Finite Group Databasebeta

← #72266339

Subgroups — #72266339

Order 4096.

Only the 2347 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 316 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72266338C214yesmaximal, complemented
2#72266337C214yesmaximal, complemented
3#72266333C218yesmaximal, complemented
4#72266330C218yesmaximal, complemented
5#72266329C22132yescomplemented
6#72266071C211yesmaximal, characteristic, complemented
7#72266070C2214yes—
8#72255646C212yesmaximal, complemented
9#72255642C2214yescomplemented
10#72251570C2214yescomplemented
11#72251492C2214yescomplemented
12#72241156C2218yes—
13#72238594C2214yescomplemented
14#72236847C211yesmaximal, characteristic, complemented
15#72236837C2218yescomplemented
16#72236834C2214yescomplemented
17#72236714C22116yescomplemented
18#72236110C211yesmaximal, characteristic, complemented
19#72236067C2212yescomplemented
20#72236058C2212yescomplemented
21#72235848C2214yescomplemented
22C2×#245233C2214yescomplemented
23#72234675C2211yescharacteristic
24#72233499C22116yescomplemented
25#72231810C2214yescomplemented
26#72226161C212yesmaximal, complemented
27#72226077C2218yescomplemented
28#72226038C2218yescomplemented
29#72225994C2218yescomplemented
30#72225881C2211yescharacteristic, complemented
31#72225737C2211yescharacteristic, complemented
32#72219377C2212yescomplemented
33C2×#7631286C2212yes—
34C2×#245225C2214yescomplemented
35C24×H(5,𝔽2)D412yescomplemented
36C2×#62817D411yescharacteristic, complemented
37C23×D42C2311yescharacteristic
38#9040981D4116yescomplemented
39#8934620D414yescomplemented
40#8934366D414yescomplemented
41#8933771D418yescomplemented
42#8390796D4132yescomplemented
43#8181078C23116yes—
44#8093110D418yescomplemented
45#8093108D418yescomplemented
46#7670333D418yescomplemented
47#7670203D418yescomplemented
48#7643002C2312yes—
49#7642273C2314yes—
50#7642244D412yescomplemented
51#7630945D414yescomplemented
52C2×#35527C2314yes—
53C2×#35527C2311yescharacteristic, complemented
54C2×#35512D414yescomplemented
55C2×#35347D414yescomplemented
56C2×#35346C2314yes—
57C2×#35346C2312yescomplemented
58C2×#34980C2314yes—
59C22×#2403D414yescomplemented
60#7626202D411yescharacteristic, complemented
61C2×#34171D412yescomplemented
62C2×#32898C2318yescomplemented
63#6628867C2314yescomplemented
64#555905C2311yescharacteristic, complemented
65#555894C2312yescomplemented
66#496225C23132yescomplemented
67#496194C23116yescomplemented
68#382422C2314yescomplemented
69#382394C2312yescomplemented
70#379759C2314yescomplemented
71#379754C2318yescomplemented
72C2×#9497C2318yescomplemented
73C2×#9495C2318yescomplemented
74#201322C2318yescomplemented
75C2×#8547C2314yescomplemented
76C2×#8546C2314yescomplemented
77C2×#8541C2314yescomplemented
78C23×H(5,𝔽2)C2×D414yes—
79C23×H(5,𝔽2)C2×D414yes—
80C23×H(5,𝔽2)C2411yescharacteristic, complemented
81C25×D4C2×D412yes—
82C25×D4C2×D412yes—
83C25×D4C2×D414yes—
84C2×#3490C2×D414yes—
85C2×#3484C2×D412yes—
86C23×#127C2411yescharacteristic
87C23×#121C2×D412yes—
