Subgroups — #10948823

Order 768.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2931	C₂	1	1	yes	maximal, characteristic
2	#72004	C₂	1	4	yes	maximal, complemented
3	#70858	C₂	1	2	yes	maximal, complemented
4	#70857	C₂	1	4	yes	maximal, complemented
5	D₄×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	#69745	C₂	1	1	yes	maximal, characteristic, complemented
7	#69744	C₂	1	1	yes	maximal, characteristic, complemented
8	#33495	—	3	1	no	maximal, Sylow
9	C₂×C₃×#131	C₂²	1	2	yes	—
10	C₃×#532	C₄	1	1	yes	characteristic
11	C₃×#436	C₄	1	2	yes	—
12	C₃×#436	C₄	1	2	yes	—
13	C₃×C₈×D₄	C₂²	1	2	yes	—
14	C₃×C₈×D₄	C₂²	1	2	yes	—
15	C₃×#423	C₄	1	1	yes	characteristic
16	C₃×#408	C₄	1	2	yes	—
17	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×#812	C₂²	1	4	yes	—
19	C₂²×C₃⋊C₁₆	C₂²	1	4	yes	—
20	#4549	C₂²	1	2	yes	—
21	#4549	C₂²	1	2	yes	—
22	#4549	C₂²	1	4	yes	—
23	#4549	C₂²	1	4	yes	—
24	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
25	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
26	#4461	C₂²	1	2	yes	—
27	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
29	#2931	D₃	1	1	yes	characteristic
30	#2169	—	3	2	no	—
31	#2168	—	3	4	no	—
32	D₄×C₁₆	—	3	2	no	—
33	#2163	—	3	1	no	—
34	#2162	—	3	1	no	—
35	#2113	—	3	4	no	—
36	C₂×C₃×#41	C₂×C₄	1	2	yes	—
37	C₃×#131	—	4	4	no	—
38	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
40	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂³	1	4	yes	—
43	C₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
44	C₃×#127	C₈	1	1	yes	characteristic
45	C₃×#124	C₈	1	2	yes	—
46	C₃×#121	C₈	1	2	yes	—
47	C₃×#121	C₈	1	2	yes	—
48	C₃×C₄×D₄	C₂×C₄	1	2	yes	—
49	C₃×C₄×D₄	C₂×C₄	1	2	yes	—
50	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
54	C₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
55	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
59	#812	—	4	4	no	—
60	C₂×C₃⋊C₁₆	—	4	4	no	—
61	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
62	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
63	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
64	C₂×#131	C₂×D₃	1	2	yes	—
65	#532	Dic₃	1	1	yes	characteristic
66	C₂×M₃₂	—	3	4	no	—
67	C₂²×C₁₆	—	3	4	no	—
68	#436	Dic₃	1	2	yes	—
69	#436	Dic₃	1	2	yes	—
70	C₈×D₄	C₂×D₃	1	2	yes	—
71	C₈×D₄	C₂×D₃	1	2	yes	—
72	#423	Dic₃	1	1	yes	characteristic
73	#408	Dic₃	1	2	yes	—
74	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
75	C₄⋊C₁₆	—	3	2	no	—
76	C₄⋊C₁₆	—	3	2	no	—
77	C₂²⋊C₁₆	—	3	2	no	—
78	C₂²⋊C₁₆	—	3	2	no	—
79	C₂²⋊C₁₆	—	3	4	no	—
80	C₂²⋊C₁₆	—	3	4	no	—
81	#346	—	3	2	no	—
82	C₄×C₁₆	—	3	1	no	—
83	C₄×C₁₆	—	3	1	no	—
84	C₃×#41	—	4	2	no	—
85	C₃×#41	—	4	2	no	—
86	C₂×C₃×Q₈	C₂×C₈	1	2	yes	—
87	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
88	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
89	C₂×C₃×D₄	C₂×C₈	1	4	yes	—
90	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
91	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
94	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₃×M₁₆	—	4	2	no	—
96	C₃×M₁₆	—	4	4	no	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	—	2	4	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	4	1	no	—
101	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
106	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
107	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
108	C₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
109	C₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
110	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
111	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
112	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₃⋊C₁₆	—	2	2	no	—
115	C₃⋊C₁₆	—	2	4	no	—
116	C₃⋊C₁₆	—	2	2	no	—
117	C₂×#41	C₂×Dic₃	1	2	yes	—
118	#131	—	4	4	no	—
119	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
120	C₂×M₁₆	C₂×Dic₃	1	4	yes	—
121	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₃	1	4	yes	—
124	C₄⋊Q₈	C₃⋊C₈	1	1	yes	characteristic
125	#127	C₃⋊C₈	1	1	yes	characteristic
126	#124	C₃⋊C₈	1	2	yes	—
