Finite Group Databasebeta

← #25251000

Subgroups — #25251000

Order 11008.

706 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583334C211yesmaximal, characteristic, complemented
2#19145082C211yesmaximal, characteristic, complemented
3#19143512C211yesmaximal, characteristic, complemented
4#19136669C214yesmaximal, complemented
5#19136666C214yesmaximal, complemented
6#19136360C211yesmaximal, characteristic, complemented
7#19136344C211yesmaximal, characteristic, complemented
8C2×#14582713C211yesmaximal, characteristic, complemented
9C43×#3414C211yesmaximal, characteristic, complemented
10C2×#13063350—21no—
11C22×D344C2211yescharacteristic
12#14583666—21no—
13#14583463—21no—
14#14583334—22no—
15#14583334C2214yes—
16C2×#13063268—21no—
17C2×#13063266—21no—
18C2×#13063265C2211yescharacteristic
19#14583285—22no—
20#14583284C2212yescomplemented
21#14583274C2212yescomplemented
22#14583021—22no—
23#14583016C2214yescomplemented
24C2×#13063261C2211yescharacteristic
25#14582916—21no—
26#14582854C2211yescharacteristic
27#14582719C2214yescomplemented
28#14582715C2212yescomplemented
29#14582713C2214yescomplemented
30#14582712C2212yescomplemented
31C2×#135×C43C2211yescharacteristic
32C2×#135×C43C2211yescharacteristic
33C43×#510—21no—
34C43×#510C2212yescomplemented
35C43×#509—21no—
36C43×#509C2212yescomplemented
37C43×#508—22no—
38C2×M32×C43C2211yescharacteristic
39C22×D172—21no—
40#13063350—42no—
41#13063350—42no—
42C2×#10621884—22no—
43C2×D344—21no—
44C2×D344—21no—
45C2×D344—21no—
46C2×D344—21no—
47C2×D344—21no—
48C2×D344C2312yes—
49C2×#10621882—22no—
50#13063332—24no—
51#13063332—42no—
52C2×#10621877—21no—
53C2×#10621877—42no—
54C2×#10621877D411yescharacteristic
55#13063321—22no—
56#13063314—22no—
57#13063299—42no—
58#13063293—42no—
59#13063284D411yescharacteristic
60#13063268—24no—
61#13063266—24no—
62#13063265—21no—
63#13063265C2312yes—
64C2×C43⋊C16—21no—
65#13063261—21no—
66#13063261C2312yes—
67#13063252—24no—
68#13063235D412yes—
69C8×Dic43C2312yes—
70C2×#41×C43—21no—
71C2×#41×C43—21no—
72#135×C43—21no—
73#135×C43—21no—
74#135×C43—21no—
75#135×C43—21no—
76#135×C43—21no—
77#135×C43—21no—
78#135×C43D412yescomplemented
79#135×C43C2312yes—
80C2×Q16×C43D411yescharacteristic, complemented
81C2×Q16×C43C2311yescharacteristic
82C2×SD16×C43—21no—
83C2×SD16×C43—21no—
84C2×D8×C43D411yescharacteristic, complemented
85C2×D8×C43C2311yescharacteristic
86C22×C8×C43C2311yescharacteristic
87Q32×C43—22no—
88SD32×C43—22no—
89SD32×C43—22no—
90D16×C43—22no—
91M32×C43—21no—
92M32×C43C2312yes—
93C2×C16×C43—21no—
94C23×D43—41no—
95#41×C43—22no—
96#41×C43—22no—
97#41×C43—41no—
98#41×C43—41no—
99#41×C43—41no—
100#41×C43—41no—
101#41×C43—41no—
102#41×C43—41no—
103C2×Q8×C43—21no—
104C2×Q8×C43—21no—
105C2×D4×C43—21no—
106C2×D4×C43—21no—
107C2×D4×C43—41no—
108C2×D4×C43—41no—
109C22×C4×C43—41no—
110C22×C4×C43—41no—
111C22×C4×C43C2×D411yescharacteristic
112C2×C22⋊D43—44no—
113C2×D172—22no—
114C2×D172—41no—
115C2×D172—41no—
116C2×D172—41no—
117C2×D172—41no—
118C2×D172—41no—
119Q16×C43—21no—
120Q16×C43—21no—
121Q16×C43—21no—
122Q16×C43C2×D412yes—
123SD16×C43—22no—
124SD16×C43—22no—
125SD16×C43—41no—
126SD16×C43—41no—
127D8×C43—21no—
128D8×C43—21no—
129D8×C43—21no—
130D8×C43C2×D412yes—
131C2×C8×C43C2×D411yescharacteristic
132C2×C8×C43C2×D411yescharacteristic
133C2×C8×C43C2×D411yescharacteristic
134C2×C8×C43#4111yescharacteristic
135C2×C8×C43#4111yescharacteristic
136C2×C8×C43C2411yescharacteristic
137C22⋊Dic43—21no—
138#10621884—44no—
139#10621882—44no—
140(C2×D43)⋊C4—44no—
