Subgroups — #79942

Order 384.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 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	#4840	C₂	1	2	yes	maximal, complemented
5	#4838	C₂	1	1	yes	maximal, characteristic, complemented
6	#4838	C₂	1	2	yes	maximal, complemented
7	#4833	C₂	1	1	yes	maximal, characteristic, complemented
8	#4708	C₂	1	2	yes	maximal, complemented
9	#4706	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	#2969	—	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	1	yes	characteristic
17	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
18	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
19	C₂×D₃×C₈	C₂²	1	2	yes	—
20	#898	C₄	1	2	yes	—
21	#895	C₄	1	1	yes	characteristic
22	#894	C₄	1	2	yes	—
23	#890	C₄	1	1	yes	characteristic
24	C₄×D₁₂	C₂²	1	1	yes	characteristic
25	C₄×D₁₂	C₂²	1	2	yes	—
26	#872	C₂²	1	2	yes	—
27	C₄²×D₃	C₂²	1	1	yes	characteristic
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	C₈×Dic₃	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	1	no	—
46	#436	—	3	2	no	—
47	#435	—	3	2	no	—
48	C₈×D₄	—	3	1	no	—
49	#433	—	3	2	no	—
50	#431	—	3	1	no	—
51	#405	—	3	2	no	—
52	C₄×M₁₆	—	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	2	yes	—
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
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	2	yes	—
80	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	ΓL(1,𝔽₂₅)	—	2	2	no	—
83	ΓL(1,𝔽₂₅)	—	2	2	no	—
84	ΓL(1,𝔽₂₅)	—	2	4	no	—
85	D₃×C₈	—	2	4	no	—
86	C₂×M₁₆	—	3	1	no	—
87	C₂×M₁₆	—	3	1	no	—
88	C₂×M₁₆	—	3	2	no	—
89	C₂²×C₈	—	3	2	no	—
90	#126	—	3	2	no	—
91	#125	—	3	1	no	—
92	#124	—	3	1	no	—
93	#123	—	3	2	no	—
94	C₂²⋊Q₈	—	3	1	no	—
95	#121	—	3	1	no	—
96	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
97	C₄×D₄	—	3	1	no	—
98	C₄×D₄	—	3	2	no	—
99	#117	—	3	2	no	—
100	C₂×C₄²	—	3	1	no	—
101	C₄⋊C₈	—	3	1	no	—
102	C₄⋊C₈	—	3	2	no	—
103	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
104	C₄⋊C₈	C₂×D₃	1	2	yes	—
105	C₂²⋊C₈	—	3	2	no	—
106	C₂²⋊C₈	—	3	2	no	—
107	C₂²⋊C₈	—	3	2	no	—
108	#97	—	3	1	no	—
109	#97	—	3	1	no	—
110	#97	C₂×D₃	1	2	yes	—
111	C₄×C₈	—	3	2	no	—
112	C₄×C₈	—	3	1	no	—
113	C₄×C₈	—	3	2	no	—
114	C₄×C₈	C₂×D₃	1	1	yes	characteristic
115	C₂²×D₃	C₂²×C₄	1	2	yes	—
116	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
117	C₃×Q₈	—	2	2	no	—
118	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
120	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂⁴	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	1	yes	characteristic
126	C₂×Dic₃	C₂²×C₄	1	2	yes	—
127	D₁₂	—	2	2	no	—
128	D₁₂	—	4	2	no	—
129	C₄×D₃	—	2	2	no	—
130	C₄×D₃	—	2	2	no	—
131	C₄×D₃	—	2	1	no	—
132	C₄×D₃	—	2	1	no	—
133	C₄×D₃	—	2	2	no	—
134	C₄×D₃	—	4	1	no	—
135	C₃×C₈	—	2	2	no	—
136	C₃×C₈	—	2	1	no	—
137	C₃×C₈	#41	1	2	yes	—
138	C₃⋊C₈	—	2	2	no	—
139	C₃⋊C₈	—	2	1	no	—
140	C₃⋊C₈	#41	1	2	yes	—
141	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
