Finite Group Databasebeta

← #11861167

Subgroups — #11861167

Order 1152.

1001 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5745C211yesmaximal, characteristic, complemented
2#10594795C211yesmaximal, characteristic
3#10594780C211yesmaximal, characteristic, complemented
4C2×#63652C211yesmaximal, characteristic, complemented
5C3×#5124C211yesmaximal, characteristic, complemented
6C2×#63568C211yesmaximal, characteristic, complemented
7#10591060C211yesmaximal, characteristic, complemented
8#10590995C214yesmaximal, complemented
9#10590994C214yesmaximal, complemented
10#83811—31nomaximal
11#79258—31nomaximal
12C2×#3828C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C2×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C3×#907C2214yescomplemented
20C3×D3×M16—22no—
21C3×#901—22no—
22#63833C412yescomplemented
23#63831C412yescomplemented
24#63830C412yescomplemented
25#63829C2214yes—
26#63825C412yescomplemented
27#63818—22no—
28#63817—22no—
29#63817C2214yescomplemented
30#63652C2212yescomplemented
31#63652C2212yescomplemented
32C3×#822C2214yescomplemented
33#63568C2212yescomplemented
34#63568C2212yescomplemented
35#63555C2212yescomplemented
36#63553C2212yescomplemented
37C3×#568—31no—
38#5822—31no—
39#5748—31no—
40#5745—31no—
41#5745—31no—
42#5745D311yescharacteristic, complemented
43C3×#430—31no—
44C2×#837—31no—
45#5145—34no—
46#5144—34no—
47C2×#825—31no—
48C2×#825—31no—
49C2×#825—31no—
50C2×#825—61no—
51#5124D311yescharacteristic, complemented
52Dic3×M16—31no—
53Dic3×M16—31no—
54#4895—34no—
55#4894—34no—
56C22×C32×C4C2311yescharacteristic
57C2×#638—21no—
58C22×C3⋊Dic3—21no—
59#3828—21no—
60#3828—21no—
61#3828C2×C412yes—
62#3828C2312yes—
63C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
64C2×C4×C3⋊D3—21no—
65C2×#634C2×C411yescharacteristic
66C2×C3×#67—21no—
67C3×#235—21no—
68C3×#235—21no—
69C3×#235C2×C412yes—
70C3×#235C2312yes—
71C2×C3×D12C2×C411yescharacteristic
72C2×C3×C4×D3—21no—
73C2×C3×Dic6C2×C411yescharacteristic
74C32×M16—21no—
75C32×M16C2312yes—
76C2×C32×C8—21no—
77C4×C3⋊Dic3—21no—
78C4×C3⋊Dic3C2311yescharacteristic
79C4×C3⋊Dic3C2311yescharacteristic
80C3×#208—21no—
81C3×#208C2×C412yes—
82C3×#208C2×C412yes—
83C3×#208D412yes—
84C3×#208D412yes—
85C3×#208C2312yes—
86C2×C3×C3⋊C8—21no—
87C2×C3×C3⋊C8C2312yes—
88C2×C3×C3⋊C8C2312yes—
89C3×ΓL(1,𝔽25)—24no—
90C3×D3×C8—24no—
91#3719C2×C414yescomplemented
92#3717C2×C414yescomplemented
93#3716C2×C414yescomplemented
94#3714C2×C414yescomplemented
95#3712—24no—
96#3710—24no—
97#2956—91noSylow
98C2×C3×#41—31no—
99C2×#235—31no—
100C2×#235—31no—
101C2×#235—61no—
102C2×#235C2×D311yescharacteristic
103C3×#131—34no—
104C3×#131—62no—
105C2×C3×M16—21no—
106C2×C3×M16—31no—
107C2×C3×M16—31no—
108C2×C3×M16—31no—
109C2×C3×M16—62no—
110C2×C3×M16C2×D311yescharacteristic
111C2×C3×M16C2×D311yescharacteristic
112#951—32no—
113#950—34no—
114#949—32no—
115#942—32no—
116#932—32no—
117C2×C4×Dic3—31no—
118C2×C4×Dic3—31no—
119C2×C4×Dic3—61no—
120C2×#208C2×D311yescharacteristic
121C2×#208C2×D311yescharacteristic
122C2×#208C2×D311yescharacteristic
123#909—32no—
124#909—32no—
125#908—32no—
126#908—32no—
127#907—34no—
128#907—62no—
129#907C2×D314yescomplemented
130D3×M16—22no—
131D3×M16—62no—
132D3×M16—62no—
