Subgroups — #78990

Order 384.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#950	C₂	1	1	yes	maximal, characteristic
2	#5819	C₂	1	1	yes	maximal, characteristic, complemented
3	#5745	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#421	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#822	C₂	1	1	yes	maximal, characteristic, complemented
6	#5005	C₂	1	1	yes	maximal, characteristic, complemented
7	#4997	C₂	1	1	yes	maximal, characteristic, complemented
8	#4824	C₂	1	4	yes	maximal, complemented
9	#4823	C₂	1	4	yes	maximal, complemented
10	#2957	—	3	1	no	maximal, Sylow
11	C₂×C₃×#41	C₂²	1	1	yes	characteristic
12	C₂×#235	C₂²	1	1	yes	characteristic
13	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₃×#117	C₂²	1	1	yes	characteristic
15	#950	C₄	1	2	yes	complemented
16	#950	C₄	1	2	yes	complemented
17	#950	C₂²	1	2	yes	—
18	#950	C₂²	1	2	yes	—
19	#948	—	2	2	no	—
20	#948	C₂²	1	4	yes	complemented
21	C₂×#216	C₄	1	1	yes	characteristic, complemented
22	C₂×#215	C₄	1	1	yes	characteristic, complemented
23	C₂×#214	C₄	1	1	yes	characteristic, complemented
24	C₂×#213	C₄	1	1	yes	characteristic, complemented
25	C₂×#208	—	2	2	no	—
26	C₂×#208	C₂²	1	1	yes	characteristic
27	C₂×#208	C₂²	1	1	yes	characteristic
28	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
29	#907	C₂²	1	4	yes	complemented
30	D₃×M₁₆	—	2	2	no	—
31	#901	—	2	2	no	—
32	C₃×#104	C₂²	1	4	yes	complemented
33	#822	C₂²	1	2	yes	complemented
34	#822	C₂²	1	2	yes	complemented
35	#805	C₂²	1	4	yes	complemented
36	#803	C₂²	1	2	yes	complemented
37	C₄×C₃⋊C₈	C₂²	1	2	yes	complemented
38	C₂×#135	—	3	1	no	—
39	#568	—	3	1	no	—
40	#568	—	3	1	no	—
41	#444	—	3	4	no	—
42	#443	—	3	4	no	—
43	C₂×#108	—	3	1	no	—
44	#421	—	3	1	no	—
45	#421	D₃	1	1	yes	characteristic, complemented
46	#405	—	3	1	no	—
47	C₃×#41	—	2	1	no	—
48	C₃×#41	—	2	1	no	—
49	C₃×#41	C₂×C₄	1	2	yes	—
50	C₃×#41	C₂³	1	2	yes	—
51	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃×D₄	—	2	1	no	—
53	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	—	2	1	no	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂×#67	—	2	1	no	—
57	#235	—	2	1	no	—
58	#235	—	2	1	no	—
59	#235	C₂×C₄	1	2	yes	—
60	#235	C₂³	1	2	yes	—
61	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₃	—	2	1	no	—
63	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
64	C₃×M₁₆	—	2	1	no	—
65	C₃×M₁₆	C₂³	1	2	yes	—
66	C₂×C₃×C₈	—	2	1	no	—
67	C₃×C₄⋊C₄	—	2	1	no	—
68	C₃×C₂²⋊C₄	—	2	1	no	—
69	C₃×C₄²	C₂³	1	2	yes	—
70	#216	C₂×C₄	1	2	yes	—
71	#216	C₂×C₄	1	2	yes	—
72	#215	C₂×C₄	1	2	yes	—
73	#215	C₂×C₄	1	2	yes	—
74	#214	C₂×C₄	1	2	yes	—
75	#214	C₂×C₄	1	2	yes	—
76	#213	C₂×C₄	1	2	yes	—
77	#213	C₂×C₄	1	2	yes	—
78	#208	—	2	1	no	—
79	#208	—	2	4	no	—
80	#208	D₄	1	2	yes	—
81	#208	D₄	1	2	yes	—
82	#208	C₂³	1	2	yes	—
83	C₂×C₃⋊C₈	—	2	1	no	—
84	C₂×C₃⋊C₈	—	2	4	no	—
85	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
86	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
