Subgroups — #10693804

Order 768.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#108	C₂	1	1	yes	maximal, characteristic
2	#69850	C₂	1	2	yes	maximal
3	C₂×#4458	C₂	1	1	yes	maximal, characteristic
4	#69724	C₂	1	1	yes	maximal, characteristic
5	#67668	C₂	1	2	yes	maximal
6	#9608	—	3	1	no	maximal, Sylow
7	C₂×C₃×#108	C₄	1	1	yes	characteristic
8	C₂×C₃×#108	C₄	1	1	yes	characteristic
9	C₃×C₄×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#4925	C₂²	1	1	yes	characteristic
12	#4925	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃×#341	C₄	1	2	yes	—
15	#4458	—	2	1	no	—
16	#4458	—	2	1	no	—
17	#4458	—	2	2	no	—
18	#4457	—	2	2	no	—
19	#4455	—	2	1	no	—
20	C₈×C₃⋊C₈	—	2	1	no	—
21	C₄×#108	D₃	1	1	yes	characteristic
22	#1565	—	3	2	no	—
23	C₂×#335	—	3	1	no	—
24	#1450	—	3	1	no	—
25	#1283	—	3	2	no	—
26	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
31	C₂²×C₃⋊C₈	—	2	2	no	—
32	C₃×#108	—	2	2	no	—
33	C₃×#108	C₈	1	4	yes	complemented
34	C₃×#97	—	2	2	no	—
35	C₃×C₄×C₈	—	2	2	no	—
36	C₃×C₄×C₈	D₄	1	1	yes	characteristic
37	C₃×C₄×C₈	D₄	1	1	yes	characteristic
38	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
40	#830	—	2	1	no	—
41	#830	Q₈	1	2	yes	—
42	C₄⋊(C₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
46	C₄⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
47	C₄×C₃⋊C₈	—	2	1	no	—
48	C₄×C₃⋊C₈	—	2	1	no	—
49	#431	—	3	1	no	—
50	#431	—	3	1	no	—
51	C₂×#108	Dic₃	1	1	yes	characteristic
52	C₂×#108	Dic₃	1	1	yes	characteristic
53	C₂×C₄⋊C₈	—	3	2	no	—
54	C₄×M₁₆	C₂×D₃	1	2	yes	—
55	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	#341	Dic₃	1	2	yes	—
57	#335	—	6	2	no	—
58	#335	—	6	1	no	—
59	#335	—	6	1	no	—
60	#334	—	6	2	no	—
61	#322	—	6	1	no	—
62	C₈²	—	6	1	no	—
63	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₃×C₄	#41	1	1	yes	characteristic
66	C₃×M₁₆	—	2	2	no	—
67	C₃×M₁₆	—	2	2	no	—
68	C₃×M₁₆	—	2	2	no	—
69	C₃×M₁₆	C₂×C₈	1	4	yes	—
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	—	2	1	no	—
72	C₂×C₃×C₈	—	2	1	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
79	C₂×C₃×C₈	M₁₆	1	2	yes	—
80	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
83	C₃×C₄²	#41	1	1	yes	characteristic
84	C₂×C₃⋊C₈	—	2	1	no	—
85	C₂×C₃⋊C₈	—	2	1	no	—
86	C₂×C₃⋊C₈	—	2	2	no	—
87	C₂×C₃⋊C₈	—	4	2	no	—
88	C₂×C₃⋊C₈	—	8	2	no	—
89	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
90	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
91	C₂²×C₈	—	6	2	no	—
92	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
95	#108	—	2	2	no	—
96	#108	C₃⋊C₈	1	4	yes	complemented
97	C₄⋊C₈	—	3	1	no	—
98	C₄⋊C₈	—	3	1	no	—
99	C₄⋊C₈	—	6	2	no	—
100	C₄⋊C₈	—	6	2	no	—
101	C₄⋊C₈	—	6	1	no	—
102	C₂²⋊C₈	—	3	2	no	—
103	C₂²⋊C₈	—	6	1	no	—
104	#97	—	2	2	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	—	6	1	no	—
107	C₄×C₈	—	6	1	no	—
108	C₄×C₈	Dic₆	1	1	yes	characteristic
109	C₄×C₈	Dic₆	1	1	yes	characteristic
110	C₄×C₈	D₁₂	1	1	yes	characteristic
111	C₄×C₈	D₁₂	1	1	yes	characteristic
112	C₂³×C₃	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	4	1	no	—
116	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₄×D₄	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	C₃×C₈	—	2	1	no	—
127	C₃×C₈	—	2	1	no	—
128	C₃×C₈	—	2	4	no	—
129	C₃×C₈	—	4	2	no	—
130	C₃×C₈	C₄⋊C₈	1	2	yes	—
