Finite Group Databasebeta

← #72286575

Subgroups — #72286575

Order 2048.

At least 6313 conjugacy classes of subgroups; only the 454 normal subgroups are shown.

The subgroup lattice has 303 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72286562C211yesmaximal, characteristic, complemented
2#72286558C211yesmaximal, characteristic, complemented
3#72286553C211yesmaximal, characteristic, complemented
4#72286548C211yesmaximal, characteristic, complemented
5#72286536C211yesmaximal, characteristic, complemented
6#72286516C211yesmaximal, characteristic, complemented
7#72286309C211yesmaximal, characteristic, complemented
8#72286294C211yesmaximal, characteristic, complemented
9#72286280C211yesmaximal, characteristic, complemented
10#72284773C211yesmaximal, characteristic, complemented
11#72284631C211yesmaximal, characteristic, complemented
12#72284480C211yesmaximal, characteristic, complemented
13#72284325C211yesmaximal, characteristic, complemented
14#72256626C211yesmaximal, characteristic, complemented
15#72214082C211yesmaximal, characteristic
16#10576177C2211yescharacteristic
17C2×#60388C2211yescharacteristic
18#7627594C2211yescharacteristic, complemented
19C2×#33359C2211yescharacteristic, complemented
20#7514141C414yescomplemented
21#7514140C414yescomplemented
22#7503371C2211yescharacteristic
23#7500918C2211yescharacteristic
24#6443311C2211yescharacteristic
25C2×#23872C2211yescharacteristic
26#6381485C2211yescharacteristic
27#512601C2211yescharacteristic, complemented
28#512594C2211yescharacteristic, complemented
29#512585C2211yescharacteristic, complemented
30#512578C2211yescharacteristic, complemented
31#512578C2211yescharacteristic, complemented
32#512571C2211yescharacteristic
33#512564C2211yescharacteristic
34#512564C2211yescharacteristic, complemented
35#494620C2211yescharacteristic, complemented
36#494599C2211yescharacteristic
37#494599C2211yescharacteristic, complemented
38#494595C2211yescharacteristic
39#494595C2211yescharacteristic, complemented
40#494595C2211yescharacteristic, complemented
41C2×#12718C2211yescharacteristic, complemented
42#201040C2211yescharacteristic, complemented
43#200841C2211yescharacteristic, complemented
44#187081C2211yescharacteristic, complemented
45#151858C2211yescharacteristic, complemented
46#151842C2211yescharacteristic, complemented
47#151794C2211yescharacteristic, complemented
48#151784C2211yescharacteristic, complemented
49#146645C2211yescharacteristic, complemented
50#146636C2211yescharacteristic, complemented
51#108381C2211yescharacteristic, complemented
52#107744C2211yescharacteristic, complemented
53C2×H(7,𝔽2)D411yescharacteristic
54C2×#3585D411yescharacteristic
55C2×#3498C2311yescharacteristic
56#61111C2×C412yescomplemented
57#61110C2×C412yescomplemented
58#60639C2311yescharacteristic
59#60415D411yescharacteristic
60C2×#3027D411yescharacteristic
61#60388C2×C414yes—
62#60388D414yes—
63C2×#3021D411yescharacteristic
64C2×#3021D411yescharacteristic
65C2×#3018C2311yescharacteristic
66C2×#3018C2311yescharacteristic
67C2×#3018C2311yescharacteristic
68C2×#2888D411yescharacteristic
69#35274D411yescharacteristic
70#34369C2311yescharacteristic
71#34346C2311yescharacteristic