87	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
88	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
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	D₃×C₈	—	2	4	no	—
94	C₂×#41	—	3	1	no	—
95	C₂×#41	C₂×D₃	1	1	yes	characteristic
96	#135	—	3	2	no	—
97	#135	—	3	2	no	—
98	#135	—	3	2	no	—
99	#135	—	3	2	no	—
100	C₂×Q₁₆	—	3	1	no	—
101	C₂×SD₁₆	—	3	1	no	—
102	C₂×SD₁₆	—	3	1	no	—
103	C₂×D₈	—	3	1	no	—
104	#131	—	3	4	no	—
105	#131	—	3	4	no	—
106	#131	—	6	2	no	—
107	#131	—	6	2	no	—
108	C₂×M₁₆	—	3	1	no	—
109	C₂×M₁₆	—	3	1	no	—
110	C₂×M₁₆	—	6	2	no	—
111	C₂×M₁₆	—	6	2	no	—
112	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
113	C₂²×C₈	—	3	1	no	—
114	#117	C₂×D₃	1	1	yes	characteristic
115	#108	—	3	2	no	—
116	#108	—	3	2	no	—
117	#104	—	3	4	no	—
118	#104	C₂×D₃	1	4	yes	complemented
119	#97	—	3	2	no	—
120	C₄×C₈	—	3	2	no	—
121	C₂³×C₃	—	2	1	no	—
122	C₂³×C₃	#41	1	1	yes	characteristic
123	C₂²×D₃	—	2	1	no	—
124	C₃×Q₈	—	2	1	no	—
125	C₃×Q₈	C₂²×C₄	1	2	yes	—
126	C₃×D₄	—	2	2	no	—
127	C₃×D₄	—	2	2	no	—
128	C₃×D₄	—	2	1	no	—
129	C₃×D₄	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄	#41	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
140	#67	—	2	2	no	—
141	#67	—	2	2	no	—
142	C₂×Dic₃	—	2	1	no	—
143	D₁₂	—	2	1	no	—
144	D₁₂	C₂²×C₄	1	2	yes	—
145	C₄×D₃	—	2	2	no	—
146	C₄×D₃	—	2	2	no	—
147	Dic₆	—	2	1	no	—
148	Dic₆	C₂²×C₄	1	2	yes	—
149	C₃×C₈	—	2	2	no	—
150	C₃⋊C₈	—	2	2	no	—
151	C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
152	C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
153	C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
154	C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
155	C₃⋊C₈	C₂×D₄	1	4	yes	—
156	#41	—	2	1	no	—
157	#41	—	2	1	no	—
158	#41	—	3	2	no	—
159	#41	—	3	2	no	—
160	#41	—	6	1	no	—
161	#41	—	6	1	no	—
162	#41	C₄×D₃	1	2	yes	—
163	#41	C₂²×D₃	1	2	yes	—
164	C₂×Q₈	—	3	1	no	—
165	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	3	1	no	—
168	C₂×D₄	—	6	1	no	—
169	C₂×D₄	C₄×D₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	6	1	no	—
172	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
173	Q₁₆	—	3	2	no	—
174	Q₁₆	—	3	2	no	—
175	SD₁₆	—	3	2	no	—
176	SD₁₆	—	3	2	no	—
177	SD₁₆	—	3	2	no	—
178	SD₁₆	—	3	2	no	—
179	D₈	—	3	2	no	—
180	D₈	—	3	2	no	—
181	M₁₆	—	2	1	no	—
182	M₁₆	—	3	2	no	—
183	M₁₆	—	3	2	no	—
184	M₁₆	—	3	2	no	—
185	M₁₆	—	6	4	no	—
186	M₁₆	—	6	4	no	—
187	M₁₆	—	6	1	no	—
188	M₁₆	C₂²×D₃	1	2	yes	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	3	1	no	—
191	C₂×C₈	—	3	1	no	—
192	C₂×C₈	—	3	1	no	—
193	C₂×C₈	—	3	1	no	—
194	C₂×C₈	—	3	1	no	—
195	C₂×C₈	—	3	2	no	—
196	C₂×C₈	—	3	1	no	—
197	C₂×C₈	—	6	4	no	—
198	C₂×C₈	—	6	4	no	—
199	C₂×C₈	—	6	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂²×D₃	1	2	yes	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	—	8	1	no	—
206	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