88C23×#121C2×D412yes—
89C23×#120C2×D411yescharacteristic
90C23×#120C2×D411yescharacteristic
91C23×#120C2412yes—
92C23×C4×D4C2×D412yes—
93#59370C2×D4132yes—
94#58990C2×D4132yes—
95#52004C2×D4116yes—
96#51940C2×D4132yes—
97#51902C2×D4116yes—
98#45727C2×D4132yes—
99#45439C2×D4132yes—
100#36703C2×D4116yes—
101#36626C2×D414yes—
102C2×#2849C2×D418yes—
103C2×#2846C2×D418yes—
104#35554C2×D414yes—
105#35528C2×D418yes—
106#35527C2×D4116yes—
107#35527C2×D418yes—
108#35527C2×D418yes—
109#35527C2×D414yes—
110#35390C2×D418yes—
111#35367C2×D418yes—
112#35346C2×D418yes—
113#35346C2×D4116yes—
114#35346C2×D418yes—
115#35222C2414yes—
116#35050C2×D418yes—
117C2×#2405C2×D418yes—
118C2×#2410C2×D4116yes—
119C2×#2403C2×D418yes—
120C2×#2403C2×D418yes—
121C2×#2403C2418yes—
122#34176C2×D414yes—
123#34171C2×D412yes—
124C2×#2309C2×D414yes—
125C2×#2309C2411yescharacteristic, complemented
126C2×#2308C2×D414yes—
127C2×#2308C2412yescomplemented
128C22×#379C2×D414yes—
129C2×#1783C2414yescomplemented
130C2×#1781C2418yescomplemented
131C27#12011yescharacteristic
132C27#12011yescharacteristic
133C27#12014yescomplemented
134C27C22×D412yes—
135C22×H(5,𝔽2)#12014yes—
136C22×H(5,𝔽2)#12018yes—
137C22×H(5,𝔽2)#12014yes—
138C22×H(5,𝔽2)#12018yescomplemented
139C22×H(5,𝔽2)#12014yescomplemented
140C22×H(5,𝔽2)#12014yescomplemented
141C22×H(5,𝔽2)#12012yescomplemented
142C24×D4#12012yes—
143C24×D4#12012yes—
144C24×D4#12012yes—
145C24×D4#12018yescomplemented
146C24×D4C22×D412yes—
147C24×D4C22×D412yes—
148C24×D4C22×D418yes—
149C24×D4C22×D412yes—
150C24×D4C22×D412yes—
151C24×D4C22×D411yescharacteristic
152C24×D4C2511yescharacteristic
153C25×C4#12012yes—
154C25×C4#12012yes—
155C25×C4#12014yescomplemented
156C2×H(3,𝔽4)#120116yescomplemented
157C2×D42#120116yescomplemented
158C2×#543#12014yescomplemented
159C2×#538#12018yescomplemented
160C2×#535#12014yescomplemented
161C2×#535C22×D418yes—
162C2×#534#12014yescomplemented
163C22×#127C22×D414yes—
164C22×#127C22×D414yes—
165C2×#525C22×D414yes—
166C22×#120C22×D418yes—
167C22×#120C22×D418yes—
168C22×#120C22×D4116yes—
169C2×#518#12012yescomplemented
170C2×#518#12014yescomplemented
171C23×C4⋊C4#12012yes—
172C23×C22⋊C4#12011yescharacteristic
173C23×C22⋊C4#12011yescharacteristic
174C23×C22⋊C4#12018yescomplemented
175C23×C22⋊C4C22×D412yes—
176C23×C22⋊C4C22×D412yes—
177C23×C22⋊C4C22×D412yes—
178C23×C42C22×D411yescharacteristic
179#2748#120132yescomplemented
180#2684#120132yescomplemented
181#2679#120132yescomplemented
182#2669#120132yescomplemented
183#2561#120116yescomplemented
184#2539#120116yescomplemented
185#2435C22×D4116yes—
186#2433C22×D4116yes—
187#2403#120116yes—
188#2403#120116yes—
189#2403#120116yescomplemented
190#2403#120116yescomplemented
191#2403C22×D4116yes—
192#2403C22×D4116yes—
193#2403C22×D4116yes—
194#2403C22×D4116yes—
195#2380#120116yescomplemented
196#2375#120132yescomplemented
197#2372#120116yescomplemented
198#2358#120132yescomplemented
199#2312#12018yescomplemented
200#2309#12014yescomplemented
201#2309#12018yescomplemented
