Subgroups — #10948326

Order 768.

524 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2873	C₂	1	1	yes	maximal, characteristic
2	C₂×#4553	C₂	1	2	yes	maximal
3	#69886	C₂	1	4	yes	maximal, complemented
4	#69885	C₂	1	2	yes	maximal, complemented
5	#69884	C₂	1	2	yes	maximal, complemented
6	#69864	C₂	1	4	yes	maximal, complemented
7	#33427	—	3	1	no	maximal, Sylow
8	C₂²×C₃×M₁₆	C₄	1	1	yes	characteristic
9	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×#117	C₄	1	1	yes	characteristic
11	C₃×#405	C₄	1	1	yes	characteristic
12	C₃×#405	C₄	1	2	yes	—
13	C₃×#405	C₄	1	1	yes	characteristic
14	C₃×#405	C₂²	1	1	yes	characteristic
15	C₃×#405	C₂²	1	2	yes	—
16	C₃×#405	C₂²	1	1	yes	characteristic
17	C₃×C₄×M₁₆	C₄	1	2	yes	—
18	C₃×C₄×M₁₆	C₂²	1	2	yes	—
19	C₂×#812	C₂²	1	4	yes	—
20	#4553	C₂²	1	4	yes	—
21	#4553	C₂²	1	4	yes	—
22	#4466	C₂²	1	8	yes	complemented
23	#4463	C₂²	1	8	yes	complemented
24	#2873	D₃	1	1	yes	characteristic
25	#2166	—	3	4	no	—
26	#2165	—	3	4	no	—
27	#2153	—	3	2	no	—
28	C₂×#349	—	3	2	no	—
29	#2110	—	3	2	no	—
30	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×M₁₆	—	2	4	no	—
32	C₂×C₃×M₁₆	—	2	4	no	—
33	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
39	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
41	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
43	C₃×#117	—	2	1	no	—
44	C₃×#117	—	2	1	no	—
45	C₃×#117	C₂×C₄	1	2	yes	—
46	C₃×#117	C₂×C₄	1	1	yes	characteristic
47	C₃×#117	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×C₄⋊C₄	—	2	1	no	—
49	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
50	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
51	C₃×#97	C₂×C₄	1	2	yes	—
52	C₃×#97	C₂×C₄	1	2	yes	—
53	C₃×#97	C₂×C₄	1	2	yes	—
54	C₃×#97	C₂³	1	2	yes	—
55	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
56	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
57	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
58	C₃×C₄×C₈	C₂³	1	2	yes	—
59	#812	—	2	4	no	—
60	#812	C₂³	1	8	yes	—
61	C₂×C₃⋊C₁₆	—	2	4	no	—
62	C₂²×M₁₆	Dic₃	1	1	yes	characteristic
63	C₂²×M₁₆	C₂×D₃	1	1	yes	characteristic
64	C₂×#117	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	Dic₃	1	1	yes	characteristic
69	#405	C₂×D₃	1	2	yes	—
70	#405	C₂×D₃	1	1	yes	characteristic
71	#405	C₂×D₃	1	1	yes	characteristic
72	C₄×M₁₆	Dic₃	1	2	yes	—
73	C₄×M₁₆	C₂×D₃	1	2	yes	—
74	#364	—	3	8	no	—
75	#349	—	3	4	no	—
76	#349	—	3	4	no	—
77	#347	—	3	8	no	—
78	C₂⁴×C₃	M₁₆	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	2	2	no	—
80	C₂²×C₃×C₄	—	2	2	no	—
81	C₂²×C₃×C₄	—	2	2	no	—
82	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
83	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
84	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₃×M₁₆	—	2	2	no	—
89	C₃×M₁₆	—	2	2	no	—
90	C₃×M₁₆	—	2	2	no	—
91	C₃×M₁₆	—	2	4	no	—
92	C₃×M₁₆	—	2	2	no	—
93	C₃×M₁₆	—	2	2	no	—
94	C₃×M₁₆	M₁₆	1	4	yes	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	4	no	—
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	2	yes	—
100	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₃×C₈	#41	1	4	yes	—
103	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	—	2	1	no	—
106	C₃×C₄⋊C₄	—	2	1	no	—
107	C₃×C₂²⋊C₄	—	2	2	no	—
108	C₃×C₂²⋊C₄	—	2	1	no	—
109	C₃×C₂²⋊C₄	—	2	1	no	—
110	C₃×C₄²	—	2	2	no	—
111	C₃×C₄²	C₂²×C₄	1	2	yes	—
112	C₃×C₄²	C₂²×C₄	1	2	yes	—
113	C₃⋊C₁₆	—	2	8	no	—
114	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
115	C₂×M₁₆	—	2	4	no	—
116	C₂×M₁₆	—	2	4	no	—
117	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
118	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
119	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
120	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
121	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
122	C₂×M₁₆	C₂²×D₃	1	1	yes	characteristic
123	C₂×M₁₆	C₂²×D₃	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₃	1	2	yes	—
125	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
