Subgroups — #10694428

Order 768.

518 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4699	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#4482	C₂	1	2	yes	maximal
4	C₂×#4458	C₂	1	1	yes	maximal, characteristic
5	#69490	C₂	1	2	yes	maximal
6	#9733	—	3	1	no	maximal, Sylow
7	C₂²×Dic₁₂	—	2	1	no	—
8	C₂²×Dic₁₂	—	2	1	no	—
9	C₂×#869	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	—	2	1	no	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	#4699	—	2	2	no	—
16	#4699	C₄	1	4	yes	complemented
17	C₃×#336	—	2	1	no	—
18	#4529	—	2	4	no	—
19	#4482	—	2	4	no	—
20	#4458	—	2	2	no	—
21	C₂×#494	—	3	1	no	—
22	#1988	—	3	2	no	—
23	C₈×C₄⋊C₄	D₃	1	1	yes	characteristic
24	C₂×#335	—	3	1	no	—
25	C₂×#332	—	3	2	no	—
26	C₂²×Dic₆	—	2	2	no	—
27	C₂²×C₃×C₈	—	2	1	no	—
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄⋊C₄	D₄	1	2	yes	—
31	C₂×C₃×C₄²	—	2	1	no	—
32	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₃	—	2	2	no	—
34	C₂×C₄⋊Dic₃	—	2	2	no	—
35	C₂²×C₃⋊C₈	—	2	2	no	—
36	C₂×Dic₁₂	—	2	4	no	—
37	C₂×Dic₁₂	—	2	4	no	—
38	C₂×Dic₁₂	—	4	2	no	—
39	C₂×Dic₁₂	—	4	2	no	—
40	C₂×Dic₁₂	—	4	2	no	—
41	C₂×Dic₁₂	—	4	2	no	—
42	#869	—	2	2	no	—
43	#869	—	2	2	no	—
44	#869	—	2	2	no	—
45	#869	C₂×C₄	1	4	yes	—
46	C₃×C₄⋊C₈	—	2	1	no	—
47	C₃×C₄⋊C₈	D₄	1	2	yes	—
48	C₃×C₄×C₈	—	4	2	no	—
49	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
50	C₃×C₄×C₈	D₄	1	2	yes	—
51	C₃×#95	—	2	1	no	—
52	C₄⋊(C₃⋊C₈)	—	2	2	no	—
53	C₄⋊(C₃⋊C₈)	—	2	2	no	—
54	C₂²×Q₁₆	—	6	1	no	—
55	C₂²×Q₁₆	—	6	1	no	—
56	C₂×C₄⋊Q₈	—	3	2	no	—
57	#494	—	3	4	no	—
58	#494	—	6	2	no	—
59	C₂×C₄⋊C₈	—	3	2	no	—
60	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
61	C₂×C₄×C₈	—	2	1	no	—
62	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
63	C₄×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
64	#340	—	6	4	no	—
65	#336	—	2	1	no	—
66	#335	—	6	2	no	—
67	#332	—	6	4	no	—
68	C₂²×C₃×C₄	—	2	2	no	—
69	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	#41	1	1	yes	characteristic
72	C₂²×Dic₃	—	4	2	no	—
73	C₂×Dic₆	—	2	4	no	—
74	C₂×Dic₆	—	4	2	no	—
75	C₂×Dic₆	—	4	2	no	—
76	C₂×Dic₆	—	4	2	no	—
77	C₂×Dic₆	—	4	2	no	—
78	C₂×Dic₆	—	4	2	no	—
79	C₂×C₃×C₈	—	2	1	no	—
80	C₂×C₃×C₈	—	2	1	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	4	1	no	—
83	C₂×C₃×C₈	—	4	1	no	—
84	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×C₈	D₈	1	2	yes	—
87	C₂×C₃×C₈	SD₁₆	1	2	yes	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄⋊C₄	Q₁₆	1	4	yes	complemented
90	C₃×C₄²	—	4	1	no	—
91	C₃×C₄²	—	4	1	no	—
92	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
93	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
