Finite Group Databasebeta

← #72263751

Subgroups — #72263751

Order 2048.

At least 6134 conjugacy classes of subgroups; only the 1450 normal subgroups are shown.

The subgroup lattice has 451 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72263750C211yesmaximal, characteristic, complemented
2#72263749C211yesmaximal, characteristic
3#72263719C212yesmaximal, complemented
4#72263713C212yesmaximal, complemented
5#72263708C212yesmaximal, complemented
6#72263701C212yesmaximal, complemented
7#72263633C211yesmaximal, characteristic, complemented
8#72263552C214yesmaximal, complemented
9#72263419C214yesmaximal, complemented
10#72251226C211yesmaximal, characteristic, complemented
11#72236740C211yesmaximal, characteristic, complemented
12#72236739C214yesmaximal, complemented
13#72236734C211yesmaximal, characteristic, complemented
14#72233659C211yesmaximal, characteristic, complemented
15#72210479C214yesmaximal, complemented
16#10586472C2211yescharacteristic
17C2×#39361C2211yescharacteristic
18#6595646C418yescomplemented
19#6573693C418yescomplemented
20#6490931C2212yes—
21#6470019C2214yes—
22#6453830C2212yes—
23#6437445C2214yes—
24#6385983C2211yescharacteristic
25#6385971C2211yescharacteristic
26#6385971C2211yescharacteristic
27#606700C2212yescomplemented
28#582593C2211yescharacteristic, complemented
29#581603C2212yescomplemented
30#559927C2211yescharacteristic, complemented
31#553150C2211yescharacteristic
32#552678C2212yescomplemented
33#537848C2211yescharacteristic, complemented
34#537841C2212yescomplemented
35#518019C2211yescharacteristic
36#514639C2211yescharacteristic, complemented
37#496306C2212yescomplemented
38#496189C2214yescomplemented
39#493956C2212yescomplemented
40#493370C2214yescomplemented
41#187944C2214yes—
42#187930C2212yes—
43#187930C2212yes—
44#187928C2214yes—
45#187928C2214yes—
46#187925C2212yes—
47#187925C2212yes—
48C2×#8534C2214yes—
49#139818C2214yescomplemented
50#139818C2214yescomplemented
51#139779C2218yescomplemented
52#139778C2218yescomplemented
53#139777C2214yescomplemented
54#139777C2214yescomplemented
55#139776C2218yescomplemented
56#139767C2218yescomplemented
57#139706C2218yescomplemented
58#139706C2218yescomplemented
59#128081C2212yescomplemented
60#128081C2212yescomplemented
61#128073C2214yescomplemented
62#128073C2214yescomplemented
63#128043C2212yescomplemented
64#128043C2212yescomplemented
65#128033C2214yescomplemented
66C2×#7304C2214yescomplemented
67#62983C2×C414yescomplemented
68D4×H(5,𝔽2)C2×C414yescomplemented
69#62773C2312yes—
70C2×#3484C2314yes—
71C22×#550D411yescharacteristic
72C22×D42C2311yescharacteristic
73C22×#535C2312yes—
74C22×#518D411yescharacteristic
75C2×#3019C2×C418yescomplemented
76#39361C2×C418yes—
77#39361D418yes—
78C2×#2867D411yescharacteristic
79C2×#2548D411yescharacteristic
80C2×#2541C2311yescharacteristic
81C2×#2431D412yes—
82C2×#2433C2312yes—
83C2×#2619D412yes—
84C2×D4×C22⋊C4D411yescharacteristic
