Subgroups — #10686159

Order 768.

528 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#4481	C₂	1	2	yes	maximal
4	#69391	C₂	1	1	yes	maximal, characteristic
5	#67702	C₂	1	2	yes	maximal
6	#8235	—	3	1	no	maximal, Sylow
7	C₂×#869	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₆	C₄	1	2	yes	—
9	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	—	2	1	no	—
11	C₃×C₄³	C₂²	1	1	yes	characteristic
12	#4935	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₃	—	2	1	no	—
14	C₄×C₄⋊Dic₃	—	2	1	no	—
15	#4929	C₄	1	2	yes	—
16	#4928	—	2	1	no	—
17	#4928	—	2	1	no	—
18	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₂×#803	—	2	1	no	—
20	C₃×#336	—	2	1	no	—
21	#4524	—	2	1	no	—
22	#4481	C₄	1	8	yes	complemented
23	C₄×C₄⋊Q₈	—	3	1	no	—
24	#1767	—	3	1	no	—
25	C₄×C₄⋊C₈	D₃	1	1	yes	characteristic
26	C₂×#333	—	3	2	no	—
27	#1296	—	3	2	no	—
28	C₂²×Dic₆	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₈	—	2	2	no	—
30	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄²	C₂×C₄	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	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₃	—	2	1	no	—
38	C₂×C₄⋊Dic₃	—	2	1	no	—
39	C₂×C₄⋊Dic₃	—	2	1	no	—
40	C₂×C₄⋊Dic₃	—	2	1	no	—
41	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
42	C₂×Dic₃⋊C₄	—	2	2	no	—
43	C₂×C₄×Dic₃	—	2	2	no	—
44	C₂²×C₃⋊C₈	—	2	2	no	—
45	#869	—	2	2	no	—
46	#869	C₂×C₄	1	4	yes	—
47	C₄×Dic₆	—	4	4	no	—
48	C₃×C₄⋊C₈	—	2	2	no	—
49	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₃×C₄×C₈	—	4	2	no	—
51	C₃⋊#95	—	2	2	no	—
52	C₃⋊#95	—	2	2	no	—
53	C₄⋊(C₃⋊C₈)	—	2	2	no	—
54	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
55	#803	—	4	2	no	—
56	C₂×C₄⋊Q₈	—	3	1	no	—
57	C₂×C₄×Q₈	—	3	2	no	—
58	C₂×C₄⋊C₈	—	3	2	no	—
59	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
60	C₂×#97	—	6	1	no	—
61	C₂×C₄×C₈	—	2	1	no	—
62	#391	—	3	2	no	—
63	#389	—	6	1	no	—
64	#389	—	6	1	no	—
65	#384	—	3	1	no	—
66	C₄×C₄⋊C₄	—	6	1	no	—
67	C₄×C₄⋊C₄	—	6	1	no	—
68	C₄³	C₂×D₃	1	1	yes	characteristic
69	#336	—	2	1	no	—
70	#336	—	6	1	no	—
71	#333	—	3	8	no	—
72	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₃	—	2	2	no	—
80	C₂²×Dic₃	—	2	2	no	—
81	C₂×Dic₆	—	2	2	no	—
82	C₂×Dic₆	—	4	2	no	—
83	C₂×Dic₆	—	4	2	no	—
84	C₂×Dic₆	C₄²	1	4	yes	—
85	C₂×C₃×C₈	—	2	4	no	—
86	C₂×C₃×C₈	—	4	2	no	—
87	C₂×C₃×C₈	—	4	2	no	—
88	C₃×C₄²	—	2	2	no	—
89	C₃×C₄²	—	2	2	no	—
90	C₃×C₄²	—	2	1	no	—
91	C₃×C₄²	—	2	1	no	—
92	C₃×C₄²	—	2	2	no	—
93	C₃×C₄²	—	4	1	no	—
94	C₃×C₄²	SD₁₆	1	2	yes	—
95	C₃×C₄²	Q₁₆	1	2	yes	—
96	C₃×C₄²	C₂²×C₄	1	2	yes	—
97	C₃×C₄²	#41	1	2	yes	—
98	C₄⋊Dic₃	—	2	4	no	—
