Subgroups — #10695326

Order 768.

430 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	#73307	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#69392	C₂	1	1	yes	maximal, characteristic
4	#67702	C₂	1	2	yes	maximal
5	#67702	C₂	1	2	yes	maximal
6	#9901	—	3	1	no	maximal, Sylow
7	C₂×#870	C₄	1	2	yes	—
8	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	—	2	1	no	—
10	C₃×C₄³	C₂²	1	1	yes	characteristic
11	#4936	C₄	1	2	yes	—
12	#4935	C₂²	1	1	yes	characteristic
13	#4935	C₂²	1	1	yes	characteristic
14	#4928	—	2	1	no	—
15	#4928	—	2	1	no	—
16	#4928	—	2	1	no	—
17	#4928	—	2	1	no	—
18	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₃×#336	—	2	1	no	—
20	#4524	—	2	1	no	—
21	#4524	—	2	1	no	—
22	#2859	—	3	1	no	—
23	#1831	—	3	1	no	—
24	C₄×C₄⋊C₈	D₃	1	1	yes	characteristic
25	#1296	—	3	2	no	—
26	#1296	—	3	2	no	—
27	C₂²×C₃×C₈	—	2	2	no	—
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊Dic₃	—	2	1	no	—
36	C₂×C₄⋊Dic₃	—	2	1	no	—
37	C₂×C₄⋊Dic₃	—	2	1	no	—
38	C₂×C₄⋊Dic₃	—	2	1	no	—
39	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
41	C₂×Dic₃⋊C₄	—	2	2	no	—
42	C₂×Dic₃⋊C₄	—	2	2	no	—
43	C₂²×C₃⋊C₈	—	2	2	no	—
44	#870	—	4	4	no	—
45	C₃×C₄⋊C₈	—	2	2	no	—
46	C₃×C₄⋊C₈	Q₈	1	4	yes	—
47	C₃×C₄×C₈	—	4	2	no	—
48	C₃⋊#95	—	2	2	no	—
49	C₃⋊#95	—	2	2	no	—
50	C₄⋊(C₃⋊C₈)	—	2	4	no	—
51	C₂×#125	—	3	2	no	—
52	C₂×C₄⋊C₈	—	3	2	no	—
53	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
54	C₂×C₄×C₈	—	2	1	no	—
55	#389	—	6	1	no	—
56	#389	—	6	1	no	—
57	#389	—	6	1	no	—
58	#389	—	6	1	no	—
59	#384	—	3	1	no	—
60	#384	—	3	1	no	—
61	#382	—	3	2	no	—
62	C₄³	C₂×D₃	1	1	yes	characteristic
63	#336	—	2	1	no	—
64	#336	—	6	1	no	—
65	#336	—	6	1	no	—
66	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₃	—	2	2	no	—
74	C₂²×Dic₃	—	2	2	no	—
75	C₂×C₃×C₈	—	2	4	no	—
76	C₂×C₃×C₈	—	4	2	no	—
77	C₂×C₃×C₈	—	4	2	no	—
78	C₃×C₄²	—	2	2	no	—
79	C₃×C₄²	—	2	2	no	—
80	C₃×C₄²	—	2	2	no	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	2	1	no	—
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	C₂×Q₈	1	2	yes	—
86	C₃×C₄²	#41	1	2	yes	—
87	C₄⋊Dic₃	—	4	2	no	—
88	C₄⋊Dic₃	—	4	2	no	—
89	C₄⋊Dic₃	—	4	2	no	—
90	C₄⋊Dic₃	—	4	1	no	—
91	C₄⋊Dic₃	—	4	1	no	—
92	C₄⋊Dic₃	—	4	2	no	—
93	C₄⋊Dic₃	—	4	2	no	—
94	Dic₃⋊C₄	—	4	4	no	—
95	Dic₃⋊C₄	—	4	4	no	—
96	C₂×C₃⋊C₈	—	4	2	no	—
97	C₂×C₃⋊C₈	—	4	2	no	—
98	C₂×C₃⋊C₈	—	4	2	no	—
99	C₂²×C₈	—	2	2	no	—
100	C₂²×C₈	—	6	2	no	—
101	#125	—	12	4	no	—
102	C₂×C₄⋊C₄	—	3	2	no	—
103	C₂×C₄⋊C₄	—	3	2	no	—
104	C₂×C₄⋊C₄	—	6	2	no	—
105	C₂×C₄⋊C₄	—	6	2	no	—
106	C₂×C₄⋊C₄	—	6	1	no	—
107	C₂×C₄⋊C₄	—	6	1	no	—
108	C₂×C₄⋊C₄	—	6	1	no	—
109	C₂×C₄⋊C₄	—	6	1	no	—
110	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
111	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
112	C₂×C₄²	D₁₂	1	1	yes	characteristic
113	C₂×C₄²	D₁₂	1	1	yes	characteristic
114	C₂×C₄²	#67	1	1	yes	characteristic
115	C₂×C₄²	#67	1	1	yes	characteristic
116	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	6	4	no	—
119	C₄⋊C₈	Dic₆	1	4	yes	—
120	C₄×C₈	—	4	2	no	—
121	#95	—	6	2	no	—
122	#95	—	6	2	no	—
123	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	#100	1	2	yes	—
137	C₂×C₃×C₄	#101	1	2	yes	—
138	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
139	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₃×C₄	#124	1	1	yes	characteristic
141	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₃×C₄	#135	1	2	yes	—
143	C₂×C₃×C₄	#135	1	2	yes	—
