Subgroups — C₃×#2324 #84188

Order 384.

668 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#390	C₂	1	2	yes	maximal, complemented
6	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#388	C₂	1	2	yes	maximal, complemented
8	C₃×#386	C₂	1	2	yes	maximal, complemented
9	C₃×#385	C₂	1	2	yes	maximal, complemented
10	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#2324	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	2	yes	—
17	C₃×#121	C₄	1	8	yes	complemented
18	C₃×C₄×D₄	—	2	4	no	—
19	C₃×#117	—	2	4	no	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
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×C₄²	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂²	1	2	yes	—
35	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
36	C₃×#95	C₂²	1	2	yes	—
37	C₃×#95	C₂²	1	2	yes	—
38	C₃×#95	C₂²	1	2	yes	—
39	C₂×#121	C₂×C₃	1	1	yes	characteristic
40	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
42	#390	C₂×C₃	1	1	yes	characteristic, complemented
43	#390	C₂×C₃	1	2	yes	complemented
44	#389	C₂×C₃	1	1	yes	characteristic, complemented
45	#388	C₂×C₃	1	2	yes	complemented
46	#386	C₂×C₃	1	2	yes	complemented
47	#385	C₂×C₃	1	2	yes	complemented
48	#380	C₂×C₃	1	1	yes	characteristic, complemented
49	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂⁴×C₃	C₂³	1	1	yes	characteristic
51	C₂⁴×C₃	C₂³	1	2	yes	—
52	C₂×C₃×D₄	—	2	2	no	—
53	C₂×C₃×D₄	—	2	2	no	—
54	C₂×C₃×D₄	—	4	2	no	—
55	C₂×C₃×D₄	—	4	2	no	—
56	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
57	C₂×C₃×D₄	C₂×C₄	1	8	yes	complemented
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	—	4	2	no	—
62	C₂²×C₃×C₄	—	4	2	no	—
63	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
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	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	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₄⋊C₄	—	2	2	no	—
76	C₃×C₄⋊C₄	—	2	2	no	—
77	C₃×C₄⋊C₄	—	4	2	no	—
78	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	2	2	no	—
81	C₃×C₂²⋊C₄	—	2	4	no	—
82	C₃×C₂²⋊C₄	—	4	2	no	—
83	C₃×C₂²⋊C₄	—	4	2	no	—
84	C₃×C₂²⋊C₄	—	4	2	no	—
85	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
86	C₃×C₄²	—	2	2	no	—
87	C₃×C₄²	—	4	2	no	—
88	C₃×C₄²	D₄	1	4	yes	complemented
89	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
90	C₂²×D₄	C₂²×C₃	1	2	yes	—
91	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
92	C₂³×C₄	C₂²×C₃	1	2	yes	—
93	#121	C₃×C₄	1	8	yes	complemented
94	C₄×D₄	—	2	4	no	—
95	#117	—	2	4	no	—
96	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
102	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
104	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
105	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
106	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
107	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
109	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
110	C₂×C₄²	C₂²×C₃	1	2	yes	—
111	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
112	#95	C₂²×C₃	1	2	yes	—
113	#95	C₂²×C₃	1	2	yes	—
114	#95	C₂²×C₃	1	2	yes	—
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	1	no	—
120	C₂³×C₃	—	2	2	no	—
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	—	2	2	no	—
123	C₂³×C₃	—	2	1	no	—
124	C₂³×C₃	—	2	1	no	—
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	—	2	2	no	—
127	C₂³×C₃	C₂²×C₄	1	4	yes	—
128	C₂³×C₃	C₂²×C₄	1	2	yes	—
129	C₂³×C₃	C₂⁴	1	1	yes	characteristic
130	C₃×D₄	—	2	8	no	—
131	C₃×D₄	—	4	8	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	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	8	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	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	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	1	no	—
153	C₂×C₃×C₄	—	4	8	no	—
154	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
155	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
156	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
157	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
158	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
159	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
160	C₂×C₃×C₄	#41	1	2	yes	—
161	C₂⁴	C₂³×C₃	1	1	yes	characteristic
162	C₂⁴	C₂³×C₃	1	2	yes	—
163	C₂×D₄	—	2	2	no	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
168	C₂×D₄	C₂×C₃×C₄	1	8	yes	complemented
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
