Subgroups — C₂×C₃×#542 #87338

Order 384.

1132 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	C₃×#542	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#126	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#124	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#123	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#542	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Q₈	C₂²	1	2	yes	—
16	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₂²×C₃×D₄	C₂²	1	2	yes	—
18	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
19	C₂³×C₃×C₄	C₂²	1	2	yes	—
20	C₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
21	C₃×#126	C₂²	1	16	yes	complemented
22	C₃×#124	C₂²	1	8	yes	complemented
23	C₃×#124	C₂²	1	16	yes	complemented
24	C₃×#123	C₂²	1	16	yes	complemented
25	C₃×C₂²⋊Q₈	C₂²	1	16	yes	complemented
26	C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
27	C₃×#121	C₂²	1	8	yes	—
28	C₃×C₄×Q₈	C₂²	1	8	yes	complemented
29	C₃×C₄×D₄	C₂²	1	8	yes	complemented
30	C₃×#117	C₂²	1	8	yes	complemented
31	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
36	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
38	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
40	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
43	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
44	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
45	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
47	C₂×C₃×C₄²	C₂²	1	2	yes	—
48	#542	C₂×C₃	1	16	yes	direct product factor
49	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#126	C₂×C₃	1	2	yes	complemented
51	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×#124	C₂×C₃	1	2	yes	complemented
53	C₂×#123	C₂×C₃	1	2	yes	complemented
54	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
55	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
56	C₂×#121	C₂×C₃	1	1	yes	characteristic
57	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
58	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
59	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂⁴×C₃	C₂³	1	1	yes	characteristic
61	C₂⁴×C₃	C₂³	1	2	yes	—
62	C₂×C₃×Q₈	—	2	4	no	—
63	C₂×C₃×Q₈	—	4	4	no	—
64	C₂×C₃×Q₈	C₂³	1	4	yes	—
65	C₂×C₃×Q₈	C₂³	1	8	yes	—
66	C₂×C₃×D₄	—	2	4	no	—
67	C₂×C₃×D₄	—	4	4	no	—
68	C₂×C₃×D₄	C₂³	1	4	yes	—
69	C₂×C₃×D₄	C₂³	1	8	yes	—
70	C₂²×C₃×C₄	—	2	4	no	—
71	C₂²×C₃×C₄	—	4	4	no	—
72	C₂²×C₃×C₄	C₂³	1	4	yes	—
73	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂³	1	2	yes	—
76	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂³	1	2	yes	—
79	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₂³	1	2	yes	—
81	C₂²×C₃×C₄	C₂³	1	8	yes	—
82	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
83	C₃×C₄⋊C₄	C₂³	1	4	yes	—
84	C₃×C₄⋊C₄	C₂³	1	4	yes	—
85	C₃×C₄⋊C₄	C₂³	1	4	yes	—
86	C₃×C₄⋊C₄	C₂³	1	4	yes	—
87	C₃×C₄⋊C₄	C₂³	1	8	yes	—
88	C₃×C₄⋊C₄	C₂³	1	8	yes	—
89	C₃×C₄⋊C₄	C₂³	1	8	yes	—
90	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
91	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
92	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
93	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
94	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
95	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
96	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
97	C₃×C₄²	C₂³	1	4	yes	—
98	C₃×C₄²	C₂³	1	4	yes	—
