Subgroups — #81221

Order 384.

338 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5690	C₂	1	1	yes	maximal, characteristic
2	#5569	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
4	#5065	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#215	C₂	1	1	yes	maximal, characteristic
6	#4890	C₂	1	1	yes	maximal, characteristic
7	#4878	C₂	1	1	yes	maximal, characteristic, complemented
8	#4871	C₂	1	1	yes	maximal, characteristic
9	#4855	C₂	1	1	yes	maximal, characteristic, complemented
10	#4847	C₂	1	1	yes	maximal, characteristic
11	#4830	C₂	1	1	yes	maximal, characteristic, complemented
12	#4799	C₂	1	1	yes	maximal, characteristic, complemented
13	#4793	C₂	1	1	yes	maximal, characteristic, complemented
14	#4788	C₂	1	1	yes	maximal, characteristic
15	#4788	C₂	1	1	yes	maximal, characteristic
16	#3381	—	3	1	no	maximal, Sylow
17	C₃×#125	C₂²	1	1	yes	characteristic
18	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
19	Q₈×Dic₃	C₂²	1	1	yes	characteristic
20	Q₈×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×#215	C₂²	1	1	yes	characteristic
22	#897	—	2	1	no	—
23	#896	—	2	1	no	—
24	#896	—	2	1	no	—
25	#893	C₂²	1	1	yes	characteristic
26	#893	C₂²	1	1	yes	characteristic
27	D₃×C₄⋊C₄	—	2	1	no	—
28	#890	—	2	1	no	—
29	#890	C₂²	1	1	yes	characteristic
30	#890	C₂²	1	1	yes	characteristic
31	#888	—	2	1	no	—
32	#888	C₂²	1	1	yes	characteristic
33	#888	C₂²	1	1	yes	characteristic
34	C₄×D₁₂	C₂²	1	1	yes	characteristic
35	#870	—	2	1	no	—
36	#870	C₂²	1	1	yes	characteristic
37	C₄×Dic₆	—	2	1	no	—
38	C₄×Dic₆	C₂²	1	1	yes	characteristic
39	C₃×#107	C₂²	1	1	yes	characteristic
40	C₃×#106	C₂²	1	1	yes	characteristic
41	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×#103	C₂²	1	1	yes	characteristic
43	C₃×#103	C₂²	1	1	yes	characteristic
44	#835	C₂²	1	1	yes	characteristic
45	#821	C₂²	1	1	yes	characteristic
46	#821	C₂²	1	1	yes	characteristic
47	#818	C₂²	1	1	yes	characteristic
48	#817	C₂²	1	1	yes	characteristic
49	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₃	C₂²	1	1	yes	characteristic
52	C₈×Dic₃	C₂²	1	1	yes	characteristic
53	#809	C₂²	1	1	yes	characteristic
54	#808	C₂²	1	1	yes	characteristic
55	#808	C₂²	1	1	yes	characteristic
56	#808	C₂²	1	1	yes	characteristic
57	#807	C₂²	1	1	yes	characteristic
58	#807	C₂²	1	1	yes	characteristic
59	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
60	#557	—	3	1	no	—
61	#554	—	3	1	no	—
62	#499	—	3	1	no	—
63	#499	—	3	1	no	—
64	#498	—	3	1	no	—
65	#479	—	3	1	no	—
66	#479	D₃	1	1	yes	characteristic, complemented
67	#478	—	3	1	no	—
68	#478	—	3	1	no	—
69	#477	—	3	1	no	—
70	#474	—	3	1	no	—
71	C₈×Q₈	—	3	1	no	—
72	C₄×SD₁₆	—	3	1	no	—
73	C₄×SD₁₆	—	3	1	no	—
74	C₄×SD₁₆	—	3	1	no	—
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	C₂×D₁₂	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₃	—	2	1	no	—
78	C₂×C₄×D₃	—	2	1	no	—
79	C₂×C₄×D₃	—	2	1	no	—
80	C₂×Dic₆	—	2	1	no	—
81	C₂×Dic₆	—	2	1	no	—
82	C₂×Dic₆	D₄	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
85	C₃×C₄⋊C₄	—	2	1	no	—
86	C₃×C₄⋊C₄	—	2	1	no	—
87	C₃×C₄⋊C₄	—	2	1	no	—
88	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₃×C₄²	—	2	1	no	—