72C2×#2132C2311yescharacteristic
73C2×#2126C2311yescharacteristic
74C2×#2126C2311yescharacteristic
75C2×#2126C2311yescharacteristic
76#31076C2×C414yescomplemented
77#30497C2×C412yescomplemented
78#30496C2×C412yescomplemented
79#25747C2×C412yes—
80#25747D412yes—
81#25744C2×C412yes—
82#25744D412yes—
83#23872C2×C414yes—
84#23872D414yes—
85#22899C2×C414yescomplemented
86C2×#1965D411yescharacteristic
87#21209C2311yescharacteristic
88C2×#1914D411yescharacteristic
89C2×#1884C2311yescharacteristic
90C2×#1795D411yescharacteristic
91C4×#421D411yescharacteristic
92#17738C2311yescharacteristic
93#17663C2311yescharacteristic
94#13049D412yescomplemented
95#13049D412yescomplemented
96#13048D414yescomplemented
97H(7,𝔽2)C22⋊C414yescomplemented
98C2×#585C2×D411yescharacteristic
99C2×#585C2×D411yescharacteristic
100C2×#585C2×D411yescharacteristic
101C22×H(5,𝔽2)C2×D411yescharacteristic
102C22×H(5,𝔽2)C2×D411yescharacteristic
103#3585C22⋊C414yescomplemented
104C2×#568C2×D411yescharacteristic
105#3498C22×C412yes—
106#3498C2×D412yes—
107#3498C2×D412yes—
108#3498C2×D412yes—
109C2×#535C2×D411yescharacteristic
110C2×#535C2×D411yescharacteristic
111C2×#532C2×D411yescharacteristic
112C2×#532C2411yescharacteristic
113C22×#127C2×D411yescharacteristic
114C2×C4×#41C2×D411yescharacteristic
115#3152C22×C411yescharacteristic
116#3152C2×D411yescharacteristic
117#3151C22×C411yescharacteristic
118#3151C2×D411yescharacteristic
119#3039C22⋊C412yescomplemented
120#3038C22⋊C412yescomplemented
121#3027C22⋊C414yescomplemented
122#3021C22⋊C414yescomplemented
123#3021C22⋊C414yescomplemented
124#3018C22×C412yes—
125#3018C22×C412yes—
126#3018C22×C412yes—
127#3018C2×D412yes—
128#3018C2×D412yes—
129#3018C2×D412yes—
130#3018#4112yes—
131#3018#4112yes—
132C2×#455C2×D412yes—
133C2×#453C2×D412yes—
134C2×#423C2×D411yescharacteristic
135#2888C22⋊C414yescomplemented
136C2×#413C2×D411yescharacteristic
137C2×#413C2×D411yescharacteristic
138C2×#413C2×D411yescharacteristic
139C2×#410C2×D411yescharacteristic
140C2×C4×M16C2×D411yescharacteristic
141#2867C22⋊C412yescomplemented
142#2858C22⋊C412yescomplemented
143#2380C2×D411yescharacteristic, complemented
144#2380C2×D411yescharacteristic, complemented
145#2372C2×D412yescomplemented
146C4×#117C2×D411yescharacteristic
147C2×#354C2×D412yescomplemented
148C2×#353C2×D412yescomplemented
149#1965C22⋊C412yescomplemented
150#1965C22⋊C412yescomplemented
151#1962C22×C412yes—
152#1962C2×D412yes—
153#1914C22⋊C414yescomplemented
154#1884C22×C412yes—
155#1884C2×D412yes—
156#1795C22⋊C414yescomplemented
157C4×#104C22⋊C412yescomplemented
158C4×#104C22⋊C412yescomplemented
159#585C2×C22⋊C412yes—
160#585C2×C22⋊C412yes—
161#585C2×C22⋊C412yescomplemented
162#585#12312yes—
163C2×H(5,𝔽2)C2×C22⋊C412yes—
164C2×H(5,𝔽2)C2×C22⋊C412yes—
165C2×H(5,𝔽2)#12012yes—
166C2×H(5,𝔽2)#12012yes—
167C2×H(5,𝔽2)#12012yes—
168C22×#41#12011yescharacteristic
169C22×#41C22×D411yescharacteristic
170C22×#41C22×D411yescharacteristic
171C23×D4#12011yescharacteristic
172C23×D4#12011yescharacteristic
173#568C2×C22⋊C412yescomplemented
174#535C2×C22⋊C412yes—
175#535C2×C22⋊C412yes—
176#535#12012yes—
177#535#12012yes—
178#532C23×C411yescharacteristic
179#532C22×D411yescharacteristic
180#532C22×D411yescharacteristic
