Subgroups — #79404

Order 384.

406 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#237	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#5111	C₂	1	2	yes	maximal, complemented
4	C₂×#815	C₂	1	2	yes	maximal, complemented
5	D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#4835	C₂	1	1	yes	maximal, characteristic, complemented
7	#4831	C₂	1	1	yes	maximal, characteristic, complemented
8	#4730	C₂	1	2	yes	maximal, complemented
9	#4719	C₂	1	2	yes	maximal, complemented
10	C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
11	#4679	C₂	1	1	yes	maximal, characteristic, complemented
12	#2970	—	3	1	no	maximal, Sylow
13	C₂×#237	C₄	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	2	yes	—
15	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
16	D₄×Dic₃	C₄	1	1	yes	characteristic
17	C₄×#67	C₂²	1	2	yes	—
18	C₂×C₄×Dic₃	C₂²	1	2	yes	—
19	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
20	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
22	#892	C₄	1	1	yes	characteristic
23	#887	C₄	1	1	yes	characteristic
24	#67⋊C₄	C₄	1	2	yes	—
25	#877	C₄	1	2	yes	—
26	C₄²×D₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₆	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
30	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
31	#830	C₂²	1	2	yes	—
32	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
33	#815	C₂²	1	2	yes	—
34	#815	C₂²	1	1	yes	characteristic
35	#815	C₂²	1	1	yes	characteristic
36	#815	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	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
42	C₄×#41	—	3	1	no	—
43	#446	—	3	1	no	—
44	#446	—	3	1	no	—
45	#435	—	3	2	no	—
46	#435	—	3	2	no	—
47	C₈×D₄	—	3	1	no	—
48	C₈×D₄	D₃	1	1	yes	characteristic, complemented
49	#432	—	3	1	no	—
50	#432	—	3	2	no	—
51	C₄×M₁₆	—	3	1	no	—
52	C₂×#97	—	3	2	no	—
53	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂×#67	C₂×C₄	1	2	yes	—
56	C₂²×Dic₃	C₂×C₄	1	2	yes	—
57	C₂²×Dic₃	C₂×C₄	1	2	yes	—
58	#237	—	2	4	no	—
59	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
62	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
63	C₂×C₃×C₈	—	2	4	no	—
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂³	1	2	yes	—
66	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
68	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
69	C₃×C₄²	C₂³	1	1	yes	characteristic
70	#217	C₂×C₄	1	2	yes	—
71	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
72	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
73	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
74	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
75	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₃	C₂×C₄	1	2	yes	—
78	C₄×Dic₃	C₂×C₄	1	2	yes	—
79	C₄×Dic₃	C₂³	1	2	yes	—
80	C₄×Dic₃	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₈	—	2	4	no	—
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₂×C₃⋊C₈	C₂³	1	2	yes	—
84	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
85	ΓL(1,𝔽₂₅)	—	2	2	no	—
86	ΓL(1,𝔽₂₅)	—	2	2	no	—
87	C₂×#41	—	3	1	no	—
88	C₂×M₁₆	—	3	1	no	—
89	C₂×M₁₆	—	3	1	no	—
90	C₂²×C₈	—	3	2	no	—
91	C₂²×C₈	C₂×D₃	1	2	yes	—
92	C₄×Q₈	—	3	1	no	—
93	C₄×Q₈	—	3	1	no	—
94	C₄×D₄	—	3	1	no	—
95	C₄×D₄	—	3	2	no	—
96	C₄×D₄	—	3	2	no	—
97	C₄×D₄	C₂×D₃	1	1	yes	characteristic
98	#117	—	3	1	no	—
99	#117	—	3	2	no	—
100	C₂×C₄²	—	3	2	no	—
101	C₂×C₄²	—	3	1	no	—
102	C₄⋊C₈	—	3	2	no	—
103	C₄⋊C₈	—	3	2	no	—
104	C₄⋊C₈	—	3	1	no	—
105	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
106	C₂²⋊C₈	—	3	2	no	—
107	C₂²⋊C₈	—	3	2	no	—
108	C₂²⋊C₈	C₂×D₃	1	2	yes	—
109	#97	—	3	1	no	—
110	#97	—	3	2	no	—
111	#97	—	3	2	no	—
112	#97	—	3	2	no	—
113	#97	—	3	1	no	—
114	C₄×C₈	—	3	1	no	—
115	C₄×C₈	C₂×D₃	1	1	yes	characteristic
116	C₂³×C₃	C₂²×C₄	1	2	yes	—
117	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
118	C₃×D₄	M₁₆	1	4	yes	complemented
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
122	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
125	#67	—	2	4	no	—
126	C₂×Dic₃	—	2	4	no	—
127	C₂×Dic₃	—	4	2	no	—
