Subgroups — #72324299

Order 4096.

Only the 451 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 276 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72324298	C₂	1	4	yes	maximal, complemented
2	#72324297	C₂	1	4	yes	maximal, complemented
3	#72323226	C₂	1	1	yes	maximal, characteristic, complemented
4	#72309645	C₂	1	1	yes	maximal, characteristic, complemented
5	#72309636	C₂²	1	4	yes	complemented
6	#72300151	C₂	1	1	yes	maximal, characteristic
7	#72300148	C₄	1	2	yes	complemented
8	#72271167	C₄	1	1	yes	characteristic, complemented
9	#72266974	C₂²	1	2	yes	—
10	#72266965	C₄	1	2	yes	complemented
11	#72266864	C₄	1	1	yes	characteristic, complemented
12	#72266864	C₄	1	1	yes	characteristic, complemented
13	#72259010	C₂	1	1	yes	maximal, characteristic, complemented
14	#72259008	C₂²	1	1	yes	characteristic, complemented
15	#72237596	C₂	1	1	yes	maximal, characteristic, complemented
16	#72237593	C₂²	1	4	yes	complemented
17	#72237593	C₂²	1	4	yes	complemented
18	#72237391	C₂	1	1	yes	maximal, characteristic, complemented
19	#72237388	C₂²	1	4	yes	complemented
20	#72237302	C₄	1	1	yes	characteristic, complemented
21	#72235718	C₂²	1	1	yes	characteristic, complemented
22	#72234516	C₂²	1	2	yes	—
23	#72226102	C₂	1	1	yes	maximal, characteristic, complemented
24	#72226097	C₂²	1	2	yes	complemented
25	#72226097	C₂²	1	2	yes	complemented
26	#72226091	C₂²	1	2	yes	complemented
27	#72226091	C₂²	1	2	yes	complemented
28	#72225870	C₂²	1	1	yes	characteristic, complemented
29	#72224136	C₂²	1	1	yes	characteristic
30	#72224136	C₂²	1	1	yes	characteristic
31	#72219311	C₂²	1	1	yes	characteristic, complemented
32	#72219264	C₂²	1	1	yes	characteristic
33	#10586797	C₂×C₄	1	2	yes	complemented
34	#10586489	C₂×C₄	1	2	yes	complemented
35	#10586404	C₂×C₄	1	1	yes	characteristic
36	#10586404	C₂×C₄	1	1	yes	characteristic
37	C₂²×#3532	C₂×C₄	1	1	yes	characteristic
38	#7630939	D₄	1	2	yes	—
39	C₂×#34980	C₂³	1	1	yes	characteristic
40	#6499871	D₄	1	2	yes	—
41	#6499871	C₂³	1	2	yes	—
42	#6499871	C₂³	1	2	yes	—
43	#6499870	D₄	1	2	yes	—
44	#6499869	C₂×C₄	1	2	yes	—
45	#6499869	C₂×C₄	1	2	yes	—
46	#6499869	D₄	1	2	yes	—
47	#6499867	D₄	1	2	yes	—
48	#6451874	D₄	1	2	yes	—
49	C₂×#23984	D₄	1	2	yes	—
50	#6374210	D₄	1	2	yes	—
51	#6374128	C₂×C₄	1	1	yes	characteristic
52	#6374128	C₂×C₄	1	1	yes	characteristic
53	#5025789	C₂³	1	1	yes	characteristic
54	#5025782	D₄	1	1	yes	characteristic
55	#5021140	C₂×C₄	1	1	yes	characteristic
56	#5021138	D₄	1	1	yes	characteristic
57	#4422949	C₂³	1	1	yes	characteristic
58	#4420586	D₄	1	1	yes	characteristic
59	#763333	D₄	1	1	yes	characteristic
60	#686187	C₂×C₄	1	1	yes	characteristic
61	#647656	D₄	1	1	yes	characteristic
62	#616961	D₄	1	1	yes	characteristic
63	#616960	D₄	1	1	yes	characteristic
64	C₂×#15947	D₄	1	1	yes	characteristic
65	#552670	C₂×C₄	1	1	yes	characteristic
66	#188065	C₂×C₄	1	2	yes	complemented
67	#188046	C₂×C₄	1	2	yes	complemented
68	#187952	C₂×C₄	1	2	yes	complemented
69	#187952	C₂×C₄	1	2	yes	complemented
70	#187943	C₂×C₄	1	2	yes	complemented
71	#187943	C₂×C₄	1	2	yes	complemented
72	#134036	C₂³	1	2	yes	complemented
73	#134036	C₂³	1	2	yes	complemented