85C2×D4×C22⋊C4D411yescharacteristic
86C2×#2309D411yescharacteristic
87C2×C4×#124C2311yescharacteristic
88C2×#2293D411yescharacteristic
89#32990C2×C418yescomplemented
90#32920C2×C418yescomplemented
91#32888C2×C418yescomplemented
92#25918C2×C414yescomplemented
93#25739C2×C414yescomplemented
94#24412C2×C4116yescomplemented
95#24314C2×C414yescomplemented
96#24224C2×C414yescomplemented
97#24018C2314yes—
98#23912C2×C4116yescomplemented
99#23907C2314yes—
100#23906C2318yes—
101D4×#100C2×C418yescomplemented
102C4×#455C2×C418yescomplemented
103#20953C2×C418yescomplemented
104#20726C2312yes—
105#20722C2312yes—
106#20722C2312yes—
107C2×#1783C2314yes—
108C2×#1781C2314yes—
109C2×#1781C2314yes—
110#14662C2312yes—
111#14661C2312yes—
112#14657D412yes—
113#14445C2312yes—
114#14419C2312yes—
115#14419C2312yes—
116#14335C2312yes—
117#14257D411yescharacteristic
118#14257D411yescharacteristic
119#14255D412yes—
120#14255D412yes—
121#14169C2311yescharacteristic
122#14169C2311yescharacteristic
123#14157C2311yescharacteristic
124#14157C2311yescharacteristic
125#14154C2312yes—
126#14136D411yescharacteristic
127#14136D411yescharacteristic
128#14134D412yes—
129#13871C2311yescharacteristic
130#13871C2311yescharacteristic
131#13867C2312yes—
132#13867C2312yes—
133#13839C2311yescharacteristic
134#13839C2311yescharacteristic
135C2×#1438C2312yes—
136#12856C2314yes—
137#12694C2314yes—
138#12693C2318yes—
139#8534C2314yes—
140#8534C2318yes—
141#8534C2314yes—
142#8534C2314yes—
143#8534C2314yes—
144#8534C2318yes—
145#7304C2314yes—
146#7304C2318yes—
147#7304C2314yes—
148#7304C2314yes—
149#7304C2314yes—
150#7304C2318yes—
151C22×H(5,𝔽2)C22×C414yes—
152C23×#41C2×D412yes—
153C24×D4C2×D411yescharacteristic
154C24×D4C2×D411yescharacteristic
155C24×D4C2412yes—
156#3501C22×C412yes—
157#3496C22×C412yes—
158#3484C22×C418yes—
159#3484C2414yes—
160C2×#550C22⋊C418yes—
161C2×D42C22×C414yes—
162C2×D42C2×D414yes—
163C2×D42#4114yes—
164C2×D42#4114yes—
165C2×#535C22×C418yes—
166C2×#535C2×D418yes—
167C22×#127C2×D412yes—
168C22×#127C2×D411yescharacteristic
169C22×#127C2×D412yes—
170C22×#127C2×D411yescharacteristic
171C22×#127C2412yes—
172C22×#124C2×D412yes—
173C22×#124C2411yescharacteristic
174C22×#121C2×D412yes—
175C22×#121C2×D411yescharacteristic
176C22×#121C2×D411yescharacteristic
177C22×#121C2×D412yes—
178C22×#120C2×D412yes—
179C22×#120C2×D412yes—
180C4×H(5,𝔽2)C22×C414yes—
181C2×#518C22×C414yes—
182C22×C4×Q8C2×D411yescharacteristic
183C22×C4×D4C2×D411yescharacteristic
184C22×C4×D4C2411yescharacteristic
185C22×#117C2×D412yes—
186C22×#117C2×D411yescharacteristic
187C22×#117C2×D411yescharacteristic
188C22×#117C2×D411yescharacteristic
189C22×#117C2×D411yescharacteristic
190C23×C22⋊C4C2×D412yes—
191C23×C22⋊C4C2×D412yes—
192C23×C22⋊C4C2×D412yes—
193C23×C22⋊C4C2×D412yes—
194C2×#455C22×C418yes—
195C2×#453C22×C4116yes—
196C22×#100C22×C418yes—
197#2867C22⋊C418yes—
198#2619C22⋊C4116yes—
199#2548C22⋊C418yes—