181#532C22×D411yescharacteristic
182#532C2×#4111yescharacteristic
183#532C2×#4111yescharacteristic
184#532H(5,𝔽2)11yescharacteristic
185#532H(5,𝔽2)11yescharacteristic
186C2×C4⋊Q8C2×C22⋊C411yescharacteristic, complemented
187C2×C4⋊Q8H(5,𝔽2)11yescharacteristic
188C2×#127C2×C22⋊C412yes—
189C2×#127C2×C22⋊C411yescharacteristic, complemented
190C2×#127#12012yes—
191C2×#127C22×D411yescharacteristic
192C2×#124C22×D411yescharacteristic
193C2×#124H(5,𝔽2)11yescharacteristic
194C4×#41C2×C22⋊C412yes—
195C4×#41#12012yes—
196C2×#117#12011yescharacteristic
197C22×C42C22×D411yescharacteristic
198#455C4×D414yes—
199#455#12114yes—
200#453C4×D414yes—
201#453#12114yes—
202#423C2×C22⋊C412yescomplemented
203#413C2×C22⋊C412yes—
204#413C2×C22⋊C412yes—
205#413C2×C22⋊C412yescomplemented
206#413#12312yes—
207C2×#100#12012yes—
208#410C2×C22⋊C412yescomplemented
209C4×M16C2×C22⋊C411yescharacteristic, complemented
210C4×M16C2×C22⋊C411yescharacteristic, complemented
211C43C2×C22⋊C412yescomplemented
212C25B(4,𝔽2)12yes—
213H(5,𝔽2)#38614yes—
214C2×#41#37912yes—
215C2×#41#45812yes—
216C2×#41C22×C22⋊C411yescharacteristic
217C2×#41C22×C22⋊C411yescharacteristic
218C2×#41C2×#12011yescharacteristic
219C2×#41C2×#12011yescharacteristic
220C2×#41C2×#12311yescharacteristic
221C2×#41#53411yescharacteristic
222C2×#41#54611yescharacteristic
223C2×#41#54611yescharacteristic
224C22×Q8B(4,𝔽2)11yescharacteristic
225C22×Q8#53411yescharacteristic
226C22×Q8D4211yescharacteristic
227C22×D4#37912yes—
228C22×D4#37912yes—
229C22×D4B(4,𝔽2)11yescharacteristic
230C22×D4#45812yes—
231C22×D4C2×#12011yescharacteristic
232C22×D4D4211yescharacteristic
233C23×C4C2×#12011yescharacteristic
234C4⋊Q8#51811yescharacteristic
235C4⋊Q8#53411yescharacteristic
236C4⋊Q8#54611yescharacteristic
237C4⋊Q8#54611yescharacteristic
238#127C2×C4×D411yescharacteristic
239#127C2×#12111yescharacteristic
240#127D4211yescharacteristic
241#127D4211yescharacteristic
242#124C2×C4×D411yescharacteristic
243#124#51811yescharacteristic
244#124C2×#12111yescharacteristic
245#124#53411yescharacteristic
246#124D4211yescharacteristic
247#124D4211yescharacteristic
248#124#54611yescharacteristic
249#124#54611yescharacteristic
250#117#37912yes—
251C2×C42C22×C22⋊C411yescharacteristic
252C2×C42C2×#12011yescharacteristic
253C2×C42D4211yescharacteristic
254C2×C42D4211yescharacteristic
255C2×C42#54611yescharacteristic
256C2×C42#54611yescharacteristic
257#100#38614yes—
258C24#179014yes—
259C24C2×B(4,𝔽2)11yescharacteristic
260C2×Q8#179012yes—
261C2×Q8#230811yescharacteristic
262C2×Q8D4×C22⋊C411yescharacteristic
263C2×Q8#240511yescharacteristic
264C2×Q8#241011yescharacteristic
265C2×D4#179012yes—
266C2×D4#219612yes—
267C2×D4#219612yes—
268C2×D4C2×#38611yescharacteristic
269C2×D4D4×C22⋊C411yescharacteristic
270C2×D4#240311yescharacteristic
271C2×D4#241011yescharacteristic
272C22×C4C2×#37911yescharacteristic
273C22×C4#240311yescharacteristic
274C22×C4#240511yescharacteristic
275C22×C4C2×B(4,𝔽2)11yescharacteristic
276C22×C4C2×#45811yescharacteristic
277C22×C4C2×#45811yescharacteristic
278C42D4×C22⋊C411yescharacteristic
279C42D4×C22⋊C411yescharacteristic
280C42#241011yescharacteristic
281C42#241011yescharacteristic
282C23C2×#179011yescharacteristic
