Subgroups — C₃×#2342 #84206

Order 384.

792 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#388	C₂	1	2	yes	maximal, complemented
5	C₃×#388	C₂	1	2	yes	maximal, complemented
6	C₃×#385	C₂	1	2	yes	maximal, complemented
7	C₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#379	C₂	1	1	yes	maximal, characteristic
10	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#2342	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂⁵×C₃	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	2	yes	—
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₃×#117	—	2	4	no	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
21	C₂×C₃×C₂²⋊C₄	C₄	1	8	yes	complemented
22	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
34	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
35	C₃×#95	C₂²	1	2	yes	complemented
36	C₃×#95	C₂²	1	2	yes	complemented
37	C₃×#95	C₂²	1	2	yes	complemented
38	C₃×#95	C₂²	1	2	yes	complemented
39	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
41	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
42	#388	C₂×C₃	1	2	yes	complemented
43	#388	C₂×C₃	1	2	yes	complemented
44	#385	C₂×C₃	1	2	yes	complemented
45	#381	C₂×C₃	1	1	yes	characteristic, complemented
46	#380	C₂×C₃	1	1	yes	characteristic, complemented
47	#379	C₂×C₃	1	1	yes	characteristic
48	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
49	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂⁴×C₃	—	2	2	no	—
51	C₂⁴×C₃	—	2	4	no	—
52	C₂⁴×C₃	—	2	2	no	—
53	C₂⁴×C₃	—	4	2	no	—
54	C₂⁴×C₃	C₂×C₄	1	4	yes	—
55	C₂⁴×C₃	C₂³	1	1	yes	characteristic
56	C₂⁴×C₃	C₂³	1	1	yes	characteristic
57	C₂⁴×C₃	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	—	2	4	no	—
61	C₂²×C₃×C₄	—	2	4	no	—
62	C₂²×C₃×C₄	—	4	2	no	—
63	C₂²×C₃×C₄	—	4	2	no	—
64	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
65	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₂²×C₃×C₄	C₂³	1	2	yes	—
68	C₂²×C₃×C₄	C₂³	1	2	yes	—
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂³	1	2	yes	—
74	C₃×C₄⋊C₄	—	2	4	no	—
75	C₃×C₄⋊C₄	—	4	2	no	—
76	C₃×C₂²⋊C₄	—	2	2	no	—
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	2	4	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	2	4	no	—
81	C₃×C₂²⋊C₄	—	4	2	no	—
82	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
83	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
84	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	complemented
85	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	complemented
86	C₃×C₂²⋊C₄	D₄	1	4	yes	—
87	C₃×C₄²	—	2	4	no	—
88	C₃×C₄²	—	4	2	no	—
89	C₂⁵	C₂²×C₃	1	1	yes	characteristic
90	C₂³×C₄	C₂²×C₃	1	2	yes	—
91	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
92	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
93	#117	—	2	4	no	—
94	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
95	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
96	C₂×C₂²⋊C₄	—	2	4	no	—
97	C₂×C₂²⋊C₄	C₃×C₄	1	8	yes	complemented
98	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
101	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
102	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
103	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
104	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
106	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
107	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
109	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
110	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
111	#95	C₂²×C₃	1	2	yes	complemented
112	#95	C₂²×C₃	1	2	yes	complemented
113	#95	C₂²×C₃	1	2	yes	complemented
114	#95	C₂²×C₃	1	2	yes	complemented
115	C₂³×C₃	—	2	1	no	—
116	C₂³×C₃	—	2	2	no	—
117	C₂³×C₃	—	2	1	no	—
118	C₂³×C₃	—	2	2	no	—
119	C₂³×C₃	—	2	2	no	—
120	C₂³×C₃	—	2	8	no	—
121	C₂³×C₃	—	2	8	no	—
122	C₂³×C₃	—	2	1	no	—
123	C₂³×C₃	—	2	2	no	—
124	C₂³×C₃	—	2	1	no	—
125	C₂³×C₃	—	2	2	no	—
126	C₂³×C₃	—	4	4	no	—
127	C₂³×C₃	—	4	8	no	—
128	C₂³×C₃	—	4	8	no	—
129	C₂³×C₃	C₂²×C₄	1	2	yes	—
130	C₂³×C₃	C₂²×C₄	1	2	yes	—
131	C₂³×C₃	C₂²×C₄	1	2	yes	—
132	C₂³×C₃	C₂×D₄	1	2	yes	—
133	C₂³×C₃	#41	1	4	yes	—
134	C₂³×C₃	#41	1	2	yes	—
135	C₂³×C₃	C₂⁴	1	1	yes	characteristic
136	C₂×C₃×C₄	—	2	8	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	4	4	no	—
155	C₂×C₃×C₄	—	4	8	no	—
156	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
157	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
158	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
159	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
160	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
161	C₂⁴	—	2	2	no	—
162	C₂⁴	—	2	4	no	—
163	C₂⁴	—	2	2	no	—
