Subgroups — #10948803

Order 768.

584 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2929	C₂	1	1	yes	maximal, characteristic
2	#72004	C₂	1	2	yes	maximal, complemented
3	C₂×#4549	C₂	1	2	yes	maximal, complemented
4	#70857	C₂	1	4	yes	maximal, complemented
5	D₄×C₃⋊C₁₆	C₂	1	4	yes	maximal, complemented
6	#69883	C₂	1	2	yes	maximal, complemented
7	#33491	—	3	1	no	maximal, Sylow
8	C₂×C₃×#131	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×#525	C₄	1	1	yes	characteristic
11	C₃×#435	C₄	1	4	yes	—
12	C₃×C₈×D₄	C₂²	1	4	yes	—
13	C₃×#424	C₄	1	1	yes	characteristic
14	C₃×#408	C₄	1	1	yes	characteristic
15	C₃×#407	C₄	1	1	yes	characteristic
16	C₃×#405	C₂²	1	1	yes	characteristic
17	C₂×#812	C₂²	1	2	yes	complemented
18	C₂²×C₃⋊C₁₆	C₂²	1	4	yes	—
19	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	complemented
20	#4549	C₂²	1	4	yes	—
21	#4549	C₂²	1	2	yes	—
22	#4549	C₂²	1	2	yes	—
23	#4549	C₂²	1	4	yes	complemented
24	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	complemented
25	#4461	C₂²	1	2	yes	complemented
26	C₄×C₃⋊C₁₆	C₂²	1	2	yes	complemented
27	#2929	D₃	1	1	yes	characteristic
28	#2168	—	3	4	no	—
29	D₄×C₁₆	—	3	4	no	—
30	#2164	—	3	2	no	—
31	#2113	—	3	2	no	—
32	C₂×C₂²⋊C₁₆	—	3	2	no	—
33	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
35	C₃×#131	—	4	2	no	—
36	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	—	4	2	no	—
40	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂³	1	2	yes	—
44	C₂²×C₃×C₈	C₂³	1	2	yes	—
45	C₃×#123	C₈	1	2	yes	—
46	C₃×C₂²⋊Q₈	C₈	1	2	yes	—
47	C₃×#121	C₈	1	2	yes	—
48	C₃×#120	C₈	1	2	yes	—
49	C₃×C₄×D₄	C₂×C₄	1	4	yes	—
50	C₃×#117	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
53	C₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
54	C₃×#97	C₂×C₄	1	2	yes	—
55	C₃×C₄×C₈	C₂³	1	2	yes	—
56	#812	—	4	2	no	—
57	C₂×C₃⋊C₁₆	—	4	4	no	—
58	C₂×C₃⋊C₁₆	—	4	2	no	—
59	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
60	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
61	C₂×#131	C₂×D₃	1	1	yes	characteristic
62	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
63	#525	Dic₃	1	1	yes	characteristic
64	C₂×M₃₂	—	3	2	no	—
65	C₂²×C₁₆	—	3	4	no	—
66	C₂²×C₁₆	—	3	2	no	—
67	#435	Dic₃	1	4	yes	—
68	C₈×D₄	C₂×D₃	1	4	yes	—
69	#424	Dic₃	1	1	yes	characteristic
70	#408	Dic₃	1	1	yes	characteristic
71	#407	Dic₃	1	1	yes	characteristic
72	#405	C₂×D₃	1	1	yes	characteristic
73	C₄⋊C₁₆	—	3	4	no	—
74	C₂²⋊C₁₆	—	3	4	no	—
75	C₂²⋊C₁₆	—	3	2	no	—
76	C₂²⋊C₁₆	—	3	2	no	—
77	C₂²⋊C₁₆	—	3	4	no	—
78	#346	—	3	2	no	—
79	C₄×C₁₆	—	3	2	no	—
80	C₂⁴×C₃	C₂×C₈	1	1	yes	characteristic
81	C₃×#41	—	4	1	no	—
82	C₃×#41	—	4	1	no	—
83	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃×D₄	C₂×C₈	1	4	yes	—
85	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
86	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₃×C₄	—	4	1	no	—
88	C₂²×C₃×C₄	—	4	1	no	—
89	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
90	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
92	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
93	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₃×M₁₆	—	4	1	no	—
95	C₃×M₁₆	—	4	2	no	—
96	C₃×M₁₆	—	4	1	no	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	4	1	no	—
102	C₂×C₃×C₈	—	4	2	no	—
103	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
108	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
109	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
110	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
111	C₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
112	C₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
113	C₃×C₄²	C₂²×C₄	1	2	yes	—
114	C₃⋊C₁₆	—	2	4	no	—
115	C₃⋊C₁₆	—	2	4	no	—
116	C₂×#41	C₂×Dic₃	1	1	yes	characteristic
117	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
118	#131	—	4	2	no	—
119	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
120	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
121	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
122	C₂²×C₈	—	4	2	no	—
123	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₃	1	2	yes	—
125	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₃	1	2	yes	—
127	C₂²×C₈	C₂²×D₃	1	2	yes	—
128	#123	C₃⋊C₈	1	2	yes	—
