Subgroups — #79960

Order 384.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5578	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
5	D₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#4831	C₂	1	1	yes	maximal, characteristic, complemented
7	#4831	C₂	1	2	yes	maximal, complemented
8	#4708	C₂	1	2	yes	maximal
9	#4701	C₂	1	2	yes	maximal, complemented
10	C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
11	C₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#2962	—	3	1	no	maximal, Sylow
13	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
14	#946	C₄	1	1	yes	characteristic
15	#944	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	#897	C₄	1	1	yes	characteristic
20	#896	C₄	1	2	yes	—
21	#889	C₄	1	2	yes	—
22	#888	C₄	1	1	yes	characteristic
23	#872	C₂²	1	2	yes	—
24	C₄²×D₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₆	C₂²	1	2	yes	—
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	#815	C₂²	1	1	yes	characteristic
32	#815	C₂²	1	1	yes	characteristic
33	#815	C₂²	1	2	yes	—
34	Dic₃⋊C₈	C₂²	1	2	yes	—
35	Dic₃⋊C₈	C₂²	1	2	yes	—
36	Dic₃⋊C₈	C₂²	1	2	yes	—
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	#533	—	3	1	no	—
43	#446	—	3	1	no	—
44	#446	—	3	2	no	—
45	#446	—	3	2	no	—
46	#446	D₃	1	1	yes	characteristic, complemented
47	C₈×Q₈	—	3	1	no	—
48	C₈×Q₈	—	3	1	no	—
49	#424	—	3	2	no	—
50	C₂×C₄⋊C₈	—	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×C₄×D₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
57	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
58	C₂×Dic₆	C₂×C₄	1	2	yes	—
59	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂³	1	2	yes	—
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
63	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
64	C₃×C₄²	C₂³	1	1	yes	characteristic
65	C₃×C₄²	C₂³	1	2	yes	—
66	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
67	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
68	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
69	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
70	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
71	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
72	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
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	C₂³	1	2	yes	—
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)	Q₈	1	4	yes	complemented
84	D₃×C₈	—	2	4	no	—
85	C₂×M₁₆	—	3	1	no	—
86	C₂×M₁₆	—	3	1	no	—
87	C₂²×C₈	—	3	2	no	—
88	C₄⋊Q₈	—	3	1	no	—
89	C₄⋊Q₈	—	3	1	no	—
90	#125	—	3	2	no	—
91	C₂²⋊Q₈	—	3	1	no	—
92	C₂²⋊Q₈	—	3	2	no	—
93	C₂²⋊Q₈	—	3	1	no	—
94	C₄×Q₈	—	3	1	no	—
95	C₄×Q₈	—	3	2	no	—
96	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
97	#117	—	3	2	no	—
98	C₂×C₄²	—	3	1	no	—
99	C₄⋊C₈	—	3	2	no	—
100	C₄⋊C₈	—	3	2	no	—
101	C₄⋊C₈	—	3	2	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	#97	—	3	1	no	—
107	#97	—	3	1	no	—
108	#97	—	3	2	no	—
109	#97	C₂×D₃	1	2	yes	—
110	C₄×C₈	—	3	2	no	—
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	1	yes	characteristic
115	C₃×Q₈	—	2	2	no	—
116	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
118	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
119	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₃	C₂²×C₄	1	2	yes	—
124	C₂×Dic₃	C₂²×C₄	1	2	yes	—
