Finite Group Databasebeta

← #26257735

Subgroups — #26257735

Order 12032.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253203C211yesmaximal, characteristic, complemented
2C2×#15689403C211yesmaximal, characteristic, complemented
3#20252054C211yesmaximal, characteristic, complemented
4#20244831C212yesmaximal, complemented
5#20244677C212yesmaximal
6#20244377C211yesmaximal, characteristic, complemented
7#20244170C212yesmaximal, complemented
8#20244163C211yesmaximal, characteristic, complemented
9#20244053C212yesmaximal, complemented
10#20243380C211yesmaximal, characteristic, complemented
11C4×AGL(1,ℤ/8)×C47C211yesmaximal, characteristic
12C2×#13094024C2211yescharacteristic
13#15689418—21no—
14#15689413—21no—
15#15689408—21no—
16#15689403—21no—
17#15689403—21no—
18#15689403—22no—
19#15689403—21no—
20#15689184—22no—
21#15689180—21no—
22#15689176—21no—
23C2×#13093959C2212yes—
24#15689166C2212yes—
25C2×#13093957C2212yes—
26#15689147C2211yescharacteristic
27#15689147C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29#15689124C2212yes—
30C2×#13093945C2212yes—
31C2×#13093941C2212yes—
32#15689090C2212yes—
33#15689079C2212yes—
34#15688892C2212yes—
35#15688891C2212yes—
36#15688891C2212yes—
37#15688887C2212yes—
38C2×AGL(1,ℤ/8)×C47C2211yescharacteristic
39C4×#41×C47C2211yescharacteristic
40C2×C4×D4×C47C2211yescharacteristic
41C47×#442—21no—
42C47×#440—21no—
43C4×SD16×C47—21no—
44C4×D8×C47—21no—
45C47×#428C2211yescharacteristic
46C47×#418C2211yescharacteristic
47C47×#417C2211yescharacteristic
48C4×M16×C47C2211yescharacteristic
49C22×Dic94—21no—
50C2×#10625329C2312yes—
51C2×#10625329C2312yes—
52C22×C47⋊C8C2312yes—
53#13094024—21no—
54#13094024—21no—
55#13094024—21no—
56#13094024—21no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47C2312yes—
59#13093959—24no—
60#13093957—22no—
61#13093957—22no—
62#13093957—22no—
63#13093955—22no—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)—21no—
66C4⋊(C47⋊C8)—21no—
67C4⋊(C47⋊C8)—21no—
68C4⋊(C47⋊C8)D411yescharacteristic
69C4⋊(C47⋊C8)D411yescharacteristic
70C4⋊(C47⋊C8)D412yes—
71#13093948—22no—
72#13093945—22no—
73#13093945—22no—
74#13093945—22no—
75#13093941—24no—
76C22⋊(C47⋊C8)—22no—
77C22⋊(C47⋊C8)—22no—
78#13093935—22no—
79C4×C47⋊C8—22no—
80C2×#41×C47C2311yescharacteristic
81C22×D4×C47C2311yescharacteristic
82C23×C4×C47—21no—
83AGL(1,ℤ/8)×C47—22no—
84AGL(1,ℤ/8)×C47—22no—
85C2×SD16×C47—21no—
86C2×D8×C47—21no—
87C2×M16×C47C2311yescharacteristic
88C2×M16×C47C2311yescharacteristic
89C4×Q8×C47D411yescharacteristic
90C4×Q8×C47D411yescharacteristic
91C4×D4×C47—21no—
92C4×D4×C47—21no—
93C4×D4×C47—21no—
94C4×D4×C47—21no—
95C4×D4×C47—21no—
96C4×D4×C47—21no—
97C4×D4×C47D411yescharacteristic
98C4×D4×C47D411yescharacteristic
99#117×C47—21no—
100#117×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C22⋊C4×C47—21no—
103C2×C42×C47—21no—
104C2×C42×C47C2311yescharacteristic
105#107×C47—21no—
106#106×C47—21no—
107#103×C47—21no—
108#102×C47—21no—
109#102×C47—21no—
110#102×C47—21no—
111#97×C47—21no—
112C4×C8×C47—21no—
113C24×C47—21no—
114#41×C47—21no—
115#41×C47—21no—
116#41×C47—21no—
117#41×C47—21no—
118C2×Q8×C47C2×D411yescharacteristic
119C2×D4×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47C2×D411yescharacteristic
126C2×D4×C47#4112yes—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—41no—
132C22×C4×C47—41no—
133C22×C4×C47—82no—
134C22×C4×C47#4111yescharacteristic
135C22×C4×C47#4111yescharacteristic
136C22×C4×C47C2411yescharacteristic
137C22×Dic47—22no—
138C2×Dic94—41no—
139C2×Dic94—41no—
140C2×Dic94—82no—
141SD16×C47—24no—
142D8×C47—24no—
143M16×C47—21no—
144M16×C47—21no—
145M16×C47—21no—
146M16×C47#4112yes—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47C2×D411yescharacteristic
154C4⋊C4×C47C2×D411yescharacteristic
