Subgroups — #83805

Order 384.

452 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	#5822	C₂	1	2	yes	maximal, complemented
3	C₂×#904	C₂	1	1	yes	maximal, characteristic, complemented
4	#5143	C₂	1	2	yes	maximal, complemented
5	#5142	C₂	1	2	yes	maximal, complemented
6	#5141	C₂	1	4	yes	maximal, complemented
7	#5132	C₂	1	2	yes	maximal, complemented
8	C₂×#819	C₂	1	1	yes	maximal, characteristic, complemented
9	#3067	—	3	1	no	maximal, Sylow
10	C₂×C₃×#41	C₂²	1	1	yes	characteristic
11	C₂×#235	C₂²	1	2	yes	—
12	C₃×#131	—	2	1	no	—
13	C₃×#131	—	2	1	no	—
14	C₃×#131	C₂²	1	2	yes	—
15	C₃×#131	C₂²	1	2	yes	—
16	C₂×C₃×M₁₆	—	2	1	no	—
17	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₈	—	2	1	no	—
19	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
20	#951	C₂²	1	4	yes	—
21	#950	—	2	4	no	—
22	#949	C₂²	1	4	yes	—
23	#942	—	2	2	no	—
24	#932	—	2	2	no	—
25	C₂×#208	C₂²	1	2	yes	—
26	C₂×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
27	#904	—	2	1	no	—
28	#904	—	2	1	no	—
29	#904	—	2	1	no	—
30	#904	—	2	1	no	—
31	C₂×D₂₄	C₂²	1	1	yes	characteristic, complemented
32	C₂×#204	C₂²	1	2	yes	complemented
33	#824	C₂²	1	4	yes	complemented
34	#823	C₂²	1	4	yes	complemented
35	#819	C₂²	1	2	yes	complemented
36	#819	C₂²	1	2	yes	complemented
37	#575	—	3	2	no	—
38	C₂×#135	—	3	1	no	—
39	C₂×#131	D₃	1	1	yes	characteristic, complemented
40	#473	—	3	2	no	—
41	#472	—	3	2	no	—
42	#471	—	3	4	no	—
43	C₂×#108	—	3	1	no	—
44	#413	—	3	2	no	—
45	C₃×#41	—	2	1	no	—
46	C₃×#41	—	2	1	no	—
47	C₃×#41	D₄	1	2	yes	—
48	C₃×#41	C₂³	1	2	yes	—
49	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
50	C₂×C₃×D₄	—	2	1	no	—
51	C₂×C₃×D₄	D₄	1	1	yes	characteristic
52	C₂²×C₃×C₄	—	2	1	no	—
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂×#67	—	2	2	no	—
55	#235	—	2	2	no	—
56	#235	—	2	2	no	—
57	#235	—	2	2	no	—
58	#235	—	2	2	no	—
59	C₂×D₁₂	C₂³	1	2	yes	—
60	C₂×C₄×D₃	—	2	2	no	—
61	C₂×Dic₆	C₂³	1	2	yes	—
62	C₃×M₁₆	—	2	1	no	—
63	C₃×M₁₆	—	2	2	no	—
64	C₃×M₁₆	—	2	2	no	—
65	C₃×M₁₆	C₂³	1	2	yes	—
66	C₂×C₃×C₈	—	2	1	no	—
67	C₂×C₃×C₈	—	2	2	no	—
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	D₄	1	1	yes	characteristic
70	C₂×C₃×C₈	D₄	1	1	yes	characteristic
71	C₂×C₃×C₈	D₄	1	1	yes	characteristic
72	C₂×C₃×C₈	D₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
75	#216	—	2	4	no	—
76	#215	—	2	4	no	—
77	#214	—	2	4	no	—
78	#213	—	2	4	no	—
79	#208	—	2	2	no	—
80	#208	C₂³	1	4	yes	—
81	C₂×C₃⋊C₈	—	2	2	no	—
82	Dic₁₂	—	2	1	no	—
83	Dic₁₂	—	2	1	no	—
84	D₂₄	—	2	1	no	—
85	D₂₄	—	2	1	no	—
86	#204	—	2	2	no	—
87	#204	—	2	2	no	—
88	C₂×#41	—	3	2	no	—
