Subgroups — #83131

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	C₃×#529	C₂	1	1	yes	maximal, characteristic
2	#5024	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#5012	C₂	1	1	yes	maximal, characteristic, complemented
5	#5012	C₂	1	2	yes	maximal, complemented
6	#5011	C₂	1	2	yes	maximal, complemented
7	D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#5000	C₂	1	2	yes	maximal, complemented
9	#4999	C₂	1	2	yes	maximal, complemented
10	#4925	C₂	1	1	yes	maximal, characteristic, complemented
11	#4924	C₂	1	1	yes	maximal, characteristic, complemented
12	#2993	—	3	1	no	maximal, Sylow
13	C₃×#126	C₄	1	2	yes	—
14	C₃×#125	C₄	1	1	yes	characteristic
15	C₃×#124	C₄	1	1	yes	characteristic
16	C₃×#123	C₄	1	2	yes	—
17	C₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
18	C₃×#121	C₄	1	1	yes	characteristic
19	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
21	C₃×C₄×D₄	C₂²	1	2	yes	—
22	C₃×#117	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
24	C₂×#208	C₂²	1	2	yes	—
25	C₂×#208	C₂²	1	2	yes	—
26	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
27	#830	C₂²	1	1	yes	characteristic
28	#830	C₂²	1	1	yes	characteristic
29	#830	C₂²	1	2	yes	—
30	#830	C₂²	1	2	yes	—
31	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
32	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
33	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
34	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
36	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
37	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
38	#803	C₂²	1	2	yes	—
39	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
41	C₄×C₃⋊C₈	C₂²	1	2	yes	—
42	#529	D₃	1	1	yes	characteristic
43	#446	—	3	1	no	—
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	#424	—	3	2	no	—
52	#423	—	3	1	no	—
53	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
55	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
56	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
62	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
63	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
64	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
65	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
66	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
67	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
68	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
69	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
70	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
71	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
72	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
73	C₃×C₄²	C₂³	1	1	yes	characteristic
74	C₃×C₄²	C₂³	1	2	yes	—
75	C₃×C₄²	C₂³	1	1	yes	characteristic
76	#208	—	4	2	no	—
77	#208	—	4	2	no	—
78	C₂×C₃⋊C₈	—	4	2	no	—
79	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₂×C₃⋊C₈	C₂³	1	2	yes	—
84	C₂×C₃⋊C₈	C₂³	1	2	yes	—
85	C₂×M₁₆	—	3	2	no	—
86	C₂×M₁₆	—	3	2	no	—
87	C₂²×C₈	—	3	2	no	—
88	#126	Dic₃	1	2	yes	—
89	#125	Dic₃	1	1	yes	characteristic
90	#124	Dic₃	1	1	yes	characteristic
91	#123	Dic₃	1	2	yes	—
92	C₂²⋊Q₈	Dic₃	1	1	yes	characteristic
93	#121	Dic₃	1	1	yes	characteristic
94	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
95	C₄×D₄	C₂×D₃	1	1	yes	characteristic
96	C₄×D₄	C₂×D₃	1	2	yes	—
97	#117	C₂×D₃	1	2	yes	—
98	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
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	—	3	2	no	—
105	C₄⋊C₈	—	3	1	no	—
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	—	3	2	no	—
111	C₄×C₈	—	3	1	no	—
112	C₄×C₈	—	3	1	no	—
113	C₄×C₈	—	3	2	no	—
114	C₂³×C₃	C₂²×C₄	1	2	yes	—
115	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
116	C₃×Q₈	—	2	2	no	—
117	C₃×D₄	—	2	2	no	—
118	C₃×D₄	—	4	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	4	1	no	—
125	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
130	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
131	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
132	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
134	C₃⋊C₈	—	2	1	no	—
135	C₃⋊C₈	—	2	2	no	—
136	C₃⋊C₈	—	2	1	no	—
137	C₃⋊C₈	—	2	2	no	—
138	C₃⋊C₈	—	2	1	no	—
139	C₃⋊C₈	—	2	1	no	—
140	C₂×Q₈	C₂×Dic₃	1	1	yes	characteristic
141	C₂×D₄	C₂×Dic₃	1	1	yes	characteristic
142	C₂×D₄	C₂×Dic₃	1	2	yes	—
143	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
146	C₂²×C₄	C₂²×D₃	1	2	yes	—
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	1	no	—
152	C₂×C₈	—	3	1	no	—
153	C₂×C₈	—	3	2	no	—
154	C₂×C₈	—	3	2	no	—
155	C₂×C₈	—	12	2	no	—
156	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
158	C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
159	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
160	C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
161	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
162	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
163	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
164	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
165	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
166	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
167	C₄²	C₂×Dic₃	1	1	yes	characteristic
168	C₄²	C₂×Dic₃	1	1	yes	characteristic
169	C₄²	C₂²×D₃	1	1	yes	characteristic
170	C₄²	C₂²×D₃	1	2	yes	—
171	C₄²	C₂²×D₃	1	1	yes	characteristic
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	#131	1	2	yes	—
187	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
188	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂³	C₂²×Dic₃	1	2	yes	—
190	C₂³	C₂²×Dic₃	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	D₄	—	2	2	no	—
193	D₄	—	4	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
201	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
202	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
203	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
204	C₂×C₄	C₂²×Dic₃	1	2	yes	—
205	C₂×C₄	C₂²×Dic₃	1	2	yes	—
206	C₂×C₄	C₂²×Dic₃	1	2	yes	—
207	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
209	C₈	—	6	1	no	—
210	C₈	—	6	2	no	—
211	C₈	—	6	1	no	—
212	C₈	—	6	2	no	—
213	C₈	—	6	1	no	—
214	C₈	—	6	1	no	—
215	C₂×C₃	—	4	1	no	—
216	C₂×C₃	—	4	2	no	—
217	C₂×C₃	#520	1	1	yes	characteristic
218	C₂×C₃	C₂×#131	1	1	yes	characteristic
219	C₂×C₃	#568	1	1	yes	characteristic
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	C₂³×Dic₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	#948	1	2	yes	—
235	C₄	#1009	1	1	yes	characteristic
236	C₄	#1010	1	1	yes	characteristic
237	C₃	#2993	1	1	yes	characteristic, complemented, Sylow
238	C₂	—	4	1	no	—
239	C₂	—	4	2	no	—
240	C₂	C₂×#948	1	1	yes	characteristic
241	C₂	#5819	1	1	yes	characteristic
242	C₂	#5827	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#529	C₂
prime radical	C₃×#121	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#529	D₃
hypocenter	C₃	#2993
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
2. #5024	C₂	semidirect
3. Q₈×C₃⋊C₈	C₂	semidirect
4. #5012	C₂	semidirect
5. #5012	C₂	semidirect
6. #5011	C₂	semidirect
7. D₄×C₃⋊C₈	C₂	semidirect
8. #5000	C₂	semidirect
9. #4999	C₂	semidirect
10. #4925	C₂	semidirect
11. #4924	C₂	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
C₃×#529	2
#5024	2
Q₈×C₃⋊C₈	2
#5012	2
#5011	2
D₄×C₃⋊C₈	2
#5000	2
#4999	2
#4925	2
#4924	2
#2993	3