Subgroups — #79406

Order 384.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 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×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#4837	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	#4685	C₂	1	1	yes	maximal, characteristic, complemented
11	#4679	C₂	1	1	yes	maximal, characteristic, complemented
12	#2990	—	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	2	yes	—
21	#892	C₄	1	1	yes	characteristic
22	#887	C₄	1	1	yes	characteristic
23	#67⋊C₄	C₄	1	2	yes	—
24	#877	C₄	1	2	yes	—
25	C₄²×D₃	C₂²	1	1	yes	characteristic
26	C₄×Dic₆	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	#830	C₂²	1	2	yes	—
31	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
33	#815	C₂²	1	2	yes	—
34	#815	C₂²	1	2	yes	—
35	Dic₃⋊C₈	C₂²	1	2	yes	—
36	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
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	2	no	—
50	#431	—	3	1	no	—
51	#423	—	3	1	no	—
52	C₂×C₄⋊C₈	—	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	—	4	2	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	1	yes	characteristic
80	C₄×Dic₃	C₂³	1	2	yes	—
81	C₂×C₃⋊C₈	—	4	2	no	—
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
84	C₂×C₃⋊C₈	C₂³	1	2	yes	—
85	ΓL(1,𝔽₂₅)	—	4	2	no	—
86	C₂×#41	—	3	1	no	—
87	C₂×M₁₆	—	3	2	no	—
88	C₂²×C₈	—	3	2	no	—
89	C₂²×C₈	C₂×D₃	1	2	yes	—
90	C₄×Q₈	—	3	1	no	—
91	C₄×Q₈	—	3	1	no	—
92	C₄×D₄	—	3	1	no	—
93	C₄×D₄	—	3	2	no	—
94	C₄×D₄	—	3	2	no	—
95	C₄×D₄	C₂×D₃	1	1	yes	characteristic
96	#117	—	3	1	no	—
97	#117	—	3	2	no	—
98	C₂×C₄²	—	3	2	no	—
99	C₂×C₄²	—	3	1	no	—
100	C₄⋊C₈	—	3	1	no	—
101	C₄⋊C₈	—	3	2	no	—
102	C₄⋊C₈	—	3	2	no	—
103	C₄⋊C₈	—	3	2	no	—
104	C₄⋊C₈	—	3	1	no	—
105	C₄⋊C₈	—	3	1	no	—
106	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
107	C₂²⋊C₈	—	3	2	no	—
108	C₂²⋊C₈	—	3	1	no	—
109	C₂²⋊C₈	—	3	1	no	—
110	C₂²⋊C₈	C₂×D₃	1	2	yes	—
111	#97	—	3	2	no	—
112	#97	—	3	2	no	—
113	C₄×C₈	—	3	1	no	—
114	C₄×C₈	C₂×D₃	1	1	yes	characteristic
115	C₂³×C₃	C₂²×C₄	1	2	yes	—
116	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
117	C₃×D₄	M₁₆	1	4	yes	complemented
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
124	#67	—	2	4	no	—
125	C₂×Dic₃	—	2	4	no	—
126	C₂×Dic₃	—	2	4	no	—
127	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₃	C₂²×C₄	1	2	yes	—
130	C₂×Dic₃	C₂²×C₄	1	2	yes	—
131	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₃	—	2	2	no	—
133	C₄×D₃	—	2	1	no	—
134	C₄×D₃	—	2	1	no	—
135	C₄×D₃	—	2	2	no	—
136	Dic₆	—	2	2	no	—
137	C₃×C₈	—	2	2	no	—
138	C₃×C₈	—	2	1	no	—
139	C₃×C₈	—	2	1	no	—
140	C₃⋊C₈	—	2	2	no	—
141	C₃⋊C₈	—	2	1	no	—
142	C₃⋊C₈	—	2	1	no	—
143	#41	—	6	4	no	—
144	C₂×Q₈	—	3	1	no	—
145	C₂×D₄	—	3	2	no	—
146	C₂×D₄	C₄×D₃	1	1	yes	characteristic
147	C₂²×C₄	—	3	2	no	—
148	C₂²×C₄	—	3	2	no	—
149	C₂²×C₄	—	3	1	no	—
150	C₂²×C₄	—	3	1	no	—
151	C₂²×C₄	—	3	1	no	—
152	C₂²×C₄	C₂²×D₃	1	2	yes	—
153	M₁₆	—	12	2	no	—
