Subgroups — #10693787

Order 768.

388 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#4457	C₂	1	2	yes	maximal
3	#69394	C₂	1	1	yes	maximal, characteristic
4	#69391	C₂	1	1	yes	maximal, characteristic
5	#67654	C₂	1	2	yes	maximal
6	#9605	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₈	C₄	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
9	C₃×C₄³	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
12	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×#803	—	2	1	no	—
16	C₃×#336	C₄	1	1	yes	characteristic
17	C₃×#336	C₄	1	1	yes	characteristic
18	#4524	—	2	1	no	—
19	#4457	—	2	4	no	—
20	#1842	—	3	1	no	—
21	#1767	—	3	1	no	—
22	C₄×C₄⋊C₈	D₃	1	1	yes	characteristic
23	C₂×#334	—	3	2	no	—
24	#1277	—	3	2	no	—
25	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
32	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
34	C₂²×C₃⋊C₈	—	2	1	no	—
35	C₂²×C₃⋊C₈	—	2	1	no	—
36	C₂²×C₃⋊C₈	—	2	2	no	—
37	C₃×C₄⋊C₈	—	2	4	no	—
38	C₃×C₄×C₈	—	2	2	no	—
39	C₃×C₄×C₈	—	2	2	no	—
40	C₄⋊(C₃⋊C₈)	—	2	1	no	—
41	C₄⋊(C₃⋊C₈)	—	2	1	no	—
42	C₄⋊(C₃⋊C₈)	—	2	2	no	—
43	C₄⋊(C₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₃⋊C₈)	—	4	2	no	—
45	C₄⋊(C₃⋊C₈)	—	4	2	no	—
46	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
47	C₄⋊(C₃⋊C₈)	Q₈	1	2	yes	—
48	#803	—	4	2	no	—
49	C₂×C₄⋊C₈	—	3	1	no	—
50	C₂×C₄⋊C₈	—	3	1	no	—
51	C₂×C₄⋊C₈	—	3	2	no	—
52	C₂×C₄⋊C₈	—	6	1	no	—
53	C₂×C₄⋊C₈	—	6	1	no	—
54	C₂×C₄⋊C₈	Dic₃	1	2	yes	—
55	C₂×#97	—	6	1	no	—
56	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
57	C₄³	C₂×D₃	1	1	yes	characteristic
58	#336	—	6	1	no	—
59	#336	Dic₃	1	1	yes	characteristic
60	#336	Dic₃	1	1	yes	characteristic
61	#334	—	6	4	no	—
62	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	M₁₆	1	4	yes	—
75	C₃×C₄²	—	2	2	no	—
76	C₃×C₄²	—	2	2	no	—
77	C₃×C₄²	—	2	2	no	—
78	C₃×C₄²	—	2	1	no	—
79	C₃×C₄²	—	2	1	no	—
80	C₃×C₄²	—	4	1	no	—
81	C₃×C₄²	SD₁₆	1	2	yes	—
82	C₃×C₄²	SD₁₆	1	2	yes	—
83	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
84	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
85	C₃×C₄²	#41	1	1	yes	characteristic
86	C₃×C₄²	#41	1	1	yes	characteristic
87	C₂×C₃⋊C₈	—	2	2	no	—
88	C₂×C₃⋊C₈	—	2	2	no	—
89	C₂×C₃⋊C₈	—	4	1	no	—
90	C₂×C₃⋊C₈	—	4	1	no	—
91	C₂×C₃⋊C₈	—	4	1	no	—
92	C₂×C₃⋊C₈	—	4	1	no	—
93	C₂×C₃⋊C₈	—	4	2	no	—
94	C₂×C₃⋊C₈	—	4	2	no	—
95	C₂×C₃⋊C₈	—	4	2	no	—
96	C₂²×C₈	—	6	1	no	—
97	C₂²×C₈	—	6	1	no	—
98	C₂²×C₈	—	6	2	no	—
99	C₂²×C₈	C₂×Dic₃	1	2	yes	—
100	C₂²×C₈	C₂×Dic₃	1	2	yes	—
101	C₂×C₄²	Dic₆	1	1	yes	characteristic
102	C₂×C₄²	Dic₆	1	1	yes	characteristic
103	C₂×C₄²	D₁₂	1	1	yes	characteristic
104	C₂×C₄²	D₁₂	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
106	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
107	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
108	C₄⋊C₈	—	2	4	no	—
109	C₄⋊C₈	—	3	2	no	—
110	C₄⋊C₈	—	3	2	no	—
111	C₄⋊C₈	—	6	2	no	—
112	C₄⋊C₈	—	6	1	no	—
113	C₄⋊C₈	—	6	1	no	—
114	C₄⋊C₈	—	6	2	no	—
115	C₄⋊C₈	—	12	2	no	—
116	C₄⋊C₈	—	12	2	no	—
117	#97	—	12	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	#106	1	2	yes	—
133	C₂×C₃×C₄	#106	1	2	yes	—
134	C₂×C₃×C₄	#108	1	2	yes	—
135	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
137	C₂×C₃×C₄	#121	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₃×C₄	#125	1	1	yes	characteristic
140	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
142	C₂×C₃×C₄	#131	1	2	yes	—
143	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₃×C₄	#137	1	1	yes	characteristic