200#2541C22×C414yes—
201#2541C2×D414yes—
202#2433C22×C418yes—
203#2433C2×D418yes—
204#2431C22⋊C4116yes—
205C4×#124C22×C414yes—
206C4×#124C2×D414yes—
207C2×#391C2×D411yescharacteristic
208#2293C22⋊C418yes—
209C2×#390C2×D412yes—
210C2×#390C2×D412yes—
211C2×#390C2411yescharacteristic
212C2×#382C2×D411yescharacteristic
213C2×#380C2×D411yescharacteristic
214C2×#380C2×D411yescharacteristic
215C2×#379C2×D412yes—
216C2×#379C2×D412yes—
217C2×#379C2×D412yes—
218C2×#379C2×D412yes—
219C2×C4×C4⋊C4C2×D411yescharacteristic
220C2×C4×C22⋊C4C2×D412yes—
221C2×C4×C22⋊C4C2×D411yescharacteristic
222C2×C4×C22⋊C4C2×D411yescharacteristic
223C2×C43C2×D411yescharacteristic
224#1887C22×C4116yes—
225#1885C22×C418yes—
226#1856C22×C418yes—
227#1800C22×C414yes—
228#1792C22×C418yes—
229#1783C22×C418yes—
230#1783C2414yes—
231#1781C2414yes—
232#1781C2414yes—
233C4×#100C22×C414yes—
234#1440C2×D411yescharacteristic
235#1440C2×D411yescharacteristic
236#1440C2×D411yescharacteristic
237#1440C2×D411yescharacteristic
238#1440C2×D412yes—
239#1440C2×D412yes—
240#1438C2×D412yes—
241#1438C2×D412yes—
242#1438C2×D412yes—
243#1438C2×D412yes—
244#1438C2×D412yes—
245#1438C2×D412yes—
246#1438C2×D412yes—
247#1438C2×D412yes—
248#1438C2411yescharacteristic
249#1438C2412yes—
250#1438C2411yescharacteristic
251#1438C2411yescharacteristic
252#1438C2411yescharacteristic
253#1438C2412yes—
254C26C22×D412yes—
255C26C22×D412yes—
256C2×H(5,𝔽2)C23×C414yes—
257C22×#41C2×C22⋊C418yes—
258C23×Q8C22×D411yescharacteristic
259C23×D4C4×D414yes—
260C23×D4C4×D414yes—
261C23×D4#12011yescharacteristic
262C23×D4#12011yescharacteristic
263C23×D4#12012yes—
264C23×D4#12012yes—
265C23×D4#12012yes—
266C23×D4#12011yescharacteristic
267C23×D4#12114yes—
268C23×D4#12114yes—
269C23×D4C23×C414yes—
270C23×D4C22×D414yes—
271C23×D4C22×D412yes—
272C23×D4C22×D412yes—
273C23×D4C22×D412yes—
274C23×D4C22×D412yes—
275C23×D4C2×#4114yes—
276C23×D4C2×#4114yes—
277C23×D4C2511yescharacteristic
278C24×C4C22×D411yescharacteristic
279C24×C4C22×D411yescharacteristic
280C24×C4C22×D412yes—
281C2×#127C2×C22⋊C418yes—
282C2×#127C2×C22⋊C414yes—
283C2×#127C2×C22⋊C414yes—
284C2×#127C2×C22⋊C418yes—
285C2×#127#12314yes—
286C2×#127C23×C414yes—
287C2×#127C22×D414yes—
288C2×#124C2×C22⋊C418yes—
289C2×#124C23×C412yes—
290C2×#124C22×D412yes—
291C2×#121C2×C22⋊C418yes—
292C2×#121C2×C22⋊C418yes—
293C2×#121C4×D414yes—
294C2×#121C4×D414yes—
295C2×#121#12114yes—
296C2×#121#12114yes—
297C2×#121#12318yes—
298C2×#120C4×D418yes—
299C2×#120C4×D418yes—
300C2×#120#12118yes—
301C2×#120#12118yes—
302#518C23×C414yes—
303C2×C4×Q8C2×C22⋊C414yes—
304C2×C4×D4C2×C22⋊C414yes—
305C2×C4×D4#12314yes—
306C2×C4×D4C23×C412yes—
307C2×C4×D4C22×D412yes—
308C2×C4×D4C2×#4112yes—
309C2×C4×D4C2×#4112yes—
310C2×#117C2×C22⋊C418yes—
311C22×C4⋊C4#12011yescharacteristic
312C22×C4⋊C4C22×D411yescharacteristic
313C22×C4⋊C4C22×D411yescharacteristic
314C22×C22⋊C4#12012yes—
315C22×C22⋊C4#12012yes—