154	C₂×C₈	—	3	1	no	—
155	C₂×C₈	—	3	1	no	—
156	C₂×C₈	—	3	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	12	2	no	—
159	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
161	C₂×C₈	C₂²×D₃	1	2	yes	—
162	C₄⋊C₄	—	3	2	no	—
163	C₄⋊C₄	—	3	2	no	—
164	C₄⋊C₄	—	3	1	no	—
165	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
166	C₂²⋊C₄	—	3	2	no	—
167	C₂²⋊C₄	—	3	2	no	—
168	C₂²⋊C₄	C₄×D₃	1	2	yes	—
169	C₄²	—	3	2	no	—
170	C₄²	—	3	1	no	—
171	C₄²	—	3	1	no	—
172	C₄²	—	3	2	no	—
173	C₄²	—	3	2	no	—
174	C₄²	—	3	1	no	—
175	C₄²	C₂²×D₃	1	1	yes	characteristic
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	C₂×M₁₆	1	4	yes	—
179	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
180	C₂×D₃	—	2	1	no	—
181	C₂×D₃	—	2	1	no	—
182	C₂×D₃	—	2	1	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	C₂×M₁₆	1	2	yes	—
186	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
187	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
188	Dic₃	—	2	1	no	—
189	Dic₃	—	2	2	no	—
190	Dic₃	—	2	1	no	—
191	Dic₃	—	2	2	no	—
192	Dic₃	#131	1	2	yes	—
193	C₂³	—	3	1	no	—
194	C₂³	C₂×C₄×D₃	1	2	yes	—
195	Q₈	—	6	2	no	—
196	D₄	—	6	4	no	—
197	D₄	ΓL(1,𝔽₂₅)	1	4	yes	complemented
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	3	1	no	—
201	C₂×C₄	—	3	1	no	—
202	C₂×C₄	—	3	2	no	—
203	C₂×C₄	—	3	1	no	—
204	C₂×C₄	—	3	2	no	—
205	C₂×C₄	—	6	2	no	—
206	C₂×C₄	—	6	4	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	C₂×C₄×D₃	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	—	6	1	no	—
219	C₈	—	6	2	no	—
220	C₈	—	6	1	no	—
221	C₂×C₃	—	2	4	no	—
222	C₂×C₃	#520	1	1	yes	characteristic
223	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
224	C₂×C₃	C₂×#131	1	1	yes	characteristic
225	D₃	—	4	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	6	1	no	—
229	C₂²	—	6	1	no	—
230	C₂²	—	6	1	no	—
231	C₂²	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
232	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	3	2	no	—
236	C₄	—	6	1	no	—
237	C₄	—	6	2	no	—
238	C₄	—	6	2	no	—
239	C₄	—	6	1	no	—
240	C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
241	C₄	#1004	1	1	yes	characteristic
242	C₄	#1010	1	1	yes	characteristic
243	C₃	#2990	1	1	yes	characteristic, complemented, Sylow
244	C₂	—	2	4	no	—
245	C₂	—	12	1	no	—
246	C₂	#5546	1	1	yes	characteristic
247	C₂	C₂²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
248	C₂	D₃×#131	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	#2990
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×Dic₃⋊C₈	C₂	semidirect
5. D₄×C₃⋊C₈	C₂	semidirect
6. #4837	C₂	semidirect
7. #4831	C₂	semidirect
8. #4730	C₂	semidirect
9. #4719	C₂	semidirect
10. #4685	C₂	semidirect
11. #4679	C₂	semidirect
48. C₈×D₄	D₃	semidirect
117. C₃×D₄	M₁₆	semidirect
197. D₄	ΓL(1,𝔽₂₅)	semidirect
243. C₃	#2990	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×Dic₃⋊C₈	2
D₄×C₃⋊C₈	2
#4837	2
#4831	2
#4730	2
#4719	2
#4685	2
#4679	2
#2990	3