89	C₂×#41	C₂×D₃	1	1	yes	characteristic
90	#137	—	3	4	no	—
91	#137	—	6	2	no	—
92	AGL(1,ℤ/8)	—	3	4	no	—
93	AGL(1,ℤ/8)	—	6	2	no	—
94	#135	—	6	4	no	—
95	#135	—	6	1	no	—
96	#135	—	6	1	no	—
97	#135	—	6	1	no	—
98	#135	—	6	1	no	—
99	C₂×Q₁₆	—	3	1	no	—
100	C₂×SD₁₆	—	3	2	no	—
101	C₂×D₈	—	3	1	no	—
102	#131	—	2	1	no	—
103	#131	—	2	1	no	—
104	#131	C₂×D₃	1	2	yes	—
105	#131	C₂×D₃	1	2	yes	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	—	3	2	no	—
108	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
111	#108	—	3	2	no	—
112	#108	—	3	2	no	—
113	#101	—	3	4	no	—
114	#100	—	3	4	no	—
115	C₂³×C₃	—	2	1	no	—
116	C₂³×C₃	#41	1	1	yes	characteristic
117	C₂²×D₃	—	2	2	no	—
118	C₃×Q₈	—	2	1	no	—
119	C₃×Q₈	C₂×D₄	1	2	yes	—
120	C₃×D₄	—	2	2	no	—
121	C₃×D₄	—	2	2	no	—
122	C₃×D₄	—	2	1	no	—
123	C₃×D₄	C₂×D₄	1	2	yes	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	#41	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
133	#67	—	4	2	no	—
134	#67	—	4	2	no	—
135	C₂×Dic₃	—	2	2	no	—
136	D₁₂	—	2	2	no	—
137	D₁₂	—	2	2	no	—
138	C₄×D₃	—	4	2	no	—
139	C₄×D₃	—	4	2	no	—
140	Dic₆	—	2	2	no	—
141	Dic₆	—	2	2	no	—
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	C₂×D₄	1	1	yes	characteristic
144	C₃×C₈	C₂×D₄	1	1	yes	characteristic
145	C₃×C₈	C₂×D₄	1	1	yes	characteristic
146	C₃×C₈	C₂×D₄	1	1	yes	characteristic
147	C₃⋊C₈	—	2	4	no	—
148	#41	—	2	1	no	—
149	#41	—	2	1	no	—
150	#41	—	6	2	no	—
151	#41	—	6	2	no	—
152	#41	—	6	2	no	—
153	#41	—	6	2	no	—
154	#41	D₁₂	1	2	yes	—
155	#41	C₂²×D₃	1	2	yes	—
156	C₂×Q₈	—	3	2	no	—
157	C₂×Q₈	D₁₂	1	1	yes	characteristic
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	3	2	no	—
160	C₂×D₄	—	6	2	no	—
161	C₂×D₄	D₁₂	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	6	2	no	—
164	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
165	Q₁₆	—	6	4	no	—
166	Q₁₆	—	6	1	no	—
167	Q₁₆	—	6	1	no	—
168	SD₁₆	—	6	4	no	—
169	SD₁₆	—	6	4	no	—
170	SD₁₆	—	6	2	no	—
171	SD₁₆	—	6	2	no	—
172	D₈	—	6	4	no	—
173	D₈	—	6	1	no	—
174	D₈	—	6	1	no	—
175	M₁₆	—	2	1	no	—
176	M₁₆	—	2	2	no	—
177	M₁₆	—	2	2	no	—
178	M₁₆	—	3	4	no	—
179	M₁₆	—	6	2	no	—
180	M₁₆	C₂²×D₃	1	2	yes	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	6	2	no	—
185	C₂×C₈	#67	1	1	yes	characteristic
186	C₂×C₈	#67	1	1	yes	characteristic
187	C₂×C₈	#67	1	1	yes	characteristic
188	C₂×C₈	#67	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
190	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	4	1	no	—
193	C₂²×C₃	—	4	2	no	—
194	C₂²×C₃	#121	1	1	yes	characteristic
195	C₂²×C₃	#121	1	1	yes	characteristic
196	C₂²×C₃	C₂×#41	1	1	yes	characteristic
