Subgroups — #78555

Order 384.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#907	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#5524	C₂	1	1	yes	maximal, characteristic
4	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	#5110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#4999	C₂	1	1	yes	maximal, characteristic, complemented
8	#4706	C₂	1	2	yes	maximal, complemented
9	#4706	C₂	1	2	yes	maximal, complemented
10	#4688	C₂	1	2	yes	maximal, complemented
11	C₈×D₁₂	C₂	1	2	yes	maximal, complemented
12	#2933	—	3	1	no	maximal, Sylow
13	C₂×#235	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
17	C₃×#117	C₂²	1	1	yes	characteristic
18	C₂×#208	C₂²	1	1	yes	characteristic
19	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#907	—	2	4	no	—
21	D₃×M₁₆	—	2	4	no	—
22	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
23	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
24	C₂×D₃×C₈	C₂²	1	2	yes	—
25	C₂×D₃×C₈	C₂²	1	2	yes	complemented
26	#897	C₄	1	1	yes	characteristic
27	#897	C₄	1	1	yes	characteristic
28	#895	C₄	1	1	yes	characteristic
29	#895	C₄	1	1	yes	characteristic
30	#886	C₄	1	2	yes	—
31	#882	C₄	1	2	yes	—
32	C₄×D₁₂	C₂²	1	2	yes	—
33	C₄×D₁₂	C₂²	1	2	yes	—
34	C₃×#97	C₂²	1	2	yes	complemented
35	C₃×C₄×C₈	C₂²	1	2	yes	complemented
36	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
37	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
38	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
39	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
40	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
41	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
42	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
43	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
44	C₂×#131	—	3	1	no	—
45	C₂²×M₁₆	—	3	1	no	—
46	#525	—	3	1	no	—
47	#436	—	3	2	no	—
48	#435	—	3	2	no	—
49	#435	—	3	2	no	—
50	C₈×D₄	—	3	2	no	—
51	#424	—	3	1	no	—
52	#408	—	3	1	no	—
53	C₂×C₂²⋊C₈	—	3	1	no	—
54	#405	D₃	1	1	yes	characteristic, complemented
55	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂×#67	C₂×C₄	1	2	yes	—
58	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
59	#235	—	2	2	no	—
60	#235	—	2	2	no	—
61	C₂×D₁₂	C₂×C₄	1	2	yes	—
62	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
63	C₂×D₁₂	C₂×C₄	1	2	yes	—
64	C₂×C₄×D₃	—	2	2	no	—
65	C₂×C₄×D₃	—	2	2	no	—
66	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₃	C₂³	1	2	yes	—
71	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
72	C₃×M₁₆	D₄	1	4	yes	complemented
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂³	1	2	yes	—
77	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
80	C₃×C₄²	C₂³	1	2	yes	—
81	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
82	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
83	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
84	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
85	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
86	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
87	#208	—	2	2	no	—
88	C₂×C₃⋊C₈	—	2	2	no	—
89	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
90	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
91	C₂×C₃⋊C₈	C₂³	1	2	yes	—
92	ΓL(1,𝔽₂₅)	—	2	4	no	—
93	ΓL(1,𝔽₂₅)	—	2	4	no	—