74	#115631	C₂³	1	2	yes	complemented
75	#115631	C₂³	1	2	yes	complemented
76	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
77	#62820	C₂²⋊C₄	1	2	yes	—
78	#62820	C₂²×C₄	1	2	yes	—
79	#62820	C₂²×C₄	1	2	yes	—
80	#62817	C₂²⋊C₄	1	2	yes	—
81	C₂×#3490	C₂²⋊C₄	1	2	yes	—
82	C₂×#3484	C₂²⋊C₄	1	2	yes	—
83	C₂²×#551	C₂²×C₄	1	1	yes	characteristic
84	C₂²×D₄²	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×H(3,𝔽₄)	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×#560	C₂²×C₄	1	1	yes	characteristic
87	C₂²×#535	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×#534	C₂²×C₄	1	1	yes	characteristic
89	C₂²×#534	C₂×D₄	1	1	yes	characteristic
90	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×#120	C₂×D₄	1	1	yes	characteristic
92	#36614	C₂²⋊C₄	1	4	yes	—
93	C₂×#2849	C₂²×C₄	1	1	yes	characteristic
94	C₂×#2849	C₂×D₄	1	1	yes	characteristic
95	C₂×#2846	C₂×D₄	1	1	yes	characteristic
96	C₂×#2846	C₂×D₄	1	1	yes	characteristic
97	#35222	C₂×D₄	1	4	yes	—
98	C₂×#2613	C₂×D₄	1	1	yes	characteristic
99	C₂×#2613	C₂×D₄	1	1	yes	characteristic
100	C₂×#2613	C₂×D₄	1	1	yes	characteristic
101	C₂×#2613	C₂⁴	1	1	yes	characteristic
102	#34171	C₂²⋊C₄	1	2	yes	—
103	C₂×#2308	C₂×D₄	1	1	yes	characteristic
104	C₂×#2308	C₂×D₄	1	1	yes	characteristic
105	C₂²×#379	C₂×D₄	1	1	yes	characteristic
106	#23985	C₂²⋊C₄	1	4	yes	—
107	#23984	C₂²⋊C₄	1	4	yes	—
108	#23984	C₂×D₄	1	4	yes	—
109	#23984	C₂×D₄	1	4	yes	—
110	#21748	C₂²⋊C₄	1	4	yes	—
111	#21748	C₂×D₄	1	4	yes	—
112	#21747	C₂²⋊C₄	1	2	yes	—
113	#21747	C₂²×C₄	1	2	yes	—
114	#21747	C₂²×C₄	1	2	yes	—
115	#21746	C₂²⋊C₄	1	2	yes	—
116	C₂×#1819	C₂²⋊C₄	1	2	yes	—
117	#15948	C₂²⋊C₄	1	1	yes	characteristic
118	#15948	C₂²⋊C₄	1	1	yes	characteristic
119	#15947	C₂²⋊C₄	1	1	yes	characteristic
120	#15947	C₂²⋊C₄	1	1	yes	characteristic
121	#15947	C₂×D₄	1	1	yes	characteristic
122	#15947	C₂×D₄	1	1	yes	characteristic
123	#15947	C₂×D₄	1	1	yes	characteristic
124	#15947	C₂×D₄	1	1	yes	characteristic
125	#15262	C₂²⋊C₄	1	1	yes	characteristic
126	#15261	C₂²⋊C₄	1	1	yes	characteristic
127	#15259	C₂²⋊C₄	1	1	yes	characteristic
128	#15259	C₂×D₄	1	1	yes	characteristic
129	#15259	C₂×D₄	1	1	yes	characteristic
130	#15258	C₂²⋊C₄	1	1	yes	characteristic
131	#15258	C₂×D₄	1	1	yes	characteristic
132	#15258	C₂×D₄	1	1	yes	characteristic
133	#14658	C₂²×C₄	1	1	yes	characteristic
134	#14657	C₂²⋊C₄	1	1	yes	characteristic
135	#14335	C₂²⋊C₄	1	1	yes	characteristic
136	#14167	C₂²×C₄	1	1	yes	characteristic
137	#14134	C₂²⋊C₄	1	1	yes	characteristic
138	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic
139	C₂⁷	#99	1	1	yes	characteristic
140	C₂⁷	#120	1	1	yes	characteristic
141	C₂⁷	#120	1	1	yes	characteristic
142	C₂²×H(5,𝔽₂)	#99	1	2	yes	—
143	C₂³×#41	#99	1	1	yes	characteristic
144	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂⁴×D₄	#120	1	1	yes	characteristic
147	C₂⁴×D₄	#120	1	1	yes	characteristic
148	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
149	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	#3490	#99	1	4	yes	—
151	#3490	C₂×C₂²⋊C₄	1	4	yes	—
152	C₂×#543	#99	1	2	yes	—
153	C₂×#535	#99	1	4	yes	—
154	C₂×#534	#99	1	2	yes	—
155	C₂²×#127	#99	1	1	yes	characteristic
