Subgroups — #83349

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#235	C₂	1	1	yes	maximal, characteristic
3	#5094	C₂	1	1	yes	maximal, characteristic, complemented
4	#5091	C₂	1	1	yes	maximal, characteristic, complemented
5	#5087	C₂	1	1	yes	maximal, characteristic, complemented
6	#5084	C₂	1	1	yes	maximal, characteristic, complemented
7	#5066	C₂	1	2	yes	maximal, complemented
8	#5063	C₂	1	2	yes	maximal, complemented
9	#5002	C₂	1	2	yes	maximal
10	#4998	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
12	#3076	—	3	1	no	maximal, Sylow
13	C₂×#235	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃×#125	C₂²	1	2	yes	—
17	C₃×C₂²⋊Q₈	—	2	1	no	—
18	C₃×C₂²⋊Q₈	—	2	1	no	—
19	C₃×C₄×Q₈	—	2	1	no	—
20	C₃×C₄×Q₈	—	2	1	no	—
21	C₃×#117	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₄×#67	—	2	1	no	—
24	C₄×#67	—	2	1	no	—
25	#926	C₂²	1	1	yes	characteristic
26	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
27	C₄×D₁₂	C₂²	1	1	yes	characteristic
28	C₄×D₁₂	C₂²	1	1	yes	characteristic
29	#872	—	2	1	no	—
30	C₄²×D₃	—	2	1	no	—
31	C₄×Dic₆	C₂²	1	1	yes	characteristic
32	C₄×Dic₆	C₂²	1	1	yes	characteristic
33	#810	C₂²	1	2	yes	—
34	#810	C₂²	1	2	yes	—
35	#809	C₂²	1	2	yes	—
36	#809	C₂²	1	2	yes	—
37	#808	C₂²	1	2	yes	—
38	#808	C₂²	1	2	yes	—
39	#807	C₂²	1	2	yes	—
40	#807	C₂²	1	2	yes	—
41	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
42	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
43	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
44	#533	D₃	1	1	yes	characteristic, complemented
45	C₄×#41	—	3	1	no	—
46	#479	—	3	2	no	—
47	#478	—	3	2	no	—
48	#477	—	3	1	no	—
49	#476	—	3	1	no	—
50	#475	—	3	1	no	—
51	#474	—	3	1	no	—
52	#427	—	3	2	no	—
53	C₂×C₄⋊C₈	—	3	1	no	—
54	#416	—	3	2	no	—
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	—	2	1	no	—
57	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂×#67	—	2	1	no	—
61	#235	—	2	2	no	—
62	#235	Q₈	1	4	yes	complemented
63	C₂×D₁₂	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₃	—	2	1	no	—
65	C₂×C₄×D₃	—	2	1	no	—
66	C₂×C₄×D₃	—	2	1	no	—
67	C₂×Dic₆	C₂³	1	1	yes	characteristic
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	1	no	—
71	C₃×C₄⋊C₄	—	2	1	no	—
72	C₃×C₄⋊C₄	C₂³	1	2	yes	—
73	C₃×C₄⋊C₄	C₂³	1	2	yes	—
74	C₃×C₂²⋊C₄	—	2	2	no	—
75	C₃×C₄²	—	2	2	no	—
76	C₃×C₄²	D₄	1	1	yes	characteristic
77	C₃×C₄²	D₄	1	1	yes	characteristic
78	C₃×C₄²	C₂³	1	1	yes	characteristic
79	C₃×C₄²	C₂³	1	1	yes	characteristic
80	#217	—	2	1	no	—
81	(C₂×D₃)⋊C₄	—	2	1	no	—
82	(C₂×D₃)⋊C₄	—	2	1	no	—
83	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
85	Dic₃⋊C₄	—	2	1	no	—
86	Dic₃⋊C₄	—	2	1	no	—
87	C₄×Dic₃	—	2	1	no	—
88	C₄×Dic₃	—	2	1	no	—
89	C₄×Dic₃	—	2	1	no	—
90	C₂×C₃⋊C₈	—	2	4	no	—
91	C₂×C₃⋊C₈	C₂³	1	2	yes	—
92	C₂×C₃⋊C₈	C₂³	1	2	yes	—
93	C₂×#41	—	3	1	no	—
94	C₂²×C₈	—	3	2	no	—
95	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
96	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
97	#125	C₂×D₃	1	2	yes	—
98	C₂²⋊Q₈	—	2	1	no	—
99	C₂²⋊Q₈	—	2	1	no	—
100	C₄×Q₈	—	2	1	no	—
101	C₄×Q₈	—	2	1	no	—
102	C₄×Q₈	—	3	1	no	—
103	C₄×Q₈	—	3	1	no	—
104	C₄×D₄	—	3	1	no	—
105	C₄×D₄	—	3	1	no	—
106	C₄×D₄	—	6	1	no	—
107	C₄×D₄	—	6	1	no	—
108	#117	—	3	1	no	—
109	#117	—	6	1	no	—
110	#117	C₂×D₃	1	2	yes	—
111	C₂×C₄²	—	6	1	no	—
112	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
113	#107	—	3	2	no	—
114	#107	—	3	2	no	—
115	#106	—	3	2	no	—
116	#106	—	3	2	no	—
117	C₄⋊C₈	—	3	1	no	—
118	C₄⋊C₈	—	3	1	no	—
119	C₄⋊C₈	—	3	2	no	—
120	#103	—	3	2	no	—
121	#103	—	3	2	no	—
122	#102	—	3	2	no	—
123	#102	—	3	2	no	—
124	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×D₃	—	2	1	no	—
126	C₃×Q₈	—	4	2	no	—
127	C₃×Q₈	—	4	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	4	1	no	—
132	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
138	C₂×C₃×C₄	#41	1	2	yes	—
139	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
140	#67	—	2	4	no	—