141C4⋊Dic43—21no—
142C4×Dic43C2×D412yes—
143#10621877—22no—
144#10621877—41no—
145#10621877—44no—
146#10621877D812yes—
147#10621877D812yes—
148C2×C43⋊C8—41no—
149C2×C43⋊C8—44no—
150C2×C43⋊C8C2×D412yes—
151D344—22no—
152D344—22no—
153D344—41no—
154D344—41no—
155C16×C43—22no—
156C43⋊C16—22no—
157C23×C43—41no—
158C23×C43—41no—
159C23×C43AGL(1,ℤ/8)11yescharacteristic
160C22×D43—42no—
161C22×D43—81no—
162C22×D43—81no—
163C22×D43—162no—
164Q8×C43—22no—
165Q8×C43—41no—
166Q8×C43—41no—
167Q8×C43—41no—
168D4×C43—22no—
169D4×C43—42no—
170D4×C43—42no—
171D4×C43—41no—
172D4×C43—41no—
173D4×C43—41no—
174D4×C43—81no—
175D4×C43—81no—
176C2×C4×C43—41no—
177C2×C4×C43—42no—
178C2×C4×C43—42no—
179C2×C4×C43—41no—
180C2×C4×C43—81no—
181C2×C4×C43—81no—
182C2×C4×C43#12111yescharacteristic
183C2×C4×C43#12111yescharacteristic
184C2×C4×C43C2×D811yescharacteristic
185C2×C4×C43C2×D811yescharacteristic
186C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C43C22×D411yescharacteristic
188C22⋊D43—164no—
189C2×Dic43—22no—
190D172—42no—
191D172—42no—
192D172—81no—
193D172—81no—
194C8×C43#12111yescharacteristic
195C8×C43#12111yescharacteristic
196C8×C43C22×D411yescharacteristic
197C8×C43C2×#4111yescharacteristic
198C43⋊C8—42no—
199C43⋊C8C2×D814yes—
200#33595—431nomaximal, Sylow
201C22×C43—41no—
202C22×C43—42no—
203C22×C43—82no—
204C22×C43—81no—
205C22×C43—161no—
206C22×C43#45911yescharacteristic
207C22×C43#45911yescharacteristic
208C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
209C2×D43—84no—
210C2×D43—82no—
211C2×D43—162no—
212C2×D43—162no—
213C4×C43—42no—
214C4×C43—81no—
215C4×C43#45911yescharacteristic
216C4×C43#45911yescharacteristic
217C4×C43C2×#12111yescharacteristic
218C4×C43C22×D811yescharacteristic
219Dic43—42no—
220#3414D4311yescharacteristic, complemented
221C2×#509—431no—
222#3155—431no—
223#2957—431no—
224#2213—434no—
225#2212—434no—
226#2153—431no—
227#2148—431no—
228C2×#361—431no—
229C2×C43—21no—
230C2×C43—21no—
231C2×C43—21no—
232C2×C43—82no—
233C2×C43—161no—
234C2×C43C2×#45911yescharacteristic
235D43—162no—
236C2×AGL(1,ℤ/8)—861no—
237C2×#135C2×D4311yescharacteristic
238C2×#135C2×D4311yescharacteristic
239C22×D8—431no—
240#568—861no—
241#535—861no—
242#510—21no—
243#510C2×D4312yescomplemented
244#509—21no—
245#509—434no—
246#509—862no—
247#509C2×D4312yescomplemented
248#508—22no—
249C2×SD32—861no—
250C2×D16—861no—
251C2×M32—431no—
252C2×M32C2×D4311yescharacteristic
253#496—862no—
254#495—432no—
255#493—432no—
256#444—862no—
257#443—434no—
258C2×#108—431no—
259#421—861no—
260#405—431no—
261#364—434no—
262#362—432no—
263#361—432no—
264#361—434no—
265C43#3359511yescharacteristic, complemented, Sylow
266C2×#41—21no—
267C2×#41—21no—
268C22×D4—861no—
269AGL(1,ℤ/8)—1722no—
270AGL(1,ℤ/8)—1722no—
271#135—21no—
272#135—21no—
273#135—21no—
274#135—21no—
275#135—21no—
276#135—21no—
277#135C22⋊D4312yescomplemented
278#135C22×D4312yes—
279C2×Q16C22⋊D4311yescharacteristic, complemented
280C2×Q16C22×D4311yescharacteristic
281C2×SD16—21no—
282C2×SD16—21no—
283C2×SD16—862no—
284C2×D8—432no—
285C2×D8—862no—
286C2×D8—861no—
287C2×D8—861no—
288C2×D8—861no—
289C2×D8—861no—
290C2×D8—861no—
291C2×D8C22⋊D4311yescharacteristic, complemented
292C2×D8C22×D4311yescharacteristic
293#131—864no—
294#131—1722no—
295C2×M16—431no—
296C2×M16—861no—
297C2×M16—1722no—
298C22×C8C22×D4311yescharacteristic
299#127—862no—
300#124—862no—
301#121—1722no—
302#120—1722no—
303#117—431no—
304Q32—22no—
305SD32—22no—
306SD32—22no—
307SD32—864no—
308D16—22no—
309D16—864no—
310M32—21no—
311M32—432no—