144	C₂×C₃×C₄	#135	1	2	yes	—
145	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₃×C₄	#137	1	1	yes	characteristic
147	C₂×Dic₃	—	4	2	no	—
148	C₂×Dic₃	—	4	2	no	—
149	C₂×Dic₃	—	4	2	no	—
150	C₂×Dic₃	—	4	2	no	—
151	C₂×Dic₃	—	4	2	no	—
152	C₂×Dic₃	—	4	2	no	—
153	C₃×C₈	—	4	4	no	—
154	C₃⋊C₈	—	8	2	no	—
155	C₂²×C₄	—	6	2	no	—
156	C₂²×C₄	—	6	2	no	—
157	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
158	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
163	C₂²×C₄	C₂×#67	1	1	yes	characteristic
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	12	2	no	—
168	C₂×C₈	—	12	2	no	—
169	C₂×C₈	—	12	2	no	—
170	C₄⋊C₄	—	12	2	no	—
171	C₄⋊C₄	—	12	2	no	—
172	C₄⋊C₄	—	12	4	no	—
173	C₄⋊C₄	—	12	4	no	—
174	C₄⋊C₄	—	12	2	no	—
175	C₄⋊C₄	—	12	2	no	—
176	C₄⋊C₄	—	12	1	no	—
177	C₄⋊C₄	—	12	1	no	—
178	C₄⋊C₄	—	12	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	2	no	—
186	C₄²	C₂×Dic₆	1	2	yes	—
187	C₄²	#235	1	2	yes	—
188	C₂²×C₃	#391	1	1	yes	characteristic
189	C₂²×C₃	C₂×#100	1	1	yes	characteristic
190	C₂²×C₃	C₂×#101	1	1	yes	characteristic
191	C₂²×C₃	#416	1	1	yes	characteristic
192	C₂²×C₃	#416	1	1	yes	characteristic
193	C₂²×C₃	#416	1	1	yes	characteristic
194	C₂²×C₃	#417	1	1	yes	characteristic
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	4	1	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	#478	1	2	yes	—
201	C₃×C₄	#479	1	2	yes	—
202	C₃×C₄	#488	1	2	yes	—
203	C₃×C₄	#499	1	2	yes	—
204	Dic₃	—	8	2	no	—
205	Dic₃	—	8	2	no	—
206	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	12	2	no	—
220	C₂×C₄	—	12	2	no	—
221	C₂×C₄	—	12	2	no	—
222	C₂×C₄	—	12	2	no	—
223	C₂×C₄	—	12	2	no	—
224	C₂×C₄	—	12	2	no	—
225	C₂×C₄	#823	1	2	yes	—
226	C₂×C₄	#824	1	2	yes	—
227	C₂×C₄	C₄×Dic₆	1	2	yes	—
228	C₂×C₄	#869	1	1	yes	characteristic
229	C₂×C₄	#875	1	1	yes	characteristic
230	C₂×C₄	#904	1	2	yes	—
231	C₂×C₄	#904	1	2	yes	—
232	C₂×C₄	#924	1	2	yes	—
233	C₂×C₄	#932	1	1	yes	characteristic
234	C₂×C₄	#942	1	1	yes	characteristic
235	C₂×C₄	#950	1	2	yes	—
236	C₈	—	4	4	no	—
237	C₈	—	24	2	no	—
238	C₂×C₃	#1543	1	1	yes	characteristic
239	C₂×C₃	#1544	1	1	yes	characteristic
240	C₂×C₃	#1551	1	1	yes	characteristic
241	C₂×C₃	#1552	1	1	yes	characteristic
242	C₂×C₃	#1989	1	1	yes	characteristic
243	C₂×C₃	#1991	1	1	yes	characteristic
244	C₂×C₃	#1997	1	1	yes	characteristic
245	C₂²	#4929	1	1	yes	characteristic
246	C₂²	#4966	1	1	yes	characteristic
247	C₂²	#5002	1	1	yes	characteristic
248	C₂²	#5113	1	1	yes	characteristic
249	C₂²	#5113	1	1	yes	characteristic
250	C₂²	C₂×#823	1	1	yes	characteristic
251	C₂²	C₂×#824	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	24	2	no	—
258	C₄	—	24	2	no	—
259	C₄	#4683	1	2	yes	—
260	C₄	#4697	1	2	yes	—
261	C₄	#5009	1	2	yes	—
262	C₄	#5022	1	2	yes	—
263	C₃	#9901	1	1	yes	characteristic, complemented, Sylow
264	C₂	#69491	1	1	yes	characteristic
265	C₂	#69778	1	1	yes	characteristic
266	C₂	#69879	1	1	yes	characteristic
267	C₂	#70433	1	1	yes	characteristic
268	C₂	#70439	1	1	yes	characteristic
269	C₂	#70773	1	1	yes	characteristic
270	C₂	#70777	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₃)⋊C₄
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊C₈	D₃
hypocenter	C₃	#9901
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
263. C₃	#9901	semidirect

Maximal subgroups

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

subgroup	index
#73307	2
C₃×C₄×C₄⋊C₈	2
#69392	2
#67702	2
#9901	3