133#904—34no—
134#901—22no—
135#901—34no—
136#901—62no—
137C3×#108—34no—
138#837—32no—
139#837—32no—
140#825—32no—
141#825—32no—
142#825—32no—
143#825—32no—
144#825—32no—
145#825—32no—
146#825—62no—
147#825—62no—
148#822C2×D314yescomplemented
149#815—32no—
150#815—32no—
151C8×Dic3—32no—
152C8×Dic3—32no—
153C23×C32#4111yescharacteristic
154C22×C3⋊D3—21no—
155C22×C3×D3—21no—
156C2×C32×C4C22×C411yescharacteristic
157C2×C32×C4C22×C411yescharacteristic
158C2×C32×C4C2×D411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4#4111yescharacteristic
161C2×C32×C4C2411yescharacteristic
162#638—22no—
163#638—22no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—42no—
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)C22×C412yes—
170C4×C3⋊D3—22no—
171C4×C3⋊D3—22no—
172#634—21no—
173#634C22×C412yes—
174C3×#67—22no—
175C3×#67—22no—
176C2×C3×Dic3—21no—
177C3×D12—21no—
178C3×D12C22×C412yes—
179C3×C4×D3—22no—
180C3×C4×D3—22no—
181C3×Dic6—21no—
182C3×Dic6C22×C412yes—
183C32×C8—22no—
184C3×C3⋊C8—22no—
185C3×C3⋊C8C22×C414yes—
186C3×C3⋊C8C2×D414yes—
187#575—91no—
188#568—31no—
189#568—91no—
190#444—94no—
191#443—94no—
192#430—31no—
193C2×#104—91no—
194C2×#104—91no—
195C4×M16—91no—
196C3×#41—32no—
197C3×#41—32no—
198C3×#41—61no—
199C3×#41—61no—
200C2×C3×Q8—31no—
201C2×C3×D4—31no—
202C2×C3×D4—61no—
203C22×C3×C4—21no—
204C22×C3×C4—61no—
205C22×C3×C4C22×D311yescharacteristic
206C22×C3×C4C22×D311yescharacteristic
207C2×#67—21no—
208C2×#67—61no—
209C2×#67—61no—
210C2×#67—121no—
211C22×Dic3—61no—
212C22×Dic3—61no—
213C22×Dic3—121no—
214#235—21no—
215#235—21no—
216#235—32no—
217#235—32no—
218#235—32no—
219#235—32no—
220#235—62no—
221#235—62no—
222#235—61no—
223#235—61no—
224#235—61no—
225#235—61no—
226#235—121no—
227#235—121no—
228#235C4×D312yes—
229#235C22×D312yes—
230C2×D12—31no—
231C2×D12—31no—
232C2×D12—61no—
233C2×D12C4×D311yescharacteristic
234C2×C4×D3—21no—
235C2×C4×D3—61no—
236C2×C4×D3—61no—
237C2×C4×D3—121no—
238C2×Dic6—31no—
239C2×Dic6—31no—
240C2×Dic6—61no—
241C2×Dic6C4×D311yescharacteristic
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—22no—
245C3×M16—32no—
246C3×M16—32no—
247C3×M16—32no—
248C3×M16—32no—
249C3×M16—32no—
250C3×M16—41no—
251C3×M16—64no—
252C3×M16—61no—
253C3×M16C22×D312yes—
254C3×M16C22×D312yes—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—32no—
258C2×C3×C8—32no—
259C2×C3×C8—41no—
260C2×C3×C8—64no—
261C2×C3×C8—61no—
262#216—34no—
263#215—34no—
264#214—34no—
265#213—34no—
266C4×Dic3—31no—
267C4×Dic3—31no—
268C4×Dic3—31no—
269C4×Dic3—31no—
270C4×Dic3—61no—
271C4×Dic3—61no—
272C4×Dic3—61no—
273C4×Dic3—61no—
274C4×Dic3—121no—
275#208—21no—
276#208C4×D312yes—
277#208C4×D312yes—
278#208#6712yes—
279#208#6712yes—
280#208C22×D312yes—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C22×D312yes—
283C2×C3⋊C8C22×D312yes—
284Dic12—34no—
285D24—34no—
286#204—34no—
287#204—34no—
288ΓL(1,𝔽25)—24no—
289ΓL(1,𝔽25)—64no—
290ΓL(1,𝔽25)—64no—
291D3×C8—24no—
292D3×C8—64no—
293D3×C8—64no—
294C22×C32—41no—
295C22×C32C4×D411yescharacteristic
296C22×C32C4×D411yescharacteristic
297C22×C32C2×#4111yescharacteristic
298C2×C3⋊D3—22no—
299C2×C3⋊D3—81no—
300C2×C3×D3—22no—
301C2×C3×D3—81no—
302C32×C4C4×D411yescharacteristic
303C32×C4C4×D411yescharacteristic
