Subgroups — #79947

Order 384.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5580	C₂	1	1	yes	maximal, characteristic
2	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#4839	C₂	1	2	yes	maximal, complemented
5	#4838	C₂	1	2	yes	maximal, complemented
6	#4837	C₂	1	1	yes	maximal, characteristic, complemented
7	#4834	C₂	1	1	yes	maximal, characteristic, complemented
8	#4709	C₂	1	2	yes	maximal, complemented
9	#4706	C₂	1	2	yes	maximal, complemented
10	#4688	C₂	1	1	yes	maximal, characteristic, complemented
11	#4685	C₂	1	1	yes	maximal, characteristic, complemented
12	#2993	—	3	1	no	maximal, Sylow
13	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
14	#947	C₄	1	1	yes	characteristic
15	#946	C₄	1	1	yes	characteristic
16	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
17	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
18	C₂×D₃×C₈	C₂²	1	2	yes	—
19	#898	C₄	1	2	yes	—
20	#895	C₄	1	1	yes	characteristic
21	#894	C₄	1	2	yes	—
22	#890	C₄	1	1	yes	characteristic
23	C₄×D₁₂	C₂²	1	1	yes	characteristic
24	C₄×D₁₂	C₂²	1	2	yes	—
25	#872	C₂²	1	2	yes	—
26	C₄²×D₃	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₃×#97	C₂²	1	2	yes	—
30	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
35	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₃⋊C₈	C₂²	1	2	yes	—
38	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
40	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
41	C₄×C₃⋊C₈	C₂²	1	2	yes	—
42	#529	—	3	1	no	—
43	#446	D₃	1	1	yes	characteristic, complemented
44	C₈×Q₈	—	3	1	no	—
45	#436	—	3	2	no	—
46	#436	—	3	1	no	—
47	#435	—	3	2	no	—
48	C₈×D₄	—	3	1	no	—
49	#433	—	3	2	no	—
50	#431	—	3	1	no	—
51	#424	—	3	2	no	—
52	#423	—	3	1	no	—
53	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
54	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
55	C₂×D₁₂	C₂×C₄	1	2	yes	—
56	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₃	C₂³	1	2	yes	—
60	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂³	1	2	yes	—
63	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
64	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
65	C₃×C₄²	C₂³	1	1	yes	characteristic
66	C₃×C₄²	C₂³	1	2	yes	—
67	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
68	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
69	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
70	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
71	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
72	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
73	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
74	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
75	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
76	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₃	C₂³	1	1	yes	characteristic
79	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₈	C₂³	1	2	yes	—
82	ΓL(1,𝔽₂₅)	—	4	2	no	—
83	ΓL(1,𝔽₂₅)	—	4	2	no	—
84	D₃×C₈	—	4	2	no	—
85	C₂×M₁₆	—	3	2	no	—
86	C₂×M₁₆	—	3	2	no	—
87	C₂²×C₈	—	3	2	no	—
88	#126	—	3	2	no	—
89	#125	—	3	1	no	—
90	#124	—	3	1	no	—
91	#123	—	3	2	no	—
92	C₂²⋊Q₈	—	3	1	no	—
93	#121	—	3	1	no	—
94	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
95	C₄×D₄	—	3	1	no	—
96	C₄×D₄	—	3	2	no	—
97	#117	—	3	2	no	—
98	C₂×C₄²	—	3	1	no	—
99	C₄⋊C₈	—	3	2	no	—
100	C₄⋊C₈	—	3	1	no	—
101	C₄⋊C₈	—	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	1	no	—
107	C₂²⋊C₈	—	3	1	no	—
108	C₂²⋊C₈	—	3	2	no	—
109	C₂²⋊C₈	—	3	2	no	—
110	#97	C₂×D₃	1	2	yes	—
111	C₄×C₈	—	3	1	no	—
112	C₄×C₈	—	3	2	no	—
113	C₄×C₈	C₂×D₃	1	1	yes	characteristic
114	C₂²×D₃	C₂²×C₄	1	2	yes	—
115	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
116	C₃×Q₈	—	2	2	no	—
