Subgroups — #79943

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×#239	C₂	1	1	yes	maximal, characteristic
2	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#5024	C₂	1	1	yes	maximal, characteristic, complemented
4	#4836	C₂	1	2	yes	maximal, complemented
5	#4835	C₂	1	1	yes	maximal, characteristic, complemented
6	#4835	C₂	1	2	yes	maximal, complemented
7	#4834	C₂	1	1	yes	maximal, characteristic, complemented
8	#4706	C₂	1	2	yes	maximal, complemented
9	D₃×#97	C₂	1	2	yes	maximal, complemented
10	C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#2970	—	3	1	no	maximal, Sylow
13	C₂×#239	C₄	1	1	yes	characteristic
14	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
15	Q₈×Dic₃	C₄	1	1	yes	characteristic
16	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
17	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
18	C₂×D₃×C₈	C₂²	1	2	yes	—
19	C₂×D₃×C₈	C₂²	1	2	yes	—
20	#893	C₄	1	2	yes	—
21	#893	C₄	1	1	yes	characteristic
22	#892	C₄	1	2	yes	—
23	#892	C₄	1	1	yes	characteristic
24	C₄×D₁₂	C₂²	1	1	yes	characteristic
25	C₄×D₁₂	C₂²	1	2	yes	—
26	C₄²×D₃	C₂²	1	1	yes	characteristic
27	C₄²×D₃	C₂²	1	2	yes	—
28	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₃×#97	C₂²	1	2	yes	—
31	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
35	#815	C₂²	1	1	yes	characteristic
36	#815	C₂²	1	1	yes	characteristic
37	#815	C₂²	1	2	yes	—
38	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
40	#803	C₂²	1	2	yes	—
41	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
42	C₄×#41	—	3	1	no	—
43	#446	—	3	1	no	—
44	#446	D₃	1	1	yes	characteristic, complemented
45	#435	—	3	2	no	—
46	#435	—	3	2	no	—
47	C₈×D₄	—	3	1	no	—
48	C₈×D₄	—	3	1	no	—
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×Q₈	C₂×C₄	1	1	yes	characteristic
54	#239	—	2	4	no	—
55	C₂×D₁₂	C₂×C₄	1	2	yes	—
56	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
58	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
61	C₂×C₄×D₃	C₂³	1	2	yes	—
62	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	2	yes	—
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
67	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
68	C₃×C₄²	C₂³	1	1	yes	characteristic
69	C₃×C₄²	C₂³	1	2	yes	—
70	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
71	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
72	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
73	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
74	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₃	C₂×C₄	1	2	yes	—
77	C₄×Dic₃	C₂×C₄	1	2	yes	—
78	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₃	C₂³	1	1	yes	characteristic
80	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₈	C₂³	1	2	yes	—
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	ΓL(1,𝔽₂₅)	—	2	2	no	—
84	ΓL(1,𝔽₂₅)	—	2	2	no	—
85	D₃×C₈	—	2	4	no	—
86	D₃×C₈	—	2	4	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	—	3	2	no	—
92	C₄×Q₈	—	3	1	no	—
93	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
94	C₄×D₄	—	3	1	no	—
95	C₄×D₄	—	3	2	no	—
96	C₄×D₄	—	3	1	no	—
97	C₄×D₄	—	3	2	no	—
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	1	no	—
103	C₄⋊C₈	—	3	2	no	—
104	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₃	1	2	yes	—
106	C₂²⋊C₈	—	3	2	no	—
107	C₂²⋊C₈	—	3	2	no	—
108	C₂²⋊C₈	—	3	2	no	—
109	#97	—	3	1	no	—
110	#97	—	3	2	no	—
111	#97	—	3	1	no	—
112	#97	—	3	2	no	—
113	#97	C₂×D₃	1	2	yes	—
114	C₄×C₈	—	3	1	no	—
115	C₄×C₈	C₂×D₃	1	1	yes	characteristic
116	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²×D₃	C₂²×C₄	1	2	yes	—
118	C₃×Q₈	—	2	2	no	—
119	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
120	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
