Subgroups — C₃×#21276 #10639883

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#2295	C₂	1	2	yes	maximal
3	C₃×#1969	C₂	1	2	yes	maximal
4	C₃×#1942	C₂	1	2	yes	maximal
5	C₃×#1933	C₂	1	4	yes	maximal
6	C₃×C₄×#107	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×#103	C₂	1	2	yes	maximal
8	C₃×C₄²×C₈	C₂	1	1	yes	maximal, characteristic
9	#21276	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×Q₈	C₂²	1	2	yes	—
12	C₃×C₄×Q₁₆	C₄	1	8	yes	complemented
13	C₂×C₃×#107	C₂²	1	2	yes	—
14	C₂×C₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₃×#103	C₂²	1	4	yes	—
16	C₂×C₃×#103	C₂²	1	2	yes	—
17	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
19	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
20	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₃×#391	C₂²	1	4	yes	—
22	C₃×#389	C₂²	1	4	yes	—
23	C₃×#387	—	2	4	no	—
24	C₃×C₄×C₄⋊C₄	—	2	2	no	—
25	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
26	C₃×#376	—	2	2	no	—
27	C₃×C₄³	C₂²	1	1	yes	characteristic
28	C₃×#340	C₂²	1	4	yes	—
29	C₃×#340	C₂²	1	2	yes	—
30	C₂×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
31	#2295	C₂×C₃	1	2	yes	—
32	#1969	C₂×C₃	1	2	yes	—
33	#1942	C₂×C₃	1	2	yes	—
34	#1933	C₂×C₃	1	4	yes	—
35	C₄×#107	C₂×C₃	1	1	yes	characteristic
36	C₄×#103	C₂×C₃	1	2	yes	—
37	C₄²×C₈	C₂×C₃	1	1	yes	characteristic
38	C₂²×C₃×Q₈	C₂³	1	2	yes	—
39	C₂×C₃×Q₁₆	—	2	4	no	—
40	C₂×C₃×Q₁₆	C₂×C₄	1	4	yes	—
41	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂³	1	2	yes	—
43	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
44	C₃×C₄×Q₈	—	2	4	no	—
45	C₃×C₄×Q₈	C₂×C₄	1	8	yes	—
46	C₂×C₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₃×C₄⋊C₄	—	2	4	no	—
48	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₃×C₄²	—	2	2	no	—
51	C₂×C₃×C₄²	—	2	2	no	—
52	C₂×C₃×C₄²	D₄	1	2	yes	—
53	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
54	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂³	1	2	yes	—
56	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
57	C₃×#107	—	2	4	no	—
58	C₃×#107	C₂×C₄	1	4	yes	—
59	C₃×#103	—	2	8	no	—
60	C₃×#103	C₂×C₄	1	8	yes	—
61	C₃×C₄×C₈	—	2	4	no	—
62	C₃×C₄×C₈	—	2	4	no	—
63	C₃×C₄×C₈	—	2	2	no	—
64	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
65	C₃×#95	—	2	4	no	—
66	C₃×#95	—	2	4	no	—
67	C₃×#95	—	2	4	no	—
68	C₃×#95	—	2	2	no	—
69	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₄×Q₈	C₂²×C₃	1	2	yes	—
71	C₄×Q₁₆	C₃×C₄	1	8	yes	complemented
72	C₂×#107	C₂²×C₃	1	2	yes	—
73	C₂×#107	C₂²×C₃	1	1	yes	characteristic
74	C₂×#103	C₂²×C₃	1	4	yes	—
75	C₂×#103	C₂²×C₃	1	2	yes	—
76	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
77	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
78	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
80	#391	C₂²×C₃	1	4	yes	—
81	#389	C₂²×C₃	1	4	yes	—
82	#387	—	2	4	no	—
83	C₄×C₄⋊C₄	—	2	2	no	—
84	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
85	#376	—	2	2	no	—
86	C₄³	C₂²×C₃	1	1	yes	characteristic
87	#340	C₂²×C₃	1	2	yes	—
88	#340	C₂²×C₃	1	4	yes	—
89	C₂×C₃×Q₈	—	2	2	no	—
90	C₂×C₃×Q₈	—	2	4	no	—
91	C₂×C₃×Q₈	—	2	4	no	—
92	C₂×C₃×Q₈	C₂²×C₄	1	4	yes	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	—	2	2	no	—
95	C₂²×C₃×C₄	—	2	4	no	—
96	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
98	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
99	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
101	C₃×Q₁₆	—	2	8	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₃×C₈	#41	1	4	yes	—
108	C₂×C₃×C₈	#41	1	4	yes	—
109	C₃×C₄⋊C₄	—	2	8	no	—
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₄⋊C₄	—	2	4	no	—
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄⋊C₄	—	4	8	no	—
114	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
115	C₃×C₄²	—	2	8	no	—
116	C₃×C₄²	—	2	4	no	—
117	C₃×C₄²	—	2	2	no	—
118	C₃×C₄²	—	2	2	no	—
119	C₃×C₄²	—	2	2	no	—
120	C₃×C₄²	—	4	4	no	—
121	C₃×C₄²	Q₁₆	1	4	yes	complemented
122	C₃×C₄²	C₂²×C₄	1	2	yes	—
123	C₃×C₄²	#41	1	2	yes	—
124	C₂²×Q₈	C₂³×C₃	1	2	yes	—
125	C₂×Q₁₆	—	2	4	no	—
126	C₂×Q₁₆	C₂×C₃×C₄	1	4	yes	—
127	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
128	C₂²×C₈	C₂³×C₃	1	2	yes	—
129	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
130	C₄×Q₈	—	2	4	no	—
131	C₄×Q₈	C₂×C₃×C₄	1	8	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
135	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	—	2	2	no	—
138	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
139	C₂×C₄²	C₃×D₄	1	2	yes	—
140	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
