Finite Group Databasebeta

← #72312950

Subgroups — #72312950

Order 4096.

At least 27485 conjugacy classes of subgroups; only the 999 normal subgroups are shown.

The subgroup lattice has 391 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72310419C211yesmaximal, characteristic, complemented
2#72309519C214yesmaximal, complemented
3#72265785C211yesmaximal, characteristic, complemented
4#72264815C214yesmaximal, complemented
5#72237032C211yesmaximal, characteristic, complemented
6#72237027C211yesmaximal, characteristic, complemented
7#72234665C211yesmaximal, characteristic
8#72234663C411yescharacteristic, complemented
9#72234619C412yescomplemented
10#72234591C411yescharacteristic, complemented
11#72234591C411yescharacteristic, complemented
12#72232884C411yescharacteristic, complemented
13#72225655C412yescomplemented
14#72225629C211yesmaximal, characteristic, complemented
15C2×#115348C2212yescomplemented
16#72224136C2212yes—
17#72219300C2212yescomplemented
18#72219247C211yesmaximal, characteristic, complemented
19#72219243C2214yescomplemented
20#72219237C2214yescomplemented
21#72219227C2211yescharacteristic, complemented
22C22×#8534C2212yescomplemented
23C2×#7670203C2211yescharacteristic
24#72218518C2211yescharacteristic, complemented
25#72218375C2211yescharacteristic, complemented
26#72218306C2212yescomplemented
27#72218291C2211yescharacteristic, complemented
28#72218069C2214yescomplemented
29#72218057C2214yescomplemented
30#72217511C2211yescharacteristic
31#72217511C2211yescharacteristic
32#72216064C2212yes—
33#10586404C2×C412yes—
34C2×#62817C2×C411yescharacteristic
35C2×#62817C2×C411yescharacteristic
36#10586066C2×C412yes—
37C22×#3484C2×C411yescharacteristic
38C22×#3484C2311yescharacteristic
39C24×#120D411yescharacteristic
40C24×#120C2311yescharacteristic
41C2×#60399D414yes—
42#7642273D414yes—
43#7642244C2×C412yes—
44#7642244C2×C412yes—
45#7642244D414yes—
46C22×#2846D411yescharacteristic
47#7630944D414yes—
48#7630939D414yes—
49#7630939C2312yes—
50#7630939C2312yes—
51C2×#35390D411yescharacteristic
52C2×#35348C2×C411yescharacteristic
53C2×#35348D411yescharacteristic
54C2×#35346C2×C411yescharacteristic
55C2×#34981D411yescharacteristic
56C2×#34981D411yescharacteristic
57C2×#34980D411yescharacteristic
58C2×#34980C2311yescharacteristic
59#7626202C2×C411yescharacteristic
60#7626202C2×C411yescharacteristic
61C2×#34198C2×C411yescharacteristic
62C2×#34198D411yescharacteristic
63#6499892D4116yes—
64#6431795D414yes—
65#6431795D414yes—
66#6374210C2×C412yes—
67#6374210C2×C412yes—
68#6374128C2×C412yes—
69#6371558D414yes—
70#6371504C2×C412yes—
71#6371504C2×C412yes—
72#6370693C2×C412yes—
73C22×#1781C2312yescomplemented
74C22×#1438C2312yescomplemented
75#392113C2312yescomplemented
