Subgroups — #80469

Order 384.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5721	C₂	1	1	yes	maximal, characteristic, complemented
2	#5568	C₂	1	1	yes	maximal, characteristic
3	C₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
4	#5190	C₂	1	1	yes	maximal, characteristic
5	#5185	C₂	1	1	yes	maximal, characteristic
6	#5183	C₂	1	1	yes	maximal, characteristic, complemented
7	#5182	C₂	1	1	yes	maximal, characteristic, complemented
8	#5086	C₂	1	1	yes	maximal, characteristic
9	#5030	C₂	1	1	yes	maximal, characteristic, complemented
10	#4845	C₂	1	1	yes	maximal, characteristic
11	#4828	C₂	1	1	yes	maximal, characteristic, complemented
12	#4818	C₂	1	1	yes	maximal, characteristic
13	#4795	C₂	1	1	yes	maximal, characteristic, complemented
14	#4789	C₂	1	1	yes	maximal, characteristic, complemented
15	#4788	C₂	1	1	yes	maximal, characteristic
16	#3398	—	3	1	no	maximal, Sylow
17	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
21	#947	C₂²	1	1	yes	characteristic
22	Q₈×Dic₃	C₂²	1	1	yes	characteristic
23	Q₈×Dic₃	C₂²	1	1	yes	characteristic
24	#944	C₂²	1	1	yes	characteristic
25	#944	C₂²	1	1	yes	characteristic
26	#944	C₂²	1	1	yes	characteristic
27	C₂×#216	C₂²	1	1	yes	characteristic
28	C₂×#216	C₂²	1	1	yes	characteristic
29	C₂×#216	C₂²	1	1	yes	characteristic
30	C₂×#215	C₂²	1	1	yes	characteristic
31	C₂×#215	C₂²	1	1	yes	characteristic
32	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
33	C₂×#204	C₂²	1	1	yes	characteristic
34	#898	—	2	1	no	—
35	#896	—	2	1	no	—
36	#893	C₂²	1	1	yes	characteristic
37	#890	—	2	1	no	—
38	#889	—	2	1	no	—
39	#889	—	2	1	no	—
40	#887	—	2	1	no	—
41	#887	C₂²	1	1	yes	characteristic
42	#876	—	2	1	no	—
43	#875	C₂²	1	1	yes	characteristic
44	#872	—	2	1	no	—
45	C₄×Dic₆	C₂²	1	1	yes	characteristic
46	C₄×Dic₆	C₂²	1	1	yes	characteristic
47	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₃×#103	C₂²	1	1	yes	characteristic
49	C₃×#103	C₂²	1	1	yes	characteristic
50	#835	C₂²	1	1	yes	characteristic
51	#821	C₂²	1	1	yes	characteristic
52	#816	C₂²	1	1	yes	characteristic
53	#815	C₂²	1	1	yes	characteristic
54	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
55	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
56	C₈×Dic₃	C₂²	1	1	yes	characteristic
57	#810	C₂²	1	1	yes	characteristic
58	#808	C₂²	1	1	yes	characteristic
59	#803	C₂²	1	1	yes	characteristic
60	#557	—	3	1	no	—
61	#556	—	3	1	no	—
62	#497	—	3	1	no	—
63	#497	—	3	1	no	—
64	#494	—	3	1	no	—
65	#464	—	3	1	no	—
66	#464	—	3	1	no	—
67	#462	—	3	1	no	—
68	#462	—	3	1	no	—
69	#462	D₃	1	1	yes	characteristic, complemented
70	#461	—	3	1	no	—
71	#446	—	3	1	no	—
72	#441	—	3	1	no	—
73	#441	—	3	1	no	—
74	C₄×SD₁₆	—	3	1	no	—
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
77	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
78	C₂×D₁₂	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₃	—	2	1	no	—
80	C₂×Dic₆	D₄	1	1	yes	characteristic
81	C₂×Dic₆	C₂³	1	1	yes	characteristic
82	C₂×Dic₆	C₂³	1	1	yes	characteristic
83	C₃×Q₁₆	—	4	1	no	—
84	C₃×Q₁₆	—	4	1	no	—
85	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
87	C₃×C₄⋊C₄	—	2	1	no	—
88	C₃×C₄⋊C₄	—	2	1	no	—
89	C₃×C₄⋊C₄	—	2	1	no	—
90	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₃×C₄²	—	2	1	no	—