202#2309#12018yescomplemented
203#2309C22×D418yes—
204#2308#12018yes—
205#2308#12018yes—
206#2297#12018yescomplemented
207#2296#12018yescomplemented
208C2×#386C22×D4116yes—
209C2×#385#120116yescomplemented
210#2280C22×D414yes—
211C2×#379#12018yescomplemented
212C2×#379#120116yescomplemented
213C2×#379C22×D418yes—
214#2273#12018yescomplemented
215C26B(4,𝔽2)14yes—
216C26C2×#12014yes—
217C26C2×#12014yes—
218C26C2×#12018yes—
219C26C2×#12012yes—
220C26C2×#12011yescharacteristic
221C26C2×#12012yes—
222C26C2×#12011yescharacteristic
223C26C23×D412yes—
224C22×#41C2×#12014yes—
225C23×D4C2×#12014yes—
226C23×D4C2×#12014yes—
227C23×D4C2×#12014yes—
228C23×D4C2×#12014yes—
229C23×D4C2×#12018yes—
230C23×D4C2×#12018yes—
231C23×D4C2×#12018yes—
232C23×D4C2×#12018yes—
233C23×D4C2×#12018yes—
234C23×D4C2×#12018yes—
235C23×D4C2×#12012yes—
236C23×D4C2×#12012yes—
237C23×D4C23×D414yes—
238C23×D4C23×D418yes—
239C24×C4C2×#12018yes—
240C24×C4C2×#12012yes—
241C24×C4C2×#12012yes—
242C24×C4C2×#12012yes—
243C24×C4C23×D411yescharacteristic
244C2×#127C2×#120116yes—
245C2×#124C2×#120116yes—
246C2×C22⋊Q8C2×#12018yes—
247C2×#121C2×#12018yes—
248C2×#120C2×#120132yes—
249C2×#120C2×#120132yes—
250C2×#117C2×#12018yes—
251C2×#117C2×#12014yes—
252C22×C22⋊C4C2×#120116yes—
253C22×C22⋊C4C2×#12018yes—
254C22×C22⋊C4C2×#12018yes—
255C22×C22⋊C4C2×#12014yes—
256C22×C22⋊C4C2×#12014yes—
257C22×C42C2×#12014yes—
258#390C2×#120132yes—
259#386C2×#120132yes—
260#386C2×#120132yes—
261#379C2×#120116yes—
262#379C2×#120116yes—
263C4×C22⋊C4C2×#120116yes—
264C25#284614yes—
265C25#284618yes—
266C25#284618yes—
267C25#284612yes—
268C25#284612yes—
269C25#284612yes—
270C25#284611yescharacteristic
271C25#284611yescharacteristic
272C25C2×B(4,𝔽2)12yes—
273C25C2×B(4,𝔽2)14yes—
274C25#302612yes—
275C25#302614yes—
276C25C22×#12014yes—
277C25C22×#12014yes—
278C25C22×#12014yes—
279C25C22×#12011yescharacteristic
280C22×Q8#284614yes—
281C22×D4#284614yes—
282C22×D4C22×#120116yes—
283C23×C4#284614yes—
284C23×C4#284614yes—
285C23×C4#284614yes—
286C23×C4#284618yes—
287C23×C4#284618yes—
288C23×C4C22×#12014yes—
289C23×C4C22×#12012yes—
290C2×C4⋊C4#284618yes—
291C2×C22⋊C4#2846132yes—
292C2×C42#2846116yes—
293C24#3289814yes—
294C24#3289812yes—
295C24#3289812yes—
296C24#3289814yes—
297C24C2×#284612yes—
298C24C2×#284611yescharacteristic
299C24C2×#284618yes—
300C24C22×B(4,𝔽2)11yescharacteristic
301C24C2×#302612yes—
302C24C2×#302611yescharacteristic
303C22×C4C2×#284614yes—
304C23#51179018yes—
305C23#634237111yescharacteristic
306C23C2×#3289812yes—
307C23C2×#3289811yescharacteristic
308C23#662889212yes—
309C23#662889211yescharacteristic
310C22#7221002512yes—
311C22C2×#51179014yes—
312C22#7221911811yescharacteristic
313C2#7221912012yes—
314C2#7226593811yescharacteristic
subgroupG/H
center, socleC22#72219118
commutator, Frattini, cosocleC24×D4C25
LCSG ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×D42 ◁ G
LECSG ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72266338C2semidirect
2. #72266337C2semidirect