197	C₂×D₃	—	4	2	no	—
198	C₂×D₃	—	8	2	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	#121	1	1	yes	characteristic
201	C₃×C₄	#121	1	1	yes	characteristic
202	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
203	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
204	Dic₃	—	4	2	no	—
205	C₂³	—	2	1	no	—
206	C₂³	—	6	2	no	—
207	C₂³	#235	1	1	yes	characteristic
208	Q₈	—	2	1	no	—
209	Q₈	—	6	2	no	—
210	Q₈	—	6	2	no	—
211	Q₈	C₂×D₁₂	1	2	yes	—
212	D₄	—	2	2	no	—
213	D₄	—	2	2	no	—
214	D₄	—	2	1	no	—
215	D₄	—	6	2	no	—
216	D₄	—	6	2	no	—
217	D₄	—	12	2	no	—
218	D₄	—	12	2	no	—
219	D₄	C₂×D₁₂	1	2	yes	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	6	2	no	—
224	C₂×C₄	—	12	2	no	—
225	C₂×C₄	—	12	2	no	—
226	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
227	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
228	C₂×C₄	#235	1	1	yes	characteristic
229	C₂×C₄	C₂×#67	1	1	yes	characteristic
230	C₂×C₄	C₂×#67	1	1	yes	characteristic
231	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	6	4	no	—
234	C₈	C₂×#67	1	1	yes	characteristic
235	C₈	C₂×#67	1	1	yes	characteristic
236	C₈	C₂×#67	1	1	yes	characteristic
237	C₈	C₂×#67	1	1	yes	characteristic
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	4	2	no	—
242	C₂×C₃	C₂×#121	1	1	yes	characteristic
243	D₃	—	8	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	12	2	no	—
248	C₂²	—	24	2	no	—
249	C₂²	#930	1	1	yes	characteristic
250	C₂²	#930	1	1	yes	characteristic
251	C₂²	C₂×#235	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	12	2	no	—
254	C₄	#930	1	1	yes	characteristic
255	C₄	#930	1	1	yes	characteristic
256	C₄	C₂²×D₁₂	1	1	yes	characteristic
257	C₄	C₂²×#67	1	1	yes	characteristic
258	C₃	#3067	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	24	2	no	—
264	C₂	C₂×#930	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#930
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#131	C₂
socle	C₂×C₃	C₂×#121
hypercenter	C₂×#131	D₃
hypocenter	C₃	#3067
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#131	C₂	semidirect
2. #5822	C₂	semidirect
3. C₂×#904	C₂	semidirect
4. #5143	C₂	semidirect
5. #5142	C₂	semidirect
6. #5141	C₂	semidirect
7. #5132	C₂	semidirect
8. C₂×#819	C₂	semidirect
26. C₂×Dic₁₂	C₂²	semidirect
31. C₂×D₂₄	C₂²	semidirect
32. C₂×#204	C₂²	semidirect
33. #824	C₂²	semidirect
34. #823	C₂²	semidirect
35. #819	C₂²	semidirect
36. #819	C₂²	semidirect
39. C₂×#131	D₃	semidirect
258. C₃	#3067	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#131	2
#5822	2
C₂×#904	2
#5143	2
#5142	2
#5141	2
#5132	2
C₂×#819	2
#3067	3