76#392109C2312yescomplemented
77C28C2×D411yescharacteristic
78C2×H(7,𝔽2)C22⋊C412yescomplemented
79C23×H(5,𝔽2)C22×C411yescharacteristic
80C25×D4C22⋊C411yescharacteristic
81C25×D4C2×D412yes—
82C25×D4C2411yescharacteristic
83#62862C22⋊C418yescomplemented
84#62842C22⋊C412yescomplemented
85#62842C22⋊C412yescomplemented
86C2×#3490C22⋊C414yes—
87C2×#3484C22⋊C414yes—
88C2×#3484C22×C412yes—
89C2×#3484C22×C412yes—
90C2×#3484C2×D414yes—
91C2×#3484C2×D414yes—
92C2×#3484C2×D414yes—
93C2×#3484C2×D414yes—
94C22×#534C22⋊C411yescharacteristic
95C22×#534C22×C411yescharacteristic
96C22×#534C2×D411yescharacteristic
97C22×#534C2×D411yescharacteristic
98C22×#534C2×D411yescharacteristic
99C23×#120C22⋊C411yescharacteristic
100C23×#120C22⋊C411yescharacteristic
101C23×#120C22⋊C411yescharacteristic
102C23×#120C22⋊C411yescharacteristic
103C23×#120C22×C411yescharacteristic
104C23×#120C22×C411yescharacteristic
105C23×#120C2×D411yescharacteristic
106C23×#120C2×D411yescharacteristic
107C23×#120C2×D411yescharacteristic
108C23×#120C2×D412yes—
109C23×#120C2×D411yescharacteristic
110C23×#120C2×D411yescharacteristic
111C23×#120C2×D412yes—
112#62652C22⋊C412yescomplemented
113C22×#518C22⋊C411yescharacteristic
114C22×#518C22×C411yescharacteristic
115C24×C22⋊C4C22⋊C411yescharacteristic
116C24×C22⋊C4C2×D411yescharacteristic
117C24×C22⋊C4C2×D412yes—
118C2×#2883C22⋊C414yescomplemented
119C2×#2883C22⋊C414yescomplemented
120C2×#2883C22⋊C414yescomplemented
121C2×#2883C22⋊C414yescomplemented
122#36614C2×D414yes—
123C2×#2846C2×D414yes—
124#35222C2×D418yes—
125#35222C2×D414yes—
126#35050C22⋊C418yes—
127#34980C22⋊C418yes—
128#34980C2×D418yes—
129#34980C2×D418yes—
130#34980C2×D414yes—
131#34176C22⋊C414yes—
132#34171C22⋊C414yes—
133#34171C22×C412yes—
134#34171C22×C412yes—
135#34171C2×D414yes—
136#34171C2×D414yes—
137C2×#2315C22⋊C411yescharacteristic
138C2×#2315C22⋊C411yescharacteristic
139C2×#2308C22⋊C411yescharacteristic
140C2×#2308C22⋊C411yescharacteristic
141C2×#2308C22⋊C411yescharacteristic
142C2×#2308C22×C411yescharacteristic
143C2×#2308C2×D411yescharacteristic
144C22×#379C22⋊C411yescharacteristic
145C22×#379C22⋊C411yescharacteristic
146C22×#379C2×D411yescharacteristic
147C2×#2273C22⋊C411yescharacteristic
148#21750C22⋊C418yescomplemented
149#21749C22⋊C418yescomplemented
150#21749C22⋊C418yescomplemented
151#20354C22⋊C414yescomplemented
152#20354C22⋊C414yescomplemented
153C27#9911yescharacteristic, complemented
154C27C2×C22⋊C411yescharacteristic
155C27C2×C22⋊C411yescharacteristic
156C27#12012yes—
157C27#12012yes—
158C27C22×D411yescharacteristic
159C27C22×D411yescharacteristic
160C22×H(5,𝔽2)#9914yescomplemented
161C22×H(5,𝔽2)C2×C22⋊C412yes—
162C22×H(5,𝔽2)C2×C22⋊C412yes—
163C22×H(5,𝔽2)C2×C22⋊C412yes—
164C22×H(5,𝔽2)C2×C22⋊C412yes—