99	C₄⋊Dic₃	—	2	4	no	—
100	C₄⋊Dic₃	—	4	2	no	—
101	C₄⋊Dic₃	—	4	1	no	—
102	C₄⋊Dic₃	—	4	1	no	—
103	C₄⋊Dic₃	—	4	2	no	—
104	Dic₃⋊C₄	—	4	4	no	—
105	C₄×Dic₃	—	4	4	no	—
106	C₂×C₃⋊C₈	—	2	4	no	—
107	C₂×C₃⋊C₈	—	4	2	no	—
108	C₂×C₃⋊C₈	—	4	2	no	—
109	C₂²×Q₈	—	3	2	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	6	2	no	—
112	C₄⋊Q₈	—	3	4	no	—
113	C₄⋊Q₈	—	6	2	no	—
114	C₄×Q₈	—	12	4	no	—
115	C₂×C₄⋊C₄	—	3	2	no	—
116	C₂×C₄⋊C₄	—	6	1	no	—
117	C₂×C₄⋊C₄	—	6	1	no	—
118	C₂×C₄⋊C₄	—	6	2	no	—
119	C₂×C₄⋊C₄	—	6	1	no	—
120	C₂×C₄⋊C₄	—	6	1	no	—
121	C₂×C₄²	—	6	2	no	—
122	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
123	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
124	C₂×C₄²	D₁₂	1	1	yes	characteristic
125	C₂×C₄²	D₁₂	1	1	yes	characteristic
126	C₂×C₄²	#67	1	1	yes	characteristic
127	C₂×C₄²	#67	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	3	4	no	—
131	C₄⋊C₈	—	6	2	no	—
132	C₄⋊C₈	C₄×D₃	1	4	yes	—
133	#97	—	12	2	no	—
134	C₄×C₈	—	4	2	no	—
135	#95	—	6	2	no	—
136	#95	—	6	2	no	—
137	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	#100	1	2	yes	—
150	C₂×C₃×C₄	#103	1	2	yes	—
151	C₂×C₃×C₄	#103	1	2	yes	—
152	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
153	C₂×C₃×C₄	#117	1	1	yes	characteristic
154	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
155	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
156	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
158	C₂×C₃×C₄	#135	1	2	yes	—
159	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂×C₃×C₄	#137	1	1	yes	characteristic
161	C₂×C₃×C₄	#137	1	2	yes	—
162	C₂×Dic₃	—	2	4	no	—
163	C₂×Dic₃	—	4	2	no	—
164	C₂×Dic₃	—	4	2	no	—
165	C₂×Dic₃	—	4	2	no	—
166	C₂×Dic₃	—	4	2	no	—
167	C₂×Dic₃	—	4	2	no	—
168	Dic₆	—	4	4	no	—
169	Dic₆	—	4	4	no	—
170	C₃×C₈	—	4	4	no	—
171	C₃⋊C₈	—	4	4	no	—
172	C₂×Q₈	—	3	4	no	—
173	C₂×Q₈	—	6	2	no	—
174	C₂×Q₈	—	12	2	no	—
175	C₂×Q₈	—	12	2	no	—
176	C₂²×C₄	—	6	2	no	—
177	C₂²×C₄	—	6	2	no	—
178	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
184	C₂²×C₄	C₂×#67	1	1	yes	characteristic
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	6	4	no	—
189	C₂×C₈	—	12	2	no	—
190	C₂×C₈	—	12	2	no	—
191	C₄⋊C₄	—	6	4	no	—
192	C₄⋊C₄	—	6	4	no	—
193	C₄⋊C₄	—	12	2	no	—
194	C₄⋊C₄	—	12	4	no	—
195	C₄⋊C₄	—	12	2	no	—
196	C₄⋊C₄	—	12	1	no	—
197	C₄⋊C₄	—	12	1	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	12	4	no	—
205	C₄²	#204	1	2	yes	—
206	C₄²	Dic₁₂	1	2	yes	—
207	C₄²	C₂×C₄×D₃	1	2	yes	—
208	C₄²	#235	1	2	yes	—
209	C₂²×C₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²×C₃	C₂×#100	1	1	yes	characteristic
211	C₂²×C₃	#413	1	1	yes	characteristic
