Subgroups — C₃×#2429 #84293

Order 384.

564 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#398	C₂	1	2	yes	maximal, complemented
6	C₃×#396	C₂	1	2	yes	maximal
7	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#389	C₂	1	2	yes	maximal, complemented
9	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#388	C₂	1	2	yes	maximal
11	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	#2429	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	—	2	4	no	—
17	C₃×C₂²⋊Q₈	—	2	4	no	—
18	C₃×C₂²⋊Q₈	—	2	4	no	—
19	C₃×C₄×Q₈	—	2	4	no	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂²	1	2	yes	—
35	C₃×#95	C₂²	1	1	yes	characteristic
36	C₃×#95	C₂²	1	1	yes	characteristic
37	C₃×#95	C₂²	1	2	yes	—
38	C₃×#95	C₂²	1	2	yes	—
39	C₃×#95	C₂²	1	2	yes	—
40	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
42	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
43	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
44	#398	C₂×C₃	1	2	yes	complemented
45	#396	C₂×C₃	1	2	yes	—
46	#391	C₂×C₃	1	1	yes	characteristic, complemented
47	#389	C₂×C₃	1	2	yes	complemented
48	#389	C₂×C₃	1	1	yes	characteristic, complemented
49	#388	C₂×C₃	1	2	yes	—
50	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
51	C₂⁴×C₃	C₂³	1	1	yes	characteristic
52	C₂×C₃×Q₈	—	2	2	no	—
53	C₂×C₃×Q₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	2	no	—
55	C₂×C₃×Q₈	—	2	2	no	—
56	C₂×C₃×Q₈	—	2	2	no	—
57	C₂×C₃×Q₈	D₄	1	4	yes	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂³	1	2	yes	—
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³	1	2	yes	—
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
71	C₃×C₄⋊C₄	—	2	4	no	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	4	no	—
74	C₃×C₄⋊C₄	—	2	4	no	—
75	C₃×C₄⋊C₄	—	2	4	no	—
76	C₃×C₄⋊C₄	—	2	4	no	—
77	C₃×C₄⋊C₄	—	2	4	no	—
78	C₃×C₄⋊C₄	D₄	1	4	yes	—
79	C₃×C₂²⋊C₄	—	2	4	no	—
80	C₃×C₂²⋊C₄	—	2	4	no	—
81	C₃×C₂²⋊C₄	—	2	2	no	—
82	C₃×C₂²⋊C₄	Q₈	1	4	yes	complemented
83	C₃×C₄²	—	2	4	no	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	—	2	2	no	—
86	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
87	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
88	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
89	C₄⋊Q₈	—	2	4	no	—
90	C₂²⋊Q₈	—	2	4	no	—
91	C₂²⋊Q₈	—	2	4	no	—
92	C₄×Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
101	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
102	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
103	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
104	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
105	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₃	1	2	yes	—
107	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
108	#95	C₂²×C₃	1	1	yes	characteristic
109	#95	C₂²×C₃	1	1	yes	characteristic
110	#95	C₂²×C₃	1	2	yes	—
111	#95	C₂²×C₃	1	2	yes	—
112	#95	C₂²×C₃	1	2	yes	—
113	C₂³×C₃	—	2	1	no	—
114	C₂³×C₃	—	2	1	no	—
115	C₂³×C₃	—	2	1	no	—
116	C₂³×C₃	—	2	1	no	—
117	C₂³×C₃	—	2	1	no	—
118	C₂³×C₃	—	2	1	no	—
119	C₂³×C₃	C₂×Q₈	1	2	yes	—
120	C₂³×C₃	C₂⁴	1	1	yes	characteristic
121	C₃×Q₈	—	2	8	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	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	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	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
147	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
148	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
149	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
150	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
151	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
152	C₂×C₃×C₄	#41	1	2	yes	—
153	C₂×C₃×C₄	#41	1	2	yes	—
154	C₂⁴	C₂³×C₃	1	1	yes	characteristic
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₃×D₄	1	4	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	C₃×D₄	1	4	yes	—
182	C₂²⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	C₃×Q₈	1	4	yes	complemented
186	C₄²	—	2	4	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	4	1	no	—
193	C₂²×C₃	—	4	1	no	—
194	C₂²×C₃	—	4	1	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
197	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
198	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
199	C₂²×C₃	C₂×#41	1	1	yes	characteristic
200	C₂²×C₃	C₂×#41	1	1	yes	characteristic
201	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
202	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	2	4	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	4	1	no	—
209	C₃×C₄	—	4	2	no	—
210	C₃×C₄	—	4	1	no	—
211	C₃×C₄	—	4	2	no	—
212	C₃×C₄	#121	1	4	yes	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	C₂×C₃×Q₈	1	2	yes	—
220	C₂³	C₂⁴×C₃	1	1	yes	characteristic
221	Q₈	—	2	8	no	—
222	Q₈	—	4	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
247	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
248	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
249	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
250	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
251	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
252	C₂×C₄	C₃×#41	1	2	yes	—
253	C₂×C₄	C₃×#41	1	2	yes	—
254	C₂×C₃	—	4	2	no	—
255	C₂×C₃	C₂×#121	1	1	yes	characteristic
256	C₂×C₃	#533	1	1	yes	characteristic
257	C₂×C₃	#541	1	1	yes	characteristic
258	C₂×C₃	#547	1	1	yes	characteristic
259	C₂×C₃	#548	1	1	yes	characteristic
260	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
261	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
270	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
271	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
272	C₂²	C₂×C₃×#41	1	1	yes	characteristic
273	C₂²	C₂×C₃×#41	1	1	yes	characteristic
274	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
275	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	2	no	—
285	C₄	C₃×#121	1	4	yes	—
286	C₃	#2429	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	4	2	no	—
288	C₂	C₂×C₃×#121	1	1	yes	characteristic
289	C₂	C₃×#533	1	1	yes	characteristic
290	C₂	C₃×#541	1	1	yes	characteristic
291	C₂	C₃×#547	1	1	yes	characteristic
292	C₂	C₃×#548	1	1	yes	characteristic
293	C₂	C₃×D₄×Q₈	1	1	yes	characteristic
294	C₂	C₃×D₄×Q₈	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄⋊Q₈	C₂	semidirect
4. C₂×C₃×C₄×Q₈	C₂	semidirect
5. C₃×#398	C₂	semidirect
7. C₃×#391	C₂	semidirect
8. C₃×#389	C₂	semidirect
9. C₃×#389	C₂	semidirect
12. #2429	C₃	direct (N × H)
40. C₂×C₄⋊Q₈	C₂×C₃	semidirect
43. C₂×C₄×Q₈	C₂×C₃	semidirect
44. #398	C₂×C₃	semidirect
46. #391	C₂×C₃	semidirect
47. #389	C₂×C₃	semidirect
48. #389	C₂×C₃	semidirect
82. C₃×C₂²⋊C₄	Q₈	semidirect
185. C₂²⋊C₄	C₃×Q₈	semidirect
286. C₃	#2429	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⋊Q₈	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₃×#398	2
C₃×#396	2
C₃×#391	2
C₃×#389	2
C₃×#388	2
C₃×C₄×C₂²⋊C₄	2
#2429	3

Subgroups — C3×#2429 #84293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2429 #84293