207	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
208	C₂²×C₃	C₂×#41	1	1	yes	characteristic
209	C₂×D₃	—	2	2	no	—
210	C₂×D₃	—	8	1	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	4	2	no	—
213	C₃×C₄	C₄×D₄	1	1	yes	characteristic
214	C₃×C₄	C₄×D₄	1	1	yes	characteristic
215	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
216	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
217	Dic₃	—	2	2	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	6	1	no	—
220	C₂³	#237	1	1	yes	characteristic
221	Q₈	—	2	1	no	—
222	Q₈	—	3	2	no	—
223	Q₈	—	6	1	no	—
224	Q₈	C₂×C₄×D₃	1	2	yes	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	1	no	—
228	D₄	—	3	2	no	—
229	D₄	—	6	2	no	—
230	D₄	—	6	2	no	—
231	D₄	—	6	1	no	—
232	D₄	C₂×C₄×D₃	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	6	1	no	—
238	C₂×C₄	—	6	2	no	—
239	C₂×C₄	—	6	2	no	—
240	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
242	C₂×C₄	D₃×D₄	1	1	yes	characteristic
243	C₂×C₄	D₃×D₄	1	1	yes	characteristic
244	C₂×C₄	#237	1	1	yes	characteristic
245	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	3	1	no	—
248	C₈	—	3	1	no	—
249	C₈	—	3	1	no	—
250	C₈	—	3	1	no	—
251	C₈	—	3	4	no	—
252	C₈	—	6	2	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	4	2	no	—
257	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
258	D₃	—	4	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	6	2	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	24	1	no	—
264	C₂²	#67⋊C₄	1	1	yes	characteristic
265	C₂²	#67⋊C₄	1	1	yes	characteristic
266	C₂²	C₂×#237	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	6	2	no	—
270	C₄	#67⋊C₄	1	1	yes	characteristic
271	C₄	#67⋊C₄	1	1	yes	characteristic
272	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
273	C₄	C₂×D₃×D₄	1	1	yes	characteristic
274	C₃	#2957	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	12	2	no	—
280	C₂	C₂×#67⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#421	C₂
socle	C₂×C₃	C₂×C₄×D₄
prime radical	C₂×#950	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#421	D₃
hypocenter	C₃	#2957
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. #5819	C₂	semidirect
3. #5745	C₂	semidirect
4. C₃×#421	C₂	semidirect
5. C₂×#822	C₂	semidirect
6. #5005	C₂	semidirect
7. #4997	C₂	semidirect
8. #4824	C₂	semidirect
9. #4823	C₂	semidirect
15. #950	C₄	semidirect
16. #950	C₄	semidirect
20. #948	C₂²	semidirect
21. C₂×#216	C₄	semidirect
22. C₂×#215	C₄	semidirect
23. C₂×#214	C₄	semidirect
24. C₂×#213	C₄	semidirect
29. #907	C₂²	semidirect
32. C₃×#104	C₂²	semidirect
33. #822	C₂²	semidirect
34. #822	C₂²	semidirect
35. #805	C₂²	semidirect
36. #803	C₂²	semidirect
37. C₄×C₃⋊C₈	C₂²	semidirect
45. #421	D₃	semidirect
118. #104	C₂×D₃	semidirect
274. C₃	#2957	semidirect

Maximal subgroups

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

subgroup	index
C₂×#950	2
#5819	2
#5745	2
C₃×#421	2
C₂×#822	2
#5005	2
#4997	2
#4824	2
#4823	2
#2957	3