Subgroups — #10804029

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#79044	C₂	1	2	yes	maximal
2	#78442	C₂	1	1	yes	maximal, characteristic
3	C₃×#1848	C₂	1	1	yes	maximal, characteristic
4	#70704	C₂	1	4	yes	maximal
5	#70702	C₂	1	4	yes	maximal
6	#70514	C₂	1	2	yes	maximal
7	C₄×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
8	#21732	—	3	1	no	maximal, Sylow
9	#5470	C₂²	1	2	yes	—
10	C₂×C₃×#108	C₂²	1	4	yes	—
11	C₃×#423	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#5122	C₂²	1	4	yes	—
14	#5119	C₂²	1	4	yes	—
15	#5106	C₂²	1	4	yes	—
16	C₂×C₈×Dic₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₃	C₂²	1	2	yes	—
18	C₄²×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#4831	—	2	4	no	—
21	#4830	—	2	4	no	—
22	#4683	C₄	1	4	yes	complemented
23	#4682	C₄	1	2	yes	complemented
24	#4680	C₄	1	2	yes	complemented
25	#4555	C₂²	1	8	yes	—
26	#3160	—	3	1	no	—
27	#2960	—	3	2	no	—
28	#1994	—	3	2	no	—
29	#1945	—	3	4	no	—
30	#1848	D₃	1	1	yes	characteristic
31	#1772	—	3	4	no	—
32	C₄²×C₈	—	3	1	no	—
33	C₂×C₃×M₁₆	C₂³	1	4	yes	—
34	C₂²×C₃×C₈	C₂³	1	2	yes	—
35	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
36	#926	C₂³	1	4	yes	—
37	#924	—	2	4	no	—
38	C₂×C₄×Dic₃	—	2	2	no	—
39	C₂×C₄×Dic₃	C₂³	1	2	yes	—
40	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
42	#870	C₂×C₄	1	4	yes	—
43	#869	C₂×C₄	1	4	yes	—
44	C₄×Dic₆	—	2	4	no	—
45	C₃×#108	—	2	8	no	—
46	C₃×C₄⋊C₈	—	2	4	no	—
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
50	#818	C₂×C₄	1	8	yes	—
51	#817	C₂×C₄	1	8	yes	—
52	#815	—	2	4	no	—
53	Dic₃⋊C₈	—	2	8	no	—
54	C₈×Dic₃	—	2	4	no	—
55	C₈×Dic₃	—	2	4	no	—
56	C₈×Dic₃	Q₈	1	8	yes	complemented
57	C₄×C₃⋊C₈	D₄	1	2	yes	—
58	C₄×C₃⋊C₈	D₄	1	2	yes	—
59	#533	—	3	2	no	—
60	#500	—	3	2	no	—
61	#499	—	3	4	no	—
62	#498	—	3	2	no	—
63	#446	—	6	4	no	—
64	C₈×Q₈	—	6	4	no	—
65	C₂×#108	C₂×D₃	1	4	yes	—
66	#427	—	3	4	no	—
67	#424	—	3	4	no	—
68	#423	C₂×D₃	1	2	yes	—
69	#405	—	3	4	no	—
70	C₂×C₄×C₈	—	3	2	no	—
71	C₂×C₄×C₈	—	3	1	no	—
72	C₂×C₄×C₈	—	3	2	no	—
73	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
74	C₄³	—	3	1	no	—
75	#339	—	3	8	no	—
76	C₂²×C₃×C₄	#41	1	2	yes	—
77	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₃	—	2	2	no	—
79	C₂×Dic₆	—	2	4	no	—
80	C₃×M₁₆	—	2	8	no	—
81	C₂×C₃×C₈	—	2	4	no	—
82	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
85	C₂×C₃×C₈	#41	1	4	yes	—
86	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
87	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₃×C₄²	C₂×D₄	1	2	yes	—
89	#217	—	2	4	no	—
90	C₄⋊Dic₃	—	2	4	no	—
91	C₄⋊Dic₃	C₂²×C₄	1	8	yes	—
92	Dic₃⋊C₄	—	2	8	no	—
93	C₄×Dic₃	—	2	8	no	—
94	C₄×Dic₃	—	2	4	no	—
95	C₄×Dic₃	—	2	2	no	—
96	C₄×Dic₃	C₂×Q₈	1	4	yes	—
97	C₂×C₃⋊C₈	—	2	2	no	—
98	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₃⋊C₈	C₂×Q₈	1	4	yes	—
102	C₂×M₁₆	C₂²×D₃	1	4	yes	—
103	C₂²×C₈	—	3	1	no	—
104	C₂²×C₈	—	3	1	no	—
105	C₂²×C₈	C₂²×D₃	1	2	yes	—
106	C₄⋊Q₈	—	3	4	no	—
107	#125	—	3	4	no	—
108	C₂²⋊Q₈	—	6	4	no	—
109	C₄×Q₈	—	6	4	no	—
110	#117	—	3	4	no	—
111	C₂×C₄²	—	3	2	no	—
112	C₂×C₄²	—	6	2	no	—
113	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
114	#108	—	2	8	no	—
115	#107	—	3	8	no	—
116	#106	—	3	8	no	—
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	—	6	8	no	—
119	#97	—	6	4	no	—
120	C₄×C₈	—	3	8	no	—
121	C₄×C₈	—	3	2	no	—
