Subgroups — #10950675

Order 768.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3160	C₂	1	1	yes	maximal, characteristic
2	#78442	C₂	1	1	yes	maximal, characteristic
3	#71385	C₂	1	2	yes	maximal, complemented
4	#71384	C₂	1	2	yes	maximal, complemented
5	#71354	C₂	1	4	yes	maximal, complemented
6	#70648	C₂	1	4	yes	maximal
7	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	#33726	—	3	1	no	maximal, Sylow
9	C₃×#533	—	2	1	no	—
10	#5470	—	2	1	no	—
11	C₃×#500	C₂²	1	2	yes	—
12	C₃×#499	—	2	1	no	—
13	C₃×#499	C₂²	1	2	yes	—
14	C₃×#498	—	2	1	no	—
15	C₃×#427	—	2	1	no	—
16	C₃×#427	C₂²	1	2	yes	—
17	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	#5106	—	2	1	no	—
19	#5106	C₂²	1	2	yes	—
20	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
21	#4683	—	2	1	no	—
22	#4683	C₂²	1	2	yes	—
23	#4682	C₂²	1	2	yes	—
24	#4680	—	2	1	no	—
25	#4490	C₂²	1	8	yes	—
26	#4489	C₂²	1	8	yes	—
27	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
30	#3160	—	3	1	no	—
31	#3160	D₃	1	1	yes	characteristic
32	#2254	—	3	2	no	—
33	#2253	—	3	4	no	—
34	#2252	—	3	2	no	—
35	#2158	—	3	4	no	—
36	C₂×C₄×C₁₆	—	3	1	no	—
37	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
39	C₃×C₄⋊Q₈	—	2	2	no	—
40	C₃×#125	—	2	2	no	—
41	C₃×C₂²⋊Q₈	—	4	2	no	—
42	C₃×C₄×Q₈	—	4	2	no	—
43	C₃×#117	—	2	2	no	—
44	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
45	#926	—	2	2	no	—
46	#924	—	4	2	no	—
47	#870	—	2	2	no	—
48	#869	—	2	2	no	—
49	C₄×Dic₆	—	4	2	no	—
50	C₃×#107	—	2	2	no	—
51	C₃×#107	C₂³	1	4	yes	—
52	C₃×#106	—	2	2	no	—
53	C₃×#106	—	2	2	no	—
54	C₃×C₄×C₈	D₄	1	1	yes	characteristic
55	C₃×C₄×C₈	D₄	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
58	#818	—	2	2	no	—
59	#818	C₂³	1	4	yes	—
60	#817	—	2	2	no	—
61	#817	—	2	2	no	—
62	C₂×C₃⋊C₁₆	Q₈	1	8	yes	complemented
63	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
64	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
65	#533	—	2	1	no	—
66	#533	—	6	1	no	—
67	C₂²×C₁₆	—	3	2	no	—
68	#500	—	3	2	no	—
69	#500	C₂×D₃	1	2	yes	—
70	#499	—	2	1	no	—
71	#499	—	3	2	no	—
72	#499	—	6	1	no	—
73	#499	C₂×D₃	1	2	yes	—
74	#498	—	2	1	no	—
75	#498	—	6	1	no	—
76	#427	—	2	1	no	—
77	#427	—	3	2	no	—
78	#427	—	6	1	no	—
79	#427	C₂×D₃	1	2	yes	—
80	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
81	#367	—	3	8	no	—
82	#366	—	3	8	no	—
83	C₄×C₁₆	—	3	1	no	—
84	C₄×C₁₆	—	3	2	no	—
85	C₄×C₁₆	—	3	1	no	—
86	C₂×C₃×Q₈	—	4	2	no	—
87	C₂²×C₃×C₄	D₈	1	2	yes	—
88	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₆	—	4	2	no	—
90	C₂×C₃×C₈	Q₁₆	1	4	yes	—
91	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
95	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
96	C₃×C₄⋊C₄	—	2	4	no	—
97	C₃×C₄⋊C₄	—	4	4	no	—
98	C₃×C₄⋊C₄	—	4	2	no	—
99	C₃×C₂²⋊C₄	—	4	2	no	—
100	C₃×C₄²	—	4	2	no	—
101	C₃×C₄²	D₈	1	2	yes	—
102	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
103	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
104	#217	—	4	2	no	—
105	C₄⋊Dic₃	—	2	4	no	—
106	C₄⋊Dic₃	—	4	2	no	—
107	Dic₃⋊C₄	—	4	4	no	—
108	C₄×Dic₃	—	4	2	no	—
109	C₃⋊C₁₆	C₂×Q₈	1	4	yes	—
110	C₃⋊C₁₆	C₂×Q₈	1	4	yes	—
111	C₂²×C₈	#67	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	—	6	2	no	—
115	#125	—	2	2	no	—
116	#125	—	6	2	no	—
117	C₂²⋊Q₈	—	4	2	no	—
118	C₂²⋊Q₈	—	12	2	no	—
119	C₄×Q₈	—	4	2	no	—
120	C₄×Q₈	—	12	2	no	—
121	#117	—	2	2	no	—
122	#117	—	6	2	no	—
123	C₂×C₄²	#67	1	1	yes	characteristic
124	C₂×C₁₆	—	3	8	no	—
125	C₂×C₁₆	—	3	2	no	—
126	C₂×C₁₆	—	3	2	no	—
