Subgroups — #10695313

Order 768.

538 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2868	C₂	1	1	yes	maximal, characteristic
2	#69394	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#67634	C₂	1	4	yes	maximal
5	#9898	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₃×C₄⋊Q₈	C₄	1	2	yes	—
9	C₃×#384	—	2	1	no	—
10	C₃×#384	—	2	1	no	—
11	C₃×#384	C₄	1	2	yes	—
12	C₃×#384	C₂²	1	2	yes	—
13	C₃×C₄³	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
15	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
16	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄×C₃⋊C₈	—	2	1	no	—
19	#4524	—	2	1	no	—
20	#2868	D₃	1	1	yes	characteristic
21	#1842	—	3	1	no	—
22	C₄×C₄⋊C₈	—	3	1	no	—
23	#1296	—	3	4	no	—
24	C₂²×C₃×Q₈	—	2	2	no	—
25	C₃×C₄⋊Q₈	—	4	4	no	—
26	C₃×C₄⋊Q₈	—	4	2	no	—
27	C₃×C₄⋊Q₈	—	4	2	no	—
28	C₃×C₄⋊Q₈	—	4	2	no	—
29	C₃×C₄⋊Q₈	—	4	2	no	—
30	C₂×C₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
43	C₂²×C₃⋊C₈	—	2	2	no	—
44	C₂²×C₃⋊C₈	—	2	2	no	—
45	C₄⋊(C₃⋊C₈)	—	2	2	no	—
46	C₄⋊(C₃⋊C₈)	—	2	4	no	—
47	C₄⋊(C₃⋊C₈)	—	4	2	no	—
48	C₄⋊(C₃⋊C₈)	—	4	2	no	—
49	C₄⋊(C₃⋊C₈)	Q₈	1	4	yes	complemented
50	C₄×C₃⋊C₈	—	4	2	no	—
51	C₂×C₄⋊Q₈	—	2	1	no	—
52	C₂×C₄⋊Q₈	—	2	1	no	—
53	C₂×C₄⋊Q₈	Dic₃	1	2	yes	—
54	C₂×C₄⋊C₈	—	3	2	no	—
55	C₂×C₄⋊C₈	—	3	2	no	—
56	C₂×C₄⋊C₈	—	6	1	no	—
57	C₂×C₄⋊C₈	—	6	1	no	—
58	C₂×C₄×C₈	—	6	1	no	—
59	#384	—	2	1	no	—
60	#384	—	2	1	no	—
61	#384	Dic₃	1	2	yes	—
62	#384	C₂×D₃	1	2	yes	—
63	C₄³	C₂×D₃	1	1	yes	characteristic
64	#336	—	6	1	no	—
65	C₂×C₃×Q₈	—	4	4	no	—
66	C₂×C₃×Q₈	—	4	4	no	—
67	C₂×C₃×Q₈	—	4	4	no	—
68	C₂²×C₃×C₄	—	2	4	no	—
69	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄⋊C₄	—	4	4	no	—
77	C₃×C₄⋊C₄	—	4	4	no	—
78	C₃×C₄⋊C₄	—	4	4	no	—
79	C₃×C₄⋊C₄	—	4	4	no	—
80	C₃×C₄⋊C₄	—	4	2	no	—
81	C₃×C₄⋊C₄	—	4	2	no	—
82	C₃×C₄⋊C₄	—	4	2	no	—
83	C₃×C₄⋊C₄	—	4	2	no	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	—	2	2	no	—
86	C₃×C₄²	—	2	2	no	—
87	C₃×C₄²	D₈	1	2	yes	—
88	C₃×C₄²	D₈	1	2	yes	—
89	C₃×C₄²	SD₁₆	1	2	yes	—
90	C₃×C₄²	SD₁₆	1	2	yes	—
91	C₃×C₄²	SD₁₆	1	2	yes	—
92	C₃×C₄²	Q₁₆	1	2	yes	—
93	C₃×C₄²	C₂×Q₈	1	2	yes	—
94	C₃×C₄²	#41	1	2	yes	—
95	C₂×C₃⋊C₈	—	2	4	no	—
96	C₂×C₃⋊C₈	—	4	2	no	—
97	C₂×C₃⋊C₈	—	4	2	no	—
98	C₂×C₃⋊C₈	—	4	2	no	—
99	C₂×C₃⋊C₈	—	4	2	no	—
100	C₂×C₃⋊C₈	—	4	2	no	—
101	C₂²×Q₈	—	2	2	no	—
102	C₂²×C₈	—	6	2	no	—
103	C₂²×C₈	—	6	2	no	—
104	C₄⋊Q₈	—	4	4	no	—
105	C₄⋊Q₈	—	4	2	no	—
106	C₄⋊Q₈	—	4	2	no	—
107	C₄⋊Q₈	—	4	2	no	—
