Subgroups — C₃×#29808 #10648415

Order 768.

914 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₃×#2857	C₂	1	2	yes	maximal
4	C₃×#2056	C₂	1	4	yes	maximal, complemented
5	C₃×#2001	C₂	1	4	yes	maximal
6	C₃×#1956	C₂	1	2	yes	maximal, complemented
7	C₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	#29808	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₁₆	C₂²	1	4	yes	—
10	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
12	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₃×#494	—	2	4	no	—
14	C₃×#431	—	2	4	no	—
15	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃×#103	C₂²	1	8	yes	—
17	C₂×C₃×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₃×#391	C₂²	1	4	yes	—
21	C₃×#389	—	2	2	no	—
22	C₃×#389	—	2	2	no	—
23	C₃×#384	C₂²	1	2	yes	—
24	C₃×#380	—	2	2	no	—
25	C₃×#380	—	2	2	no	—
26	C₃×#340	C₂²	1	4	yes	—
27	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×#431	C₂×C₃	1	1	yes	characteristic
29	#2857	C₂×C₃	1	2	yes	—
30	#2056	C₂×C₃	1	4	yes	complemented
31	#2001	C₂×C₃	1	4	yes	—
32	#1956	C₂×C₃	1	2	yes	complemented
33	#1834	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂²×C₃×Q₈	C₂³	1	4	yes	—
35	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
36	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
37	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂×C₃×Q₁₆	—	2	8	no	—
39	C₂×C₃×Q₁₆	—	4	8	no	—
40	C₂²×C₃×C₈	C₂³	1	2	yes	—
41	C₂²×C₃×C₈	C₂³	1	2	yes	—
42	C₃×C₄⋊Q₈	—	2	4	no	—
43	C₃×C₄⋊Q₈	—	2	4	no	—
44	C₃×C₂²⋊Q₈	—	4	8	no	—
45	C₂×C₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₃×C₄⋊C₄	—	2	4	no	—
50	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
51	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
52	C₂×C₃×C₄²	—	2	2	no	—
53	C₂×C₃×C₄²	—	2	2	no	—
54	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
55	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
58	C₃×C₄⋊C₈	—	2	4	no	—
59	C₃×#103	—	4	8	no	—
60	C₃×C₂²⋊C₈	—	2	4	no	—
61	C₃×C₄×C₈	—	2	2	no	—
62	C₃×C₄×C₈	D₄	1	4	yes	—
63	C₃×#95	—	2	4	no	—
64	C₃×#95	—	2	4	no	—
65	C₂²×Q₁₆	C₂²×C₃	1	4	yes	—
66	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
67	C₂×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
68	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
69	#494	—	2	4	no	—
70	#431	—	2	4	no	—
71	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
72	C₂×#103	C₂²×C₃	1	8	yes	—
73	C₂×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
74	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
76	#391	C₂²×C₃	1	4	yes	—
77	#389	—	2	2	no	—
78	#389	—	2	2	no	—
79	#384	C₂²×C₃	1	2	yes	—
80	#380	—	2	2	no	—
81	#380	—	2	2	no	—
82	#340	C₂²×C₃	1	4	yes	—
83	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×Q₈	—	2	4	no	—
85	C₂×C₃×Q₈	—	2	4	no	—
86	C₂×C₃×Q₈	—	2	8	no	—
87	C₂×C₃×Q₈	—	2	8	no	—
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	2	4	no	—
92	C₂²×C₃×C₄	—	2	1	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	—	2	4	no	—
95	C₂²×C₃×C₄	—	2	4	no	—
96	C₂²×C₃×C₄	—	2	4	no	—
97	C₂²×C₃×C₄	Q₁₆	1	4	yes	complemented
98	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
104	C₃×Q₁₆	—	4	16	no	—
105	C₂×C₃×C₈	—	2	4	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	4	no	—
108	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
109	C₃×C₄⋊C₄	—	2	8	no	—
110	C₃×C₄⋊C₄	—	4	8	no	—
111	C₃×C₄⋊C₄	—	4	4	no	—
112	C₃×C₄⋊C₄	—	4	4	no	—
113	C₃×C₄⋊C₄	—	4	4	no	—
114	C₃×C₄⋊C₄	—	4	4	no	—
115	C₃×C₂²⋊C₄	—	4	8	no	—
116	C₃×C₄²	—	2	2	no	—
117	C₃×C₄²	—	2	2	no	—
118	C₃×C₄²	—	2	1	no	—
119	C₃×C₄²	—	2	1	no	—
120	C₃×C₄²	C₂×D₄	1	2	yes	—
121	C₃×C₄²	#41	1	2	yes	—
122	C₂²×Q₈	C₂³×C₃	1	4	yes	—
123	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
124	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
125	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₂×Q₁₆	—	2	8	no	—
127	C₂×Q₁₆	—	4	8	no	—
128	C₂²×C₈	C₂³×C₃	1	2	yes	—
129	C₂²×C₈	C₂³×C₃	1	2	yes	—
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₂²⋊Q₈	—	4	8	no	—
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
139	C₂×C₂²⋊C₄	—	2	4	no	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
143	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
144	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