165C24×D4#9911yescharacteristic, complemented
166C24×D4#9911yescharacteristic, complemented
167C24×D4#9912yescomplemented
168C24×D4C2×C22⋊C412yes—
169C24×D4C2×C22⋊C412yes—
170C24×D4C2×C22⋊C411yescharacteristic
171C24×D4C2×C22⋊C411yescharacteristic
172C24×D4C2×C22⋊C411yescharacteristic
173C24×D4#12012yes—
174C24×D4#12012yes—
175C24×D4C23×C411yescharacteristic
176C24×D4C22×D411yescharacteristic
177C24×D4C22×D414yes—
178C24×D4C22×D411yescharacteristic
179C24×D4C22×D411yescharacteristic
180C24×D4C22×D414yes—
181C25×C4#9911yescharacteristic, complemented
182C25×C4#12012yes—
183C25×C4C22×D411yescharacteristic
184C2×#543#9914yescomplemented
185C2×#538#9918yescomplemented
186C2×#535C2×C22⋊C414yes—
187C2×#535#12018yes—
188C2×#534#9914yescomplemented
189C2×#534#9918yescomplemented
190C2×#534#9914yescomplemented
191C2×#534C2×C22⋊C414yes—
192C2×#534C2×C22⋊C414yes—
193C2×#534#12012yes—
194C2×#534#12014yes—
195C2×#534#12012yes—
196C2×#534#12018yes—
197C2×#534#12014yes—
198C2×#525C2×C22⋊C414yes—
199C2×#525#12018yes—
200C22×#120C2×C22⋊C414yes—
201C22×#120#12012yes—
202C22×#120#12014yes—
203C22×#120#12012yes—
204C22×#120#12014yes—
205C22×#120#12018yes—
206C2×#518#9914yescomplemented
207C2×#518#9912yescomplemented
208C2×#518#9912yescomplemented
209C2×#518C2×C22⋊C412yes—
210C2×#518C2×C22⋊C414yes—
211C2×#518C2×C22⋊C412yes—
212C2×#518C2×C22⋊C414yes—
213C2×#518C2×C22⋊C414yes—
214C23×C22⋊C4#9912yescomplemented
215C23×C22⋊C4C2×C22⋊C412yes—
216C23×C22⋊C4C2×C22⋊C411yescharacteristic
217C23×C22⋊C4C2×C22⋊C411yescharacteristic
218C23×C22⋊C4C2×C22⋊C412yes—
219C23×C22⋊C4C2×C22⋊C411yescharacteristic
220C23×C22⋊C4C2×C22⋊C411yescharacteristic
221C23×C22⋊C4C2×C22⋊C411yescharacteristic
222C23×C22⋊C4C2×C22⋊C411yescharacteristic
223C23×C22⋊C4C2×C22⋊C411yescharacteristic
224C23×C22⋊C4#12012yes—
225C23×C22⋊C4#12012yes—
226C23×C22⋊C4#12012yes—
227#2309C2×C22⋊C414yes—
228#2309#12018yes—
229#2308C2×C22⋊C414yes—
230#2308C2×C22⋊C414yes—
231#2308#12014yes—
232#2308#12018yes—
233#2280C2×C22⋊C414yes—
234#2280#12018yes—
235C2×#379C2×C22⋊C414yes—
236C2×#379#12014yes—
237C2×#379#12018yes—
238C26#37912yes—
239C26#37912yes—
240C26C2×#9911yescharacteristic
241C26C2×#9912yes—
242C26C2×#9911yescharacteristic
243C26C2×#9911yescharacteristic
244C26C2×#9911yescharacteristic
245C26C2×#9911yescharacteristic
246C26C22×C22⋊C411yescharacteristic
247C26C22×C22⋊C411yescharacteristic
248C26C2×#12011yescharacteristic
249C26C2×#12011yescharacteristic
250C26C2×#12012yes—
251C26C2×#12011yescharacteristic
252C26C2×#12014yes—
253C26C2×#12012yes—
254C22×#41#37914yes—
255C23×D4#37911yescharacteristic
256C23×D4#37912yes—
257C23×D4#37911yescharacteristic
258C23×D4#37912yes—
259C23×D4#37914yes—
260C23×D4C2×#9914yes—
261C23×D4C2×#9914yes—
262C23×D4C2×#9911yescharacteristic