92	C₃×C₄²	C₂³	1	1	yes	characteristic
93	#216	—	2	2	no	—
94	#216	—	2	2	no	—
95	#216	—	4	1	no	—
96	#215	—	2	2	no	—
97	#215	—	4	1	no	—
98	(C₂×D₃)⋊C₄	—	2	1	no	—
99	(C₂×D₃)⋊C₄	—	2	1	no	—
100	(C₂×D₃)⋊C₄	—	2	1	no	—
101	(C₂×D₃)⋊C₄	—	2	1	no	—
102	(C₂×D₃)⋊C₄	—	2	1	no	—
103	C₄⋊Dic₃	—	2	1	no	—
104	C₄⋊Dic₃	—	2	1	no	—
105	C₄⋊Dic₃	D₄	1	1	yes	characteristic
106	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
112	Dic₃⋊C₄	—	2	1	no	—
113	Dic₃⋊C₄	—	2	1	no	—
114	Dic₃⋊C₄	—	2	1	no	—
115	Dic₃⋊C₄	D₄	1	1	yes	characteristic
116	Dic₃⋊C₄	D₄	1	1	yes	characteristic
117	C₄×Dic₃	—	2	1	no	—
118	C₄×Dic₃	—	2	1	no	—
119	C₄×Dic₃	—	2	1	no	—
120	C₄×Dic₃	—	2	1	no	—
121	C₄×Dic₃	C₂³	1	1	yes	characteristic
122	C₄×Dic₃	C₂³	1	1	yes	characteristic
123	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
124	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
125	Dic₁₂	—	4	1	no	—
126	#204	—	4	1	no	—
127	C₂×Q₁₆	—	3	1	no	—
128	C₂×Q₁₆	—	3	1	no	—
129	C₂×Q₁₆	—	3	1	no	—
130	C₂×Q₁₆	—	3	1	no	—
131	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic
132	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic
133	C₂×SD₁₆	—	3	1	no	—
134	C₂×SD₁₆	—	3	1	no	—
135	C₂×SD₁₆	—	3	1	no	—
136	C₄⋊Q₈	—	3	1	no	—
137	C₄⋊Q₈	—	3	1	no	—
138	C₄⋊Q₈	—	3	1	no	—
139	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
140	#126	—	6	1	no	—
141	#126	—	6	1	no	—
142	#125	—	6	1	no	—
143	#125	—	6	1	no	—
144	#125	—	6	1	no	—
145	#124	—	3	1	no	—
146	#124	—	3	1	no	—
147	C₂²⋊Q₈	—	6	1	no	—
148	C₄×Q₈	—	3	1	no	—
149	C₄×Q₈	—	3	1	no	—
150	C₄×Q₈	—	3	1	no	—
151	C₄×Q₈	—	3	1	no	—
152	C₄×Q₈	—	3	1	no	—
153	C₄×Q₈	—	6	1	no	—
154	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
155	C₄×D₄	—	3	1	no	—
156	#117	—	6	1	no	—
157	#106	—	3	1	no	—
158	C₄⋊C₈	—	3	1	no	—
159	C₄⋊C₈	—	3	1	no	—
160	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
161	#103	—	3	1	no	—
162	#103	—	3	1	no	—
163	#103	—	3	1	no	—
164	#103	C₂×D₃	1	1	yes	characteristic
165	#103	C₂×D₃	1	1	yes	characteristic
166	#102	—	3	1	no	—
167	#97	—	3	1	no	—
168	#97	—	3	1	no	—
169	C₄×C₈	—	3	1	no	—
170	C₂²×D₃	—	2	1	no	—
171	C₃×Q₈	—	2	1	no	—
172	C₃×Q₈	—	2	1	no	—
173	C₃×Q₈	—	2	1	no	—
174	C₃×Q₈	—	2	1	no	—
175	C₃×Q₈	—	2	1	no	—
176	C₃×Q₈	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
184	C₂×Dic₃	—	2	1	no	—
185	C₂×Dic₃	—	2	1	no	—
186	C₂×Dic₃	—	2	1	no	—
187	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
188	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
189	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
190	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
191	D₁₂	—	2	1	no	—
192	D₁₂	—	2	1	no	—
193	C₄×D₃	—	4	2	no	—
194	Dic₆	—	2	2	no	—
195	Dic₆	—	2	1	no	—
196	Dic₆	—	2	1	no	—
197	Dic₆	—	2	1	no	—
198	Dic₆	—	2	1	no	—
199	C₃×C₈	—	2	1	no	—
200	C₃×C₈	—	2	1	no	—
201	C₃⋊C₈	—	2	1	no	—
202	C₃⋊C₈	#41	1	2	yes	—
203	C₂×Q₈	—	3	1	no	—
204	C₂×Q₈	—	3	1	no	—
205	C₂×Q₈	—	3	1	no	—
206	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
207	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
208	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
209	C₂×D₄	—	3	1	no	—
210	C₂²×C₄	—	6	1	no	—
211	Q₁₆	—	4	1	no	—
212	Q₁₆	—	4	1	no	—
213	Q₁₆	—	6	2	no	—
214	Q₁₆	—	6	2	no	—
215	Q₁₆	—	12	1	no	—
216	Q₁₆	—	12	1	no	—