145	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
146	C₄⋊C₈	—	2	4	no	—
147	#103	—	4	8	no	—
148	C₂²⋊C₈	—	2	4	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	C₃×D₄	1	4	yes	—
151	#95	—	2	4	no	—
152	#95	—	2	4	no	—
153	C₂³×C₃	—	2	1	no	—
154	C₂³×C₃	—	2	2	no	—
155	C₂³×C₃	—	2	1	no	—
156	C₂³×C₃	—	2	2	no	—
157	C₂³×C₃	C₂×Q₁₆	1	2	yes	—
158	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
159	C₃×Q₈	—	4	8	no	—
160	C₃×Q₈	—	4	8	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	8	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	4	2	no	—
169	C₂×C₃×C₄	—	4	2	no	—
170	C₂×C₃×C₄	—	4	4	no	—
171	C₂×C₃×C₄	—	4	4	no	—
172	C₂×C₃×C₄	—	4	4	no	—
173	C₂×C₃×C₄	—	4	4	no	—
174	C₂×C₃×C₄	—	4	4	no	—
175	C₂×C₃×C₄	—	4	4	no	—
176	C₂×C₃×C₄	—	4	4	no	—
177	C₂×C₃×C₄	#121	1	2	yes	—
178	C₂×C₃×C₄	#121	1	2	yes	—
179	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
180	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
181	C₂×C₃×C₄	#135	1	2	yes	—
182	C₂×C₃×C₄	#137	1	4	yes	—
183	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
185	C₂×C₃×C₄	H(5,𝔽₂)	1	4	yes	—
186	C₃×C₈	—	2	4	no	—
187	C₃×C₈	—	4	2	no	—
188	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	8	no	—
192	C₂×Q₈	—	2	8	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₃×Q₁₆	1	4	yes	complemented
203	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
209	Q₁₆	—	4	16	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	4	8	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₂²⋊C₄	—	4	8	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	C₂×C₃×D₄	1	2	yes	—
226	C₄²	C₃×#41	1	2	yes	—
227	C₂²×C₃	—	2	4	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	4	1	no	—
230	C₂²×C₃	—	4	2	no	—
231	C₂²×C₃	—	4	1	no	—
232	C₂²×C₃	C₂×#121	1	1	yes	characteristic
233	C₂²×C₃	#535	1	2	yes	—
234	C₂²×C₃	C₂²×Q₁₆	1	1	yes	characteristic
235	C₂²×C₃	C₂×#135	1	1	yes	characteristic
236	C₂²×C₃	C₂×#137	1	2	yes	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	4	1	no	—
239	C₃×C₄	—	4	1	no	—
240	C₃×C₄	—	8	4	no	—
241	C₃×C₄	—	8	4	no	—
242	C₃×C₄	#467	1	4	yes	—
243	C₃×C₄	#470	1	4	yes	—
244	C₃×C₄	#535	1	2	yes	—
245	C₃×C₄	#540	1	2	yes	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₃×Q₁₆	1	2	yes	—
251	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
252	Q₈	—	4	8	no	—
253	Q₈	—	4	8	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	C₃×#121	1	2	yes	—
271	C₂×C₄	C₃×#121	1	2	yes	—
272	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
273	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
274	C₂×C₄	C₃×#135	1	2	yes	—
275	C₂×C₄	C₃×#137	1	4	yes	—
276	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
278	C₂×C₄	C₃×H(5,𝔽₂)	1	4	yes	—
279	C₈	—	2	4	no	—
280	C₈	—	4	2	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	#2619	1	1	yes	characteristic
283	C₂×C₃	C₂×#467	1	1	yes	characteristic
284	C₂×C₃	C₂×#470	1	1	yes	characteristic
285	C₂×C₃	#3209	1	2	yes	—
286	C₂×C₃	#3210	1	2	yes	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	C₂×C₃×#121	1	1	yes	characteristic
293	C₂²	C₃×#535	1	2	yes	—
294	C₂²	C₂²×C₃×Q₁₆	1	1	yes	characteristic
295	C₂²	C₂×C₃×#135	1	1	yes	characteristic
296	C₂²	C₂×C₃×#137	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	8	4	no	—
301	C₄	—	8	4	no	—
302	C₄	C₃×#467	1	4	yes	—
303	C₄	C₃×#470	1	4	yes	—
304	C₄	C₃×#535	1	2	yes	—
305	C₄	C₃×#540	1	2	yes	—
306	C₃	#29808	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	4	2	no	—
308	C₂	C₃×#2619	1	1	yes	characteristic
309	C₂	C₂×C₃×#467	1	1	yes	characteristic
310	C₂	C₂×C₃×#470	1	1	yes	characteristic
311	C₂	C₃×#3209	1	2	yes	—
312	C₂	C₃×#3210	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#494	C₂	semidirect
4. C₃×#2056	C₂	semidirect
6. C₃×#1956	C₂	semidirect
7. C₃×#1834	C₂	semidirect
8. #29808	C₃	direct (N × H)
27. C₂×#494	C₂×C₃	semidirect
30. #2056	C₂×C₃	semidirect
32. #1956	C₂×C₃	semidirect
33. #1834	C₂×C₃	semidirect
97. C₂²×C₃×C₄	Q₁₆	semidirect
202. C₂²×C₄	C₃×Q₁₆	semidirect
306. C₃	#29808	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×#494	2
C₂×C₃×#431	2
C₃×#2857	2
C₃×#2056	2
C₃×#2001	2
C₃×#1956	2
C₃×#1834	2
#29808	3

Subgroups — C3×#29808 #10648415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29808 #10648415