94	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
95	C₃×C₄²	#41	1	1	yes	characteristic
96	C₄⋊Dic₃	—	2	4	no	—
97	C₄⋊Dic₃	—	2	4	no	—
98	C₄⋊Dic₃	—	4	2	no	—
99	C₄⋊Dic₃	—	4	2	no	—
100	C₂×C₃⋊C₈	—	4	2	no	—
101	C₂×C₃⋊C₈	—	4	4	no	—
102	Dic₁₂	—	4	8	no	—
103	Dic₁₂	—	4	8	no	—
104	C₂²×Q₈	—	6	2	no	—
105	C₂×Q₁₆	—	6	4	no	—
106	C₂×Q₁₆	—	6	4	no	—
107	C₂×Q₁₆	—	12	2	no	—
108	C₂×Q₁₆	—	12	2	no	—
109	C₂×Q₁₆	—	12	2	no	—
110	C₂×Q₁₆	—	12	2	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	6	2	no	—
113	C₂²×C₈	#67	1	1	yes	characteristic
114	C₂²×C₈	#67	1	1	yes	characteristic
115	C₄⋊Q₈	—	3	4	no	—
116	C₄⋊Q₈	—	6	2	no	—
117	C₄⋊Q₈	—	6	2	no	—
118	C₄⋊Q₈	—	6	2	no	—
119	C₂×C₄⋊C₄	—	6	2	no	—
120	C₂×C₄⋊C₄	—	6	2	no	—
121	C₂×C₄⋊C₄	D₁₂	1	2	yes	—
122	C₂×C₄²	—	2	1	no	—
123	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
124	C₄⋊C₈	—	2	1	no	—
125	C₄⋊C₈	—	6	2	no	—
126	C₄⋊C₈	—	6	2	no	—
127	C₄⋊C₈	D₁₂	1	2	yes	—
128	C₄×C₈	—	4	2	no	—
129	C₄×C₈	C₄×D₃	1	2	yes	—
130	C₄×C₈	D₁₂	1	2	yes	—
131	#95	—	2	1	no	—
132	C₂³×C₃	#121	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	#121	1	1	yes	characteristic
141	C₂×C₃×C₄	#124	1	1	yes	characteristic
142	C₂×C₃×C₄	#127	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
146	C₂×C₃×C₄	#135	1	2	yes	—
147	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×Dic₃	—	4	4	no	—
150	C₂×Dic₃	—	8	2	no	—
151	C₂×Dic₃	—	8	2	no	—
152	Dic₆	—	4	4	no	—
153	Dic₆	—	4	4	no	—
154	Dic₆	—	8	2	no	—
155	Dic₆	—	8	2	no	—
156	C₃×C₈	—	2	2	no	—
157	C₃×C₈	—	4	2	no	—
158	C₃×C₈	#102	1	4	yes	—
159	C₃⋊C₈	—	8	2	no	—
160	C₂×Q₈	—	6	4	no	—
161	C₂×Q₈	—	12	2	no	—
162	C₂×Q₈	—	12	2	no	—
163	C₂×Q₈	—	12	2	no	—
164	C₂×Q₈	—	12	2	no	—
165	C₂×Q₈	—	12	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	12	2	no	—
168	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
169	C₂²×C₄	#235	1	1	yes	characteristic
170	C₂²×C₄	C₂×#67	1	1	yes	characteristic
171	Q₁₆	—	12	8	no	—
172	Q₁₆	—	12	8	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	12	2	no	—
179	C₂×C₈	—	12	4	no	—
180	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
181	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
182	C₂×C₈	#213	1	2	yes	—
183	C₂×C₈	#215	1	2	yes	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	6	4	no	—
186	C₄⋊C₄	—	6	4	no	—
187	C₄⋊C₄	—	12	2	no	—
188	C₄⋊C₄	—	12	2	no	—
189	C₄⋊C₄	Dic₁₂	1	4	yes	complemented
190	C₄²	—	4	1	no	—
191	C₄²	—	4	1	no	—
192	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
193	C₄²	C₂×D₁₂	1	1	yes	characteristic
194	C₄²	C₂×D₁₂	1	1	yes	characteristic
195	C₄²	#235	1	1	yes	characteristic
196	C₂²×C₃	#390	1	1	yes	characteristic
197	C₂²×C₃	#459	1	1	yes	characteristic