304C32×C4C23×C411yescharacteristic
305C32×C4C22×D411yescharacteristic
306C3⋊Dic3—22no—
307C3⋊Dic3—22no—
308C3⋊Dic3—41no—
309C3×Dic3—22no—
310C2×#41—31no—
311C2×#41—91no—
312#137—92no—
313#137—92no—
314AGL(1,ℤ/8)—92no—
315AGL(1,ℤ/8)—92no—
316#135—94no—
317#131—34no—
318#131—62no—
319#131—94no—
320#131—182no—
321C2×M16—31no—
322C2×M16—31no—
323C2×M16—31no—
324C2×M16—62no—
325C2×M16—182no—
326C2×M16D3211yescharacteristic
327C2×C42—91no—
328#108—34no—
329#104—92no—
330#104—92no—
331#104—92no—
332#104—92no—
333#97—92no—
334C4×C8—92no—
335C23×C3—21no—
336C23×C3—61no—
337C23×C3#23511yescharacteristic
338C23×C3#23911yescharacteristic
339C22×D3—21no—
340C22×D3—61no—
341C22×D3—61no—
342C22×D3—121no—
343C3×Q8—32no—
344C3×Q8—61no—
345C3×D4—32no—
346C3×D4—62no—
347C3×D4—62no—
348C3×D4—61no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—61no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4C2×C4×D311yescharacteristic
359C2×C3×C4C2×C4×D311yescharacteristic
360C2×C3×C4C2×C4×D311yescharacteristic
361C2×C3×C4C2×C4×D311yescharacteristic
362C2×C3×C4#23511yescharacteristic
363C2×C3×C4D3×D411yescharacteristic
364C2×C3×C4D3×D411yescharacteristic
365C2×C3×C4#23911yescharacteristic
366C2×C3×C4C2×#6711yescharacteristic
367C2×C3×C4C2×#6711yescharacteristic
368C2×C3×C4C23×D311yescharacteristic
369C2×C3×C4C23×D311yescharacteristic
370#67—22no—
371#67—22no—
372#67—62no—
373#67—62no—
374#67—62no—
375#67—62no—
376#67—122no—
377#67—122no—
378C2×Dic3—21no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—121no—
386C2×Dic3—121no—
387C2×Dic3—121no—
388C2×Dic3—122no—
389C2×Dic3—122no—
390C2×Dic3—124no—
391D12—21no—
392D12—32no—
393D12—32no—
394D12—62no—
395D12—61no—
396D12—61no—
397D12—121no—
398D12C2×C4×D312yes—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—62no—
402C4×D3—62no—
403C4×D3—62no—
404C4×D3—62no—
405C4×D3—122no—
406C4×D3—122no—
407Dic6—21no—
408Dic6—32no—
409Dic6—32no—
410Dic6—62no—
411Dic6—61no—
412Dic6—61no—
413Dic6—121no—
414Dic6C2×C4×D312yes—
415C3×C8—22no—
416C3×C8—22no—
417C3×C8—34no—
418C3×C8—34no—
419C3×C8—42no—
420C3×C8—62no—
421C3⋊C8—22no—
422C3⋊C8C2×C4×D314yes—
423C3⋊C8C2×#6714yes—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32C2×C4×D411yescharacteristic
428C3⋊D3—42no—
429C3×D3—42no—
430#41—32no—
431#41—32no—
432#41—61no—
433#41—61no—
434#41—92no—
435#41—92no—
436#41—181no—
437#41—181no—
438C2×Q8—31no—
439C2×Q8—91no—
440C2×D4—31no—
441C2×D4—61no—
442C2×D4—91no—
443C2×D4—181no—
444C22×C4—61no—
445C22×C4—181no—
446C22×C4—181no—
447C22×C4C2×D3211yescharacteristic
448Q16—94no—
449SD16—94no—
450SD16—94no—
451D8—94no—
452M16—21no—
453M16—32no—
454M16—32no—
455M16—32no—
456M16—32no—
457M16—32no—
458M16—64no—
459M16—61no—
460M16—184no—
461M16C2×D3212yes—
462C2×C8—21no—
463C2×C8—32no—
464C2×C8—32no—
465C2×C8—64no—
466C2×C8—61no—
467C2×C8—184no—
468C42—91no—
469C42—91no—
470C42—181no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—42no—
477C22×C3—62no—
478C22×C3—241no—
479C22×C3#89311yescharacteristic
480C22×C3#89311yescharacteristic
481C22×C3C4×#6711yescharacteristic
482C22×C3C4×#6711yescharacteristic
483C22×C3C2×#23511yescharacteristic
484C22×C3C2×#23911yescharacteristic
485C2×D3—22no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—81no—