108	C₄⋊Q₈	—	4	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
115	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
117	C₂×C₄²	#67	1	1	yes	characteristic
118	C₂×C₄²	#67	1	1	yes	characteristic
119	C₂×C₄²	#67	1	1	yes	characteristic
120	C₂×C₄²	#67	1	1	yes	characteristic
121	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
122	C₄⋊C₈	—	3	4	no	—
123	C₄⋊C₈	—	6	2	no	—
124	C₄⋊C₈	—	6	4	no	—
125	C₄⋊C₈	—	12	2	no	—
126	C₄⋊C₈	—	12	2	no	—
127	C₄×C₈	—	12	2	no	—
128	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₃×Q₈	—	8	8	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	—	4	4	no	—
137	C₂×C₃×C₄	—	4	4	no	—
138	C₂×C₃×C₄	—	4	4	no	—
139	C₂×C₃×C₄	#100	1	2	yes	—
140	C₂×C₃×C₄	#101	1	2	yes	—
141	C₂×C₃×C₄	#102	1	2	yes	—
142	C₂×C₃×C₄	#102	1	2	yes	—
143	C₂×C₃×C₄	#102	1	2	yes	—
144	C₂×C₃×C₄	#102	1	2	yes	—
145	C₂×C₃×C₄	#103	1	2	yes	—
146	C₂×C₃×C₄	#103	1	2	yes	—
147	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
148	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
149	C₂×C₃×C₄	#124	1	1	yes	characteristic
150	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
157	C₂×C₃×C₄	#137	1	2	yes	—
158	C₃⋊C₈	—	4	4	no	—
159	C₃⋊C₈	—	8	2	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂×Q₈	—	4	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	#217	1	1	yes	characteristic
165	C₂²×C₄	#217	1	1	yes	characteristic
166	C₂²×C₄	#217	1	1	yes	characteristic
167	C₂²×C₄	#217	1	1	yes	characteristic
168	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×#67	1	1	yes	characteristic
170	C₂²×C₄	C₂×#67	1	1	yes	characteristic
171	C₂×C₈	—	6	4	no	—
172	C₂×C₈	—	12	2	no	—
173	C₂×C₈	—	12	2	no	—
174	C₂×C₈	—	12	2	no	—
175	C₂×C₈	—	12	2	no	—
176	C₂×C₈	—	12	2	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	#213	1	2	yes	—
189	C₄²	#213	1	2	yes	—
190	C₄²	#214	1	2	yes	—
191	C₄²	#214	1	2	yes	—
192	C₄²	#215	1	2	yes	—
193	C₄²	#216	1	2	yes	—
194	C₄²	D₃×Q₈	1	2	yes	—
195	C₄²	#239	1	2	yes	—
196	C₂²×C₃	#391	1	1	yes	characteristic
197	C₂²×C₃	C₂×#100	1	1	yes	characteristic
198	C₂²×C₃	C₂×#101	1	1	yes	characteristic
199	C₂²×C₃	C₂×#102	1	1	yes	characteristic
200	C₂²×C₃	C₂×#102	1	1	yes	characteristic
201	C₂²×C₃	C₂×#103	1	1	yes	characteristic
202	C₂²×C₃	#419	1	1	yes	characteristic
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	8	4	no	—
207	C₃×C₄	#331	1	4	yes	—
208	C₃×C₄	#332	1	4	yes	—
209	C₃×C₄	#474	1	2	yes	—
210	C₃×C₄	#475	1	2	yes	—
211	C₃×C₄	#486	1	1	yes	characteristic