164	C₂⁴	—	4	2	no	—
165	C₂⁴	C₂×C₃×C₄	1	4	yes	—
166	C₂⁴	C₂³×C₃	1	1	yes	characteristic
167	C₂⁴	C₂³×C₃	1	1	yes	characteristic
168	C₂⁴	C₂³×C₃	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
176	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
177	C₂²×C₄	C₂³×C₃	1	2	yes	—
178	C₂²×C₄	C₂³×C₃	1	2	yes	—
179	C₂²×C₄	C₂³×C₃	1	2	yes	—
180	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
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	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
194	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
195	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
196	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
197	C₂²⋊C₄	C₃×D₄	1	4	yes	—
198	C₄²	—	2	4	no	—
199	C₄²	—	4	2	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	2	4	no	—
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	4	16	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	—	4	2	no	—
214	C₂²×C₃	#117	1	4	yes	complemented
215	C₂²×C₃	C₄×D₄	1	4	yes	—
216	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
217	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
218	C₂²×C₃	C₂×#41	1	2	yes	—
219	C₂²×C₃	C₂×#41	1	1	yes	characteristic
220	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	4	2	no	—
227	C₃×C₄	—	4	2	no	—
228	C₃×C₄	—	4	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	8	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	4	4	no	—
241	C₂³	—	4	8	no	—
242	C₂³	—	4	8	no	—
243	C₂³	C₂²×C₃×C₄	1	2	yes	—
244	C₂³	C₂²×C₃×C₄	1	2	yes	—
245	C₂³	C₂²×C₃×C₄	1	2	yes	—
246	C₂³	C₂×C₃×D₄	1	2	yes	—
247	C₂³	C₃×#41	1	4	yes	—
248	C₂³	C₃×#41	1	2	yes	—
249	C₂³	C₂⁴×C₃	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
271	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
272	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
273	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
274	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
275	C₂×C₃	—	2	4	no	—
276	C₂×C₃	—	2	4	no	—
277	C₂×C₃	—	4	2	no	—
278	C₂×C₃	C₂×#117	1	1	yes	characteristic
279	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
280	C₂×C₃	#518	1	1	yes	characteristic
281	C₂×C₃	#546	1	2	yes	—
282	C₂×C₃	#551	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	16	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	C₃×#117	1	4	yes	complemented
298	C₂²	C₃×C₄×D₄	1	4	yes	—
299	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
300	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
301	C₂²	C₂×C₃×#41	1	2	yes	—
302	C₂²	C₂×C₃×#41	1	1	yes	characteristic
303	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₃	#2342	1	1	yes	characteristic, direct product factor, Sylow
313	C₂	—	2	4	no	—
314	C₂	—	2	4	no	—
315	C₂	—	4	2	no	—
316	C₂	C₂×C₃×#117	1	1	yes	characteristic
317	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
318	C₂	C₃×#518	1	1	yes	characteristic
319	C₂	C₃×#546	1	2	yes	—
320	C₂	C₃×#551	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²
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. C₂×C₃×#117	C₂	semidirect
4. C₃×#388	C₂	semidirect
5. C₃×#388	C₂	semidirect
6. C₃×#385	C₂	semidirect
7. C₃×#381	C₂	semidirect
8. C₃×#380	C₂	semidirect
10. C₃×C₄×C₂²⋊C₄	C₂	semidirect
11. C₃×C₄×C₂²⋊C₄	C₂	semidirect
12. #2342	C₃	direct (N × H)
18. C₂×C₃×C₄⋊C₄	C₂²	semidirect
19. C₂×C₃×C₄⋊C₄	C₂²	semidirect
21. C₂×C₃×C₂²⋊C₄	C₄	semidirect
33. C₂×C₃×C₄²	C₂²	semidirect
34. C₂×C₃×C₄²	C₂²	semidirect
35. C₃×#95	C₂²	semidirect
36. C₃×#95	C₂²	semidirect
37. C₃×#95	C₂²	semidirect
38. C₃×#95	C₂²	semidirect
39. C₂×#117	C₂×C₃	semidirect
42. #388	C₂×C₃	semidirect
43. #388	C₂×C₃	semidirect
44. #385	C₂×C₃	semidirect
45. #381	C₂×C₃	semidirect
46. #380	C₂×C₃	semidirect
48. C₄×C₂²⋊C₄	C₂×C₃	semidirect
49. C₄×C₂²⋊C₄	C₂×C₃	semidirect
64. C₂²×C₃×C₄	C₂×C₄	semidirect
65. C₂²×C₃×C₄	C₂×C₄	semidirect
84. C₃×C₂²⋊C₄	C₂×C₄	semidirect
85. C₃×C₂²⋊C₄	C₂×C₄	semidirect
94. C₂×C₄⋊C₄	C₂²×C₃	semidirect
95. C₂×C₄⋊C₄	C₂²×C₃	semidirect
97. C₂×C₂²⋊C₄	C₃×C₄	semidirect
109. C₂×C₄²	C₂²×C₃	semidirect
110. C₂×C₄²	C₂²×C₃	semidirect
111. #95	C₂²×C₃	semidirect
112. #95	C₂²×C₃	semidirect
113. #95	C₂²×C₃	semidirect
114. #95	C₂²×C₃	semidirect
175. C₂²×C₄	C₂×C₃×C₄	semidirect
176. C₂²×C₄	C₂×C₃×C₄	semidirect
195. C₂²⋊C₄	C₂×C₃×C₄	semidirect
196. C₂²⋊C₄	C₂×C₃×C₄	semidirect
214. C₂²×C₃	#117	semidirect
297. C₂²	C₃×#117	semidirect
312. C₃	#2342	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×#117	2
C₂²×C₃×C₂²⋊C₄	2
C₃×#388	2
C₃×#385	2
C₃×#381	2
C₃×#380	2
C₃×#379	2
C₃×C₄×C₂²⋊C₄	2
#2342	3

Subgroups — C3×#2342 #84206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2342 #84206