147	C₃×C₈	—	2	4	no	—
148	C₃×C₈	—	4	2	no	—
149	C₃⋊C₈	—	4	2	no	—
150	C₃⋊C₈	—	4	2	no	—
151	C₃⋊C₈	—	8	2	no	—
152	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
153	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
154	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
155	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	6	2	no	—
165	C₂×C₈	—	6	2	no	—
166	C₂×C₈	—	12	1	no	—
167	C₂×C₈	—	12	1	no	—
168	C₂×C₈	—	12	1	no	—
169	C₂×C₈	—	12	1	no	—
170	C₂×C₈	—	12	2	no	—
171	C₂×C₈	—	12	2	no	—
172	C₂×C₈	—	12	2	no	—
173	C₂×C₈	#208	1	4	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	4	1	no	—
180	C₄²	#204	1	2	yes	—
181	C₄²	#204	1	2	yes	—
182	C₄²	D₃×D₄	1	1	yes	characteristic
183	C₄²	#237	1	1	yes	characteristic
184	C₄²	D₃×Q₈	1	1	yes	characteristic
185	C₄²	#239	1	1	yes	characteristic
186	C₂²×C₃	#389	1	1	yes	characteristic
187	C₂²×C₃	#423	1	1	yes	characteristic
188	C₂²×C₃	#424	1	1	yes	characteristic
189	C₂²×C₃	C₂×#106	1	1	yes	characteristic
190	C₂²×C₃	#428	1	1	yes	characteristic
191	C₂²×C₃	C₂×#108	1	1	yes	characteristic
192	C₂²×C₃	#430	1	1	yes	characteristic
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	4	1	no	—
199	C₃×C₄	#435	1	2	yes	—
200	C₃×C₄	#446	1	2	yes	—
201	C₃×C₄	#460	1	1	yes	characteristic
202	C₃×C₄	#461	1	1	yes	characteristic
203	C₃×C₄	#475	1	1	yes	characteristic
204	C₃×C₄	#476	1	1	yes	characteristic
205	C₂³	C₂×C₄⋊Dic₃	1	1	yes	characteristic
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	#817	1	2	yes	—
218	C₂×C₄	#817	1	2	yes	—
219	C₂×C₄	#819	1	2	yes	—
220	C₂×C₄	#888	1	1	yes	characteristic
221	C₂×C₄	#890	1	1	yes	characteristic
222	C₂×C₄	#895	1	1	yes	characteristic
223	C₂×C₄	#897	1	1	yes	characteristic
224	C₂×C₄	C₂×#204	1	1	yes	characteristic
225	C₂×C₄	C₂×#204	1	1	yes	characteristic
226	C₂×C₄	#908	1	1	yes	characteristic
227	C₂×C₄	#909	1	1	yes	characteristic
228	C₂×C₄	C₂×#208	1	2	yes	—
229	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
230	C₂×C₄	Q₈×Dic₃	1	1	yes	characteristic
231	C₂×C₄	#948	1	2	yes	—
232	C₈	—	2	4	no	—
233	C₈	—	4	2	no	—
234	C₈	—	12	2	no	—
235	C₈	—	12	2	no	—
236	C₈	—	24	2	no	—
237	C₂×C₃	#1566	1	1	yes	characteristic
238	C₂×C₃	#1569	1	1	yes	characteristic
239	C₂×C₃	#1570	1	1	yes	characteristic
240	C₂×C₃	#1573	1	1	yes	characteristic
241	C₂×C₃	#1940	1	1	yes	characteristic
242	C₂×C₃	#1948	1	1	yes	characteristic
243	C₂×C₃	#1950	1	1	yes	characteristic
244	C₂²	#4924	1	1	yes	characteristic
245	C₂²	#4984	1	1	yes	characteristic
246	C₂²	#4999	1	1	yes	characteristic
247	C₂²	C₂×#817	1	1	yes	characteristic
248	C₂²	C₂×#819	1	1	yes	characteristic
249	C₂²	#5121	1	1	yes	characteristic
250	C₂²	#5125	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	#4841	1	1	yes	characteristic
258	C₄	#4842	1	1	yes	characteristic
259	C₄	#4848	1	1	yes	characteristic
260	C₄	#4851	1	1	yes	characteristic
261	C₄	#5011	1	2	yes	—
262	C₄	#5024	1	2	yes	—
263	C₃	#9605	1	1	yes	characteristic, complemented, Sylow
264	C₂	#69485	1	1	yes	characteristic
265	C₂	#69730	1	1	yes	characteristic
266	C₂	#69851	1	1	yes	characteristic
267	C₂	#70454	1	1	yes	characteristic
268	C₂	#70474	1	1	yes	characteristic
269	C₂	#70749	1	1	yes	characteristic
270	C₂	#70752	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊C₈	D₃
hypocenter	C₃	#9605
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
263. C₃	#9605	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₄⋊C₈	2
C₂×#4457	2
#69394	2
#69391	2
#67654	2
#9605	3