Subgroups — #10683402

Order 768.

462 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#69393	C₂	1	1	yes	maximal, characteristic
3	#69391	C₂	1	1	yes	maximal, characteristic
4	#67642	C₂	1	4	yes	maximal
5	#7624	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×Q₈	C₄	1	2	yes	—
8	C₃×#391	C₂²	1	2	yes	—
9	C₃×#389	—	2	1	no	—
10	C₃×#389	—	2	1	no	—
11	C₃×#384	C₄	1	1	yes	characteristic
12	C₃×C₄×C₄⋊C₄	—	2	1	no	—
13	C₃×C₄×C₄⋊C₄	—	2	1	no	—
14	C₃×C₄³	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
16	C₂×#803	—	2	1	no	—
17	#4524	—	2	1	no	—
18	#4524	—	2	1	no	—
19	#4524	C₂²	1	2	yes	—
20	#4524	C₂²	1	2	yes	—
21	C₄×C₄⋊Q₈	D₃	1	1	yes	characteristic
22	#1839	—	3	1	no	—
23	#1767	—	3	1	no	—
24	#1272	—	3	4	no	—
25	C₂²×C₃×Q₈	C₂×C₄	1	2	yes	—
26	C₃×C₄⋊Q₈	—	2	4	no	—
27	C₃×C₄×Q₈	—	4	4	no	—
28	C₂×C₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄²	—	2	2	no	—
35	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
42	C₂²×C₃⋊C₈	—	2	2	no	—
43	C₂²×C₃⋊C₈	—	2	2	no	—
44	C₃×#95	—	2	2	no	—
45	C₃×#95	—	2	2	no	—
46	C₄⋊(C₃⋊C₈)	—	2	4	no	—
47	#803	—	4	2	no	—
48	C₂×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
49	C₂×C₄×Q₈	Dic₃	1	2	yes	—
50	C₂×C₄⋊C₈	—	6	1	no	—
51	C₂×#97	—	6	1	no	—
52	#391	C₂×D₃	1	2	yes	—
53	#389	—	2	1	no	—
54	#389	—	2	1	no	—
55	#384	Dic₃	1	1	yes	characteristic
56	C₄×C₄⋊C₄	—	2	1	no	—
57	C₄×C₄⋊C₄	—	2	1	no	—
58	C₄³	C₂×D₃	1	1	yes	characteristic
59	#336	—	3	2	no	—
60	#336	—	3	2	no	—
61	#336	—	6	1	no	—
62	#336	—	6	1	no	—
63	C₂×C₃×Q₈	—	2	2	no	—
64	C₂×C₃×Q₈	—	4	2	no	—
65	C₂×C₃×Q₈	—	4	2	no	—
66	C₂×C₃×Q₈	M₁₆	1	4	yes	—
67	C₂²×C₃×C₄	—	2	2	no	—
68	C₂²×C₃×C₄	—	2	2	no	—
69	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄⋊C₄	—	2	4	no	—
77	C₃×C₄⋊C₄	—	2	4	no	—
78	C₃×C₄⋊C₄	—	4	2	no	—
79	C₃×C₄⋊C₄	—	4	2	no	—
80	C₃×C₄⋊C₄	—	4	4	no	—
81	C₃×C₄⋊C₄	—	4	2	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	2	2	no	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	—	2	2	no	—
86	C₃×C₄²	—	4	1	no	—
87	C₃×C₄²	—	4	1	no	—
88	C₃×C₄²	—	4	4	no	—
89	C₃×C₄²	SD₁₆	1	2	yes	—
90	C₃×C₄²	Q₁₆	1	2	yes	—
91	C₂×C₃⋊C₈	—	2	4	no	—
92	C₂×C₃⋊C₈	—	4	2	no	—
93	C₂×C₃⋊C₈	—	4	2	no	—
94	C₂×C₃⋊C₈	—	4	2	no	—
95	C₂×C₃⋊C₈	—	4	2	no	—
96	C₂×C₃⋊C₈	—	4	2	no	—
97	C₂²×Q₈	C₂×Dic₃	1	2	yes	—
98	C₂²×C₈	—	6	2	no	—
99	C₂²×C₈	—	6	2	no	—
100	C₄⋊Q₈	—	2	4	no	—
101	C₄×Q₈	—	4	4	no	—
102	C₂×C₄⋊C₄	—	2	1	no	—
103	C₂×C₄⋊C₄	—	2	1	no	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
108	C₂×C₄²	—	2	2	no	—
109	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₄²	#67	1	1	yes	characteristic
112	C₂×C₄²	#67	1	1	yes	characteristic
113	C₂×C₄²	#67	1	1	yes	characteristic
114	C₂×C₄²	#67	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
116	C₄⋊C₈	—	6	4	no	—
117	#97	—	12	2	no	—
118	#95	—	2	2	no	—
119	#95	—	2	2	no	—
120	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₃×Q₈	—	4	4	no	—
122	C₃×Q₈	—	4	4	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	4	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	—	4	2	no	—
141	C₂×C₃×C₄	—	4	2	no	—
142	C₂×C₃×C₄	—	4	2	no	—