489C2×D3—122no—
490C2×D3—241no—
491C2×D3—241no—
492C2×D3—242no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—62no—
498C3×C4#89311yescharacteristic
499C3×C4#89311yescharacteristic
500C3×C4C4×#6711yescharacteristic
501C3×C4C4×#6711yescharacteristic
502C3×C4C22×C4×D311yescharacteristic
503C3×C4C22×C4×D311yescharacteristic
504C3×C4C2×D3×D411yescharacteristic
505C3×C4C22×#6711yescharacteristic
506Dic3—22no—
507Dic3—62no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—122no—
512Dic3—122no—
513Dic3—121no—
514Dic3—121no—
515Dic3—122no—
516C32#295611yescharacteristic, complemented, Sylow
517C23—61no—
518C23—181no—
519C23#379911yescharacteristic
520Q8—32no—
521Q8—61no—
522Q8—92no—
523Q8—181no—
524D4—32no—
525D4—62no—
526D4—62no—
527D4—61no—
528D4—92no—
529D4—182no—
530D4—182no—
531D4—181no—
532C2×C4—61no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—181no—
536C2×C4—181no—
537C2×C4—182no—
538C2×C4—182no—
539C2×C4—181no—
540C2×C4—362no—
541C2×C4#379911yescharacteristic
542C2×C4C4×D3211yescharacteristic
543C2×C4C4×D3211yescharacteristic
544C2×C4D3×#6711yescharacteristic
545C2×C4D3×#6711yescharacteristic
546C2×C4C22×D3211yescharacteristic
547C8—22no—
548C8—34no—
549C8—34no—
550C8—62no—
551C2×C3—22no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—41no—
560C2×C3—41no—
561C2×C3—122no—
562C2×C3C2×#89311yescharacteristic
563C2×C3C2×C4×#6711yescharacteristic
564D3—42no—
565D3—122no—
566D3—122no—
567D3—242no—
568C22—41no—
569C22—62no—
570C22—182no—
571C22—241no—
572C22—721no—
573C22#6390411yescharacteristic
574C22#6390411yescharacteristic
575C22C2×#379911yescharacteristic
576C4—62no—
577C4—182no—
578C4—182no—
579C4—361no—
580C4#6390411yescharacteristic
581C4#6390411yescharacteristic
582C4C2×C4×D3211yescharacteristic
583C4C2×D3×#6711yescharacteristic
584C3—21no—
585C3#7925811yescharacteristic, complemented
586C3#8381111yescharacteristic, complemented
587C2—21no—
588C2—21no—
589C2—21no—
590C2—122no—
591C2—362no—
592C2C2×#6390411yescharacteristic
subgroupG/H
centerC4C2×D3×#67
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC2×C32C2×C4×D4
prime radical#10594795C2
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32#2956
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C3×#5745C2semidirect
3. #10594780C2semidirect
4. C2×#63652C2semidirect
5. C3×#5124C2semidirect
6. C2×#63568C2semidirect
7. #10591060C2semidirect
8. #10590995C2semidirect
9. #10590994C2semidirect
19. C3×#907C22semidirect
22. #63833C4semidirect
23. #63831C4semidirect
24. #63830C4semidirect
26. #63825C4semidirect
29. #63817C22semidirect
30. #63652C22semidirect
31. #63652C22semidirect
32. C3×#822C22semidirect
33. #63568C22semidirect
34. #63568C22semidirect
35. #63555C22semidirect
36. #63553C22semidirect
42. #5745D3semidirect
51. #5124D3semidirect
91. #3719C2×C4semidirect
92. #3717C2×C4semidirect
93. #3716C2×C4semidirect
94. #3714C2×C4semidirect
129. #907C2×D3semidirect
148. #822C2×D3semidirect
516. C32#2956semidirect
585. C3#79258semidirect
586. C3#83811semidirect

Maximal subgroups

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

subgroupindex
C3×#57452
#105947952
#105947802
C2×#636522
C3×#51242
C2×#635682
#105910602
#105909952
#105909942
#838113
#792583