127	#121	C₃⋊C₈	1	2	yes	—
128	#121	C₃⋊C₈	1	2	yes	—
129	C₄×D₄	C₂×Dic₃	1	2	yes	—
130	C₄×D₄	C₂×Dic₃	1	2	yes	—
131	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
132	M₃₂	—	12	4	no	—
133	C₂×C₁₆	—	3	2	no	—
134	C₂×C₁₆	—	3	2	no	—
135	C₂×C₁₆	—	3	4	no	—
136	C₂×C₁₆	—	12	4	no	—
137	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
138	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
139	C₂²⋊C₈	C₂×Dic₃	1	4	yes	—
140	C₂²⋊C₈	C₂×Dic₃	1	4	yes	—
141	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
142	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
143	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
144	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
145	C₂³×C₃	C₂²×C₈	1	4	yes	—
146	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
147	C₃×Q₈	—	4	2	no	—
148	C₃×D₄	—	2	4	no	—
149	C₃×D₄	—	4	2	no	—
150	C₃×D₄	—	4	2	no	—
151	C₃×D₄	—	4	4	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
163	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
165	C₃×C₈	—	2	4	no	—
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	1	no	—
169	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
170	#41	—	4	2	no	—
171	#41	—	4	2	no	—
172	C₂×Q₈	C₂×C₃⋊C₈	1	2	yes	—
173	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
174	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
175	C₂×D₄	C₂×C₃⋊C₈	1	4	yes	—
176	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
177	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
179	C₂²×C₄	C₂²×Dic₃	1	4	yes	—
180	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
181	M₁₆	—	4	4	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
188	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
189	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
190	C₂×C₈	C₂²×Dic₃	1	4	yes	—
191	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
192	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
193	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
194	C₂²⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
195	C₂²⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
196	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
197	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
198	C₄²	C₂²×Dic₃	1	1	yes	characteristic
199	C₄²	C₂²×Dic₃	1	1	yes	characteristic
200	C₁₆	—	6	2	no	—
201	C₁₆	—	6	4	no	—
202	C₁₆	—	6	2	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	4	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	4	no	—
209	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	#504	1	2	yes	—
216	C₃×C₄	#518	1	1	yes	characteristic
217	C₃×C₄	#568	1	1	yes	characteristic
218	C₂³	C₂²×C₃⋊C₈	1	4	yes	—
219	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	D₄	—	2	4	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₃⋊C₈	1	4	yes	—
236	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
237	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	4	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	#1004	1	2	yes	—
243	C₂×C₃	—	4	1	no	—
244	C₂×C₃	—	4	4	no	—
245	C₂×C₃	#2972	1	1	yes	characteristic
246	C₂×C₃	C₂×#504	1	1	yes	characteristic
247	C₂×C₃	#3407	1	1	yes	characteristic
248	C₂²	—	2	1	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	4	no	—
260	C₄	#5140	1	2	yes	—
261	C₄	#5798	1	1	yes	characteristic
262	C₄	#5819	1	1	yes	characteristic
263	C₃	#33495	1	1	yes	characteristic, complemented, Sylow
264	C₂	—	4	1	no	—
265	C₂	—	4	4	no	—
266	C₂	#82301	1	1	yes	characteristic
267	C₂	C₂×#5140	1	1	yes	characteristic
268	C₂	#83780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2931	C₂
prime radical	C₃×#532	C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2931	D₃
hypocenter	C₃	#33495
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #72004	C₂	semidirect
3. #70858	C₂	semidirect
4. #70857	C₂	semidirect
5. D₄×C₃⋊C₁₆	C₂	semidirect
6. #69745	C₂	semidirect
7. #69744	C₂	semidirect
263. C₃	#33495	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2931	2
#72004	2
#70858	2
#70857	2
D₄×C₃⋊C₁₆	2
#69745	2
#69744	2
#33495	3