175	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₃	1	2	yes	—
178	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
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	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	4	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
197	C₄²	—	2	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	C₃×D₄	1	4	yes	complemented
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	4	no	—
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	—	2	4	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	4	2	no	—
207	C₂²×C₃	—	4	1	no	—
208	C₂²×C₃	—	4	1	no	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	—	4	1	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	1	no	—
213	C₂²×C₃	—	4	2	no	—
214	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
215	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
216	C₂²×C₃	C₂×#41	1	1	yes	characteristic
217	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
219	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
220	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	4	1	no	—
226	C₃×C₄	—	4	2	no	—
227	C₃×C₄	—	4	1	no	—
228	C₃×C₄	—	4	2	no	—
229	C₃×C₄	C₄×D₄	1	4	yes	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	C₂²×C₃×C₄	1	4	yes	—
243	C₂³	C₂²×C₃×C₄	1	2	yes	—
244	C₂³	C₂⁴×C₃	1	1	yes	characteristic
245	D₄	—	2	8	no	—
246	D₄	—	4	8	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
270	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
271	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
272	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
273	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
274	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
275	C₂×C₄	C₃×#41	1	2	yes	—
276	C₂×C₃	—	4	2	no	—
277	C₂×C₃	—	4	4	no	—
278	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
279	C₂×C₃	#518	1	1	yes	characteristic
280	C₂×C₃	#520	1	1	yes	characteristic
281	C₂×C₃	#535	1	1	yes	characteristic
282	C₂×C₃	#537	1	1	yes	characteristic
283	C₂×C₃	#540	1	1	yes	characteristic
284	C₂×C₃	#542	1	1	yes	characteristic
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	2	no	—
299	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
300	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
301	C₂²	C₂×C₃×#41	1	1	yes	characteristic
302	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
303	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
304	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
305	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	4	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	2	no	—
314	C₄	C₃×C₄×D₄	1	4	yes	—
315	C₃	#2324	1	1	yes	characteristic, direct product factor, Sylow
316	C₂	—	4	2	no	—
317	C₂	—	4	4	no	—
318	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
319	C₂	C₃×#518	1	1	yes	characteristic
320	C₂	C₃×#520	1	1	yes	characteristic
321	C₂	C₃×#535	1	1	yes	characteristic
322	C₂	C₃×#537	1	1	yes	characteristic
323	C₂	C₃×#540	1	1	yes	characteristic
324	C₂	C₃×#542	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×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
2. C₂×C₃×C₄×D₄	C₂	semidirect
3. C₂×C₃×#117	C₂	semidirect
4. C₃×#390	C₂	semidirect
5. C₃×#390	C₂	semidirect
6. C₃×#389	C₂	semidirect
7. C₃×#388	C₂	semidirect
8. C₃×#386	C₂	semidirect
9. C₃×#385	C₂	semidirect
10. C₃×#380	C₂	semidirect
11. C₃×C₄×C₄⋊C₄	C₂	semidirect
12. #2324	C₃	direct (N × H)
17. C₃×#121	C₄	semidirect
40. C₂×C₄×D₄	C₂×C₃	semidirect
41. C₂×#117	C₂×C₃	semidirect
42. #390	C₂×C₃	semidirect
43. #390	C₂×C₃	semidirect
44. #389	C₂×C₃	semidirect
45. #388	C₂×C₃	semidirect
46. #386	C₂×C₃	semidirect
47. #385	C₂×C₃	semidirect
48. #380	C₂×C₃	semidirect
49. C₄×C₄⋊C₄	C₂×C₃	semidirect
57. C₂×C₃×D₄	C₂×C₄	semidirect
63. C₂²×C₃×C₄	C₂×C₄	semidirect
78. C₃×C₄⋊C₄	C₂×C₄	semidirect
85. C₃×C₂²⋊C₄	C₂×C₄	semidirect
88. C₃×C₄²	D₄	semidirect
93. #121	C₃×C₄	semidirect
168. C₂×D₄	C₂×C₃×C₄	semidirect
174. C₂²×C₄	C₂×C₃×C₄	semidirect
189. C₄⋊C₄	C₂×C₃×C₄	semidirect
196. C₂²⋊C₄	C₂×C₃×C₄	semidirect
199. C₄²	C₃×D₄	semidirect
315. C₃	#2324	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×#121	2
C₂×C₃×C₄×D₄	2
C₂×C₃×#117	2
C₃×#390	2
C₃×#389	2
C₃×#388	2
C₃×#386	2
C₃×#385	2
C₃×#380	2
C₃×C₄×C₄⋊C₄	2
#2324	3

Subgroups — C3×#2324 #84188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2324 #84188