156	C₂²×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×#124	#99	1	1	yes	characteristic
158	C₂²×#124	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×#525	C₂×C₂²⋊C₄	1	4	yes	—
160	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×#123	#120	1	1	yes	characteristic
163	C₂²×#123	C₂³×C₄	1	1	yes	characteristic
164	C₂²×#120	#99	1	1	yes	characteristic
165	C₂²×#120	#99	1	1	yes	characteristic
166	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×#120	#120	1	1	yes	characteristic
172	C₂²×#120	#120	1	1	yes	characteristic
173	C₂²×#120	#120	1	1	yes	characteristic
174	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
175	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
176	C₂×#518	#99	1	2	yes	—
177	C₂²×#117	#99	1	1	yes	characteristic
178	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic
179	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
181	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
182	#2280	C₂×C₂²⋊C₄	1	4	yes	—
183	C₂×#381	C₂×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×#381	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×#381	#120	1	1	yes	characteristic
186	C₂×#381	C₂²×D₄	1	1	yes	characteristic
187	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×#379	#120	1	1	yes	characteristic
189	C₂×#379	#120	1	1	yes	characteristic
190	C₂×#379	#120	1	1	yes	characteristic
191	C₂×#379	#120	1	1	yes	characteristic
192	C₂×#379	#120	1	1	yes	characteristic
193	C₂×#379	C₂²×D₄	1	1	yes	characteristic
194	C₂×#379	C₂²×D₄	1	1	yes	characteristic
195	#1819	C₂×C₂²⋊C₄	1	4	yes	—
196	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂⁶	#379	1	1	yes	characteristic
201	C₂⁶	#379	1	1	yes	characteristic
202	C₂⁶	#379	1	1	yes	characteristic
203	C₂⁶	C₂×#99	1	1	yes	characteristic
204	C₂⁶	C₂×#99	1	1	yes	characteristic
205	C₂⁶	C₂×#99	1	1	yes	characteristic
206	C₂⁶	C₂×#120	1	1	yes	characteristic
207	C₂⁶	C₂×#120	1	1	yes	characteristic
208	C₂²×#41	C₂×#99	1	4	yes	—
209	C₂³×Q₈	C₂×#99	1	1	yes	characteristic
210	C₂³×D₄	#379	1	1	yes	characteristic
211	C₂³×D₄	C₂×#99	1	1	yes	characteristic
212	C₂³×D₄	C₂×#99	1	2	yes	—
213	C₂³×D₄	C₂×#99	1	2	yes	—
214	C₂³×D₄	C₂×#99	1	1	yes	characteristic
215	C₂³×D₄	C₂×#99	1	1	yes	characteristic
216	C₂³×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
217	C₂⁴×C₄	#379	1	1	yes	characteristic
218	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
219	C₂⁴×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
220	#543	C₂×#99	1	4	yes	—
221	#534	C₂×#99	1	4	yes	—
222	C₂×#127	AΣL(1,𝔽₁₆)	1	8	yes	complemented
223	C₂×#124	AΣL(1,𝔽₁₆)	1	8	yes	complemented
224	C₂×#117	C₂×#99	1	4	yes	—
225	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
226	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
227	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
228	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
229	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
230	C₂²×C₂²⋊C₄	C₂×#99	1	1	yes	characteristic
231	C₂²×C₂²⋊C₄	C₂×#99	1	1	yes	characteristic
232	C₂²×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
233	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
234	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