263C23×D4C2×#9911yescharacteristic
264C23×D4C2×#9912yes—
265C23×D4C2×#9912yes—
266C23×D4C2×#9918yes—
267C23×D4C22×C22⋊C412yes—
268C23×D4C22×C22⋊C412yes—
269C23×D4C2×#12012yes—
270C23×D4C2×#12014yes—
271C23×D4C2×#12012yes—
272C23×D4C2×#12014yes—
273C24×C4#37912yes—
274C24×C4#37912yes—
275C24×C4C2×#9911yescharacteristic
276C24×C4C2×#9911yescharacteristic
277C24×C4C2×#9911yescharacteristic
278C24×C4C2×#9912yes—
279C24×C4C22×C22⋊C411yescharacteristic
280C24×C4C2×#12011yescharacteristic
281C24×C4C2×#12014yes—
282C2×C22⋊Q8C2×#9918yes—
283C2×#121C2×#9918yes—
284C2×#120B(4,𝔽2)14yes—
285C2×#120#45814yes—
286C2×#117#37918yes—
287C2×#117#37914yes—
288C22×C22⋊C4#37912yes—
289C22×C22⋊C4#37912yes—
290C22×C22⋊C4#37912yes—
291C22×C22⋊C4#37912yes—
292C22×C22⋊C4#37918yes—
293C22×C22⋊C4#37912yes—
294C22×C22⋊C4#37914yes—
295C22×C22⋊C4C2×#9914yes—
296C22×C22⋊C4C2×#9914yes—
297C22×C22⋊C4C2×#9912yes—
298C22×C22⋊C4C2×#9912yes—
299C22×C22⋊C4C2×#9912yes—
300C22×C22⋊C4C2×#9918yes—
301C25#178112yes—
302C25#178112yes—
303C25#178112yes—
304C25#178111yescharacteristic
305C25#178112yes—
306C25#178112yes—
307C25#178112yes—
308C25#178112yes—
309C25#178111yescharacteristic
310C25#178112yes—
311C25#178114yescomplemented
312C25C2×#37912yes—
313C25C2×#37911yescharacteristic
314C25C2×#37911yescharacteristic
315C25C2×#37911yescharacteristic
316C25#284612yes—
317C25#284612yes—
318C25#284612yes—
319C25C22×#9911yescharacteristic
320C25C22×#9911yescharacteristic
321C25C22×#9911yescharacteristic
322C25C22×#9911yescharacteristic
323C25C22×#9914yes—
324C25C2×B(4,𝔽2)12yes—
325C25C2×#45812yes—
326C25#302612yes—
327C25#302612yes—
328C22×Q8#178114yescomplemented
329C22×D4#178114yescomplemented
330C22×D4#179014yes—
331C23×C4#178112yes—
332C23×C4#178111yescharacteristic
333C23×C4#178111yescharacteristic
334C23×C4#178114yes—
335C23×C4#178114yes—
336C23×C4#178114yes—
337C23×C4#178112yes—
338C23×C4#178114yes—
339C23×C4#178114yescomplemented
340C23×C4#178118yescomplemented
341C23×C4C2×#37912yes—
342C23×C4#284612yes—
343C23×C4C22×#9914yes—
344C2×C4⋊C4#178118yescomplemented
345C2×C22⋊C4#179014yes—
346C24#853412yescomplemented
347C24#854112yes—
348C24#854112yes—
349C24#854112yes—
350C24#854112yes—
351C24#1467911yescharacteristic
352C24C2×#178111yescharacteristic
353C24C2×#178112yes—
354C24C2×#178112yes—
355C24C2×#178111yescharacteristic
356C24C2×#178114yes—
357C24C2×#178111yescharacteristic
358C24C2×#178111yescharacteristic
359C24C2×#178111yescharacteristic
360C24C2×#178111yescharacteristic
361C24C2×#179012yes—
362C24#2060912yes—
363C24#3289812yes—
364C24#3289812yes—
365C24#3290012yes—
366C24#3290012yes—
367C22×C4#853412yescomplemented
368C22×C4#854118yes—
369C22×C4C2×#178114yes—
370C23#13383914yescomplemented
371C23C2×#853411yescharacteristic
