Subgroups — C₃×#2534 #84398

Order 384.

800 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#120	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#396	C₂	1	2	yes	maximal, complemented
6	C₃×#394	C₂	1	2	yes	maximal, complemented
7	C₃×#388	C₂	1	2	yes	maximal, complemented
8	C₃×#386	C₂	1	2	yes	maximal, complemented
9	C₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#379	C₂	1	1	yes	maximal, characteristic, complemented
12	#2534	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂⁵×C₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
18	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
19	C₃×#123	—	2	4	no	—
20	C₃×#120	—	2	4	no	—
21	C₃×C₄×D₄	—	2	4	no	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
23	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
25	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
34	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
35	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
37	C₃×#95	C₂²	1	2	yes	complemented
38	C₃×#95	C₂²	1	2	yes	complemented
39	C₃×#95	C₂²	1	2	yes	complemented
40	C₂×#123	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×#120	C₂×C₃	1	1	yes	characteristic
42	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
44	#396	C₂×C₃	1	2	yes	complemented
45	#394	C₂×C₃	1	2	yes	complemented
46	#388	C₂×C₃	1	2	yes	complemented
47	#386	C₂×C₃	1	2	yes	complemented
48	#381	C₂×C₃	1	1	yes	characteristic, complemented
49	#380	C₂×C₃	1	1	yes	characteristic, complemented
50	#379	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂⁴×C₃	—	2	2	no	—
52	C₂⁴×C₃	—	2	4	no	—
53	C₂⁴×C₃	—	2	2	no	—
54	C₂⁴×C₃	—	2	2	no	—
55	C₂⁴×C₃	—	4	2	no	—
56	C₂⁴×C₃	C₂³	1	1	yes	characteristic
57	C₂⁴×C₃	C₂³	1	2	yes	—
58	C₂⁴×C₃	C₂³	1	1	yes	characteristic
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	2	4	no	—
62	C₂×C₃×D₄	—	2	4	no	—
63	C₂×C₃×D₄	—	4	2	no	—
64	C₂×C₃×D₄	D₄	1	4	yes	—
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	—	2	2	no	—
67	C₂²×C₃×C₄	—	2	2	no	—
68	C₂²×C₃×C₄	—	2	2	no	—
69	C₂²×C₃×C₄	—	4	2	no	—
70	C₂²×C₃×C₄	—	4	2	no	—
71	C₂²×C₃×C₄	D₄	1	4	yes	complemented
72	C₂²×C₃×C₄	C₂³	1	2	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	2	yes	—
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	2	4	no	—
82	C₃×C₄⋊C₄	—	4	2	no	—
83	C₃×C₂²⋊C₄	—	2	2	no	—
84	C₃×C₂²⋊C₄	—	2	2	no	—
85	C₃×C₂²⋊C₄	—	2	2	no	—
86	C₃×C₂²⋊C₄	—	2	4	no	—
87	C₃×C₂²⋊C₄	—	2	4	no	—
88	C₃×C₂²⋊C₄	—	2	4	no	—
89	C₃×C₂²⋊C₄	—	2	4	no	—
90	C₃×C₂²⋊C₄	—	2	4	no	—
91	C₃×C₂²⋊C₄	—	2	2	no	—
92	C₃×C₂²⋊C₄	—	4	2	no	—
93	C₃×C₄²	—	2	2	no	—
94	C₂⁵	C₂²×C₃	1	1	yes	characteristic
95	C₂²×D₄	C₂²×C₃	1	2	yes	—
96	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
97	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
98	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
99	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic, complemented
100	#123	—	2	4	no	—
101	#120	—	2	4	no	—
102	C₄×D₄	—	2	4	no	—
103	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
104	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
105	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
106	C₂×C₂²⋊C₄	—	2	4	no	—
107	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
113	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
116	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
117	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
118	#95	C₂²×C₃	1	2	yes	complemented
119	#95	C₂²×C₃	1	2	yes	complemented
120	#95	C₂²×C₃	1	2	yes	complemented
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	—	2	2	no	—
123	C₂³×C₃	—	2	8	no	—
124	C₂³×C₃	—	2	2	no	—
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	—	2	2	no	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	—	2	2	no	—
129	C₂³×C₃	—	2	2	no	—
130	C₂³×C₃	—	2	2	no	—
131	C₂³×C₃	—	2	2	no	—
132	C₂³×C₃	—	2	2	no	—
133	C₂³×C₃	—	2	2	no	—
134	C₂³×C₃	—	4	8	no	—
135	C₂³×C₃	—	4	8	no	—
136	C₂³×C₃	—	4	4	no	—
137	C₂³×C₃	—	4	4	no	—
138	C₂³×C₃	C₂×D₄	1	2	yes	—
139	C₂³×C₃	C₂×D₄	1	2	yes	—
140	C₂³×C₃	C₂×D₄	1	2	yes	—
141	C₂³×C₃	#41	1	4	yes	—
142	C₂³×C₃	#41	1	2	yes	—
143	C₂³×C₃	C₂⁴	1	1	yes	characteristic
144	C₃×D₄	—	2	8	no	—
145	C₃×D₄	—	4	8	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	1	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	—	2	2	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	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	—	4	4	no	—
164	C₂×C₃×C₄	—	4	4	no	—
165	C₂×C₃×C₄	—	4	4	no	—
166	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
167	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