99	C₃×C₄²	C₂³	1	8	yes	—
100	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
101	C₂²×Q₈	C₂²×C₃	1	2	yes	—
102	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
103	C₂²×D₄	C₂²×C₃	1	2	yes	—
104	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
105	C₂³×C₄	C₂²×C₃	1	2	yes	—
106	C₄⋊Q₈	C₂²×C₃	1	8	yes	complemented
107	#126	C₂²×C₃	1	16	yes	complemented
108	#124	C₂²×C₃	1	8	yes	complemented
109	#124	C₂²×C₃	1	16	yes	complemented
110	#123	C₂²×C₃	1	16	yes	complemented
111	C₂²⋊Q₈	C₂²×C₃	1	16	yes	complemented
112	C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
113	#121	C₂²×C₃	1	8	yes	—
114	C₄×Q₈	C₂²×C₃	1	8	yes	complemented
115	C₄×D₄	C₂²×C₃	1	8	yes	complemented
116	#117	C₂²×C₃	1	8	yes	complemented
117	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
125	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
131	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²×C₃	1	2	yes	—
134	C₂³×C₃	—	2	2	no	—
135	C₂³×C₃	—	2	4	no	—
136	C₂³×C₃	—	2	4	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	—	2	2	no	—
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	C₂⁴	1	2	yes	—
141	C₂³×C₃	C₂⁴	1	4	yes	—
142	C₂³×C₃	C₂⁴	1	1	yes	characteristic
143	C₃×Q₈	—	2	8	no	—
144	C₃×Q₈	—	4	8	no	—
145	C₃×D₄	—	2	8	no	—
146	C₃×D₄	—	4	8	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	4	8	no	—
157	C₂×C₃×C₄	#41	1	4	yes	—
158	C₂×C₃×C₄	C₂⁴	1	2	yes	—
159	C₂×C₃×C₄	C₂⁴	1	2	yes	—
160	C₂×C₃×C₄	C₂⁴	1	4	yes	—
161	C₂×C₃×C₄	C₂⁴	1	2	yes	—
162	C₂×C₃×C₄	C₂⁴	1	2	yes	—
163	C₂×C₃×C₄	C₂⁴	1	4	yes	—
164	C₂×C₃×C₄	C₂⁴	1	2	yes	—
165	C₂×C₃×C₄	C₂⁴	1	4	yes	—
166	C₂×C₃×C₄	C₂⁴	1	2	yes	—
167	C₂⁴	C₂³×C₃	1	1	yes	characteristic
168	C₂⁴	C₂³×C₃	1	2	yes	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	C₂³×C₃	1	4	yes	—
172	C₂×Q₈	C₂³×C₃	1	8	yes	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	C₂³×C₃	1	4	yes	—
176	C₂×D₄	C₂³×C₃	1	8	yes	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₃	1	2	yes	—
182	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₃	1	2	yes	—
185	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₃	1	2	yes	—
187	C₂²×C₄	C₂³×C₃	1	4	yes	—
188	C₂²×C₄	C₂³×C₃	1	8	yes	—
189	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₃	1	4	yes	—
191	C₄⋊C₄	C₂³×C₃	1	4	yes	—
192	C₄⋊C₄	C₂³×C₃	1	4	yes	—
193	C₄⋊C₄	C₂³×C₃	1	4	yes	—
194	C₄⋊C₄	C₂³×C₃	1	8	yes	—
195	C₄⋊C₄	C₂³×C₃	1	8	yes	—
196	C₄⋊C₄	C₂³×C₃	1	8	yes	—
197	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
198	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
199	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
200	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
201	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
202	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
203	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
204	C₄²	C₂³×C₃	1	4	yes	—
205	C₄²	C₂³×C₃	1	4	yes	—
206	C₄²	C₂³×C₃	1	8	yes	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	4	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	4	no	—
213	C₂²×C₃	—	4	4	no	—
214	C₂²×C₃	—	4	8	no	—
215	C₂²×C₃	C₂×#41	1	2	yes	—
216	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
217	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
218	C₂²×C₃	C₂⁵	1	1	yes	characteristic
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	2	4	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	C₂×#41	1	4	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	4	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	4	no	—