212	C₃×C₄	#487	1	1	yes	characteristic
213	C₃×C₄	#498	1	1	yes	characteristic
214	C₃×C₄	#500	1	1	yes	characteristic
215	C₂³	C₂×#217	1	1	yes	characteristic
216	Q₈	—	8	8	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	#832	1	2	yes	—
227	C₂×C₄	#832	1	2	yes	—
228	C₂×C₄	#832	1	2	yes	—
229	C₂×C₄	#832	1	2	yes	—
230	C₂×C₄	#833	1	2	yes	—
231	C₂×C₄	#835	1	2	yes	—
232	C₂×C₄	#835	1	2	yes	—
233	C₂×C₄	#836	1	2	yes	—
234	C₂×C₄	C₂×#213	1	1	yes	characteristic
235	C₂×C₄	C₂×#213	1	1	yes	characteristic
236	C₂×C₄	C₂×#214	1	1	yes	characteristic
237	C₂×C₄	C₂×#214	1	1	yes	characteristic
238	C₂×C₄	C₂×#215	1	1	yes	characteristic
239	C₂×C₄	#942	1	2	yes	—
240	C₂×C₄	C₂×#216	1	1	yes	characteristic
241	C₂×C₄	#944	1	1	yes	characteristic
242	C₂×C₄	Q₈×Dic₃	1	2	yes	—
243	C₂×C₄	#946	1	2	yes	—
244	C₂×C₄	#947	1	1	yes	characteristic
245	C₈	—	12	4	no	—
246	C₈	—	24	2	no	—
247	C₂×C₃	C₂×#331	1	1	yes	characteristic
248	C₂×C₃	C₂×#332	1	1	yes	characteristic
249	C₂×C₃	#1546	1	1	yes	characteristic
250	C₂×C₃	#1548	1	1	yes	characteristic
251	C₂×C₃	#1988	1	1	yes	characteristic
252	C₂×C₃	#1990	1	1	yes	characteristic
253	C₂×C₃	#1997	1	1	yes	characteristic
254	C₂²	C₂×#832	1	1	yes	characteristic
255	C₂²	C₂×#832	1	1	yes	characteristic
256	C₂²	C₂×#833	1	1	yes	characteristic
257	C₂²	C₂×#835	1	1	yes	characteristic
258	C₂²	#5225	1	1	yes	characteristic
259	C₂²	C₂×#836	1	1	yes	characteristic
260	C₂²	#5229	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	8	4	no	—
265	C₄	#4544	1	4	yes	—
266	C₄	#4547	1	4	yes	—
267	C₄	#5078	1	1	yes	characteristic
268	C₄	#5080	1	1	yes	characteristic
269	C₄	#5087	1	2	yes	—
270	C₄	#5088	1	1	yes	characteristic
271	C₄	#5093	1	1	yes	characteristic
272	C₄	#5094	1	2	yes	—
273	C₃	#9898	1	1	yes	characteristic, complemented, Sylow
274	C₂	C₂×#4544	1	1	yes	characteristic
275	C₂	C₂×#4547	1	1	yes	characteristic
276	C₂	#71773	1	1	yes	characteristic
277	C₂	#71774	1	1	yes	characteristic
278	C₂	#71780	1	1	yes	characteristic
279	C₂	#71787	1	1	yes	characteristic
280	C₂	#71790	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#217
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#2868	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#2868	D₃
hypocenter	C₃	#9898
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
49. C₄⋊(C₃⋊C₈)	Q₈	semidirect
273. C₃	#9898	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2868	2
#69394	2
C₄×C₄⋊(C₃⋊C₈)	2
#67634	2
#9898	3