄×#67	C₂²	1	1	yes	characteristic
20	#926	C₄	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
22	C₂×D₃×C₈	C₂²	1	2	yes	—
23	C₄×D₁₂	C₄	1	1	yes	characteristic
24	C₄×D₁₂	C₂²	1	1	yes	characteristic
25	#872	C₄	1	1	yes	characteristic
26	#872	C₄	1	1	yes	characteristic
27	C₄²×D₃	C₂²	1	1	yes	characteristic
28	C₄²×D₃	C₂²	1	1	yes	characteristic
29	C₄×Dic₆	C₄	1	1	yes	characteristic
30	C₄×Dic₆	C₂²	1	1	yes	characteristic
31	C₃×#97	C₂²	1	2	yes	—
32	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
33	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
34	#830	C₂²	1	2	yes	—
35	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
36	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
37	#815	C₂²	1	2	yes	—
38	Dic₃⋊C₈	C₂²	1	2	yes	—
39	Dic₃⋊C₈	C₂²	1	2	yes	—
40	C₈×Dic₃	C₂²	1	2	yes	—
41	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
42	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
43	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
44	C₄×#41	—	3	1	no	—
45	#446	—	3	2	no	—
46	#436	—	3	2	no	—
47	#435	—	3	2	no	—
48	#435	—	3	2	no	—
49	#431	—	3	1	no	—
50	#431	—	3	1	no	—
51	#431	—	3	1	no	—
52	C₄×M₁₆	—	3	1	no	—
53	C₄×M₁₆	D₃	1	1	yes	characteristic, complemented
54	C₂×C₄×C₈	—	3	1	no	—
55	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂×#67	C₂×C₄	1	1	yes	characteristic
59	C₂×#67	C₂×C₄	1	1	yes	characteristic
60	#235	—	2	4	no	—
61	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
68	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
69	C₃×M₁₆	—	2	4	no	—
70	C₂×C₃×C₈	C₂³	1	2	yes	—
71	C₂×C₃×C₈	C₂³	1	2	yes	—
72	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
73	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
74	C₃×C₄²	C₂³	1	1	yes	characteristic
75	C₃×C₄²	C₂³	1	1	yes	characteristic
76	#217	C₂×C₄	1	1	yes	characteristic
77	#217	C₂×C₄	1	1	yes	characteristic
78	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
79	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
80	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
81	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
83	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
84	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
85	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
86	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
87	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
89	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₃	C₂³	1	1	yes	characteristic
93	C₄×Dic₃	C₂³	1	1	yes	characteristic
94	C₂×C₃⋊C₈	C₂³	1	2	yes	—
95	C₂×C₃⋊C₈	C₂³	1	2	yes	—
96	ΓL(1,𝔽₂₅)	—	2	4	no	—
97	D₃×C₈	—	2	4	no	—
98	C₂×#41	—	3	1	no	—
99	C₂×M₁₆	—	3	2	no	—
100	C₂×M₁₆	C₂×D₃	1	2	yes	—
101	C₂²×C₈	—	3	2	no	—
102	C₄×Q₈	—	3	1	no	—
103	C₄×Q₈	—	3	1	no	—
104	C₄×D₄	—	3	1	no	—
105	C₄×D₄	—	3	1	no	—
106	C₄×D₄	—	3	1	no	—
107	C₄×D₄	—	3	1	no	—
108	C₄×D₄	—	3	1	no	—
109	C₄×D₄	—	3	1	no	—
110	#117	—	3	1	no	—
111	#117	—	3	1	no	—
112	#117	—	3	1	no	—
113	C₂×C₄²	—	3	1	no	—
114	C₂×C₄²	—	3	1	no	—
115	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
116	C₄⋊C₈	—	3	2	no	—
117	C₄⋊C₈	—	3	2	no	—
118	C₄⋊C₈	—	3	2	no	—
119	C₂²⋊C₈	—	3	2	no	—
120	C₂²⋊C₈	—	3	2	no	—
121	C₂²⋊C₈	—	3	2	no	—
122	#97	—	3	2	no	—
123	#97	C₂×D₃	1	2	yes	—
124	C₄×C₈	—	3	2	no	—
125	C₄×C₈	—	3	1	no	—
126	C₄×C₈	—	3	1	no	—
127	C₄×C₈	C₂×D₃	1	1	yes	characteristic
128	C₄×C₈	C₂×D₃	1	1	yes	characteristic
129	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
141	#67	—	2	2	no	—
142	#67	—	2	2	no	—
143	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
144	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
145	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
149	D₁₂	—	2	2	no	—
150	C₄×D₃	—	2	2	no	—
151	C₄×D₃	—	2	2	no	—
152	C₄×D₃	—	2	2	no	—
153	C₄×D₃	—	2	2	no	—
154	C₄×D₃	—	2	2	no	—
155	C₄×D₃	M₁₆	1	4	yes	complemented