235	C₂²×C₄²	C₂×#99	1	1	yes	characteristic
236	C₂²×C₄²	C₂×#99	1	1	yes	characteristic
237	C₂⁵	#1781	1	1	yes	characteristic
238	C₂⁵	#1781	1	1	yes	characteristic
239	C₂⁵	#1781	1	1	yes	characteristic
240	C₂⁵	#1781	1	1	yes	characteristic
241	C₂⁵	#1781	1	1	yes	characteristic
242	C₂⁵	#1781	1	1	yes	characteristic
243	C₂⁵	#1781	1	1	yes	characteristic
244	C₂⁵	#1781	1	2	yes	—
245	C₂⁵	C₂×#379	1	1	yes	characteristic
246	C₂⁵	C₂²×#99	1	1	yes	characteristic
247	C₂²×Q₈	C₂×AΣL(1,𝔽₁₆)	1	4	yes	complemented
248	C₂²×D₄	C₂×AΣL(1,𝔽₁₆)	1	8	yes	—
249	C₂²×D₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	complemented
250	C₂³×C₄	#1781	1	1	yes	characteristic
251	C₂³×C₄	C₂²×#99	1	2	yes	—
252	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
253	C₂×C₂²⋊C₄	#1781	1	2	yes	—
254	C₂×C₂²⋊C₄	C₂²×#99	1	2	yes	—
255	C₂×C₄²	C₂×AΣL(1,𝔽₁₆)	1	8	yes	complemented
256	C₂⁴	#8541	1	1	yes	characteristic
257	C₂⁴	#8541	1	1	yes	characteristic
258	C₂⁴	#8541	1	1	yes	characteristic
259	C₂⁴	#8541	1	1	yes	characteristic
260	C₂⁴	#12687	1	4	yes	—
261	C₂⁴	#12687	1	4	yes	—
262	C₂⁴	C₂×#1781	1	1	yes	characteristic
263	C₂⁴	C₂×#1781	1	1	yes	characteristic
264	C₂⁴	C₂×#1781	1	1	yes	characteristic
265	C₂²×C₄	C₂²×AΣL(1,𝔽₁₆)	1	4	yes	—
266	C₂³	#133822	1	2	yes	—
267	C₂³	#133822	1	2	yes	—
268	C₂³	#133839	1	2	yes	—
269	C₂³	C₂×#8541	1	1	yes	characteristic
270	C₂³	C₂×#12687	1	2	yes	—
271	C₂²	C₂×#133822	1	1	yes	characteristic
272	C₂²	C₂×#133839	1	1	yes	characteristic
273	C₂²	#72240419	1	1	yes	characteristic
274	C₂	#72240423	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72240423
commutator	C₂²×#123	C₂³×C₄
Frattini, cosocle	C₂×#2613	C₂⁴
prime radical	#72300151	C₂
LCS	G ▷ C₂²×#123 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72324298	C₂	semidirect
2. #72324297	C₂	semidirect
3. #72323226	C₂	semidirect
4. #72309645	C₂	semidirect
5. #72309636	C₂²	semidirect
7. #72300148	C₄	semidirect
8. #72271167	C₄	semidirect
10. #72266965	C₄	semidirect
11. #72266864	C₄	semidirect
12. #72266864	C₄	semidirect
13. #72259010	C₂	semidirect
14. #72259008	C₂²	semidirect
15. #72237596	C₂	semidirect
16. #72237593	C₂²	semidirect
17. #72237593	C₂²	semidirect
18. #72237391	C₂	semidirect
19. #72237388	C₂²	semidirect
20. #72237302	C₄	semidirect
21. #72235718	C₂²	semidirect
23. #72226102	C₂	semidirect
24. #72226097	C₂²	semidirect
25. #72226097	C₂²	semidirect
26. #72226091	C₂²	semidirect
27. #72226091	C₂²	semidirect
28. #72225870	C₂²	semidirect
31. #72219311	C₂²	semidirect
33. #10586797	C₂×C₄	semidirect
34. #10586489	C₂×C₄	semidirect
66. #188065	C₂×C₄	semidirect
67. #188046	C₂×C₄	semidirect
68. #187952	C₂×C₄	semidirect
69. #187952	C₂×C₄	semidirect
70. #187943	C₂×C₄	semidirect
71. #187943	C₂×C₄	semidirect
72. #134036	C₂³	semidirect
73. #134036	C₂³	semidirect
74. #115631	C₂³	semidirect
75. #115631	C₂³	semidirect
222. C₂×#127	AΣL(1,𝔽₁₆)	semidirect
223. C₂×#124	AΣL(1,𝔽₁₆)	semidirect
247. C₂²×Q₈	C₂×AΣL(1,𝔽₁₆)	semidirect
249. C₂²×D₄	C₂×AΣL(1,𝔽₁₆)	semidirect
255. C₂×C₄²	C₂×AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
#72324298	2
#72324297	2
#72323226	2
#72309645	2
#72300151	2
#72259010	2
#72237596	2
#72237391	2
#72226102	2