92	C₃×C₄²	C₂³	1	1	yes	characteristic
93	#215	Q₈	1	4	yes	complemented
94	(C₂×D₃)⋊C₄	—	2	1	no	—
95	(C₂×D₃)⋊C₄	—	2	1	no	—
96	(C₂×D₃)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₃	—	2	1	no	—
98	C₄⋊Dic₃	—	2	1	no	—
99	C₄⋊Dic₃	—	2	1	no	—
100	C₄⋊Dic₃	—	2	1	no	—
101	C₄⋊Dic₃	D₄	1	1	yes	characteristic
102	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
103	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
104	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
105	Dic₃⋊C₄	—	2	1	no	—
106	Dic₃⋊C₄	—	2	1	no	—
107	Dic₃⋊C₄	—	2	1	no	—
108	Dic₃⋊C₄	—	2	1	no	—
109	Dic₃⋊C₄	—	2	1	no	—
110	Dic₃⋊C₄	—	2	1	no	—
111	Dic₃⋊C₄	D₄	1	1	yes	characteristic
112	Dic₃⋊C₄	D₄	1	1	yes	characteristic
113	C₄×Dic₃	—	2	1	no	—
114	C₄×Dic₃	—	2	1	no	—
115	C₄×Dic₃	C₂³	1	1	yes	characteristic
116	C₄×Dic₃	C₂³	1	1	yes	characteristic
117	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
118	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
119	C₂×SD₁₆	—	3	1	no	—
120	C₄⋊Q₈	—	3	1	no	—
121	C₄⋊Q₈	—	3	1	no	—
122	C₄⋊Q₈	—	6	1	no	—
123	#125	—	3	1	no	—
124	#125	—	3	1	no	—
125	#125	—	3	1	no	—
126	#125	—	6	1	no	—
127	#125	—	6	1	no	—
128	#125	C₂×D₃	1	1	yes	characteristic
129	C₂²⋊Q₈	—	6	1	no	—
130	C₂²⋊Q₈	—	6	1	no	—
131	C₂²⋊Q₈	—	6	1	no	—
132	C₄×Q₈	—	3	1	no	—
133	C₄×Q₈	—	3	1	no	—
134	C₄×Q₈	—	3	1	no	—
135	C₄×Q₈	—	6	1	no	—
136	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
137	C₄×D₄	—	3	1	no	—
138	C₄×D₄	—	3	1	no	—
139	C₄×D₄	—	3	1	no	—
140	C₂×C₄⋊C₄	—	6	1	no	—
141	#107	—	3	1	no	—
142	#107	—	3	1	no	—
143	#107	—	3	1	no	—
144	#107	C₂×D₃	1	1	yes	characteristic
145	#106	—	3	1	no	—
146	#106	—	3	1	no	—
147	#106	—	3	1	no	—
148	#106	—	3	1	no	—
149	#106	C₂×D₃	1	1	yes	characteristic
150	C₄⋊C₈	—	3	1	no	—
151	C₄⋊C₈	—	3	1	no	—
152	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
153	#103	—	3	1	no	—
154	#103	C₂×D₃	1	1	yes	characteristic
155	#103	C₂×D₃	1	1	yes	characteristic
156	#102	—	3	1	no	—
157	#102	—	3	1	no	—
158	#102	—	3	1	no	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	—	3	1	no	—
161	C₄×C₈	—	3	1	no	—
162	C₂²×D₃	—	2	1	no	—
163	C₃×Q₈	—	2	1	no	—
164	C₃×Q₈	C₂×Q₈	1	2	yes	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
172	C₂×Dic₃	—	2	1	no	—
173	C₂×Dic₃	—	2	1	no	—
174	C₂×Dic₃	—	2	1	no	—
175	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
176	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
177	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
178	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
179	D₁₂	—	2	1	no	—
180	D₁₂	C₂×Q₈	1	2	yes	—
181	C₄×D₃	—	4	2	no	—
182	C₄×D₃	—	8	1	no	—
183	C₄×D₃	—	8	1	no	—
184	Dic₆	—	2	2	no	—
185	Dic₆	—	4	2	no	—
186	Dic₆	—	8	1	no	—
187	C₃×C₈	—	2	1	no	—
188	C₃×C₈	—	2	1	no	—
189	C₃⋊C₈	—	2	1	no	—
190	C₃⋊C₈	C₂×Q₈	1	2	yes	—
191	C₂×Q₈	—	3	1	no	—
192	C₂×Q₈	—	6	1	no	—
193	C₂×Q₈	—	6	1	no	—
194	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
195	C₂×D₄	—	3	1	no	—
196	C₂²×C₄	—	6	1	no	—
197	C₂²×C₄	—	6	1	no	—
198	C₂²×C₄	—	6	1	no	—
199	SD₁₆	—	3	4	no	—
200	C₂×C₈	—	3	1	no	—
201	C₂×C₈	—	3	1	no	—
202	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	3	1	no	—
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	3	1	no	—