122	C₄×C₈	—	3	2	no	—
123	C₄×C₈	—	6	4	no	—
124	C₄×C₈	—	6	4	no	—
125	C₄×C₈	C₄×D₃	1	1	yes	characteristic
126	C₄×C₈	C₄×D₃	1	2	yes	—
127	C₄×C₈	C₄×D₃	1	1	yes	characteristic
128	C₂³×C₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
131	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
132	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
135	C₂×C₃×C₄	C₂×#41	1	2	yes	—
136	C₂×Dic₃	—	2	8	no	—
137	C₂×Dic₃	—	2	4	no	—
138	C₂×Dic₃	—	2	8	no	—
139	Dic₆	—	4	8	no	—
140	C₃×C₈	—	4	4	no	—
141	C₃×C₈	C₄×Q₈	1	4	yes	—
142	C₃×C₈	C₄×Q₈	1	4	yes	—
143	C₃⋊C₈	—	2	2	no	—
144	C₃⋊C₈	C₄⋊Q₈	1	2	yes	—
145	C₃⋊C₈	C₄⋊Q₈	1	2	yes	—
146	C₂×Q₈	—	6	4	no	—
147	C₂²×C₄	—	6	2	no	—
148	C₂²×C₄	#239	1	2	yes	—
149	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
150	M₁₆	—	2	8	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	3	1	no	—
153	C₂×C₈	—	3	1	no	—
154	C₂×C₈	—	3	4	no	—
155	C₂×C₈	—	3	2	no	—
156	C₂×C₈	—	6	2	no	—
157	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
158	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
159	C₂×C₈	#237	1	4	yes	—
160	C₂×C₈	D₃×Q₈	1	4	yes	—
161	C₄⋊C₄	—	3	8	no	—
162	C₄⋊C₄	—	6	8	no	—
163	C₄⋊C₄	—	6	4	no	—
164	C₂²⋊C₄	—	6	4	no	—
165	C₄²	—	3	4	no	—
166	C₄²	—	6	8	no	—
167	C₄²	—	6	4	no	—
168	C₄²	—	6	2	no	—
169	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
170	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
171	C₄²	D₃×D₄	1	2	yes	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂²×C₃	#533	1	2	yes	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	#443	1	4	yes	—
177	C₃×C₄	C₂×C₄×Q₈	1	2	yes	—
178	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
179	C₃×C₄	#532	1	1	yes	characteristic
180	Dic₃	—	2	8	no	—
181	Dic₃	—	4	8	no	—
182	C₂³	C₂×#239	1	1	yes	characteristic
183	Q₈	—	12	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	6	8	no	—
186	C₂×C₄	—	6	4	no	—
187	C₂×C₄	—	6	8	no	—
188	C₂×C₄	#893	1	2	yes	—
189	C₂×C₄	#893	1	2	yes	—
190	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
191	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
192	C₂×C₄	C₂×#237	1	2	yes	—
193	C₂×C₄	C₂×D₃×Q₈	1	2	yes	—
194	C₈	—	3	2	no	—
195	C₈	—	3	2	no	—
196	C₈	—	4	4	no	—
197	C₈	—	6	2	no	—
198	C₈	#887	1	4	yes	—
199	C₈	#887	1	4	yes	—
200	C₂×C₃	—	2	2	no	—
201	C₂×C₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
202	C₂×C₃	C₂×#443	1	2	yes	—
203	C₂²	—	2	2	no	—
204	C₂²	C₂×#893	1	1	yes	characteristic
205	C₂²	#5728	1	2	yes	—
206	C₄	—	2	2	no	—
207	C₄	—	6	8	no	—
208	C₄	—	12	8	no	—
209	C₄	#4895	1	4	yes	—
210	C₄	C₂×#887	1	2	yes	—
211	C₄	#5716	1	1	yes	characteristic
212	C₄	D₃×C₄⋊Q₈	1	1	yes	characteristic
213	C₃	#21732	1	1	yes	characteristic, complemented, Sylow
214	C₂	—	2	2	no	—
215	C₂	#74320	1	1	yes	characteristic
216	C₂	C₂×#4895	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₃×#1848	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₂×#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#1848	D₃
hypocenter	C₃	#21732
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
22. #4683	C₄	semidirect
23. #4682	C₄	semidirect
24. #4680	C₄	semidirect
56. C₈×Dic₃	Q₈	semidirect
213. C₃	#21732	semidirect

Maximal subgroups

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

subgroup	index
#79044	2
#78442	2
C₃×#1848	2
#70704	2
#70702	2
#70514	2
C₄×C₈×Dic₃	2
#21732	3