155C22⋊C4×C47—21no—
156C22⋊C4×C47—21no—
157C22⋊C4×C47—21no—
158C22⋊C4×C47—41no—
159C22⋊C4×C47—41no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—21no—
163C42×C47C2×D411yescharacteristic
164C42×C47C2×D411yescharacteristic
165C42×C47C2×D411yescharacteristic
166C42×C47C2×D411yescharacteristic
167C4⋊Dic47—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47—41no—
170C4⋊Dic47—41no—
171#10625329—22no—
172#10625329—22no—
173#10625329—22no—
174#10625329—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8C2×D412yes—
180C2×C47⋊C8C2×D412yes—
181C23×C47—21no—
182C23×C47—22no—
183C23×C47—41no—
184C23×C47—41no—
185C23×C47—81no—
186C23×C47—81no—
187C23×C47C2×#4111yescharacteristic
188Q8×C47—21no—
189Q8×C47—21no—
190D4×C47—21no—
191D4×C47—22no—
192D4×C47—21no—
193D4×C47—22no—
194D4×C47—41no—
195D4×C47—41no—
196D4×C47—41no—
197D4×C47—41no—
198C2×C4×C47—22no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—42no—
206C2×C4×C47—41no—
207C2×C4×C47—81no—
208C2×C4×C47—82no—
209C2×C4×C47—82no—
210C2×C4×C47—81no—
211C2×C4×C47#12111yescharacteristic
212C2×C4×C47#12111yescharacteristic
213C2×C4×C47#12111yescharacteristic
214C2×C4×C47#12111yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C472−1+411yescharacteristic
220C2×C4×C472−1+411yescharacteristic
221C2×Dic47—42no—
222C2×Dic47—82no—
223Dic94—82no—
224Dic94—82no—
225C8×C47—22no—
226C8×C47—41no—
227C47⋊C8—22no—
228C47⋊C8—22no—
229C47⋊C8—42no—
230C47⋊C8—42no—
231#24720—471nomaximal, Sylow
232C22×C47—21no—
233C22×C47—21no—
234C22×C47—41no—
235C22×C47—42no—
236C22×C47—41no—
237C22×C47—41no—
238C22×C47—82no—
239C22×C47—82no—
240C22×C47—81no—
241C22×C47—81no—
242C22×C47C2×#12111yescharacteristic
243C22×C47#55211yescharacteristic
244C22×C47#55611yescharacteristic
245C4×C47—21no—
246C4×C47—21no—
247C4×C47—21no—
248C4×C47—41no—
249C4×C47—41no—
250C4×C47—41no—
251C4×C47—81no—
252C4×C47#47212yes—
253C4×C47#52511yescharacteristic
254C4×C47#53611yescharacteristic
255C4×C47#54711yescharacteristic
256C4×C47#54811yescharacteristic
257Dic47—82no—
258#3079—471no—
259C2×#474—471no—
260C4×AGL(1,ℤ/8)D4711yescharacteristic
261#2925—471no—
262#2078—472no—
263#2051—472no—
264#1997—471no—
265#1952—472no—
266#1950—471no—
267#1931—472no—
268#1800—471no—
269C2×C47—21no—
270C2×C47—21no—
271C2×C47—81no—
272C2×C47—82no—
273C2×C47#241511yescharacteristic
274C2×C47C2×#47211yescharacteristic
275C2×C47#306911yescharacteristic
276C2×AGL(1,ℤ/8)C2×D4711yescharacteristic
277C2×C4⋊Q8—471no—
278C4×#41C2×D4711yescharacteristic
279C2×C4×D4C2×D4711yescharacteristic
280#477—941no—
281#476—941no—
282#475—941no—
283#474—941no—
284#474—941no—
285#474—942no—
286#474—941no—
287#442—21no—
288#440—21no—
289C4×SD16—21no—
290C4×D8—21no—
291#433—942no—
292#432—941no—
293#431—941no—
294C2×#108—472no—
295#428—472no—
296#428C2×D4711yescharacteristic
297C2×#107—472no—
298#423—471no—
299#423—471no—
300C2×C4⋊C8—471no—
301#418C2×D4711yescharacteristic
302#417—472no—
303#417C2×D4711yescharacteristic
304C2×#102—472no—
305C2×#100—472no—
306#408—472no—
307#405—472no—
308C4×M16C2×D4711yescharacteristic
309#343—472no—
310#343—472no—
311#343—472no—
312#341—472no—
313C47#2472011yescharacteristic, complemented, Sylow
314C2×#41C22×D4711yescharacteristic
315C22×Q8—941no—
316C22×D4C22×D4711yescharacteristic
317C23×C4—21no—
318AGL(1,ℤ/8)—22no—
319AGL(1,ℤ/8)—22no—
320C2×SD16—21no—
321C2×D8—21no—
322C2×M16—472no—
323C2×M16—472no—
324C2×M16C22×D4711yescharacteristic
325C2×M16C22×D4711yescharacteristic
326C22×C8—472no—
327C4⋊Q8—941no—
328C4⋊Q8—941no—
329C4⋊Q8—941no—
330C4⋊Q8—941no—
331C4×Q8C22⋊D4711yescharacteristic
332C4×Q8C22⋊D4711yescharacteristic
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4C22⋊D4711yescharacteristic
340C4×D4C22⋊D4711yescharacteristic
341#117—21no—
342#117C22×D4711yescharacteristic