316C22×C22⋊C4#12012yes—
317C22×C22⋊C4C22×D411yescharacteristic
318C22×C22⋊C4C22×D411yescharacteristic
319C22×C22⋊C4C22×D411yescharacteristic
320C22×C22⋊C4C22×D411yescharacteristic
321C22×C22⋊C4C22×D411yescharacteristic
322C22×C22⋊C4C22×D411yescharacteristic
323C22×C22⋊C4C22×D412yes—
324C22×C22⋊C4C22×D412yes—
325C22×C22⋊C4C22×D412yes—
326C22×C22⋊C4C22×D411yescharacteristic
327C22×C22⋊C4C22×D411yescharacteristic
328C22×C22⋊C4C22×D412yes—
329C22×C22⋊C4C22×D411yescharacteristic
330C22×C22⋊C4C22×D411yescharacteristic
331C22×C42C22×D411yescharacteristic
332C22×C42C22×D411yescharacteristic
333C22×C42C22×D412yes—
334C22×C42C22×D411yescharacteristic
335C2×#100C23×C418yes—
336#391C2×C22⋊C414yes—
337#390C2×C22⋊C418yes—
338#390C2×C22⋊C418yes—
339#390C23×C412yes—
340#390C22×D412yes—
341#382C2×C22⋊C414yes—
342C4×C4⋊C4C2×C22⋊C414yes—
343C4×C22⋊C4C2×C22⋊C418yes—
344C43C2×C22⋊C414yes—
345C25C2×C4×D414yes—
346C25C2×C4×D414yes—
347C25C2×#12012yes—
348C25C2×#12012yes—
349C25C2×#12012yes—
350C25C2×#12114yes—
351C25C2×#12114yes—
352C25C23×D411yescharacteristic
353C25C23×D411yescharacteristic
354C22×Q8C22×C22⋊C412yes—
355C22×D4#38614yes—
356C22×D4#38614yes—
357C22×D4#38618yes—
358C22×D4#38618yes—
359C22×D4#38614yes—
360C22×D4#38618yes—
361C22×D4B(4,𝔽2)11yescharacteristic
362C22×D4B(4,𝔽2)11yescharacteristic
363C22×D4B(4,𝔽2)12yes—
364C22×D4B(4,𝔽2)14yes—
365C22×D4C22×C22⋊C414yes—
366C22×D4C22×C22⋊C414yes—
367C22×D4C2×C4×D414yes—
368C22×D4C2×C4×D414yes—
369C22×D4C2×#12114yes—
370C22×D4C2×#12114yes—
371C22×D4C2×#12314yes—
372C22×D4C24×C411yescharacteristic
373C22×D4C23×D411yescharacteristic
374C22×D4C22×#4111yescharacteristic
375C22×D4C22×#4111yescharacteristic
376C23×C4C22×C22⋊C414yes—
377C23×C4C2×C4×D412yes—
378C23×C4C2×C4×D412yes—
379C23×C4C2×#12011yescharacteristic
380C23×C4C2×#12011yescharacteristic
381C23×C4C2×#12011yescharacteristic
382C23×C4C2×#12012yes—
383C23×C4C2×#12011yescharacteristic
384C23×C4C2×#12112yes—
385C23×C4C2×#12112yes—
386C23×C4C2×#12314yes—
387C23×C4C23×D411yescharacteristic
388C2×C4⋊C4#38614yes—
389C2×C4⋊C4C22×C22⋊C412yes—
390C2×C4⋊C4C22×C22⋊C412yes—
391C2×C4⋊C4C2×#12312yes—
392C2×C22⋊C4#38618yes—
393C2×C22⋊C4#38618yes—
394C2×C22⋊C4#38618yes—
395C2×C22⋊C4C2×C4×D412yes—
396C2×C22⋊C4C2×C4×D412yes—
397C2×C22⋊C4C2×#12112yes—
398C2×C22⋊C4C2×#12112yes—
399C2×C42C22×C22⋊C414yes—
400C2×C42C22×C22⋊C412yes—
401C2×C42C22×C22⋊C412yes—
402C2×C42C22×C22⋊C412yes—
403C2×C42C2×#12312yes—
404C24C2×#38614yes—
405C24C2×#38614yes—
406C24C2×#38614yes—
407C24C2×B(4,𝔽2)12yes—
408C24C2×B(4,𝔽2)14yes—
409C24C2×B(4,𝔽2)12yes—
410C24C22×C4×D411yescharacteristic
411C24C22×C4×D411yescharacteristic
412C24C22×#12011yescharacteristic
413C24C22×#12111yescharacteristic
414C24C22×#12111yescharacteristic
415C2×D4#188914yes—
416C2×D4#188914yes—
417C2×D4#188918yes—
418C22×C4C2×#38614yes—
419C22×C4C2×#38612yes—
420C22×C4C2×#38612yes—
421C22×C4C2×#38612yes—
422C22×C4C2×#38612yes—