117	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
119	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
120	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₃	C₂²×C₄	1	2	yes	—
126	D₁₂	—	2	2	no	—
127	D₁₂	—	4	2	no	—
128	C₄×D₃	—	2	2	no	—
129	C₄×D₃	—	2	2	no	—
130	C₄×D₃	—	2	1	no	—
131	C₄×D₃	—	2	1	no	—
132	C₄×D₃	—	2	2	no	—
133	C₄×D₃	—	4	1	no	—
134	C₃×C₈	—	2	2	no	—
135	C₃×C₈	—	2	1	no	—
136	C₃×C₈	—	2	1	no	—
137	C₃⋊C₈	—	2	1	no	—
138	C₃⋊C₈	—	2	2	no	—
139	C₃⋊C₈	—	2	1	no	—
140	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
141	C₂×D₄	—	3	1	no	—
142	C₂×D₄	—	3	2	no	—
143	C₂²×C₄	—	3	1	no	—
144	C₂²×C₄	—	3	1	no	—
145	C₂²×C₄	—	3	2	no	—
146	C₂²×C₄	—	3	1	no	—
147	M₁₆	—	12	2	no	—
148	M₁₆	—	12	2	no	—
149	C₂×C₈	—	3	1	no	—
150	C₂×C₈	—	3	1	no	—
151	C₂×C₈	—	3	2	no	—
152	C₂×C₈	—	12	2	no	—
153	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
154	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
155	C₂×C₈	C₂²×D₃	1	2	yes	—
156	C₄⋊C₄	—	3	2	no	—
157	C₄⋊C₄	—	3	2	no	—
158	C₄⋊C₄	—	3	1	no	—
159	C₄⋊C₄	—	3	2	no	—
160	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
161	C₄⋊C₄	C₄×D₃	1	2	yes	—
162	C₂²⋊C₄	—	3	2	no	—
163	C₂²⋊C₄	—	3	2	no	—
164	C₂²⋊C₄	—	3	2	no	—
165	C₂²⋊C₄	—	3	2	no	—
166	C₂²⋊C₄	—	3	2	no	—
167	C₄²	—	3	1	no	—
168	C₄²	—	3	1	no	—
169	C₄²	—	3	1	no	—
170	C₄²	C₂²×D₃	1	1	yes	characteristic
171	C₄²	C₂²×D₃	1	2	yes	—
172	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
173	C₂×D₃	—	2	2	no	—
174	C₂×D₃	—	2	1	no	—
175	C₂×D₃	—	2	1	no	—
176	C₂×D₃	—	2	2	no	—
177	C₂×D₃	—	2	1	no	—
178	C₂×D₃	—	2	2	no	—
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	#131	1	2	yes	—
183	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
184	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
185	Dic₃	—	2	2	no	—
186	Dic₃	—	2	1	no	—
187	Dic₃	—	2	1	no	—
188	Dic₃	—	2	1	no	—
189	C₂³	—	3	2	no	—
190	C₂³	—	3	1	no	—
191	Q₈	—	2	2	no	—
192	D₄	—	6	2	no	—
193	D₄	—	12	2	no	—
194	C₂×C₄	—	3	1	no	—
195	C₂×C₄	—	3	1	no	—
196	C₂×C₄	—	3	1	no	—
197	C₂×C₄	—	3	2	no	—
198	C₂×C₄	—	6	2	no	—
199	C₂×C₄	—	6	1	no	—
200	C₂×C₄	—	6	2	no	—
201	C₂×C₄	—	6	2	no	—
202	C₂×C₄	—	6	1	no	—
203	C₂×C₄	—	12	1	no	—
204	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
207	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₈	—	6	1	no	—
213	C₈	—	6	1	no	—
214	C₈	—	6	2	no	—
215	C₂×C₃	#520	1	1	yes	characteristic
216	C₂×C₃	C₂×#131	1	1	yes	characteristic
217	C₂×C₃	#568	1	1	yes	characteristic
218	D₃	—	4	1	no	—
219	D₃	—	4	2	no	—
220	C₂²	—	6	1	no	—
221	C₂²	—	6	2	no	—
222	C₂²	—	6	1	no	—
223	C₂²	—	6	2	no	—
224	C₂²	—	6	2	no	—
225	C₂²	—	6	1	no	—
226	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	6	1	no	—
231	C₄	—	6	1	no	—
232	C₄	—	6	1	no	—
233	C₄	—	6	2	no	—
234	C₄	#907	1	2	yes	—
235	C₄	#1007	1	1	yes	characteristic
236	C₄	#1009	1	1	yes	characteristic
237	C₃	#2993	1	1	yes	characteristic, complemented, Sylow
238	C₂	—	12	1	no	—
239	C₂	—	12	2	no	—
240	C₂	#5574	1	1	yes	characteristic
241	C₂	C₂×#907	1	1	yes	characteristic
242	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	#895	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#446	D₃
hypocenter	C₃	#2993
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. Q₈×C₃⋊C₈	C₂	semidirect
4. #4839	C₂	semidirect
5. #4838	C₂	semidirect
6. #4837	C₂	semidirect
7. #4834	C₂	semidirect
8. #4709	C₂	semidirect
9. #4706	C₂	semidirect
10. #4688	C₂	semidirect
11. #4685	C₂	semidirect
43. #446	D₃	semidirect
237. C₃	#2993	semidirect

Maximal subgroups

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

subgroup	index
#5580	2
C₃×#446	2
Q₈×C₃⋊C₈	2
#4839	2
#4838	2
#4837	2
#4834	2
#4709	2
#4706	2
#4688	2
#4685	2
#2993	3