127	#107	—	2	2	no	—
128	#107	—	3	4	no	—
129	#107	—	6	2	no	—
130	#107	C₂²×D₃	1	4	yes	—
131	#106	—	2	2	no	—
132	#106	—	2	2	no	—
133	#106	—	6	2	no	—
134	#106	—	6	2	no	—
135	C₄×C₈	#67	1	1	yes	characteristic
136	C₄×C₈	#67	1	1	yes	characteristic
137	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
138	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
139	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
140	C₃×Q₈	—	8	4	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	4	4	no	—
143	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
144	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
145	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
146	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
148	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₃	—	4	4	no	—
150	Dic₆	—	8	4	no	—
151	C₃×C₈	C₄⋊Q₈	1	2	yes	—
152	C₃×C₈	C₂×Q₁₆	1	2	yes	—
153	C₃×C₈	C₂×Q₁₆	1	2	yes	—
154	C₃×C₈	C₂²×Q₈	1	2	yes	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	12	2	no	—
157	C₂²×C₄	#213	1	2	yes	—
158	C₂²×C₄	C₂×#67	1	1	yes	characteristic
159	C₂×C₈	#216	1	4	yes	—
160	C₂×C₈	D₃×Q₈	1	4	yes	—
161	C₂×C₈	C₂×#67	1	1	yes	characteristic
162	C₂×C₈	C₂×#67	1	1	yes	characteristic
163	C₂×C₈	C₂×#67	1	1	yes	characteristic
164	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	6	4	no	—
169	C₄⋊C₄	—	12	4	no	—
170	C₄⋊C₄	—	12	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	12	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	12	2	no	—
175	C₄²	#213	1	2	yes	—
176	C₄²	C₂×#67	1	1	yes	characteristic
177	C₄²	C₂×#67	1	1	yes	characteristic
178	C₁₆	—	3	4	no	—
179	C₁₆	—	3	4	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	#500	1	2	yes	—
182	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	8	4	no	—
185	C₃×C₄	#500	1	2	yes	—
186	C₃×C₄	#508	1	4	yes	—
187	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
188	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
189	Dic₃	—	8	4	no	—
190	C₂³	C₂×#213	1	1	yes	characteristic
191	Q₈	—	8	4	no	—
192	Q₈	—	24	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	12	4	no	—
196	C₂×C₄	C₂×#213	1	2	yes	—
197	C₂×C₄	C₂×#213	1	2	yes	—
198	C₂×C₄	C₂×#213	1	1	yes	characteristic
199	C₂×C₄	C₂×#216	1	2	yes	—
200	C₂×C₄	#944	1	2	yes	—
201	C₂×C₄	C₂²×#67	1	1	yes	characteristic
202	C₈	C₂×#216	1	2	yes	—
203	C₈	C₂×#216	1	2	yes	—
204	C₈	#944	1	2	yes	—
205	C₈	C₂×D₃×Q₈	1	2	yes	—
206	C₂×C₃	—	2	2	no	—
207	C₂×C₃	C₂×#500	1	1	yes	characteristic
208	C₂×C₃	C₂×#508	1	2	yes	—
209	C₂²	—	2	2	no	—
210	C₂²	#5078	1	2	yes	—
211	C₂²	C₂²×#213	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	8	4	no	—
214	C₄	—	24	4	no	—
215	C₄	#5078	1	2	yes	—
216	C₄	#5194	1	4	yes	—
217	C₄	C₂²×#216	1	1	yes	characteristic
218	C₄	C₂×#944	1	1	yes	characteristic
219	C₃	#33726	1	1	yes	characteristic, complemented, Sylow
220	C₂	—	2	2	no	—
221	C₂	C₂×#5078	1	1	yes	characteristic
222	C₂	C₂×#5194	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#213
commutator, cosocle	C₂×C₃×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×#3160	C₂
socle	C₂²×C₃	C₂²×D₈
prime radical	C₂³×C₃	C₂×D₈
hypercenter	#3160	D₃
hypocenter	C₃	#33726
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #71385	C₂	semidirect
4. #71384	C₂	semidirect
5. #71354	C₂	semidirect
62. C₂×C₃⋊C₁₆	Q₈	semidirect
219. C₃	#33726	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3160	2
#78442	2
#71385	2
#71384	2
#71354	2
#70648	2
C₂×C₄×C₃⋊C₁₆	2
#33726	3