Subgroups — #10693729

Order 768.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈²	C₂	1	1	yes	maximal, characteristic
2	#69786	C₂	1	2	yes	maximal
3	C₄×#818	C₂	1	1	yes	maximal, characteristic
4	C₂×#4458	C₂	1	1	yes	maximal, characteristic
5	#67655	C₂	1	2	yes	maximal
6	#9594	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#818	C₄	1	1	yes	characteristic
11	C₂×#818	C₄	1	1	yes	characteristic
12	C₂×Dic₃⋊C₈	—	2	2	no	—
13	C₄×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₃×C₈²	—	2	2	no	—
16	#4551	C₄	1	2	yes	—
17	#4550	—	2	2	no	—
18	#4458	—	2	2	no	—
19	#1939	—	3	2	no	—
20	C₄×#107	—	3	1	no	—
21	C₂×#335	—	3	1	no	—
22	C₂×C₈²	D₃	1	1	yes	characteristic
23	#1279	—	3	2	no	—
24	C₂²×C₃×C₈	—	2	1	no	—
25	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₃	—	2	2	no	—
33	C₂²×C₃⋊C₈	—	2	2	no	—
34	C₃×C₄×C₈	—	2	1	no	—
35	C₃×C₄×C₈	—	2	1	no	—
36	C₃×C₄×C₈	—	2	1	no	—
37	C₃×C₄×C₈	D₄	1	2	yes	—
38	C₃×C₄×C₈	Q₈	1	2	yes	—
39	C₃×C₄×C₈	Q₈	1	2	yes	—
40	C₃⋊#95	—	2	2	no	—
41	#818	—	2	2	no	—
42	#818	C₈	1	4	yes	complemented
43	Dic₃⋊C₈	—	4	4	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₃⋊C₈)	—	2	2	no	—
46	C₂×#107	—	3	1	no	—
47	C₂×#107	—	3	1	no	—
48	C₂×C₄⋊C₈	—	3	2	no	—
49	C₂×C₄⋊C₈	—	6	2	no	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	C₄×C₄⋊C₄	—	3	2	no	—
54	#340	—	3	2	no	—
55	#336	—	6	2	no	—
56	#335	—	6	2	no	—
57	C₈²	—	2	2	no	—
58	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	#41	1	1	yes	characteristic
61	C₂²×Dic₃	—	2	2	no	—
62	C₂²×Dic₃	—	2	2	no	—
63	C₂×C₃×C₈	—	2	2	no	—
64	C₂×C₃×C₈	—	2	2	no	—
65	C₂×C₃×C₈	—	2	1	no	—
66	C₂×C₃×C₈	—	2	1	no	—
67	C₂×C₃×C₈	—	2	2	no	—
68	C₂×C₃×C₈	—	2	1	no	—
69	C₂×C₃×C₈	—	2	1	no	—
70	C₂×C₃×C₈	—	2	1	no	—
71	C₂×C₃×C₈	—	2	1	no	—
72	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
74	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
75	C₂×C₃×C₈	M₁₆	1	2	yes	—
76	C₂×C₃×C₈	D₈	1	2	yes	—
77	C₂×C₃×C₈	Q₁₆	1	2	yes	—
78	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
80	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
81	C₃×C₄²	#41	1	1	yes	characteristic
82	C₄⋊Dic₃	—	2	2	no	—
83	C₄⋊Dic₃	—	2	2	no	—
84	C₄⋊Dic₃	—	2	2	no	—
85	C₄⋊Dic₃	C₂×C₈	1	4	yes	—
86	C₄×Dic₃	—	4	2	no	—
87	C₄×Dic₃	—	4	2	no	—
88	C₂×C₃⋊C₈	—	4	2	no	—
89	C₂×C₃⋊C₈	—	4	2	no	—
90	C₂×C₃⋊C₈	—	4	2	no	—
91	C₂²×C₈	—	2	1	no	—
92	C₂²×C₈	—	6	2	no	—
93	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₈	D₁₂	1	1	yes	characteristic
96	C₂²×C₈	D₁₂	1	1	yes	characteristic
97	C₂×C₄⋊C₄	—	3	2	no	—
98	C₂×C₄⋊C₄	—	3	2	no	—
99	C₂×C₄²	—	6	2	no	—
100	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
101	#107	—	3	4	no	—
102	#107	—	6	2	no	—
103	C₄⋊C₈	—	6	2	no	—
104	C₄⋊C₈	—	6	2	no	—
105	C₄⋊C₈	—	12	4	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	Dic₆	1	2	yes	—
110	C₄×C₈	Dic₆	1	2	yes	—
111	C₄×C₈	#67	1	2	yes	—
112	#95	—	6	2	no	—
113	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
118	C₂×C₃×C₄	#121	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₃×C₄	#125	1	1	yes	characteristic
121	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₃×C₄	#131	1	1	yes	characteristic
125	C₂×C₃×C₄	#131	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×C₃×C₄	#135	1	1	yes	characteristic
129	C₂×C₃×C₄	#135	1	1	yes	characteristic
130	C₂×Dic₃	—	2	4	no	—
131	C₂×Dic₃	—	4	2	no	—
132	C₂×Dic₃	—	4	2	no	—
133	C₂×Dic₃	—	4	2	no	—
134	C₂×Dic₃	—	4	2	no	—
135	C₂×Dic₃	—	4	2	no	—
136	C₃×C₈	—	2	4	no	—
137	C₃×C₈	—	2	2	no	—
138	C₃×C₈	—	2	2	no	—
139	C₃×C₈	—	2	2	no	—
140	C₃×C₈	C₄⋊C₈	1	4	yes	complemented
141	C₃⋊C₈	—	8	2	no	—
142	C₂²×C₄	—	6	2	no	—
143	C₂²×C₄	—	6	2	no	—
144	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
146	C₂²×C₄	#235	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	12	2	no	—
157	C₂×C₈	—	12	2	no	—
158	C₂×C₈	—	12	2	no	—
159	C₂×C₈	D₃×C₈	1	2	yes	—