212	C₂²×C₃	C₂×#103	1	1	yes	characteristic
213	C₂²×C₃	#416	1	1	yes	characteristic
214	C₂²×C₃	#417	1	1	yes	characteristic
215	C₂²×C₃	#419	1	1	yes	characteristic
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	4	1	no	—
222	C₃×C₄	C₄×SD₁₆	1	2	yes	—
223	C₃×C₄	C₄×Q₁₆	1	2	yes	—
224	C₃×C₄	#440	1	2	yes	—
225	C₃×C₄	#441	1	2	yes	—
226	Dic₃	—	4	4	no	—
227	Dic₃	—	8	2	no	—
228	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
229	Q₈	—	12	4	no	—
230	Q₈	—	12	4	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	6	4	no	—
243	C₂×C₄	—	12	2	no	—
244	C₂×C₄	—	12	2	no	—
245	C₂×C₄	—	12	2	no	—
246	C₂×C₄	—	12	2	no	—
247	C₂×C₄	—	12	2	no	—
248	C₂×C₄	#816	1	2	yes	—
249	C₂×C₄	#816	1	2	yes	—
250	C₂×C₄	#823	1	2	yes	—
251	C₂×C₄	C₄²×D₃	1	1	yes	characteristic
252	C₂×C₄	#872	1	1	yes	characteristic
253	C₂×C₄	C₄×D₁₂	1	2	yes	—
254	C₂×C₄	C₂×#204	1	1	yes	characteristic
255	C₂×C₄	#904	1	2	yes	—
256	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
257	C₂×C₄	C₄×#67	1	2	yes	—
258	C₂×C₄	#932	1	1	yes	characteristic
259	C₂×C₄	#942	1	1	yes	characteristic
260	C₂×C₄	#951	1	2	yes	—
261	C₈	—	4	4	no	—
262	C₈	—	12	4	no	—
263	C₂×C₃	#1548	1	1	yes	characteristic
264	C₂×C₃	#1550	1	1	yes	characteristic
265	C₂×C₃	#1551	1	1	yes	characteristic
266	C₂×C₃	#1554	1	1	yes	characteristic
267	C₂×C₃	C₄×#100	1	1	yes	characteristic
268	C₂×C₃	C₄×#103	1	1	yes	characteristic
269	C₂×C₃	#1763	1	1	yes	characteristic
270	C₂²	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
271	C₂²	#4966	1	1	yes	characteristic
272	C₂²	#5004	1	1	yes	characteristic
273	C₂²	C₂×#816	1	1	yes	characteristic
274	C₂²	#5113	1	1	yes	characteristic
275	C₂²	C₂×#823	1	1	yes	characteristic
276	C₂²	#5132	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	12	4	no	—
284	C₄	—	24	2	no	—
285	C₄	C₄×#204	1	2	yes	—
286	C₄	C₄×Dic₁₂	1	2	yes	—
287	C₄	#5018	1	2	yes	—
288	C₄	#5030	1	2	yes	—
289	C₃	#8235	1	1	yes	characteristic, complemented, Sylow
290	C₂	#69475	1	1	yes	characteristic
291	C₂	C₄×#816	1	1	yes	characteristic
292	C₂	C₄×#823	1	1	yes	characteristic
293	C₂	#70433	1	1	yes	characteristic
294	C₂	#70438	1	1	yes	characteristic
295	C₂	#70779	1	1	yes	characteristic
296	C₂	#70781	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₄²
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	#8235
LCS	G ▷ C₂×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
22. #4481	C₄	semidirect
289. C₃	#8235	semidirect

Maximal subgroups

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

subgroup	index
C₄×#869	2
C₃×C₄×C₄⋊C₈	2
C₂×#4481	2
#69391	2
#67702	2
#8235	3