423C22×C4C2×B(4,𝔽2)12yes—
424C22×C4C2×B(4,𝔽2)11yescharacteristic
425C22×C4C2×B(4,𝔽2)11yescharacteristic
426C22×C4C23×C22⋊C411yescharacteristic
427C22×C4C22×#12311yescharacteristic
428C42#288314yescomplemented
429C23C2×#188914yes—
430C23C2×#188914yes—
431C23C22×#38611yescharacteristic
432C23C22×B(4,𝔽2)11yescharacteristic
433C23C22×B(4,𝔽2)11yescharacteristic
434C2×C4C2×#188912yes—
435C2×C4C2×#188912yes—
436C2×C4#3355112yes—
437C2×C4#3355212yes—
438C2×C4C2×#288312yes—
439C22C22×#188911yescharacteristic
440C22#637150411yescharacteristic
441C22#637155111yescharacteristic
442C22C2×#3355111yescharacteristic
443C22C2×#3355211yescharacteristic
444C22#762288011yescharacteristic
445C22#762288211yescharacteristic
446C4#49460014yes—
447C2C2×#49460011yescharacteristic
448C2#7225123211yescharacteristic
449C2#7225662711yescharacteristic
subgroupG/H
center, socleC22C22×#1889
commutatorC22×D4C24×C4
Frattini, cosocleC23×D4C25
prime radical#72263749C2
LCSG ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×D4 ◁ C22×D42 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72263750C2semidirect
3. #72263719C2semidirect
4. #72263713C2semidirect
5. #72263708C2semidirect
6. #72263701C2semidirect
7. #72263633C2semidirect
8. #72263552C2semidirect
9. #72263419C2semidirect
10. #72251226C2semidirect
11. #72236740C2semidirect
12. #72236739C2semidirect
13. #72236734C2semidirect
14. #72233659C2semidirect
15. #72210479C2semidirect
18. #6595646C4semidirect
19. #6573693C4semidirect
27. #606700C22semidirect
28. #582593C22semidirect
29. #581603C22semidirect
30. #559927C22semidirect
32. #552678C22semidirect
33. #537848C22semidirect
34. #537841C22semidirect
36. #514639C22semidirect
37. #496306C22semidirect
38. #496189C22semidirect
39. #493956C22semidirect
40. #493370C22semidirect
49. #139818C22semidirect
50. #139818C22semidirect
51. #139779C22semidirect
52. #139778C22semidirect
53. #139777C22semidirect
54. #139777C22semidirect
55. #139776C22semidirect
56. #139767C22semidirect
57. #139706C22semidirect
58. #139706C22semidirect
59. #128081C22semidirect
60. #128081C22semidirect
61. #128073C22semidirect
62. #128073C22semidirect
63. #128043C22semidirect
64. #128043C22semidirect
65. #128033C22semidirect
66. C2×#7304C22semidirect
67. #62983C2×C4semidirect
68. D4×H(5,𝔽2)C2×C4semidirect
75. C2×#3019C2×C4semidirect
89. #32990C2×C4semidirect
90. #32920C2×C4semidirect
91. #32888C2×C4semidirect
92. #25918C2×C4semidirect
93. #25739C2×C4semidirect
94. #24412C2×C4semidirect
95. #24314C2×C4semidirect
96. #24224C2×C4semidirect
98. #23912C2×C4semidirect
101. D4×#100C2×C4semidirect
102. C4×#455C2×C4semidirect
103. #20953C2×C4semidirect
428. C42#2883semidirect

Maximal subgroups

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

subgroupindex
#722637502
#722637492
#722637192
#722637132
#722637082
#722637012
#722636332
#722635522
#722634192
#722512262
#722367402
#722367392
#722367342
#722336592
#722104792