160	C₂×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
161	C₂×C₈	D₂₄	1	2	yes	—
162	C₂×C₈	Dic₁₂	1	2	yes	—
163	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
164	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
165	C₄⋊C₄	—	3	4	no	—
166	C₄⋊C₄	—	6	2	no	—
167	C₄⋊C₄	—	6	2	no	—
168	C₄⋊C₄	—	6	2	no	—
169	C₄²	—	12	2	no	—
170	C₄²	—	12	2	no	—
171	C₄²	C₂×Dic₆	1	1	yes	characteristic
172	C₄²	C₂×Dic₆	1	1	yes	characteristic
173	C₄²	#235	1	1	yes	characteristic
174	C₄²	C₂×#67	1	1	yes	characteristic
175	C₂²×C₃	#389	1	1	yes	characteristic
176	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
177	C₂²×C₃	#436	1	1	yes	characteristic
178	C₂²×C₃	C₄×D₈	1	1	yes	characteristic
179	C₂²×C₃	C₄×Q₁₆	1	1	yes	characteristic
180	C₂²×C₃	#443	1	1	yes	characteristic
181	C₂²×C₃	#443	1	1	yes	characteristic
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
185	C₃×C₄	#424	1	1	yes	characteristic
186	C₃×C₄	C₈×Q₈	1	1	yes	characteristic
187	C₃×C₄	#446	1	1	yes	characteristic
188	C₃×C₄	#466	1	1	yes	characteristic
189	C₃×C₄	#467	1	1	yes	characteristic
190	C₃×C₄	#499	1	1	yes	characteristic
191	C₃×C₄	#500	1	1	yes	characteristic
192	Dic₃	—	4	4	no	—
193	Dic₃	—	8	2	no	—
194	C₂³	C₄×D₁₂	1	1	yes	characteristic
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	6	4	no	—
198	C₂×C₄	—	12	2	no	—
199	C₂×C₄	—	12	2	no	—
200	C₂×C₄	—	12	2	no	—
201	C₂×C₄	—	12	2	no	—
202	C₂×C₄	—	12	2	no	—
203	C₂×C₄	C₄×Dic₆	1	1	yes	characteristic
204	C₂×C₄	#869	1	1	yes	characteristic
205	C₂×C₄	#870	1	1	yes	characteristic
206	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
208	C₂×C₄	#901	1	1	yes	characteristic
209	C₂×C₄	#901	1	1	yes	characteristic
210	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
211	C₂×C₄	#904	1	1	yes	characteristic
212	C₂×C₄	#904	1	1	yes	characteristic
213	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
214	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
215	C₂×C₄	#924	1	1	yes	characteristic
216	C₂×C₄	#930	1	1	yes	characteristic
217	C₈	—	2	4	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	24	2	no	—
222	C₈	Dic₃⋊C₈	1	4	yes	complemented
223	C₂×C₃	C₈×D₈	1	1	yes	characteristic
224	C₂×C₃	C₈×Q₁₆	1	1	yes	characteristic
225	C₂×C₃	#1589	1	1	yes	characteristic
226	C₂×C₃	#1591	1	1	yes	characteristic
227	C₂×C₃	#1939	1	1	yes	characteristic
228	C₂×C₃	#1942	1	1	yes	characteristic
229	C₂×C₃	#1945	1	1	yes	characteristic
230	C₂²	C₈×D₁₂	1	1	yes	characteristic
231	C₂²	#4688	1	1	yes	characteristic
232	C₂²	C₄×D₂₄	1	1	yes	characteristic
233	C₂²	C₄×Dic₁₂	1	1	yes	characteristic
234	C₂²	#4700	1	1	yes	characteristic
235	C₂²	#4700	1	1	yes	characteristic
236	C₂²	#4928	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	12	4	no	—
240	C₄	—	24	2	no	—
241	C₄	C₈×Dic₆	1	1	yes	characteristic
242	C₄	#4679	1	1	yes	characteristic
243	C₄	#4682	1	1	yes	characteristic
244	C₄	#4683	1	1	yes	characteristic
245	C₄	C₂×Dic₃⋊C₈	1	1	yes	characteristic
246	C₄	#5102	1	1	yes	characteristic
247	C₄	#5115	1	1	yes	characteristic
248	C₄	#5116	1	1	yes	characteristic
249	C₃	#9594	1	1	yes	characteristic, complemented, Sylow
250	C₂	C₈×D₂₄	1	1	yes	characteristic
251	C₂	#68052	1	1	yes	characteristic
252	C₂	C₈×Dic₁₂	1	1	yes	characteristic
253	C₂	#68056	1	1	yes	characteristic
254	C₂	#69770	1	1	yes	characteristic
255	C₂	#69786	1	1	yes	characteristic
256	C₂	#69794	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×C₈²	C₂
socle, prime radical	C₂³×C₃	C₄×D₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×C₈²	D₃
hypocenter	C₃	#9594
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
42. #818	C₈	semidirect
140. C₃×C₈	C₄⋊C₈	semidirect
222. C₈	Dic₃⋊C₈	semidirect
249. C₃	#9594	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈²	2
#69786	2
C₄×#818	2
C₂×#4458	2
#67655	2
#9594	3