128	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₃	C₂²×C₄	1	2	yes	—
132	C₂×Dic₃	C₂²×C₄	1	2	yes	—
133	C₄×D₃	—	2	2	no	—
134	C₄×D₃	—	2	1	no	—
135	C₄×D₃	—	2	1	no	—
136	C₄×D₃	—	4	1	no	—
137	Dic₆	—	2	2	no	—
138	C₃×C₈	—	2	1	no	—
139	C₃×C₈	—	2	2	no	—
140	C₃×C₈	#41	1	2	yes	complemented
141	C₃⋊C₈	—	2	1	no	—
142	C₃⋊C₈	—	2	2	no	—
143	C₃⋊C₈	#41	1	2	yes	complemented
144	#41	—	6	4	no	—
145	C₂×Q₈	—	3	1	no	—
146	C₂×D₄	—	3	2	no	—
147	C₂×D₄	C₄×D₃	1	1	yes	characteristic
148	C₂²×C₄	—	3	2	no	—
149	C₂²×C₄	—	3	2	no	—
150	C₂²×C₄	—	3	1	no	—
151	C₂²×C₄	—	3	1	no	—
152	C₂²×C₄	—	3	1	no	—
153	C₂²×C₄	C₂²×D₃	1	2	yes	—
154	M₁₆	—	6	2	no	—
155	M₁₆	—	6	2	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	3	1	no	—
158	C₂×C₈	—	3	1	no	—
159	C₂×C₈	—	3	2	no	—
160	C₂×C₈	—	6	4	no	—
161	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
162	C₂×C₈	C₂²×D₃	1	2	yes	—
163	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
164	C₄⋊C₄	—	3	2	no	—
165	C₄⋊C₄	—	3	2	no	—
166	C₄⋊C₄	—	3	1	no	—
167	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
168	C₂²⋊C₄	—	3	2	no	—
169	C₂²⋊C₄	—	3	2	no	—
170	C₂²⋊C₄	C₄×D₃	1	2	yes	—
171	C₄²	—	3	2	no	—
172	C₄²	—	3	1	no	—
173	C₄²	—	3	1	no	—
174	C₄²	—	3	2	no	—
175	C₄²	—	3	1	no	—
176	C₄²	—	3	2	no	—
177	C₄²	C₂²×D₃	1	1	yes	characteristic
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	C₂×M₁₆	1	4	yes	—
181	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
182	C₂×D₃	—	2	1	no	—
183	C₂×D₃	—	2	1	no	—
184	C₂×D₃	—	2	1	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	C₂×M₁₆	1	2	yes	—
188	C₃×C₄	C₂×#41	1	1	yes	characteristic
189	C₃×C₄	C₂×#41	1	1	yes	characteristic
190	Dic₃	—	2	1	no	—
191	Dic₃	—	2	2	no	—
192	Dic₃	—	2	1	no	—
193	Dic₃	—	2	1	no	—
194	Dic₃	—	2	2	no	—
195	C₂³	—	3	1	no	—
196	C₂³	C₂×C₄×D₃	1	2	yes	—
197	Q₈	—	6	2	no	—
198	D₄	—	6	4	no	—
199	D₄	ΓL(1,𝔽₂₅)	1	4	yes	complemented
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	3	1	no	—
203	C₂×C₄	—	3	1	no	—
204	C₂×C₄	—	3	2	no	—
205	C₂×C₄	—	3	1	no	—
206	C₂×C₄	—	3	2	no	—
207	C₂×C₄	—	6	4	no	—
208	C₂×C₄	—	6	2	no	—
209	C₂×C₄	—	6	1	no	—
210	C₂×C₄	—	6	1	no	—
211	C₂×C₄	—	12	1	no	—
212	C₂×C₄	—	12	2	no	—
213	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	—	3	2	no	—
220	C₈	—	6	1	no	—
221	C₈	—	6	2	no	—
222	C₈	#237	1	2	yes	complemented
223	C₂×C₃	—	2	4	no	—
224	C₂×C₃	C₄×#41	1	1	yes	characteristic
225	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
226	C₂×C₃	#568	1	1	yes	characteristic
227	D₃	—	4	1	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	6	1	no	—
231	C₂²	—	6	1	no	—
232	C₂²	—	6	1	no	—
233	C₂²	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
234	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	2	no	—
237	C₄	—	6	1	no	—
238	C₄	—	6	2	no	—
239	C₄	—	6	1	no	—
240	C₄	—	6	2	no	—
241	C₄	—	6	1	no	—
242	C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
243	C₄	C₂×#237	1	1	yes	characteristic
244	C₄	D₃×#41	1	1	yes	characteristic
245	C₃	#2970	1	1	yes	characteristic, complemented, Sylow
246	C₂	—	2	4	no	—
247	C₂	—	12	1	no	—
248	C₂	C₄×#237	1	1	yes	characteristic
249	C₂	C₂²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
250	C₂	#5750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₈×D₄	C₂
prime radical	C₂×#237	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₈×D₄	D₃
hypocenter	C₃	#2970
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₈×D₄	C₂	semidirect
3. #5111	C₂	semidirect
4. C₂×#815	C₂	semidirect
5. D₄×C₃⋊C₈	C₂	semidirect
6. #4835	C₂	semidirect
7. #4831	C₂	semidirect
8. #4730	C₂	semidirect
9. #4719	C₂	semidirect
10. C₄×ΓL(1,𝔽₂₅)	C₂	semidirect
11. #4679	C₂	semidirect
48. C₈×D₄	D₃	semidirect
118. C₃×D₄	M₁₆	semidirect
140. C₃×C₈	#41	semidirect
143. C₃⋊C₈	#41	semidirect
199. D₄	ΓL(1,𝔽₂₅)	semidirect
222. C₈	#237	semidirect
245. C₃	#2970	semidirect

Maximal subgroups

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

subgroup	index
C₄×#237	2
C₃×C₈×D₄	2
#5111	2
C₂×#815	2
D₄×C₃⋊C₈	2
#4835	2
#4831	2
#4730	2
#4719	2
C₄×ΓL(1,𝔽₂₅)	2
#4679	2
#2970	3