343C2×C4⋊C4—472no—
344C2×C4⋊C4—941no—
345C2×C4⋊C4C22×D4711yescharacteristic
346C2×C22⋊C4—21no—
347C2×C42—21no—
348C2×C42C22×D4711yescharacteristic
349#108—944no—
350#107—21no—
351#107—942no—
352#107—942no—
353#107—942no—
354#106—21no—
355#106—942no—
356C4⋊C8—471no—
357C4⋊C8—471no—
358C4⋊C8—472no—
359C4⋊C8—941no—
360C4⋊C8—941no—
361C4⋊C8—941no—
362C4⋊C8—941no—
363#103—21no—
364#103—942no—
365#102—21no—
366#102—21no—
367#102—21no—
368#102—942no—
369#102—942no—
370#102—942no—
371#100—944no—
372C22⋊C8—942no—
373C22⋊C8—942no—
374#97—21no—
375#97—942no—
376C4×C8—21no—
377C4×C8—942no—
378C24—21no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383C2×Q8—3762no—
384C2×Q8—1881no—
385C2×Q8—1881no—
386C2×Q8C2×C22⋊D4711yescharacteristic
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4#1062535012yes—
394C2×D4C2×C22⋊D4711yescharacteristic
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—82no—
402C22×C4—942no—
403C22×C4#1062535211yescharacteristic
404C22×C4#1062535211yescharacteristic
405C22×C4C23×D4711yescharacteristic
406SD16—24no—
407D8—24no—
408M16—21no—
409M16—21no—
410M16—21no—
411M16—942no—
412M16—942no—
413M16—942no—
414M16—942no—
415M16#1062535012yes—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—472no—
419C2×C8—472no—
420C2×C8—942no—
421C2×C8—942no—
422C2×C8—942no—
423C2×C8—942no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—942no—
429C4⋊C4—942no—
430C4⋊C4—1881no—
431C4⋊C4—1881no—
432C4⋊C4C2×C22⋊D4711yescharacteristic
433C4⋊C4C2×C22⋊D4711yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42D4×D4711yescharacteristic
443C42D4×D4711yescharacteristic
444C42C2×C22⋊D4711yescharacteristic
445C42C2×C22⋊D4711yescharacteristic
446C23—21no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—81no—
451C23—81no—
452C23C2×#1062535211yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—3762no—
456Q8—3762no—
457D4—22no—
458D4—21no—
459D4—22no—
460D4—21no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—3762no—
479C2×C4—1882no—
480C2×C4#1309399611yescharacteristic
481C2×C4#1309399611yescharacteristic
482C2×C4#1309399611yescharacteristic
483C2×C4#1309399611yescharacteristic
484C2×C4C2×D4×D4711yescharacteristic
485C2×C4C22×C22⋊D4711yescharacteristic
486C2×C4C2×#1062535011yescharacteristic
487C2×C4C2×#1062535011yescharacteristic
488C2×C4#1309407111yescharacteristic
489C2×C4#1309407111yescharacteristic
490C8—22no—
491C8—41no—
492C8—942no—
493C8—942no—
494C8—1882no—
495C8—1882no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—42no—
500C22—41no—
501C22—41no—
502C22—82no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22C2×#1309399611yescharacteristic
507C22#1568981311yescharacteristic
508C22#1568984311yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—3762no—
517C4#1568939112yes—
518C4#1568963411yescharacteristic
519C4#1568969411yescharacteristic
520C4#1568977811yescharacteristic
521C4#1568978211yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2—82no—
526C2#2024681711yescharacteristic
527C2C2×#1568939111yescharacteristic
528C2#2025312711yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC4×AGL(1,ℤ/8)×C47C2
socleC22×C47C2×#121
prime radicalC2×AGL(1,ℤ/8)×C47C22
hypercenterC4×AGL(1,ℤ/8)D47
hypocenterC47#24720
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20253203C2semidirect
2. C2×#15689403C2semidirect
3. #20252054C2semidirect
4. #20244831C2semidirect
6. #20244377C2semidirect
7. #20244170C2semidirect
8. #20244163C2semidirect
9. #20244053C2semidirect
10. #20243380C2semidirect
313. C47#24720semidirect

Maximal subgroups

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

subgroupindex
#202532032
C2×#156894032
#202520542
#202448312
#202446772
#202443772
#202441702
#202441632
#202440532
#202433802
C4×AGL(1,ℤ/8)×C472
#2472047