127	#117	—	2	1	no	—
128	#117	—	2	1	no	—
129	#117	C₂×Dic₃	1	2	yes	—
130	#117	C₂×Dic₃	1	1	yes	characteristic
131	#117	C₂×Dic₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₂²⋊C₄	—	2	1	no	—
134	C₂×C₄²	C₂×Dic₃	1	2	yes	—
135	M₃₂	—	3	8	no	—
136	M₃₂	—	6	4	no	—
137	C₂×C₁₆	—	6	4	no	—
138	#97	C₂×Dic₃	1	2	yes	—
139	#97	C₂×Dic₃	1	2	yes	—
140	#97	C₂×Dic₃	1	2	yes	—
141	#97	C₂²×D₃	1	2	yes	—
142	C₄×C₈	C₂×Dic₃	1	2	yes	—
143	C₄×C₈	C₂×Dic₃	1	2	yes	—
144	C₄×C₈	C₂×Dic₃	1	2	yes	—
145	C₄×C₈	C₂²×D₃	1	2	yes	—
146	C₂³×C₃	—	2	2	no	—
147	C₂³×C₃	—	8	1	no	—
148	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
149	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
150	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	8	2	no	—
161	C₂×C₃×C₄	#117	1	2	yes	—
162	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
165	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	1	no	—
169	C₃×C₈	—	2	2	no	—
170	C₃×C₈	#117	1	2	yes	—
171	C₃×C₈	C₂×M₁₆	1	4	yes	—
172	C₃×C₈	C₂×#41	1	2	yes	—
173	C₂⁴	#208	1	1	yes	characteristic
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	#208	1	1	yes	characteristic
178	C₂²×C₄	#208	1	1	yes	characteristic
179	C₂²×C₄	#208	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
182	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
183	M₁₆	—	2	2	no	—
184	M₁₆	—	2	4	no	—
185	M₁₆	—	2	2	no	—
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	2	no	—
188	M₁₆	—	2	2	no	—
189	M₁₆	#208	1	4	yes	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	#235	1	4	yes	—
193	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
194	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
195	C₂×C₈	C₂²×Dic₃	1	2	yes	—
196	C₂×C₈	C₂²×Dic₃	1	2	yes	—
197	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
198	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	C₂²×Dic₃	1	2	yes	—
207	C₄²	C₂²×Dic₃	1	2	yes	—
208	C₁₆	—	6	8	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	4	1	no	—
212	C₂²×C₃	—	4	1	no	—
213	C₂²×C₃	—	4	1	no	—
214	C₂²×C₃	—	8	2	no	—
215	C₂²×C₃	#432	1	2	yes	—
216	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	#432	1	2	yes	—
222	C₃×C₄	C₂×#117	1	1	yes	characteristic
223	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
224	C₂³	—	2	2	no	—
225	C₂³	—	8	1	no	—
226	C₂³	C₂×#208	1	1	yes	characteristic
227	C₂³	C₂×#208	1	1	yes	characteristic
228	C₂³	C₂×#208	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	C₂×#208	1	1	yes	characteristic
240	C₂×C₄	C₂×#208	1	1	yes	characteristic
241	C₂×C₄	C₂×#208	1	1	yes	characteristic
242	C₂×C₄	C₂×#208	1	2	yes	—
243	C₂×C₄	#926	1	2	yes	—
244	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	C₂×#208	1	4	yes	—
249	C₈	#926	1	2	yes	—
250	C₈	C₂×#235	1	2	yes	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	2	no	—
253	C₂×C₃	—	4	1	no	—
254	C₂×C₃	—	4	1	no	—
255	C₂×C₃	C₂×#432	1	1	yes	characteristic
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	8	2	no	—
262	C₂²	#4926	1	2	yes	—
263	C₂²	C₂²×#208	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	#4926	1	2	yes	—
269	C₄	C₂²×#208	1	1	yes	characteristic
270	C₄	C₂×#926	1	1	yes	characteristic
271	C₃	#33427	1	1	yes	characteristic, complemented, Sylow
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	4	1	no	—
275	C₂	—	4	1	no	—
276	C₂	C₂×#4926	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#926
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×#2873	C₂
socle	C₂×C₃	C₂×#432
prime radical	C₂⁴×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2873	D₃
hypocenter	C₃	#33427
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2873

Complements

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

N	H ≅ G/N	type
3. #69886	C₂	semidirect
4. #69885	C₂	semidirect
5. #69884	C₂	semidirect
6. #69864	C₂	semidirect
22. #4466	C₂²	semidirect
23. #4463	C₂²	semidirect
271. C₃	#33427	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2873	2
C₂×#4553	2
#69886	2
#69885	2
#69884	2
#69864	2
#33427	3