312M32—861no—
313M32C22×D4312yes—
314C2×C16—21no—
315C2×C16—861no—
316#108—432no—
317#108—861no—
318#104—864no—
319#97—432no—
320C4×C8—432no—
321C24—1721no—
322#41—22no—
323#41—22no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—862no—
337C2×D4—1724no—
338C2×D4—1721no—
339C2×D4—1721no—
340C2×D4—1721no—
341C2×D4—1721no—
342C2×D4—1721no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4D4×D4311yescharacteristic
346Q16—21no—
347Q16—21no—
348Q16—21no—
349Q16C2×C22⋊D4312yes—
350SD16—22no—
351SD16—22no—
352SD16—41no—
353SD16—41no—
354SD16—1724no—
355D8—21no—
356D8—21no—
357D8—21no—
358D8—862no—
359D8—862no—
360D8—1724no—
361D8—1721no—
362D8—1721no—
363D8C2×C22⋊D4312yes—
364M16—432no—
365M16—432no—
366M16—862no—
367M16—1724no—
368M16—1721no—
369C2×C8—432no—
370C2×C8—1724no—
371C2×C8—1721no—
372C2×C8D4×D4311yescharacteristic
373C2×C8#1062190011yescharacteristic
374C2×C8#1062190011yescharacteristic
375C2×C8C2×C22⋊D4311yescharacteristic
376C2×C8C2×C22⋊D4311yescharacteristic
377C2×C8C23×D4311yescharacteristic
378C4⋊C4—861no—
379C22⋊C4—861no—
380C22⋊C4—1724no—
381C42—432no—
382C16—22no—
383C16—862no—
384C23—41no—
385C23—41no—
386C23—3441no—
387C23—3441no—
388C23—1722no—
389C23—6882no—
390C23#1306334911yescharacteristic
391Q8—22no—
392Q8—41no—
393Q8—41no—
394Q8—41no—
395D4—22no—
396D4—42no—
397D4—42no—
398D4—41no—
399D4—41no—
400D4—41no—
401D4—81no—
402D4—81no—
403D4—3441no—
404D4—3441no—
405D4—1722no—
406D4—1722no—
407D4—6884no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—862no—
415C2×C4#1306329711yescharacteristic
416C2×C4#1306329711yescharacteristic
417C2×C4D8×D4311yescharacteristic
418C2×C4D8×D4311yescharacteristic
419C2×C4#1306334911yescharacteristic
420C2×C4C2×D4×D4311yescharacteristic
421C8—434no—
422C8—1722no—
423C8#1306329711yescharacteristic
424C8#1306329711yescharacteristic
425C8C22×C22⋊D4311yescharacteristic
426C8C2×#1062190011yescharacteristic
427C22—41no—
428C22—42no—
429C22—82no—
430C22—81no—
431C22—161no—
432C22—3444no—
433C22—3442no—
434C22—6882no—
435C22—6882no—
436C22#1458309211yescharacteristic
437C22#1458309211yescharacteristic
438C22C2×#1306334911yescharacteristic
439C4—42no—
440C4—81no—
441C4—1722no—
442C4#1458309211yescharacteristic
443C4#1458309211yescharacteristic
444C4C2×#1306329711yescharacteristic
445C4C2×D8×D4311yescharacteristic
446C2—21no—
447C2—21no—
448C2—21no—
449C2—82no—
450C2—161no—
451C2—6882no—
452C2C2×#1458309211yescharacteristic
subgroupG/H
centerC2C2×#14583092
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingC43×#3414C2
socleC2×C43C2×#459
hypercenter#3414D43
hypocenterC43#33595
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. C2×#14583334C2semidirect
2. #19145082C2semidirect
3. #19143512C2semidirect
4. #19136669C2semidirect
5. #19136666C2semidirect
6. #19136360C2semidirect
7. #19136344C2semidirect
8. C2×#14582713C2semidirect
9. C43×#3414C2semidirect
20. #14583284C22semidirect
21. #14583274C22semidirect
23. #14583016C22semidirect
27. #14582719C22semidirect
28. #14582715C22semidirect
29. #14582713C22semidirect
30. #14582712C22semidirect
34. C43×#510C22semidirect
36. C43×#509C22semidirect
78. #135×C43D4semidirect
80. C2×Q16×C43D4semidirect
84. C2×D8×C43D4semidirect
220. #3414D43semidirect
243. #510C2×D43semidirect
247. #509C2×D43semidirect
265. C43#33595semidirect
277. #135C22⋊D43semidirect
279. C2×Q16C22⋊D43semidirect
291. C2×D8C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145833342
#191450822
#191435122
#191366692
#191366662
#191363602
#191363442
C2×#145827132
C43×#34142
#3359543