141	C₂×C₄²	C₂³×C₃	1	2	yes	—
142	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
143	#107	—	2	4	no	—
144	#107	C₂×C₃×C₄	1	4	yes	—
145	#103	—	2	8	no	—
146	#103	C₂×C₃×C₄	1	8	yes	—
147	C₄×C₈	—	2	4	no	—
148	C₄×C₈	—	2	4	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	C₂×C₃×C₄	1	4	yes	—
151	#95	—	2	4	no	—
152	#95	—	2	4	no	—
153	#95	—	2	4	no	—
154	#95	—	2	2	no	—
155	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
156	C₃×Q₈	—	2	4	no	—
157	C₃×Q₈	—	2	4	no	—
158	C₃×Q₈	—	4	4	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	4	4	no	—
167	C₂×C₃×C₄	—	4	4	no	—
168	C₂×C₃×C₄	—	4	4	no	—
169	C₂×C₃×C₄	—	4	4	no	—
170	C₂×C₃×C₄	—	4	4	no	—
171	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
172	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
173	C₂×C₃×C₄	C₂×Q₁₆	1	4	yes	—
174	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
175	C₂×C₃×C₄	#135	1	4	yes	—
176	C₂×C₃×C₄	#135	1	2	yes	—
177	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×#41	1	2	yes	—
179	C₂×C₃×C₄	C₂×#41	1	2	yes	—
180	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
181	C₃×C₈	—	2	2	no	—
182	C₃×C₈	—	2	4	no	—
183	C₃×C₈	#117	1	4	yes	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	C₂²×C₃×C₄	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
193	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
194	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
196	Q₁₆	—	2	8	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
201	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
202	C₂×C₈	C₃×#41	1	4	yes	—
203	C₂×C₈	C₃×#41	1	4	yes	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	8	no	—
209	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
210	C₄²	—	2	8	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	4	no	—
216	C₄²	C₃×Q₁₆	1	4	yes	complemented
217	C₄²	C₂²×C₃×C₄	1	2	yes	—
218	C₄²	C₃×#41	1	2	yes	—
219	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂²×C₃	#532	1	2	yes	—
221	C₂²×C₃	C₂²×Q₁₆	1	1	yes	characteristic
222	C₂²×C₃	C₂×#135	1	1	yes	characteristic
223	C₂²×C₃	C₂×#135	1	2	yes	—
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	4	4	no	—
228	C₃×C₄	—	4	4	no	—
229	C₃×C₄	—	8	4	no	—
230	C₃×C₄	C₄×Q₁₆	1	4	yes	—
231	C₃×C₄	C₂×#117	1	1	yes	characteristic
232	C₃×C₄	C₄×#41	1	1	yes	characteristic
233	C₃×C₄	#529	1	2	yes	—
234	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
235	Q₈	—	2	4	no	—
236	Q₈	—	2	4	no	—
237	Q₈	—	4	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
251	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
252	C₂×C₄	C₂×C₃×Q₁₆	1	4	yes	—
253	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
254	C₂×C₄	C₃×#135	1	4	yes	—
255	C₂×C₄	C₃×#135	1	2	yes	—
256	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×#41	1	2	yes	—
258	C₂×C₄	C₂×C₃×#41	1	2	yes	—
259	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	4	no	—
262	C₈	C₃×#117	1	4	yes	—
263	C₂×C₃	#2293	1	1	yes	characteristic
264	C₂×C₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
265	C₂×C₃	C₄×#135	1	1	yes	characteristic
266	C₂×C₃	#3168	1	2	yes	—
267	C₂×C₃	#3170	1	2	yes	—
268	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
269	C₂²	C₃×#532	1	2	yes	—
270	C₂²	C₂²×C₃×Q₁₆	1	1	yes	characteristic
271	C₂²	C₂×C₃×#135	1	1	yes	characteristic
272	C₂²	C₂×C₃×#135	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	—	8	4	no	—
279	C₄	C₃×C₄×Q₁₆	1	4	yes	—
280	C₄	C₂×C₃×#117	1	1	yes	characteristic
281	C₄	C₃×C₄×#41	1	1	yes	characteristic
282	C₄	C₃×#529	1	2	yes	—
283	C₃	#21276	1	1	yes	characteristic, direct product factor, Sylow
284	C₂	C₃×#2293	1	1	yes	characteristic
285	C₂	C₂×C₃×C₄×Q₁₆	1	1	yes	characteristic
286	C₂	C₃×C₄×#135	1	1	yes	characteristic
287	C₂	C₃×#3168	1	2	yes	—
288	C₂	C₃×#3170	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₄
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle, prime radical	C₂³×C₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×C₄³ ◁ G

Complements

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

N	H ≅ G/N	type
9. #21276	C₃	direct (N × H)
12. C₃×C₄×Q₁₆	C₄	semidirect
71. C₄×Q₁₆	C₃×C₄	semidirect
121. C₃×C₄²	Q₁₆	semidirect
216. C₄²	C₃×Q₁₆	semidirect
283. C₃	#21276	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×Q₁₆	2
C₃×#2295	2
C₃×#1969	2
C₃×#1942	2
C₃×#1933	2
C₃×C₄×#107	2
C₃×C₄×#103	2
C₃×C₄²×C₈	2
#21276	3

Subgroups — C3×#21276 #10639883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21276 #10639883