168	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
169	C₂⁴	—	2	2	no	—
170	C₂⁴	—	2	4	no	—
171	C₂⁴	—	2	2	no	—
172	C₂⁴	—	2	2	no	—
173	C₂⁴	—	4	2	no	—
174	C₂⁴	C₂³×C₃	1	1	yes	characteristic
175	C₂⁴	C₂³×C₃	1	2	yes	—
176	C₂⁴	C₂³×C₃	1	1	yes	characteristic
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	C₃×D₄	1	4	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₃×D₄	1	4	yes	complemented
190	C₂²×C₄	C₂³×C₃	1	2	yes	—
191	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₃	1	2	yes	—
194	C₂²×C₄	C₂³×C₃	1	2	yes	—
195	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	2	4	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	4	no	—
218	C₂²×C₃	—	2	4	no	—
219	C₂²×C₃	—	2	4	no	—
220	C₂²×C₃	—	2	4	no	—
221	C₂²×C₃	—	4	16	no	—
222	C₂²×C₃	—	4	2	no	—
223	C₂²×C₃	—	4	1	no	—
224	C₂²×C₃	—	4	1	no	—
225	C₂²×C₃	—	4	2	no	—
226	C₂²×C₃	—	4	2	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	—	4	2	no	—
229	C₂²×C₃	#121	1	4	yes	—
230	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
231	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
232	C₂²×C₃	C₂×#41	1	1	yes	characteristic
233	C₂²×C₃	C₂×#41	1	2	yes	—
234	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	4	2	no	—
239	C₃×C₄	—	4	1	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	—	4	2	no	—
242	C₃×C₄	—	4	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	8	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	4	8	no	—
257	C₂³	—	4	8	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	4	4	no	—
260	C₂³	C₂×C₃×D₄	1	2	yes	—
261	C₂³	C₂×C₃×D₄	1	2	yes	—
262	C₂³	C₂×C₃×D₄	1	2	yes	—
263	C₂³	C₃×#41	1	4	yes	—
264	C₂³	C₃×#41	1	2	yes	—
265	C₂³	C₂⁴×C₃	1	1	yes	characteristic
266	D₄	—	2	8	no	—
267	D₄	—	4	8	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
289	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
290	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
291	C₂×C₃	—	2	4	no	—
292	C₂×C₃	—	4	2	no	—
293	C₂×C₃	—	4	4	no	—
294	C₂×C₃	C₂×#121	1	1	yes	characteristic
295	C₂×C₃	#525	1	1	yes	characteristic
296	C₂×C₃	#534	1	1	yes	characteristic
297	C₂×C₃	#546	1	2	yes	—
298	C₂×C₃	#551	1	2	yes	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	4	16	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₃×#121	1	4	yes	—
317	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
318	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
319	C₂²	C₂×C₃×#41	1	1	yes	characteristic
320	C₂²	C₂×C₃×#41	1	2	yes	—
321	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₃	#2534	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	2	4	no	—
332	C₂	—	4	4	no	—
333	C₂	—	4	2	no	—
334	C₂	C₂×C₃×#121	1	1	yes	characteristic
335	C₂	C₃×#525	1	1	yes	characteristic
336	C₂	C₃×#534	1	1	yes	characteristic
337	C₂	C₃×#546	1	2	yes	—
338	C₂	C₃×#551	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂×C₃×#120	C₂
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×#123	C₂	semidirect
3. C₂×C₃×C₄×D₄	C₂	semidirect
4. C₂²×C₃×C₂²⋊C₄	C₂	semidirect
5. C₃×#396	C₂	semidirect
6. C₃×#394	C₂	semidirect
7. C₃×#388	C₂	semidirect
8. C₃×#386	C₂	semidirect
9. C₃×#381	C₂	semidirect
10. C₃×#380	C₂	semidirect
11. C₃×#379	C₂	semidirect
12. #2534	C₃	direct (N × H)
18. C₂³×C₃×C₄	C₂²	semidirect
22. C₂×C₃×C₄⋊C₄	C₂²	semidirect
23. C₂×C₃×C₄⋊C₄	C₂²	semidirect
24. C₂×C₃×C₄⋊C₄	C₂²	semidirect
34. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
35. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
36. C₂×C₃×C₄²	C₂²	semidirect
37. C₃×#95	C₂²	semidirect
38. C₃×#95	C₂²	semidirect
39. C₃×#95	C₂²	semidirect
40. C₂×#123	C₂×C₃	semidirect
42. C₂×C₄×D₄	C₂×C₃	semidirect
43. C₂²×C₂²⋊C₄	C₂×C₃	semidirect
44. #396	C₂×C₃	semidirect
45. #394	C₂×C₃	semidirect
46. #388	C₂×C₃	semidirect
47. #386	C₂×C₃	semidirect
48. #381	C₂×C₃	semidirect
49. #380	C₂×C₃	semidirect
50. #379	C₂×C₃	semidirect
71. C₂²×C₃×C₄	D₄	semidirect
99. C₂³×C₄	C₂²×C₃	semidirect
103. C₂×C₄⋊C₄	C₂²×C₃	semidirect
104. C₂×C₄⋊C₄	C₂²×C₃	semidirect
105. C₂×C₄⋊C₄	C₂²×C₃	semidirect
115. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
116. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
117. C₂×C₄²	C₂²×C₃	semidirect
118. #95	C₂²×C₃	semidirect
119. #95	C₂²×C₃	semidirect
120. #95	C₂²×C₃	semidirect
189. C₂²×C₄	C₃×D₄	semidirect
330. C₃	#2534	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×#123	2
C₂×C₃×#120	2
C₂×C₃×C₄×D₄	2
C₂²×C₃×C₂²⋊C₄	2
C₃×#396	2
C₃×#394	2
C₃×#388	2
C₃×#386	2
C₃×#381	2
C₃×#380	2
C₃×#379	2
#2534	3

Subgroups — C3×#2534 #84398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2534 #84398