3. #72266333C2semidirect
4. #72266330C2semidirect
5. #72266329C22semidirect
6. #72266071C2semidirect
8. #72255646C2semidirect
9. #72255642C22semidirect
10. #72251570C22semidirect
11. #72251492C22semidirect
13. #72238594C22semidirect
14. #72236847C2semidirect
15. #72236837C22semidirect
16. #72236834C22semidirect
17. #72236714C22semidirect
18. #72236110C2semidirect
19. #72236067C22semidirect
20. #72236058C22semidirect
21. #72235848C22semidirect
22. C2×#245233C22semidirect
24. #72233499C22semidirect
25. #72231810C22semidirect
26. #72226161C2semidirect
27. #72226077C22semidirect
28. #72226038C22semidirect
29. #72225994C22semidirect
30. #72225881C22semidirect
31. #72225737C22semidirect
32. #72219377C22semidirect
34. C2×#245225C22semidirect
35. C24×H(5,𝔽2)D4semidirect
36. C2×#62817D4semidirect
38. #9040981D4semidirect
39. #8934620D4semidirect
40. #8934366D4semidirect
41. #8933771D4semidirect
42. #8390796D4semidirect
44. #8093110D4semidirect
45. #8093108D4semidirect
46. #7670333D4semidirect
47. #7670203D4semidirect
50. #7642244D4semidirect
51. #7630945D4semidirect
53. C2×#35527C23semidirect
54. C2×#35512D4semidirect
55. C2×#35347D4semidirect
57. C2×#35346C23semidirect
59. C22×#2403D4semidirect
60. #7626202D4semidirect
61. C2×#34171D4semidirect
62. C2×#32898C23semidirect
63. #6628867C23semidirect
64. #555905C23semidirect
65. #555894C23semidirect
66. #496225C23semidirect
67. #496194C23semidirect
68. #382422C23semidirect
69. #382394C23semidirect
70. #379759C23semidirect
71. #379754C23semidirect
72. C2×#9497C23semidirect
73. C2×#9495C23semidirect
74. #201322C23semidirect
75. C2×#8547C23semidirect
76. C2×#8546C23semidirect
77. C2×#8541C23semidirect
80. C23×H(5,𝔽2)C24semidirect
125. C2×#2309C24semidirect
127. C2×#2308C24semidirect
129. C2×#1783C24semidirect
130. C2×#1781C24semidirect
133. C27#120semidirect
138. C22×H(5,𝔽2)#120semidirect
139. C22×H(5,𝔽2)#120semidirect
140. C22×H(5,𝔽2)#120semidirect
141. C22×H(5,𝔽2)#120semidirect
145. C24×D4#120semidirect
155. C25×C4#120semidirect
156. C2×H(3,𝔽4)#120semidirect
157. C2×D42#120semidirect
158. C2×#543#120semidirect
159. C2×#538#120semidirect
160. C2×#535#120semidirect
162. C2×#534#120semidirect
169. C2×#518#120semidirect
170. C2×#518#120semidirect
174. C23×C22⋊C4#120semidirect
179. #2748#120semidirect
180. #2684#120semidirect
181. #2679#120semidirect
182. #2669#120semidirect
183. #2561#120semidirect
184. #2539#120semidirect
189. #2403#120semidirect
190. #2403#120semidirect
195. #2380#120semidirect
196. #2375#120semidirect
197. #2372#120semidirect
198. #2358#120semidirect
199. #2312#120semidirect
200. #2309#120semidirect
201. #2309#120semidirect
202. #2309#120semidirect
206. #2297#120semidirect
207. #2296#120semidirect
209. C2×#385#120semidirect
211. C2×#379#120semidirect
212. C2×#379#120semidirect
214. #2273#120semidirect

Maximal subgroups

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

subgroupindex
#722663382
#722663372
#722663332
#722663302
#722660712
#722556462
#722368472
#722361102
#722261612