141	C₂×Dic₃	—	2	1	no	—
142	C₂×Dic₃	—	2	1	no	—
143	C₂×Dic₃	—	2	1	no	—
144	D₁₂	—	2	1	no	—
145	D₁₂	C₂×Q₈	1	2	yes	—
146	C₄×D₃	—	2	4	no	—
147	C₄×D₃	—	4	2	no	—
148	C₄×D₃	—	4	2	no	—
149	Dic₆	—	2	1	no	—
150	Dic₆	C₂×Q₈	1	2	yes	—
151	C₃⋊C₈	—	2	2	no	—
152	C₃⋊C₈	—	2	2	no	—
153	#41	—	3	4	no	—
154	#41	—	6	2	no	—
155	C₂×Q₈	—	2	1	no	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	3	1	no	—
158	C₂×D₄	—	3	1	no	—
159	C₂×D₄	—	6	1	no	—
160	C₂²×C₄	—	6	1	no	—
161	C₂²×C₄	—	6	1	no	—
162	C₂²×C₄	—	6	1	no	—
163	C₂²×C₄	#67	1	1	yes	characteristic
164	C₂²×C₄	#67	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
166	C₂×C₈	—	3	2	no	—
167	C₂×C₈	—	3	2	no	—
168	C₂×C₈	—	6	4	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	3	1	no	—
174	C₄⋊C₄	—	3	1	no	—
175	C₄⋊C₄	—	6	1	no	—
176	C₄⋊C₄	—	6	1	no	—
177	C₄⋊C₄	C₂²×D₃	1	2	yes	—
178	C₄⋊C₄	C₂²×D₃	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	6	1	no	—
181	C₂²⋊C₄	—	6	1	no	—
182	C₂²⋊C₄	—	6	1	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	6	1	no	—
185	C₄²	—	6	1	no	—
186	C₄²	—	6	1	no	—
187	C₄²	#67	1	1	yes	characteristic
188	C₄²	#67	1	1	yes	characteristic
189	C₄²	C₂²×D₃	1	1	yes	characteristic
190	C₄²	C₂²×D₃	1	1	yes	characteristic
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
194	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×D₃	—	2	2	no	—
196	C₂×D₃	—	4	1	no	—
197	C₂×D₃	—	4	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	4	2	no	—
202	C₃×C₄	—	4	2	no	—
203	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
204	C₃×C₄	#135	1	2	yes	—
205	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
206	C₃×C₄	C₂×#41	1	1	yes	characteristic
207	Dic₃	—	2	2	no	—
208	Dic₃	—	4	1	no	—
209	Dic₃	—	4	1	no	—
210	C₂³	—	6	1	no	—
211	C₂³	C₂×#67	1	1	yes	characteristic
212	Q₈	—	3	2	no	—
213	Q₈	—	4	2	no	—
214	Q₈	—	4	2	no	—
215	Q₈	—	6	1	no	—
216	D₄	—	3	2	no	—
217	D₄	—	6	4	no	—
218	D₄	—	6	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	6	1	no	—
224	C₂×C₄	—	6	4	no	—
225	C₂×C₄	—	6	1	no	—
226	C₂×C₄	—	6	1	no	—
227	C₂×C₄	—	12	2	no	—
228	C₂×C₄	—	12	2	no	—
229	C₂×C₄	D₃×Q₈	1	2	yes	—
230	C₂×C₄	#239	1	2	yes	—
231	C₂×C₄	C₂×#67	1	1	yes	characteristic
232	C₂×C₄	C₂×#67	1	1	yes	characteristic
233	C₂×C₄	C₂×#67	1	1	yes	characteristic
234	C₂×C₄	C₂×#67	1	1	yes	characteristic
235	C₂×C₄	C₂×#67	1	1	yes	characteristic
236	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
237	C₈	—	6	2	no	—
238	C₈	—	6	2	no	—
239	C₂×C₃	—	2	2	no	—
240	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
241	C₂×C₃	C₂×#135	1	1	yes	characteristic
242	C₂×C₃	#575	1	1	yes	characteristic
243	D₃	—	4	2	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	6	2	no	—
247	C₂²	—	12	1	no	—
248	C₂²	—	12	1	no	—
249	C₂²	#946	1	2	yes	—
250	C₂²	C₂²×#67	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	—	6	2	no	—
257	C₄	—	12	1	no	—
258	C₄	—	12	1	no	—
259	C₄	#946	1	2	yes	—
260	C₄	#950	1	2	yes	—
261	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
262	C₄	C₂×#239	1	1	yes	characteristic
263	C₃	#3076	1	1	yes	characteristic, complemented, Sylow
264	C₂	—	2	2	no	—
265	C₂	—	12	2	no	—
266	C₂	C₂×#946	1	1	yes	characteristic
267	C₂	C₂×#950	1	1	yes	characteristic
268	C₂	#5822	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#533	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#235	C₂²
hypercenter	#533	D₃
hypocenter	C₃	#3076
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₃×#533	C₂	semidirect
3. #5094	C₂	semidirect
4. #5091	C₂	semidirect
5. #5087	C₂	semidirect
6. #5084	C₂	semidirect
7. #5066	C₂	semidirect
8. #5063	C₂	semidirect
10. #4998	C₂	semidirect
11. C₂×C₄⋊(C₃⋊C₈)	C₂	semidirect
44. #533	D₃	semidirect
62. #235	Q₈	semidirect
263. C₃	#3076	semidirect

Maximal subgroups

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

subgroup	index
C₃×#533	2
C₄×#235	2
#5094	2
#5091	2
#5087	2
#5084	2
#5066	2
#5063	2
#5002	2
#4998	2
C₂×C₄⋊(C₃⋊C₈)	2
#3076	3