Subgroups — #10948332

Order 768.

528 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#131	C₂	1	1	yes	maximal, characteristic
2	#72007	C₂	1	2	yes	maximal
3	#69886	C₂	1	4	yes	maximal, complemented
4	#69884	C₂	1	2	yes	maximal, complemented
5	#69864	C₂	1	4	yes	maximal, complemented
6	C₂×#4463	C₂	1	2	yes	maximal, complemented
7	#33428	—	3	1	no	maximal, Sylow
8	C₂×C₃×#131	C₄	1	1	yes	characteristic
9	C₂×C₃×#131	C₂²	1	1	yes	characteristic
10	C₃×C₄×#41	C₄	1	1	yes	characteristic
11	C₃×#405	C₄	1	1	yes	characteristic
12	C₃×#405	C₄	1	2	yes	—
13	C₃×#405	C₂²	1	1	yes	characteristic
14	C₃×#405	C₂²	1	2	yes	—
15	C₃×C₄×M₁₆	C₄	1	1	yes	characteristic
16	C₃×C₄×M₁₆	C₂²	1	2	yes	—
17	C₂×C₃×C₄×C₈	C₄	1	2	yes	—
18	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×#812	C₂²	1	4	yes	—
20	#4553	C₂²	1	8	yes	—
21	#4466	C₂²	1	8	yes	—
22	#4463	C₂²	1	4	yes	—
23	#4463	C₂²	1	4	yes	—
24	C₄×#131	D₃	1	1	yes	characteristic
25	#2166	—	3	4	no	—
26	#2165	—	3	4	no	—
27	C₂×#364	—	3	2	no	—
28	#2115	—	3	2	no	—
29	#2110	—	3	2	no	—
30	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
31	C₃×#131	—	2	4	no	—
32	C₃×#131	—	2	2	no	—
33	C₃×#131	—	2	2	no	—
34	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×M₁₆	C₂³	1	2	yes	—
38	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
42	C₃×C₄×Q₈	—	2	1	no	—
43	C₃×C₄×D₄	—	2	2	no	—
44	C₃×C₄×D₄	—	2	1	no	—
45	C₃×#117	C₂×C₄	1	2	yes	—
46	C₃×#117	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
48	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
49	C₃×#97	C₂×C₄	1	2	yes	—
50	C₃×#97	C₂×C₄	1	2	yes	—
51	C₃×#97	C₂³	1	2	yes	—
52	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
54	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
55	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
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	—	2	4	no	—
60	#812	C₂³	1	8	yes	—
61	C₂×C₃⋊C₁₆	—	2	4	no	—
62	C₂×#131	Dic₃	1	1	yes	characteristic
63	C₂×#131	C₂×D₃	1	1	yes	characteristic
64	C₄×#41	Dic₃	1	1	yes	characteristic
65	C₂×M₃₂	—	3	4	no	—
66	#405	Dic₃	1	1	yes	characteristic
67	#405	Dic₃	1	2	yes	—
68	#405	C₂×D₃	1	2	yes	—
69	#405	C₂×D₃	1	1	yes	characteristic
70	C₄×M₁₆	Dic₃	1	1	yes	characteristic
71	C₄×M₁₆	C₂×D₃	1	2	yes	—
72	C₂×C₄×C₈	Dic₃	1	2	yes	—
73	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
74	#364	—	3	4	no	—
75	#364	—	3	4	no	—
76	#349	—	3	8	no	—
77	#347	—	3	8	no	—
78	C₃×#41	—	2	2	no	—
79	C₃×#41	—	2	2	no	—
80	C₂×C₃×Q₈	M₁₆	1	1	yes	characteristic
81	C₂×C₃×D₄	M₁₆	1	2	yes	—
82	C₂×C₃×D₄	M₁₆	1	1	yes	characteristic
83	C₂²×C₃×C₄	—	2	2	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
86	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₃×M₁₆	—	2	2	no	—
88	C₃×M₁₆	—	2	2	no	—
89	C₃×M₁₆	—	2	4	no	—
90	C₃×M₁₆	—	2	4	no	—
91	C₂×C₃×C₈	—	2	2	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	4	no	—
94	C₂×C₃×C₈	M₁₆	1	4	yes	—
95	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₃×C₈	#41	1	2	yes	—
101	C₂×C₃×C₈	#41	1	2	yes	—
102	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
103	C₃×C₄⋊C₄	—	2	2	no	—
104	C₃×C₄⋊C₄	—	2	1	no	—
105	C₃×C₂²⋊C₄	—	2	2	no	—
106	C₃×C₂²⋊C₄	—	2	1	no	—
107	C₃×C₄²	—	2	2	no	—
108	C₃×C₄²	—	2	1	no	—
109	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₃×C₄²	C₂²×C₄	1	2	yes	—
111	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
112	C₃⋊C₁₆	—	2	8	no	—
113	C₂×#41	C₂×Dic₃	1	1	yes	characteristic
114	#131	—	2	4	no	—
115	#131	—	2	2	no	—
116	#131	—	2	2	no	—
117	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
118	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
119	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
120	C₂×M₁₆	C₂²×D₃	1	2	yes	—
121	C₂²×C₈	C₂×Dic₃	1	2	yes	—
122	C₂²×C₈	C₂×Dic₃	1	2	yes	—
123	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₄×Q₈	—	2	1	no	—