234	C₂³	C₂⁴×C₃	1	2	yes	—
235	C₂³	C₂⁴×C₃	1	4	yes	—
236	C₂³	C₂⁴×C₃	1	1	yes	characteristic
237	Q₈	—	2	8	no	—
238	Q₈	—	4	8	no	—
239	D₄	—	2	8	no	—
240	D₄	—	4	8	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	C₃×#41	1	4	yes	—
252	C₂×C₄	C₂⁴×C₃	1	2	yes	—
253	C₂×C₄	C₂⁴×C₃	1	2	yes	—
254	C₂×C₄	C₂⁴×C₃	1	4	yes	—
255	C₂×C₄	C₂⁴×C₃	1	2	yes	—
256	C₂×C₄	C₂⁴×C₃	1	2	yes	—
257	C₂×C₄	C₂⁴×C₃	1	4	yes	—
258	C₂×C₄	C₂⁴×C₃	1	2	yes	—
259	C₂×C₄	C₂⁴×C₃	1	4	yes	—
260	C₂×C₄	C₂⁴×C₃	1	2	yes	—
261	C₂×C₃	—	4	2	no	—
262	C₂×C₃	—	4	4	no	—
263	C₂×C₃	#542	1	4	yes	direct product factor
264	C₂×C₃	C₂²×#41	1	1	yes	characteristic
265	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	4	no	—
274	C₂²	—	4	8	no	—
275	C₂²	C₂×C₃×#41	1	2	yes	—
276	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
277	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
278	C₂²	C₂⁵×C₃	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	4	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	4	no	—
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	C₂×C₃×#41	1	4	yes	—
288	C₃	C₂×#542	1	1	yes	characteristic, direct product factor, Sylow
289	C₂	—	4	2	no	—
290	C₂	—	4	4	no	—
291	C₂	C₃×#542	1	4	yes	direct product factor
292	C₂	C₂²×C₃×#41	1	1	yes	characteristic
293	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
294	C₂	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂×C₃×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#542	C₂	direct (N × H)
2. C₂×C₃×C₄⋊Q₈	C₂	semidirect
3. C₂×C₃×#126	C₂	semidirect
4. C₂×C₃×#124	C₂	semidirect
5. C₂×C₃×#124	C₂	semidirect
6. C₂×C₃×#123	C₂	semidirect
7. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
8. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
10. C₂×C₃×C₄×Q₈	C₂	semidirect
11. C₂×C₃×C₄×D₄	C₂	semidirect
12. C₂×C₃×#117	C₂	semidirect
13. C₂×#542	C₃	direct (N × H)
20. C₃×C₄⋊Q₈	C₂²	semidirect
21. C₃×#126	C₂²	semidirect
22. C₃×#124	C₂²	semidirect
23. C₃×#124	C₂²	semidirect
24. C₃×#123	C₂²	semidirect
25. C₃×C₂²⋊Q₈	C₂²	semidirect
26. C₃×C₂²⋊Q₈	C₂²	semidirect
28. C₃×C₄×Q₈	C₂²	semidirect
29. C₃×C₄×D₄	C₂²	semidirect
30. C₃×#117	C₂²	semidirect
48. #542	C₂×C₃	direct (N × H)
49. C₂×C₄⋊Q₈	C₂×C₃	semidirect
50. C₂×#126	C₂×C₃	semidirect
51. C₂×#124	C₂×C₃	semidirect
52. C₂×#124	C₂×C₃	semidirect
53. C₂×#123	C₂×C₃	semidirect
54. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
55. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
57. C₂×C₄×Q₈	C₂×C₃	semidirect
58. C₂×C₄×D₄	C₂×C₃	semidirect
59. C₂×#117	C₂×C₃	semidirect
106. C₄⋊Q₈	C₂²×C₃	semidirect
107. #126	C₂²×C₃	semidirect
108. #124	C₂²×C₃	semidirect
109. #124	C₂²×C₃	semidirect
110. #123	C₂²×C₃	semidirect
111. C₂²⋊Q₈	C₂²×C₃	semidirect
112. C₂²⋊Q₈	C₂²×C₃	semidirect
114. C₄×Q₈	C₂²×C₃	semidirect
115. C₄×D₄	C₂²×C₃	semidirect
116. #117	C₂²×C₃	semidirect
263. C₂×C₃	#542	direct (N × H)
288. C₃	C₂×#542	direct (N × H)
291. C₂	C₃×#542	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#542	2
C₂×C₃×C₄⋊Q₈	2
C₂×C₃×#126	2
C₂×C₃×#124	2
C₂×C₃×#123	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×#121	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×C₄×D₄	2
C₂×C₃×#117	2
C₂×#542	3

Subgroups — C2×C3×#542 #87338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#542 #87338