142	C₂×D₄	—	3	1	no	—
143	C₂×D₄	—	3	2	no	—
144	C₂²×C₄	—	3	1	no	—
145	C₂²×C₄	—	3	1	no	—
146	C₂²×C₄	—	3	2	no	—
147	C₂²×C₄	—	3	1	no	—
148	M₁₆	—	6	2	no	—
149	M₁₆	—	6	4	no	—
150	M₁₆	—	6	2	no	—
151	C₂×C₈	—	3	1	no	—
152	C₂×C₈	—	3	2	no	—
153	C₂×C₈	—	3	1	no	—
154	C₂×C₈	—	6	4	no	—
155	C₂×C₈	C₂²×D₃	1	2	yes	—
156	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
158	C₄⋊C₄	—	3	2	no	—
159	C₄⋊C₄	—	3	2	no	—
160	C₄⋊C₄	—	3	1	no	—
161	C₄⋊C₄	—	3	2	no	—
162	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
163	C₄⋊C₄	C₄×D₃	1	2	yes	—
164	C₂²⋊C₄	—	3	2	no	—
165	C₂²⋊C₄	—	3	2	no	—
166	C₂²⋊C₄	—	3	2	no	—
167	C₂²⋊C₄	—	3	2	no	—
168	C₂²⋊C₄	—	3	2	no	—
169	C₄²	—	3	1	no	—
170	C₄²	—	3	1	no	—
171	C₄²	—	3	1	no	—
172	C₄²	C₂²×D₃	1	1	yes	characteristic
173	C₄²	C₂²×D₃	1	2	yes	—
174	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
175	C₂×D₃	—	2	2	no	—
176	C₂×D₃	—	2	1	no	—
177	C₂×D₃	—	2	1	no	—
178	C₂×D₃	—	2	2	no	—
179	C₂×D₃	—	2	1	no	—
180	C₂×D₃	—	2	2	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	#131	1	2	yes	—
185	C₃×C₄	C₂×#41	1	1	yes	characteristic
186	C₃×C₄	C₂×#41	1	1	yes	characteristic
187	Dic₃	—	2	2	no	—
188	Dic₃	—	2	1	no	—
189	Dic₃	—	2	1	no	—
190	Dic₃	—	2	1	no	—
191	C₂³	—	3	2	no	—
192	C₂³	—	3	1	no	—
193	Q₈	—	2	2	no	—
194	D₄	—	6	2	no	—
195	D₄	—	12	2	no	—
196	C₂×C₄	—	3	1	no	—
197	C₂×C₄	—	3	1	no	—
198	C₂×C₄	—	3	1	no	—
199	C₂×C₄	—	3	2	no	—
200	C₂×C₄	—	6	2	no	—
201	C₂×C₄	—	6	2	no	—
202	C₂×C₄	—	6	1	no	—
203	C₂×C₄	—	6	1	no	—
204	C₂×C₄	—	6	2	no	—
205	C₂×C₄	—	12	1	no	—
206	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
208	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
209	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	1	no	—
213	C₈	—	3	2	no	—
214	C₈	—	6	2	no	—
215	C₈	—	6	1	no	—
216	C₈	#239	1	2	yes	—
217	C₂×C₃	C₄×#41	1	1	yes	characteristic
218	C₂×C₃	C₂×#131	1	1	yes	characteristic
219	C₂×C₃	#568	1	1	yes	characteristic
220	D₃	—	4	1	no	—
221	D₃	—	4	2	no	—
222	C₂²	—	6	1	no	—
223	C₂²	—	6	2	no	—
224	C₂²	—	6	1	no	—
225	C₂²	—	6	2	no	—
226	C₂²	—	6	2	no	—
227	C₂²	—	6	1	no	—
228	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	6	1	no	—
233	C₄	—	6	1	no	—
234	C₄	—	6	1	no	—
235	C₄	—	6	2	no	—
236	C₄	#907	1	2	yes	—
237	C₄	C₂×#239	1	1	yes	characteristic
238	C₄	D₃×#41	1	1	yes	characteristic
239	C₃	#2969	1	1	yes	characteristic, complemented, Sylow
240	C₂	—	12	1	no	—
241	C₂	—	12	2	no	—
242	C₂	C₄×#239	1	1	yes	characteristic
243	C₂	C₂×#907	1	1	yes	characteristic
244	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	#2969
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. #4840	C₂	semidirect
5. #4838	C₂	semidirect
6. #4838	C₂	semidirect
7. #4833	C₂	semidirect
8. #4708	C₂	semidirect
9. #4706	C₂	semidirect
10. C₄×ΓL(1,𝔽₂₅)	C₂	semidirect
11. C₈×D₁₂	C₂	semidirect
43. #446	D₃	semidirect
239. C₃	#2969	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
#4840	2
#4838	2
#4833	2
#4708	2
#4706	2
C₄×ΓL(1,𝔽₂₅)	2
C₈×D₁₂	2
#2969	3