129	C₂²⋊Q₈	C₃⋊C₈	1	2	yes	—
130	#121	C₃⋊C₈	1	2	yes	—
131	#120	C₃⋊C₈	1	2	yes	—
132	C₄×D₄	C₂×Dic₃	1	4	yes	—
133	#117	C₂×Dic₃	1	1	yes	characteristic
134	M₃₂	—	12	2	no	—
135	C₂×C₁₆	—	3	4	no	—
136	C₂×C₁₆	—	3	4	no	—
137	C₂×C₁₆	—	12	2	no	—
138	C₂×C₁₆	—	12	4	no	—
139	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
140	C₂²⋊C₈	C₂×Dic₃	1	4	yes	—
141	C₂²⋊C₈	C₂×Dic₃	1	4	yes	—
142	#97	C₂×Dic₃	1	2	yes	—
143	C₄×C₈	C₂²×D₃	1	2	yes	—
144	C₂³×C₃	—	2	4	no	—
145	C₂³×C₃	—	4	1	no	—
146	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
147	C₂³×C₃	C₂²×C₈	1	2	yes	—
148	C₂³×C₃	C₂²×C₈	1	2	yes	—
149	C₃×Q₈	—	4	1	no	—
150	C₃×D₄	—	2	8	no	—
151	C₃×D₄	—	4	1	no	—
152	C₃×D₄	—	4	2	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
162	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
163	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
164	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
166	C₃×C₈	—	2	2	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	—	2	2	no	—
170	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
171	C₂⁴	C₂×C₃⋊C₈	1	1	yes	characteristic
172	#41	—	4	1	no	—
173	#41	—	4	1	no	—
174	C₂×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
175	C₂×D₄	C₂×C₃⋊C₈	1	4	yes	—
176	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
177	C₂×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
181	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
182	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
185	M₁₆	—	4	2	no	—
186	M₁₆	—	4	1	no	—
187	M₁₆	—	4	1	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	C₂²×Dic₃	1	2	yes	—
195	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
196	C₂×C₈	C₂²×Dic₃	1	2	yes	—
197	C₂×C₈	C₂²×Dic₃	1	2	yes	—
198	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
199	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
200	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
201	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
202	C₂²⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
203	C₂²⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
204	C₄²	C₂²×Dic₃	1	2	yes	—
205	C₁₆	—	6	4	no	—
206	C₁₆	—	6	4	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	4	4	no	—
214	C₂²×C₃	#504	1	4	yes	—
215	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	#518	1	1	yes	characteristic
221	C₃×C₄	#568	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	4	1	no	—
224	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
225	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
226	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
227	Q₈	—	4	1	no	—
228	D₄	—	2	8	no	—
229	D₄	—	4	1	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	C₂²×C₃⋊C₈	1	4	yes	—
240	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
241	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
242	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
243	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	#1004	1	2	yes	—
249	C₂×C₃	—	2	4	no	—
250	C₂×C₃	—	4	2	no	—
251	C₂×C₃	—	4	1	no	—
252	C₂×C₃	#2972	1	1	yes	characteristic
253	C₂×C₃	C₂×#504	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	4	no	—
261	C₂²	#5140	1	4	yes	—
262	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	#5798	1	1	yes	characteristic
268	C₄	#5819	1	1	yes	characteristic
269	C₃	#33491	1	1	yes	characteristic, complemented, Sylow
270	C₂	—	2	4	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	2	no	—
273	C₂	#82301	1	1	yes	characteristic
274	C₂	C₂×#5140	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2929	C₂
prime radical	C₃×#525	C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2929	D₃
hypocenter	C₃	#33491
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
2. #72004	C₂	semidirect
3. C₂×#4549	C₂	semidirect
4. #70857	C₂	semidirect
5. D₄×C₃⋊C₁₆	C₂	semidirect
6. #69883	C₂	semidirect
17. C₂×#812	C₂²	semidirect
19. C₂²×C₃⋊C₁₆	C₂²	semidirect
23. #4549	C₂²	semidirect
24. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
25. #4461	C₂²	semidirect
26. C₄×C₃⋊C₁₆	C₂²	semidirect
269. C₃	#33491	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2929	2
#72004	2
C₂×#4549	2
#70857	2
D₄×C₃⋊C₁₆	2
#69883	2
#33491	3