283C23#2390011yescharacteristic
284C23#2405611yescharacteristic
285D4#1275914yes—
286C2×C4#1386711yescharacteristic
287C2×C4C2×#179011yescharacteristic
288C2×C4#2390111yescharacteristic
289C2×C4#2405611yescharacteristic
290C2×C4C2×#219611yescharacteristic
291C2×C4C2×#219611yescharacteristic
292C2×C4#3355111yescharacteristic
293C2×C4#3355111yescharacteristic
294C22#35955211yescharacteristic
295C22#49978711yescharacteristic
296C22#49978711yescharacteristic
297C4C2×#1275911yescharacteristic
298C4#49358211yescharacteristic
299C4#50021811yescharacteristic
300C4#50021811yescharacteristic
301C2#7226742711yescharacteristic
subgroupG/H
center, socleC2#72267427
commutator#532C23×C4
Frattini, cosocleC2×#532C24
prime radical#72214082C2
LCSG ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ C2×#3498 ◁ G
LECSG ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72286562C2semidirect
2. #72286558C2semidirect
3. #72286553C2semidirect
4. #72286548C2semidirect
5. #72286536C2semidirect
6. #72286516C2semidirect
7. #72286309C2semidirect
8. #72286294C2semidirect
9. #72286280C2semidirect
10. #72284773C2semidirect
11. #72284631C2semidirect
12. #72284480C2semidirect
13. #72284325C2semidirect
14. #72256626C2semidirect
18. #7627594C22semidirect
19. C2×#33359C22semidirect
20. #7514141C4semidirect
21. #7514140C4semidirect
27. #512601C22semidirect
28. #512594C22semidirect
29. #512585C22semidirect
30. #512578C22semidirect
31. #512578C22semidirect
34. #512564C22semidirect
35. #494620C22semidirect
37. #494599C22semidirect
39. #494595C22semidirect
40. #494595C22semidirect
41. C2×#12718C22semidirect
42. #201040C22semidirect
43. #200841C22semidirect
44. #187081C22semidirect
45. #151858C22semidirect
46. #151842C22semidirect
47. #151794C22semidirect
48. #151784C22semidirect
49. #146645C22semidirect
50. #146636C22semidirect
51. #108381C22semidirect
52. #107744C22semidirect
56. #61111C2×C4semidirect
57. #61110C2×C4semidirect
76. #31076C2×C4semidirect
77. #30497C2×C4semidirect
78. #30496C2×C4semidirect
85. #22899C2×C4semidirect
94. #13049D4semidirect
95. #13049D4semidirect
96. #13048D4semidirect
97. H(7,𝔽2)C22⋊C4semidirect
103. #3585C22⋊C4semidirect
119. #3039C22⋊C4semidirect
120. #3038C22⋊C4semidirect
121. #3027C22⋊C4semidirect
122. #3021C22⋊C4semidirect
123. #3021C22⋊C4semidirect
135. #2888C22⋊C4semidirect
141. #2867C22⋊C4semidirect
142. #2858C22⋊C4semidirect
143. #2380C2×D4semidirect
144. #2380C2×D4semidirect
145. #2372C2×D4semidirect
147. C2×#354C2×D4semidirect
148. C2×#353C2×D4semidirect
149. #1965C22⋊C4semidirect
150. #1965C22⋊C4semidirect
153. #1914C22⋊C4semidirect
156. #1795C22⋊C4semidirect
157. C4×#104C22⋊C4semidirect
158. C4×#104C22⋊C4semidirect
161. #585C2×C22⋊C4semidirect
173. #568C2×C22⋊C4semidirect
186. C2×C4⋊Q8C2×C22⋊C4semidirect
189. C2×#127C2×C22⋊C4semidirect
202. #423C2×C22⋊C4semidirect
205. #413C2×C22⋊C4semidirect
208. #410C2×C22⋊C4semidirect
209. C4×M16C2×C22⋊C4semidirect
210. C4×M16C2×C22⋊C4semidirect
211. C43C2×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#722865622
#722865582
#722865532
#722865482
#722865362
#722865162
#722863092
#722862942
#722862802
#722847732
#722846312
#722844802
#722843252
#722566262
#722140822