94	D₃×C₈	—	2	4	no	—
95	D₃×C₈	—	2	4	no	—
96	C₂×#41	—	3	1	no	—
97	C₂³×C₄	—	3	1	no	—
98	#131	—	6	4	no	—
99	C₂×M₁₆	—	3	1	no	—
100	C₂×M₁₆	—	3	2	no	—
101	C₂×M₁₆	—	3	2	no	—
102	C₂×M₁₆	—	6	4	no	—
103	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
104	C₂²×C₈	—	3	1	no	—
105	C₂²×C₈	—	3	2	no	—
106	C₂²×C₈	—	3	2	no	—
107	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
108	#123	—	3	2	no	—
109	C₂²⋊Q₈	—	3	1	no	—
110	C₂²⋊Q₈	—	3	1	no	—
111	#121	—	3	1	no	—
112	#121	—	3	1	no	—
113	#120	—	3	2	no	—
114	C₄×D₄	—	3	2	no	—
115	C₄×D₄	—	3	2	no	—
116	#117	C₂×D₃	1	1	yes	characteristic
117	C₄⋊C₈	—	3	2	no	—
118	C₄⋊C₈	—	3	2	no	—
119	C₂²⋊C₈	—	3	1	no	—
120	C₂²⋊C₈	—	3	1	no	—
121	C₂²⋊C₈	—	3	1	no	—
122	C₂²⋊C₈	—	3	1	no	—
123	C₂²⋊C₈	—	3	2	no	—
124	C₂²⋊C₈	—	3	2	no	—
125	#97	C₂×D₃	1	2	yes	complemented
126	C₄×C₈	C₂×D₃	1	2	yes	complemented
127	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
128	C₂²×D₃	—	2	2	no	—
129	C₂²×D₃	—	2	2	no	—
130	C₂²×D₃	—	4	1	no	—
131	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
132	C₂²×D₃	C₂²×C₄	1	2	yes	—
133	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
135	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
136	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
138	C₂×C₃×C₄	#41	1	2	yes	—
139	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
140	#67	—	2	4	no	—
141	C₂×Dic₃	—	2	2	no	—
142	C₂×Dic₃	C₂²×C₄	1	2	yes	—
143	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
144	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
145	D₁₂	—	2	4	no	—
146	D₁₂	—	2	2	no	—
147	D₁₂	—	4	2	no	—
148	C₄×D₃	—	2	2	no	—
149	C₄×D₃	—	2	2	no	—
150	C₄×D₃	—	2	1	no	—
151	C₄×D₃	—	2	2	no	—
152	C₄×D₃	—	2	1	no	—
153	C₄×D₃	—	2	1	no	—
154	C₄×D₃	—	2	1	no	—
155	C₄×D₃	—	2	2	no	—
156	Dic₆	—	2	2	no	—
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	C₂×D₄	1	4	yes	—
160	C₃⋊C₈	—	2	1	no	—
161	C₃⋊C₈	—	2	2	no	—
162	C₃⋊C₈	—	2	1	no	—
163	C₂⁴	—	3	1	no	—
164	#41	—	6	2	no	—
165	#41	—	6	2	no	—
166	C₂×Q₈	—	3	1	no	—
167	C₂×D₄	—	3	2	no	—
168	C₂×D₄	—	3	2	no	—
169	C₂×D₄	—	3	1	no	—
170	C₂×D₄	—	3	2	no	—
171	C₂²×C₄	—	3	1	no	—
172	C₂²×C₄	—	3	1	no	—
173	C₂²×C₄	—	3	1	no	—
174	C₂²×C₄	—	3	1	no	—
175	C₂²×C₄	—	3	2	no	—
176	C₂²×C₄	—	3	1	no	—
177	C₂²×C₄	—	6	2	no	—
178	C₂²×C₄	—	6	2	no	—
179	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
180	M₁₆	—	6	2	no	—
181	M₁₆	—	6	4	no	—
182	M₁₆	—	6	4	no	—
183	M₁₆	D₁₂	1	4	yes	complemented
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	3	1	no	—
186	C₂×C₈	—	3	1	no	—
187	C₂×C₈	—	3	2	no	—
188	C₂×C₈	—	6	2	no	—
189	C₂×C₈	—	6	4	no	—
190	C₂×C₈	—	6	4	no	—
191	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
192	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₃	1	2	yes	—
194	C₄⋊C₄	—	3	2	no	—
195	C₄⋊C₄	—	3	2	no	—
196	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
197	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
198	C₂²⋊C₄	—	3	2	no	—
199	C₂²⋊C₄	—	3	2	no	—
200	C₂²⋊C₄	—	3	2	no	—
201	C₂²⋊C₄	—	3	2	no	—
202	C₂²⋊C₄	C₄×D₃	1	2	yes	—
203	C₄²	C₂²×D₃	1	2	yes	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	C₄×D₄	1	2	yes	—
207	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
208	C₂×D₃	—	2	1	no	—
209	C₂×D₃	—	2	1	no	—
210	C₂×D₃	—	2	1	no	—
211	C₂×D₃	—	2	2	no	—
212	C₂×D₃	—	2	2	no	—
213	C₂×D₃	—	2	1	no	—