126	C₄×D₄	—	2	2	no	—
127	C₄×D₄	—	2	1	no	—
128	#117	C₂×Dic₃	1	2	yes	—
129	#117	C₂×Dic₃	1	1	yes	characteristic
130	C₂×C₄²	C₂×Dic₃	1	2	yes	—
131	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
132	M₃₂	—	3	8	no	—
133	M₃₂	—	6	4	no	—
134	C₂×C₁₆	—	6	4	no	—
135	#97	C₂×Dic₃	1	2	yes	—
136	#97	C₂×Dic₃	1	2	yes	—
137	#97	C₂²×D₃	1	2	yes	—
138	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
139	C₄×C₈	C₂×Dic₃	1	2	yes	—
140	C₄×C₈	C₂×Dic₃	1	2	yes	—
141	C₄×C₈	C₂×Dic₃	1	2	yes	—
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	C₂×M₁₆	1	2	yes	—
146	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
147	C₃×Q₈	—	2	2	no	—
148	C₃×D₄	—	2	4	no	—
149	C₃×D₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	4	4	no	—
157	C₂×C₃×C₄	#117	1	2	yes	—
158	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
159	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
160	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
162	C₃×C₈	—	2	2	no	—
163	C₃×C₈	—	2	2	no	—
164	C₃×C₈	#117	1	1	yes	characteristic
165	C₃×C₈	#117	1	1	yes	characteristic
166	C₃×C₈	C₂×M₁₆	1	2	yes	—
167	C₃×C₈	C₂×M₁₆	1	2	yes	—
168	C₃×C₈	C₂×#41	1	2	yes	—
169	#41	—	2	2	no	—
170	#41	—	2	2	no	—
171	C₂×Q₈	#208	1	1	yes	characteristic
172	C₂×D₄	#208	1	2	yes	—
173	C₂×D₄	#208	1	1	yes	characteristic
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
178	M₁₆	—	2	2	no	—
179	M₁₆	—	2	2	no	—
180	M₁₆	—	2	4	no	—
181	M₁₆	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	#208	1	4	yes	—
186	C₂×C₈	#235	1	2	yes	—
187	C₂×C₈	#235	1	2	yes	—
188	C₂×C₈	C₂²×Dic₃	1	2	yes	—
189	C₂×C₈	C₂²×Dic₃	1	2	yes	—
190	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
191	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
192	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
193	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	1	no	—
200	C₄²	C₂²×Dic₃	1	1	yes	characteristic
201	C₄²	C₂²×Dic₃	1	2	yes	—
202	C₄²	C₂²×Dic₃	1	1	yes	characteristic
203	C₁₆	—	6	8	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	—	4	2	no	—
207	C₂²×C₃	#432	1	2	yes	—
208	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	4	2	no	—
212	C₃×C₄	—	4	1	no	—
213	C₃×C₄	#432	1	2	yes	—
214	C₃×C₄	C₂×#117	1	1	yes	characteristic
215	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
216	C₂³	C₂×#208	1	2	yes	—
217	C₂³	C₂×#208	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	C₂×#208	1	2	yes	—
229	C₂×C₄	C₂×#208	1	1	yes	characteristic
230	C₂×C₄	C₂×#208	1	2	yes	—
231	C₂×C₄	#926	1	2	yes	—
232	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	C₂×#208	1	2	yes	—
236	C₈	C₂×#208	1	2	yes	—
237	C₈	#926	1	1	yes	characteristic
238	C₈	#926	1	1	yes	characteristic
239	C₈	C₂×#235	1	2	yes	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	—	4	2	no	—
243	C₂×C₃	C₂×#432	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	2	no	—
247	C₂²	#4926	1	2	yes	—
248	C₂²	C₂²×#208	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	1	no	—
253	C₄	#4926	1	2	yes	—
254	C₄	C₂²×#208	1	1	yes	characteristic
255	C₄	C₂×#926	1	1	yes	characteristic
256	C₃	#33428	1	1	yes	characteristic, complemented, Sylow
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#4926	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#926
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×C₄×#131	C₂
socle	C₂×C₃	C₂×#432
prime radical	C₂×C₃×#41	C₂×C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₄×#131	D₃
hypocenter	C₃	#33428
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ #405 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #69886	C₂	semidirect
4. #69884	C₂	semidirect
5. #69864	C₂	semidirect
6. C₂×#4463	C₂	semidirect
256. C₃	#33428	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#131	2
#72007	2
#69886	2
#69884	2
#69864	2
C₂×#4463	2
#33428	3