156	Dic₆	—	2	2	no	—
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	#41	1	4	yes	complemented
159	C₃⋊C₈	—	2	2	no	—
160	C₃⋊C₈	—	2	2	no	—
161	#41	—	6	4	no	—
162	C₂×Q₈	—	3	1	no	—
163	C₂×D₄	—	3	1	no	—
164	C₂×D₄	—	3	1	no	—
165	C₂×D₄	—	3	1	no	—
166	C₂²×C₄	—	3	1	no	—
167	C₂²×C₄	—	3	1	no	—
168	C₂²×C₄	—	3	1	no	—
169	C₂²×C₄	—	3	1	no	—
170	C₂²×C₄	—	3	1	no	—
171	C₂²×C₄	—	3	1	no	—
172	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
173	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
175	M₁₆	—	2	4	no	—
176	M₁₆	—	6	4	no	—
177	C₂×C₈	—	3	2	no	—
178	C₂×C₈	—	3	2	no	—
179	C₂×C₈	—	6	4	no	—
180	C₂×C₈	C₂²×D₃	1	2	yes	—
181	C₂×C₈	C₂²×D₃	1	2	yes	—
182	C₄⋊C₄	—	3	1	no	—
183	C₄⋊C₄	—	3	1	no	—
184	C₄⋊C₄	—	3	1	no	—
185	C₄⋊C₄	—	3	1	no	—
186	C₄⋊C₄	—	3	1	no	—
187	C₄⋊C₄	—	3	1	no	—
188	C₂²⋊C₄	—	3	1	no	—
189	C₂²⋊C₄	—	3	1	no	—
190	C₂²⋊C₄	—	3	1	no	—
191	C₂²⋊C₄	—	3	1	no	—
192	C₂²⋊C₄	—	3	1	no	—
193	C₂²⋊C₄	—	3	1	no	—
194	C₄²	—	3	1	no	—
195	C₄²	—	3	1	no	—
196	C₄²	—	3	1	no	—
197	C₄²	—	3	1	no	—
198	C₄²	—	3	1	no	—
199	C₄²	—	3	1	no	—
200	C₄²	C₄×D₃	1	1	yes	characteristic
201	C₄²	C₄×D₃	1	1	yes	characteristic
202	C₄²	C₂²×D₃	1	1	yes	characteristic
203	C₄²	C₂²×D₃	1	1	yes	characteristic
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
208	C₂×D₃	—	2	1	no	—
209	C₂×D₃	—	2	1	no	—
210	C₂×D₃	—	2	1	no	—
211	C₂×D₃	—	2	1	no	—
212	C₂×D₃	—	2	1	no	—
213	C₂×D₃	C₂×M₁₆	1	2	yes	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	C₂×M₁₆	1	2	yes	—
218	C₃×C₄	#131	1	2	yes	—
219	C₃×C₄	C₂×#41	1	2	yes	—
220	Dic₃	—	2	1	no	—
221	Dic₃	—	2	1	no	—
222	Dic₃	—	2	1	no	—
223	Dic₃	—	2	1	no	—
224	Dic₃	—	2	1	no	—
225	Dic₃	C₂×M₁₆	1	2	yes	—
226	C₂³	—	3	1	no	—
227	C₂³	—	3	1	no	—
228	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
229	Q₈	—	6	2	no	—
230	D₄	—	6	2	no	—
231	D₄	—	6	2	no	—
232	D₄	—	6	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	3	1	no	—
237	C₂×C₄	—	3	4	no	—
238	C₂×C₄	—	3	1	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	—	6	2	no	—
245	C₂×C₄	—	6	2	no	—
246	C₂×C₄	—	6	2	no	—
247	C₂×C₄	—	6	2	no	—
248	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	—	6	2	no	—
256	C₈	—	6	2	no	—
257	C₈	#235	1	4	yes	complemented
258	C₂×C₃	—	4	1	no	—
259	C₂×C₃	C₄×#41	1	1	yes	characteristic
260	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
261	C₂×C₃	C₂×#131	1	1	yes	characteristic
262	D₃	—	2	2	no	—
263	D₃	—	4	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	3	2	no	—
268	C₂²	—	6	1	no	—
269	C₂²	—	6	1	no	—
270	C₂²	—	6	1	no	—
271	C₂²	—	6	1	no	—
272	C₂²	—	6	1	no	—
273	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	3	2	no	—
278	C₄	—	6	1	no	—
279	C₄	—	6	1	no	—
280	C₄	—	6	1	no	—
281	C₄	—	6	1	no	—
282	C₄	—	6	1	no	—
283	C₄	D₃×M₁₆	1	2	yes	—
284	C₄	#907	1	2	yes	—
285	C₄	C₂×#235	1	2	yes	—
286	C₃	#2965	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	4	1	no	—
288	C₂	—	6	2	no	—
289	C₂	—	12	1	no	—
290	C₂	C₄×#235	1	1	yes	characteristic
291	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic
292	C₂	C₂×#907	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×M₁₆	C₂
prime radical	C₂×#235	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	#2965
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×M₁₆	C₂	semidirect
3. #5128	C₂	semidirect
4. #5127	C₂	semidirect
5. #4925	C₂	semidirect
6. #4706	C₂	semidirect
7. #4701	C₂	semidirect
8. C₄×ΓL(1,𝔽₂₅)	C₂	semidirect
9. #4685	C₂	semidirect
10. #4685	C₂	semidirect
11. C₄×D₃×C₈	C₂	semidirect
53. C₄×M₁₆	D₃	semidirect
155. C₄×D₃	M₁₆	semidirect
158. C₃×C₈	#41	semidirect
257. C₈	#235	semidirect
286. C₃	#2965	semidirect

Maximal subgroups

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

subgroup	index
C₄×#235	2
C₃×C₄×M₁₆	2
#5128	2
#5127	2
#4925	2
#4706	2
#4701	2
C₄×ΓL(1,𝔽₂₅)	2
#4685	2
C₄×D₃×C₈	2
#2965	3