143	C₂×C₃×C₄	#100	1	2	yes	—
144	C₂×C₃×C₄	#103	1	2	yes	—
145	C₂×C₃×C₄	#103	1	2	yes	—
146	C₂×C₃×C₄	#104	1	2	yes	—
147	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
148	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
150	C₂×C₃×C₄	#137	1	2	yes	—
151	C₃⋊C₈	—	4	4	no	—
152	C₃⋊C₈	—	8	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	#208	1	4	yes	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	#217	1	1	yes	characteristic
160	C₂²×C₄	#217	1	1	yes	characteristic
161	C₂²×C₄	#217	1	1	yes	characteristic
162	C₂²×C₄	#217	1	1	yes	characteristic
163	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×#67	1	1	yes	characteristic
165	C₂²×C₄	C₂×#67	1	1	yes	characteristic
166	C₂×C₈	—	6	4	no	—
167	C₂×C₈	—	12	2	no	—
168	C₂×C₈	—	12	2	no	—
169	C₂×C₈	—	12	2	no	—
170	C₂×C₈	—	12	2	no	—
171	C₂×C₈	—	12	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	4	1	no	—
183	C₄²	—	4	1	no	—
184	C₄²	—	4	4	no	—
185	C₄²	#215	1	2	yes	—
186	C₄²	#216	1	2	yes	—
187	C₂²×C₃	#407	1	1	yes	characteristic
188	C₂²×C₃	#410	1	1	yes	characteristic
189	C₂²×C₃	C₂×#100	1	1	yes	characteristic
190	C₂²×C₃	C₂×#103	1	1	yes	characteristic
191	C₂²×C₃	#419	1	1	yes	characteristic
192	C₂²×C₃	C₂×#104	1	1	yes	characteristic
193	C₂²×C₃	#421	1	1	yes	characteristic
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	2	no	—
201	C₃×C₄	—	2	2	no	—
202	C₃×C₄	—	4	4	no	—
203	C₃×C₄	—	8	2	no	—
204	C₂³	C₂×#217	1	1	yes	characteristic
205	Q₈	—	4	4	no	—
206	Q₈	—	4	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	#833	1	2	yes	—
228	C₂×C₄	#835	1	2	yes	—
229	C₂×C₄	#835	1	2	yes	—
230	C₂×C₄	#837	1	2	yes	—
231	C₂×C₄	C₂×#208	1	2	yes	—
232	C₂×C₄	C₂×#215	1	1	yes	characteristic
233	C₂×C₄	#942	1	2	yes	—
234	C₂×C₄	C₂×#216	1	1	yes	characteristic
235	C₈	—	12	4	no	—
236	C₈	—	24	2	no	—
237	C₂×C₃	#1466	1	1	yes	characteristic
238	C₂×C₃	#1487	1	1	yes	characteristic
239	C₂×C₃	#1491	1	1	yes	characteristic
240	C₂×C₃	#1503	1	1	yes	characteristic
241	C₂×C₃	#1515	1	1	yes	characteristic
242	C₂×C₃	#1530	1	1	yes	characteristic
243	C₂×C₃	#1548	1	1	yes	characteristic
244	C₂²	#5207	1	1	yes	characteristic
245	C₂²	C₂×#833	1	1	yes	characteristic
246	C₂²	C₂×#835	1	1	yes	characteristic
247	C₂²	#5225	1	1	yes	characteristic
248	C₂²	C₂×#837	1	1	yes	characteristic
249	C₂²	#5236	1	1	yes	characteristic
250	C₂²	#5240	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	4	no	—
260	C₄	—	8	2	no	—
261	C₃	#7624	1	1	yes	characteristic, complemented, Sylow
262	C₂	#71603	1	1	yes	characteristic
263	C₂	#71622	1	1	yes	characteristic
264	C₂	#71623	1	1	yes	characteristic
265	C₂	#71774	1	1	yes	characteristic
266	C₂	#71798	1	1	yes	characteristic
267	C₂	#71835	1	1	yes	characteristic
268	C₂	#71837	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#217
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊Q₈	D₃
hypocenter	C₃	#7624
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
261. C₃	#7624	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₄⋊Q₈	2
#69393	2
#69391	2
#67642	2
#7624	3