122	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₃	C₂²×C₄	1	2	yes	—
127	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
128	D₁₂	—	2	4	no	—
129	D₁₂	M₁₆	1	4	yes	complemented
130	C₄×D₃	—	2	4	no	—
131	C₄×D₃	—	2	2	no	—
132	C₄×D₃	—	2	2	no	—
133	C₄×D₃	—	2	2	no	—
134	C₄×D₃	—	2	1	no	—
135	C₄×D₃	—	2	1	no	—
136	C₄×D₃	—	4	2	no	—
137	C₄×D₃	—	4	1	no	—
138	C₃×C₈	—	2	2	no	—
139	C₃×C₈	—	2	1	no	—
140	C₃×C₈	#41	1	2	yes	complemented
141	C₃⋊C₈	—	2	2	no	—
142	C₃⋊C₈	—	2	1	no	—
143	C₃⋊C₈	#41	1	2	yes	complemented
144	#41	—	6	4	no	—
145	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
146	C₂×D₄	—	3	1	no	—
147	C₂×D₄	—	3	2	no	—
148	C₂²×C₄	—	3	2	no	—
149	C₂²×C₄	—	3	1	no	—
150	C₂²×C₄	—	3	2	no	—
151	C₂²×C₄	—	3	1	no	—
152	C₂²×C₄	—	3	2	no	—
153	C₂²×C₄	—	3	1	no	—
154	M₁₆	—	6	2	no	—
155	M₁₆	—	6	2	no	—
156	C₂×C₈	—	3	1	no	—
157	C₂×C₈	—	3	1	no	—
158	C₂×C₈	—	3	2	no	—
159	C₂×C₈	—	6	4	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	1	no	—
166	C₄⋊C₄	C₄×D₃	1	2	yes	—
167	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
168	C₂²⋊C₄	—	3	2	no	—
169	C₂²⋊C₄	—	3	2	no	—
170	C₂²⋊C₄	—	3	2	no	—
171	C₄²	—	3	2	no	—
172	C₄²	—	3	1	no	—
173	C₄²	—	3	2	no	—
174	C₄²	—	3	1	no	—
175	C₄²	—	3	1	no	—
176	C₄²	C₂²×D₃	1	1	yes	characteristic
177	C₄²	C₂²×D₃	1	2	yes	—
178	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
179	C₂×D₃	—	2	2	no	—
180	C₂×D₃	—	2	2	no	—
181	C₂×D₃	—	2	1	no	—
182	C₂×D₃	—	2	1	no	—
183	C₂×D₃	—	2	1	no	—
184	C₂×D₃	C₂×M₁₆	1	4	yes	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	C₂×M₁₆	1	2	yes	—
189	C₃×C₄	C₂×#41	1	1	yes	characteristic
190	C₃×C₄	C₂×#41	1	1	yes	characteristic
191	Dic₃	—	2	2	no	—
192	Dic₃	—	2	1	no	—
193	Dic₃	—	2	1	no	—
194	Dic₃	—	2	1	no	—
195	C₂³	—	3	2	no	—
196	C₂³	—	3	1	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	3	4	no	—
199	D₄	—	6	4	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	—	6	4	no	—
205	C₂×C₄	—	6	2	no	—
206	C₂×C₄	—	6	2	no	—
207	C₂×C₄	—	6	1	no	—
208	C₂×C₄	—	6	2	no	—
209	C₂×C₄	—	6	1	no	—
210	C₂×C₄	—	12	2	no	—
211	C₂×C₄	—	12	1	no	—
212	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
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	2	no	—
221	C₈	—	6	1	no	—
222	C₈	#239	1	2	yes	complemented
223	C₂×C₃	C₄×#41	1	1	yes	characteristic
224	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
225	C₂×C₃	#568	1	1	yes	characteristic
226	D₃	—	2	4	no	—
227	D₃	—	4	1	no	—
228	C₂²	—	3	4	no	—
229	C₂²	—	6	2	no	—
230	C₂²	—	6	2	no	—
231	C₂²	—	6	1	no	—
232	C₂²	—	6	1	no	—
233	C₂²	—	6	1	no	—
234	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	6	1	no	—
239	C₄	—	6	1	no	—
240	C₄	—	6	2	no	—
241	C₄	—	6	1	no	—
242	C₄	D₃×M₁₆	1	2	yes	—
243	C₄	C₂×#239	1	1	yes	characteristic
244	C₄	D₃×#41	1	1	yes	characteristic
245	C₃	#2970	1	1	yes	characteristic, complemented, Sylow
246	C₂	—	6	4	no	—
247	C₂	—	12	1	no	—
248	C₂	C₄×#239	1	1	yes	characteristic
249	C₂	C₂×D₃×M₁₆	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×#446	C₂
prime radical	C₂×#239	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#446	D₃
hypocenter	C₃	#2970
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₃×#446	C₂	semidirect
3. #5024	C₂	semidirect
4. #4836	C₂	semidirect
5. #4835	C₂	semidirect
6. #4835	C₂	semidirect
7. #4834	C₂	semidirect
8. #4706	C₂	semidirect
9. D₃×#97	C₂	semidirect
10. C₄×ΓL(1,𝔽₂₅)	C₂	semidirect
11. C₈×D₁₂	C₂	semidirect
44. #446	D₃	semidirect
129. D₁₂	M₁₆	semidirect
140. C₃×C₈	#41	semidirect
143. C₃⋊C₈	#41	semidirect
222. C₈	#239	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×#239	2
C₃×#446	2
#5024	2
#4836	2
#4835	2
#4834	2
#4706	2
D₃×#97	2
C₄×ΓL(1,𝔽₂₅)	2
C₈×D₁₂	2
#2970	3