125	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₃	—	2	2	no	—
127	C₄×D₃	—	4	1	no	—
128	C₄×D₃	C₂×Q₈	1	2	yes	—
129	C₄×D₃	#41	1	2	yes	—
130	Dic₆	—	2	2	no	—
131	Dic₆	—	4	2	no	—
132	C₃×C₈	—	2	2	no	—
133	C₃×C₈	—	2	1	no	—
134	C₃×C₈	C₂×Q₈	1	2	yes	—
135	C₃⋊C₈	—	2	2	no	—
136	C₃⋊C₈	—	2	1	no	—
137	C₃⋊C₈	C₂×Q₈	1	2	yes	—
138	C₂×Q₈	—	3	1	no	—
139	C₂×Q₈	—	3	2	no	—
140	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
141	C₂²×C₄	—	3	1	no	—
142	C₂²×C₄	—	3	1	no	—
143	C₂²×C₄	—	3	1	no	—
144	M₁₆	—	3	4	no	—
145	M₁₆	—	6	2	no	—
146	C₂×C₈	—	3	1	no	—
147	C₂×C₈	—	3	2	no	—
148	C₂×C₈	—	3	1	no	—
149	C₂×C₈	—	6	4	no	—
150	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₃	1	2	yes	—
152	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
153	C₄⋊C₄	—	3	2	no	—
154	C₄⋊C₄	—	3	2	no	—
155	C₄⋊C₄	—	3	2	no	—
156	C₄⋊C₄	—	3	2	no	—
157	C₄⋊C₄	—	3	2	no	—
158	C₄⋊C₄	—	3	2	no	—
159	C₄⋊C₄	—	3	1	no	—
160	C₄⋊C₄	C₄×D₃	1	2	yes	—
161	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
162	C₂²⋊C₄	—	3	2	no	—
163	C₂²⋊C₄	—	3	2	no	—
164	C₄²	—	3	1	no	—
165	C₄²	—	3	1	no	—
166	C₄²	—	3	2	no	—
167	C₄²	—	3	1	no	—
168	C₄²	C₂²×D₃	1	1	yes	characteristic
169	C₄²	C₂²×D₃	1	2	yes	—
170	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
171	C₂×D₃	—	2	1	no	—
172	C₂×D₃	—	2	1	no	—
173	C₂×D₃	C₄×Q₈	1	2	yes	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	#131	1	2	yes	—
178	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
179	C₃×C₄	C₂×#41	1	1	yes	characteristic
180	Dic₃	—	2	2	no	—
181	Dic₃	—	2	1	no	—
182	Dic₃	—	2	1	no	—
183	Dic₃	—	2	2	no	—
184	Dic₃	C₄×Q₈	1	2	yes	—
185	C₂³	—	3	1	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	6	2	no	—
188	Q₈	—	12	2	no	—
189	C₂×C₄	—	3	1	no	—
190	C₂×C₄	—	3	2	no	—
191	C₂×C₄	—	3	2	no	—
192	C₂×C₄	—	3	1	no	—
193	C₂×C₄	—	3	1	no	—
194	C₂×C₄	—	3	2	no	—
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	—	6	2	no	—
197	C₂×C₄	—	12	1	no	—
198	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
201	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	1	no	—
205	C₈	—	3	2	no	—
206	C₈	—	6	2	no	—
207	C₈	—	6	1	no	—
208	C₈	D₃×Q₈	1	2	yes	—
209	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
210	C₂×C₃	C₂×#131	1	1	yes	characteristic
211	C₂×C₃	#568	1	1	yes	characteristic
212	D₃	—	2	2	no	—
213	C₂²	—	3	2	no	—
214	C₂²	—	6	1	no	—
215	C₂²	—	6	1	no	—
216	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	3	2	no	—
221	C₄	—	6	1	no	—
222	C₄	—	6	2	no	—
223	C₄	—	6	1	no	—
224	C₄	—	6	2	no	—
225	C₄	#907	1	2	yes	—
226	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
227	C₄	D₃×#41	1	1	yes	characteristic
228	C₃	#2962	1	1	yes	characteristic, complemented, Sylow
229	C₂	—	6	2	no	—
230	C₂	C₄×D₃×Q₈	1	1	yes	characteristic
231	C₂	C₂×#907	1	1	yes	characteristic
232	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	C₂²×D₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#446	D₃
hypocenter	C₃	#2962
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
6. #4831	C₂	semidirect
7. #4831	C₂	semidirect
9. #4701	C₂	semidirect
11. C₈×Dic₆	C₂	semidirect
46. #446	D₃	semidirect
83. ΓL(1,𝔽₂₅)	Q₈	semidirect
228. C₃	#2962	semidirect

Maximal subgroups

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

subgroup	index
#5578	2
C₃×#446	2
Q₈×C₃⋊C₈	2
#4839	2
D₃×C₄⋊C₈	2
#4831	2
#4708	2
#4701	2
C₄×ΓL(1,𝔽₂₅)	2
C₈×Dic₆	2
#2962	3