Subgroups — C₃×#2323 #84187

Order 384.

556 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#389	C₂	1	2	yes	maximal, complemented
6	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#388	C₂	1	2	yes	maximal
8	C₃×#388	C₂	1	2	yes	maximal
9	C₃×#387	C₂	1	2	yes	maximal, complemented
10	C₃×#380	C₂	1	1	yes	maximal, characteristic
11	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#2323	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×C₂²⋊Q₈	C₄	1	8	yes	complemented
16	C₃×C₄×Q₈	—	2	4	no	—
17	C₃×#117	—	2	4	no	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
25	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
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	2	yes	—
30	C₂×C₃×C₄²	C₂²	1	2	yes	—
31	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄²	C₂²	1	2	yes	—
33	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
34	C₃×#95	C₂²	1	2	yes	—
35	C₃×#95	C₂²	1	2	yes	—
36	C₃×#95	C₂²	1	2	yes	—
37	C₃×#95	C₂²	1	2	yes	—
38	C₃×#95	C₂²	1	2	yes	—
39	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
40	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×#117	C₂×C₃	1	1	yes	characteristic
42	#391	C₂×C₃	1	1	yes	characteristic, complemented
43	#389	C₂×C₃	1	2	yes	complemented
44	#389	C₂×C₃	1	1	yes	characteristic, complemented
45	#388	C₂×C₃	1	2	yes	—
46	#388	C₂×C₃	1	2	yes	—
47	#387	C₂×C₃	1	2	yes	complemented
48	#380	C₂×C₃	1	1	yes	characteristic
49	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂⁴×C₃	C₂³	1	1	yes	characteristic
51	C₂×C₃×Q₈	—	2	2	no	—
52	C₂×C₃×Q₈	—	2	2	no	—
53	C₂×C₃×Q₈	C₂×C₄	1	4	yes	—
54	C₂²×C₃×C₄	—	2	2	no	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂³	1	2	yes	—
61	C₂²×C₃×C₄	C₂³	1	2	yes	—
62	C₂²×C₃×C₄	C₂³	1	2	yes	—
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	4	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	—	4	2	no	—
73	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
74	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
75	C₃×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₂²⋊C₄	—	2	2	no	—
77	C₃×C₂²⋊C₄	—	4	2	no	—
78	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
79	C₃×C₄²	—	2	2	no	—
80	C₃×C₄²	—	2	4	no	—
81	C₃×C₄²	—	4	2	no	—
82	C₃×C₄²	D₄	1	4	yes	—
83	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
84	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂²⋊Q₈	C₃×C₄	1	8	yes	complemented
86	C₄×Q₈	—	2	4	no	—
87	#117	—	2	4	no	—
88	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
92	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
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×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
97	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₃	1	2	yes	—
102	C₂×C₄²	C₂²×C₃	1	2	yes	—
103	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
104	#95	C₂²×C₃	1	2	yes	—
105	#95	C₂²×C₃	1	2	yes	—
106	#95	C₂²×C₃	1	2	yes	—
107	#95	C₂²×C₃	1	2	yes	—
108	#95	C₂²×C₃	1	2	yes	—
109	C₂³×C₃	—	2	1	no	—
110	C₂³×C₃	—	2	1	no	—
111	C₂³×C₃	—	2	1	no	—
112	C₂³×C₃	—	2	1	no	—
113	C₂³×C₃	—	2	1	no	—
114	C₂³×C₃	—	2	1	no	—
115	C₂³×C₃	C₂²×C₄	1	2	yes	—
116	C₂³×C₃	C₂⁴	1	1	yes	characteristic
117	C₃×Q₈	—	2	8	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	8	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
142	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
143	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
144	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
145	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
146	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
147	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
148	C₂×C₃×C₄	#41	1	2	yes	—
149	C₂⁴	C₂³×C₃	1	1	yes	characteristic
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	C₂×C₃×C₄	1	4	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
156	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₃	1	2	yes	—
158	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
173	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	4	no	—
180	C₄²	—	4	2	no	—
181	C₄²	C₃×D₄	1	4	yes	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	4	1	no	—
186	C₂²×C₃	—	4	1	no	—
187	C₂²×C₃	—	4	1	no	—
188	C₂²×C₃	—	4	1	no	—
189	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
190	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
191	C₂²×C₃	C₂×#41	1	1	yes	characteristic
192	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
194	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
195	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	2	2	no	—
201	C₃×C₄	—	4	1	no	—
202	C₃×C₄	—	4	1	no	—
203	C₃×C₄	—	4	2	no	—
204	C₃×C₄	—	4	2	no	—
205	C₃×C₄	C₄×D₄	1	4	yes	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂²×C₃×C₄	1	2	yes	—
213	C₂³	C₂⁴×C₃	1	1	yes	characteristic
214	Q₈	—	2	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
239	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
240	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
241	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
242	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
243	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
244	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
245	C₂×C₄	C₃×#41	1	2	yes	—
246	C₂×C₃	—	4	2	no	—
247	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
248	C₂×C₃	#519	1	1	yes	characteristic
249	C₂×C₃	#520	1	1	yes	characteristic
250	C₂×C₃	#536	1	1	yes	characteristic
251	C₂×C₃	#537	1	1	yes	characteristic
252	C₂×C₃	#541	1	1	yes	characteristic
253	C₂×C₃	#542	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
262	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
263	C₂²	C₂×C₃×#41	1	1	yes	characteristic
264	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
266	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
267	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	4	no	—
270	C₄	—	2	4	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	C₃×C₄×D₄	1	4	yes	—
278	C₃	#2323	1	1	yes	characteristic, direct product factor, Sylow
279	C₂	—	4	2	no	—
280	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
281	C₂	C₃×#519	1	1	yes	characteristic
282	C₂	C₃×#520	1	1	yes	characteristic
283	C₂	C₃×#536	1	1	yes	characteristic
284	C₂	C₃×#537	1	1	yes	characteristic
285	C₂	C₃×#541	1	1	yes	characteristic
286	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	C₂⁴×C₃	C₂³
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×Q₈	C₂	semidirect
4. C₃×#391	C₂	semidirect
5. C₃×#389	C₂	semidirect
6. C₃×#389	C₂	semidirect
9. C₃×#387	C₂	semidirect
11. C₃×C₄×C₄⋊C₄	C₂	semidirect
12. #2323	C₃	direct (N × H)
15. C₃×C₂²⋊Q₈	C₄	semidirect
40. C₂×C₄×Q₈	C₂×C₃	semidirect
42. #391	C₂×C₃	semidirect
43. #389	C₂×C₃	semidirect
44. #389	C₂×C₃	semidirect
47. #387	C₂×C₃	semidirect
49. C₄×C₄⋊C₄	C₂×C₃	semidirect
85. C₂²⋊Q₈	C₃×C₄	semidirect
278. C₃	#2323	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×C₂²⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×#117	2
C₃×#391	2
C₃×#389	2
C₃×#388	2
C₃×#387	2
C₃×#380	2
C₃×C₄×C₄⋊C₄	2
#2323	3

Subgroups — C3×#2323 #84187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2323 #84187