217	SD₁₆	—	6	2	no	—
218	SD₁₆	—	12	1	no	—
219	SD₁₆	—	12	1	no	—
220	C₂×C₈	—	3	1	no	—
221	C₂×C₈	—	3	1	no	—
222	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	—	3	1	no	—
229	C₄⋊C₄	—	3	1	no	—
230	C₄⋊C₄	—	3	1	no	—
231	C₄⋊C₄	—	6	1	no	—
232	C₄⋊C₄	—	6	1	no	—
233	C₄⋊C₄	—	6	1	no	—
234	C₄⋊C₄	—	6	1	no	—
235	C₄⋊C₄	—	6	1	no	—
236	C₄⋊C₄	—	6	1	no	—
237	C₄⋊C₄	—	6	1	no	—
238	C₄⋊C₄	—	6	1	no	—
239	C₄⋊C₄	—	6	1	no	—
240	C₄⋊C₄	—	6	1	no	—
241	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
242	C₂²⋊C₄	—	6	1	no	—
243	C₂²⋊C₄	—	6	1	no	—
244	C₂²⋊C₄	—	6	1	no	—
245	C₂²⋊C₄	—	6	1	no	—
246	C₂²⋊C₄	—	6	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	3	1	no	—
249	C₄²	—	3	1	no	—
250	C₄²	—	6	1	no	—
251	C₄²	—	6	1	no	—
252	C₄²	—	6	1	no	—
253	C₄²	—	6	1	no	—
254	C₄²	C₂²×D₃	1	1	yes	characteristic
255	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
256	C₂×D₃	—	4	1	no	—
257	C₂×D₃	—	4	1	no	—
258	C₂×D₃	—	4	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	4	1	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	4	1	no	—
264	C₃×C₄	—	4	1	no	—
265	C₃×C₄	C₂×#41	1	1	yes	characteristic
266	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
267	Dic₃	—	2	1	no	—
268	Dic₃	—	2	1	no	—
269	Dic₃	—	2	1	no	—
270	Dic₃	—	4	1	no	—
271	Dic₃	—	4	1	no	—
272	Dic₃	—	4	1	no	—
273	Dic₃	#137	1	2	yes	—
274	C₂³	—	6	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	6	2	no	—
282	Q₈	—	6	1	no	—
283	Q₈	—	6	1	no	—
284	Q₈	—	6	1	no	—
285	Q₈	—	6	1	no	—
286	D₄	—	6	1	no	—
287	D₄	—	6	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	3	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	3	1	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	6	1	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	—	12	2	no	—
300	C₂×C₄	D₃×D₄	1	1	yes	characteristic
301	C₂×C₄	D₃×D₄	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	3	2	no	—
306	C₈	—	6	1	no	—
307	C₂×C₃	#550	1	1	yes	characteristic
308	C₂×C₃	C₂×#137	1	1	yes	characteristic
309	C₂×C₃	#578	1	1	yes	characteristic
310	D₃	—	8	1	no	—
311	C₂²	—	12	1	no	—
312	C₂²	—	12	1	no	—
313	C₂²	—	12	1	no	—
314	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	3	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	6	1	no	—
323	C₄	—	6	1	no	—
324	C₄	—	6	1	no	—
325	C₄	—	12	1	no	—
326	C₄	—	12	1	no	—
327	C₄	—	12	1	no	—
328	C₄	#1004	1	1	yes	characteristic
329	C₄	D₃×#41	1	1	yes	characteristic
330	C₃	#3398	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	24	1	no	—
332	C₂	#5599	1	1	yes	characteristic
333	C₂	#5766	1	1	yes	characteristic
334	C₂	D₃×#137	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×#462	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×D₁₂	C₂³
hypercenter	#462	D₃
hypocenter	C₃	#3398
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
1. #5721	C₂	semidirect
3. C₃×#462	C₂	semidirect
6. #5183	C₂	semidirect
7. #5182	C₂	semidirect
9. #5030	C₂	semidirect
11. #4828	C₂	semidirect
13. #4795	C₂	semidirect
14. #4789	C₂	semidirect
69. #462	D₃	semidirect
330. C₃	#3398	semidirect

Maximal subgroups

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

subgroup	index
#5721	2
#5568	2
C₃×#462	2
#5190	2
#5185	2
#5183	2
#5182	2
#5086	2
#5030	2
#4845	2
#4828	2
#4818	2
#4795	2
#4789	2
#4788	2
#3398	3