131	C₃×C₈	C₄⋊C₈	1	2	yes	—
132	C₃⋊C₈	—	4	2	no	—
133	C₃⋊C₈	—	4	1	no	—
134	C₃⋊C₈	—	4	1	no	—
135	C₃⋊C₈	—	8	2	no	—
136	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
137	C₂²×C₄	#239	1	1	yes	characteristic
138	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	—	2	2	no	—
142	M₁₆	C₂×C₃⋊C₈	1	4	yes	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	6	1	no	—
148	C₂×C₈	—	6	1	no	—
149	C₂×C₈	—	6	2	no	—
150	C₂×C₈	—	12	2	no	—
151	C₂×C₈	—	24	2	no	—
152	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
153	C₂×C₈	#208	1	2	yes	—
154	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
155	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
156	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
157	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
158	C₄²	C₂×Dic₆	1	1	yes	characteristic
159	C₄²	C₂×D₁₂	1	1	yes	characteristic
160	C₄²	D₃×D₄	1	1	yes	characteristic
161	C₄²	#237	1	1	yes	characteristic
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	#389	1	1	yes	characteristic
165	C₂²×C₃	C₈×Q₈	1	1	yes	characteristic
166	C₂²×C₃	#446	1	1	yes	characteristic
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
171	C₃×C₄	#424	1	1	yes	characteristic
172	C₃×C₄	C₈×D₄	1	1	yes	characteristic
173	C₃×C₄	#436	1	1	yes	characteristic
174	C₃×C₄	#472	1	1	yes	characteristic
175	C₃×C₄	#473	1	1	yes	characteristic
176	C₂³	Q₈×Dic₃	1	1	yes	characteristic
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	#888	1	1	yes	characteristic
181	C₂×C₄	#890	1	1	yes	characteristic
182	C₂×C₄	#895	1	1	yes	characteristic
183	C₂×C₄	#897	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
185	C₂×C₄	C₂×#208	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
188	C₂×C₄	#948	1	1	yes	characteristic
189	C₂×C₄	#948	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	4	no	—
193	C₈	—	4	2	no	—
194	C₈	—	12	2	no	—
195	C₈	—	12	1	no	—
196	C₈	—	12	1	no	—
197	C₈	—	24	2	no	—
198	C₈	C₄⋊(C₃⋊C₈)	1	2	yes	—
199	C₈	C₄⋊(C₃⋊C₈)	1	2	yes	—
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	#1939	1	1	yes	characteristic
203	C₂×C₃	#1952	1	1	yes	characteristic
204	C₂×C₃	#1953	1	1	yes	characteristic
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	#4984	1	1	yes	characteristic
208	C₂²	Q₈×C₃⋊C₈	1	1	yes	characteristic
209	C₂²	#5024	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	#4896	1	1	yes	characteristic
214	C₄	#4897	1	1	yes	characteristic
215	C₄	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
216	C₄	#4999	1	1	yes	characteristic
217	C₄	D₄×C₃⋊C₈	1	1	yes	characteristic
218	C₄	#5012	1	1	yes	characteristic
219	C₃	#9608	1	1	yes	characteristic, complemented, Sylow
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#69484	1	1	yes	characteristic
223	C₂	#70701	1	1	yes	characteristic
224	C₂	#70707	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#895
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×#108	C₂
socle	C₂²×C₃	#389
prime radical	C₂³×C₃	C₄×Q₈
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×#108	D₃
hypocenter	C₃	#9608
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
33. C₃×#108	C₈	semidirect
96. #108	C₃⋊C₈	semidirect
219. C₃	#9608	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#108	2
#69850	2
C₂×#4458	2
#69724	2
#67668	2
#9608	3