198	C₂²×C₃	#460	1	1	yes	characteristic
199	C₂²×C₃	#467	1	2	yes	—
200	C₂²×C₃	#473	1	2	yes	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	4	4	no	—
204	C₃×C₄	C₂×#102	1	1	yes	characteristic
205	C₃×C₄	#418	1	1	yes	characteristic
206	C₃×C₄	C₄×Q₁₆	1	2	yes	—
207	C₃×C₄	#459	1	1	yes	characteristic
208	C₃×C₄	#460	1	1	yes	characteristic
209	C₃×C₄	#494	1	1	yes	characteristic
210	C₃×C₄	#495	1	1	yes	characteristic
211	Dic₃	—	8	4	no	—
212	C₂³	#930	1	1	yes	characteristic
213	Q₈	—	12	4	no	—
214	Q₈	—	12	4	no	—
215	Q₈	—	24	2	no	—
216	Q₈	—	24	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	12	4	no	—
222	C₂×C₄	—	24	2	no	—
223	C₂×C₄	—	24	2	no	—
224	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
225	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
226	C₂×C₄	#874	1	1	yes	characteristic
227	C₂×C₄	#875	1	1	yes	characteristic
228	C₂×C₄	#904	1	2	yes	—
229	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
230	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	#930	1	1	yes	characteristic
232	C₂×C₄	C₂×#213	1	1	yes	characteristic
233	C₂×C₄	C₂×#215	1	1	yes	characteristic
234	C₂×C₄	#949	1	1	yes	characteristic
235	C₂×C₄	#949	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	4	2	no	—
238	C₈	—	24	2	no	—
239	C₈	#809	1	4	yes	—
240	C₂×C₃	#1669	1	2	yes	—
241	C₂×C₃	#1670	1	2	yes	—
242	C₂×C₃	#1969	1	1	yes	characteristic
243	C₂×C₃	#1979	1	1	yes	characteristic
244	C₂×C₃	#1982	1	1	yes	characteristic
245	C₂²	#4939	1	1	yes	characteristic
246	C₂²	#5015	1	1	yes	characteristic
247	C₂²	#5027	1	1	yes	characteristic
248	C₂²	#5116	1	2	yes	—
249	C₂²	#5143	1	2	yes	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	24	4	no	—
254	C₄	#4696	1	1	yes	characteristic
255	C₄	C₄×Dic₁₂	1	2	yes	—
256	C₄	#4699	1	1	yes	characteristic
257	C₄	C₂×#809	1	1	yes	characteristic
258	C₄	#5001	1	1	yes	characteristic
259	C₄	#5015	1	1	yes	characteristic
260	C₄	#5027	1	1	yes	characteristic
261	C₃	#9733	1	1	yes	characteristic, complemented, Sylow
262	C₂	#69487	1	1	yes	characteristic
263	C₂	#69775	1	1	yes	characteristic
264	C₂	#69904	1	1	yes	characteristic
265	C₂	#70850	1	2	yes	—
266	C₂	#70942	1	2	yes	—

	subgroup	G/H
center	C₂³	#930
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₈×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	#121
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₈×C₄⋊C₄	D₃
hypocenter	C₃	#9733
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
16. #4699	C₄	semidirect
89. C₃×C₄⋊C₄	Q₁₆	semidirect
189. C₄⋊C₄	Dic₁₂	semidirect
261. C₃	#9733	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4699	2
C₃×C₈×C₄⋊C₄	2
C₂×#4482	2
C₂×#4458	2
#69490	2
#9733	3