210	C₄⋊C₄	—	3	1	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	3	1	no	—
213	C₄⋊C₄	—	6	1	no	—
214	C₄⋊C₄	—	6	1	no	—
215	C₄⋊C₄	—	6	1	no	—
216	C₄⋊C₄	—	6	1	no	—
217	C₄⋊C₄	—	6	1	no	—
218	C₄⋊C₄	—	6	1	no	—
219	C₄⋊C₄	—	6	1	no	—
220	C₄⋊C₄	—	6	1	no	—
221	C₄⋊C₄	—	6	1	no	—
222	C₄⋊C₄	—	6	1	no	—
223	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
224	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
225	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
226	C₂²⋊C₄	—	6	1	no	—
227	C₂²⋊C₄	—	6	1	no	—
228	C₂²⋊C₄	—	6	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	3	1	no	—
231	C₄²	—	3	1	no	—
232	C₄²	—	6	1	no	—
233	C₄²	—	6	1	no	—
234	C₄²	C₂²×D₃	1	1	yes	characteristic
235	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
236	C₂×D₃	—	2	2	no	—
237	C₂×D₃	—	4	1	no	—
238	C₂×D₃	—	4	1	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	4	1	no	—
243	C₃×C₄	—	4	1	no	—
244	C₃×C₄	—	4	1	no	—
245	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
246	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
247	Dic₃	—	2	1	no	—
248	Dic₃	—	2	1	no	—
249	Dic₃	—	2	1	no	—
250	Dic₃	—	4	1	no	—
251	Dic₃	—	4	1	no	—
252	Dic₃	—	4	1	no	—
253	Dic₃	#135	1	2	yes	—
254	C₂³	—	6	1	no	—
255	Q₈	—	2	1	no	—
256	Q₈	—	6	2	no	—
257	Q₈	—	12	2	no	—
258	Q₈	—	24	1	no	—
259	Q₈	D₃×Q₈	1	2	yes	—
260	D₄	—	3	2	no	—
261	D₄	—	6	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	3	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	6	1	no	—
271	C₂×C₄	—	6	1	no	—
272	C₂×C₄	—	6	1	no	—
273	C₂×C₄	—	12	2	no	—
274	C₂×C₄	—	24	1	no	—
275	C₂×C₄	—	24	1	no	—
276	C₂×C₄	D₃×D₄	1	1	yes	characteristic
277	C₂×C₄	D₃×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	3	2	no	—
282	C₈	—	6	1	no	—
283	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
284	C₂×C₃	C₂×#135	1	1	yes	characteristic
285	C₂×C₃	#577	1	1	yes	characteristic
286	D₃	—	4	2	no	—
287	C₂²	—	6	2	no	—
288	C₂²	—	12	1	no	—
289	C₂²	—	12	1	no	—
290	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	3	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	6	1	no	—
299	C₄	—	6	1	no	—
300	C₄	—	6	1	no	—
301	C₄	—	12	1	no	—
302	C₄	—	12	1	no	—
303	C₄	—	12	1	no	—
304	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
305	C₄	#1010	1	1	yes	characteristic
306	C₃	#3381	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	12	2	no	—
308	C₂	#5632	1	1	yes	characteristic
309	C₂	D₃×#135	1	1	yes	characteristic
310	C₂	#5778	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#479	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×D₁₂	C₂³
hypercenter	#479	D₃
hypocenter	C₃	#3381
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
2. #5569	C₂	semidirect
3. C₃×#479	C₂	semidirect
4. #5065	C₂	semidirect
7. #4878	C₂	semidirect
9. #4855	C₂	semidirect
11. #4830	C₂	semidirect
12. #4799	C₂	semidirect
13. #4793	C₂	semidirect
66. #479	D₃	semidirect
93. #215	Q₈	semidirect
306. C₃	#3381	semidirect

Maximal subgroups

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

subgroup	index
#5690	2
#5569	2
C₃×#479	2
#5065	2
C₄×#215	2
#4890	2
#4878	2
#4871	2
#4855	2
#4847	2
#4830	2
#4799	2
#4793	2
#4788	2
#3381	3