214	C₂×D₃	—	2	1	no	—
215	C₂×D₃	—	2	2	no	—
216	C₂×D₃	—	4	4	no	—
217	C₂×D₃	#131	1	2	yes	complemented
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	C₄×D₄	1	2	yes	—
221	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
222	C₃×C₄	C₂×#41	1	1	yes	characteristic
223	Dic₃	—	2	2	no	—
224	Dic₃	—	2	1	no	—
225	Dic₃	—	2	1	no	—
226	C₂³	—	3	1	no	—
227	C₂³	—	3	1	no	—
228	C₂³	—	3	2	no	—
229	C₂³	—	6	2	no	—
230	C₂³	—	6	2	no	—
231	C₂³	—	12	1	no	—
232	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
233	Q₈	—	6	2	no	—
234	D₄	—	6	4	no	—
235	D₄	—	6	4	no	—
236	D₄	—	6	2	no	—
237	D₄	—	12	2	no	—
238	C₂×C₄	—	3	1	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	2	no	—
241	C₂×C₄	—	6	2	no	—
242	C₂×C₄	—	6	2	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	—	6	1	no	—
245	C₂×C₄	—	6	2	no	—
246	C₂×C₄	—	6	1	no	—
247	C₂×C₄	—	6	1	no	—
248	C₂×C₄	—	6	1	no	—
249	C₂×C₄	—	6	2	no	—
250	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
251	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
252	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₁₂	1	2	yes	—
254	C₂×C₄	#235	1	2	yes	—
255	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	6	1	no	—
259	C₈	—	6	1	no	—
260	C₈	—	6	2	no	—
261	C₈	C₂×D₁₂	1	4	yes	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
264	C₂×C₃	C₂×#131	1	1	yes	characteristic
265	C₂×C₃	#568	1	1	yes	characteristic
266	D₃	—	2	2	no	—
267	D₃	—	4	2	no	—
268	D₃	—	4	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	3	2	no	—
272	C₂²	—	6	1	no	—
273	C₂²	—	6	1	no	—
274	C₂²	—	6	1	no	—
275	C₂²	—	6	2	no	—
276	C₂²	—	6	2	no	—
277	C₂²	—	6	1	no	—
278	C₂²	—	6	1	no	—
279	C₂²	—	6	2	no	—
280	C₂²	—	12	4	no	—
281	C₂²	C₄×D₁₂	1	2	yes	—
282	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	6	1	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	2	no	—
288	C₄	C₄×D₁₂	1	2	yes	—
289	C₄	C₂²×D₁₂	1	1	yes	characteristic
290	C₄	C₂×#235	1	1	yes	characteristic
291	C₃	#2933	1	1	yes	characteristic, complemented, Sylow
292	C₂	—	2	2	no	—
293	C₂	—	6	2	no	—
294	C₂	—	12	1	no	—
295	C₂	—	12	2	no	—
296	C₂	C₂×C₄×D₁₂	1	1	yes	characteristic
297	C₂	D₃×#131	1	1	yes	characteristic
298	C₂	#5750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#405	C₂
prime radical	#5524	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#405	D₃
hypocenter	C₃	#2933
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#907	C₂	semidirect
2. C₂×D₃×M₁₆	C₂	semidirect
4. C₃×#405	C₂	semidirect
5. #5110	C₂	semidirect
6. C₂×(C₂×D₃)⋊C₈	C₂	semidirect
7. #4999	C₂	semidirect
8. #4706	C₂	semidirect
9. #4706	C₂	semidirect
10. #4688	C₂	semidirect
11. C₈×D₁₂	C₂	semidirect
23. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
25. C₂×D₃×C₈	C₂²	semidirect
34. C₃×#97	C₂²	semidirect
35. C₃×C₄×C₈	C₂²	semidirect
40. (C₂×D₃)⋊C₈	C₂²	semidirect
41. (C₂×D₃)⋊C₈	C₂²	semidirect
42. C₄⋊(C₃⋊C₈)	C₂²	semidirect
43. C₄⋊(C₃⋊C₈)	C₂²	semidirect
54. #405	D₃	semidirect
72. C₃×M₁₆	D₄	semidirect
125. #97	C₂×D₃	semidirect
126. C₄×C₈	C₂×D₃	semidirect
183. M₁₆	D₁₂	semidirect
217. C₂×D₃	#131	semidirect
291. C₃	#2933	semidirect

Maximal subgroups

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

subgroup	index
C₂×#907	2
C₂×D₃×M₁₆	2
#5524	2
C₃×#405	2
#5110	2
C₂×(C₂×D₃)⋊C₈	2
#4999	2
#4706	2
#4688	2
C₈×D₁₂	2
#2933	3