372C23#18792812yes—
373C23#18794411yescharacteristic
374C23C2×#854112yes—
375C23C2×#854112yes—
376C23#39210911yescharacteristic
377C23#39210911yescharacteristic
378C23#39210911yescharacteristic
379C23#39212512yes—
380C23#39213512yes—
381C22C2×#13383912yes—
382C22#7221846711yescharacteristic
383C22#7221851811yescharacteristic
384C22#7222612311yescharacteristic
385C22#7223416412yes—
386C22#7223416412yes—
387C22#7224041912yes—
388C2#7224042012yes—
389C2#7226613411yescharacteristic
subgroupG/H
center, socleC22#72218518
commutatorC24×D4C23×C4
Frattini, cosocleC25×D4C24
prime radical#72234665C2
LCSG ▷ C24×D4 ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C24×#120 ◁ G
LECSG ▷ C25×D4 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72310419C2semidirect
2. #72309519C2semidirect
3. #72265785C2semidirect
4. #72264815C2semidirect
5. #72237032C2semidirect
6. #72237027C2semidirect
8. #72234663C4semidirect
9. #72234619C4semidirect
10. #72234591C4semidirect
11. #72234591C4semidirect
12. #72232884C4semidirect
13. #72225655C4semidirect
14. #72225629C2semidirect
15. C2×#115348C22semidirect
17. #72219300C22semidirect
18. #72219247C2semidirect
19. #72219243C22semidirect
20. #72219237C22semidirect
21. #72219227C22semidirect
22. C22×#8534C22semidirect
24. #72218518C22semidirect
25. #72218375C22semidirect
26. #72218306C22semidirect
27. #72218291C22semidirect
28. #72218069C22semidirect
29. #72218057C22semidirect
73. C22×#1781C23semidirect
74. C22×#1438C23semidirect
75. #392113C23semidirect
76. #392109C23semidirect
78. C2×H(7,𝔽2)C22⋊C4semidirect
83. #62862C22⋊C4semidirect
84. #62842C22⋊C4semidirect
85. #62842C22⋊C4semidirect
112. #62652C22⋊C4semidirect
118. C2×#2883C22⋊C4semidirect
119. C2×#2883C22⋊C4semidirect
120. C2×#2883C22⋊C4semidirect
121. C2×#2883C22⋊C4semidirect
148. #21750C22⋊C4semidirect
149. #21749C22⋊C4semidirect
150. #21749C22⋊C4semidirect
151. #20354C22⋊C4semidirect
152. #20354C22⋊C4semidirect
153. C27#99semidirect
160. C22×H(5,𝔽2)#99semidirect
165. C24×D4#99semidirect
166. C24×D4#99semidirect
167. C24×D4#99semidirect
181. C25×C4#99semidirect
184. C2×#543#99semidirect
185. C2×#538#99semidirect
188. C2×#534#99semidirect
189. C2×#534#99semidirect
190. C2×#534#99semidirect
206. C2×#518#99semidirect
207. C2×#518#99semidirect
208. C2×#518#99semidirect
214. C23×C22⋊C4#99semidirect
311. C25#1781semidirect
328. C22×Q8#1781semidirect
329. C22×D4#1781semidirect
339. C23×C4#1781semidirect
340. C23×C4#1781semidirect
344. C2×C4⋊C4#1781semidirect
346. C24#8534semidirect
367. C22×C4#8534semidirect
370. C23#133839semidirect

Maximal subgroups

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

subgroupindex
#723104192
#723095192
#722657852
#722648152
#722370322
#722370272
#722346652
#722256292
#722192472