Finite Group Databasebeta

← #11861170

Subgroups — #11861170

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×#5822C211yesmaximal, characteristic
2C3×#5819C211yesmaximal, characteristic, complemented
3#10594776C211yesmaximal, characteristic, complemented
4C2×#63664C211yesmaximal, characteristic, complemented
5#10591674C214yesmaximal, complemented
6#10591673C214yesmaximal, complemented
7#10591141C211yesmaximal, characteristic, complemented
8C2×#63568C211yesmaximal, characteristic, complemented
9#10591060C211yesmaximal, characteristic, complemented
10#84000—31nomaximal
11#83811—31nomaximal
12C2×C32×#41C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×C4×C3⋊Dic3C2211yescharacteristic
15C2×#3748C2211yescharacteristic
16C3×#951C412yescomplemented
17C3×#950C2214yes—
18C3×#949C412yescomplemented
19C3×#948—22no—
20C3×#948C2214yescomplemented
21C3×#942C412yescomplemented
22C3×#932C412yescomplemented
23C2×C3×#208—22no—
24C2×C3×#208C2211yescharacteristic
25C2×C3×#208C2211yescharacteristic
26C2×C3×#208C2211yescharacteristic
27#63815—22no—
28#63814C2214yescomplemented
29D3×#208—22no—
30#63664C2212yescomplemented
31#63664C2212yescomplemented
32#63577C2214yescomplemented
33#63568C2212yescomplemented
34#63568C2212yescomplemented
35#63555C2212yescomplemented
36#63553C2212yescomplemented
37C3×#575—31no—
38C3×#568—31no—
39#5822D311yescharacteristic
40#5819—31no—
41#5819D311yescharacteristic, complemented
42#5745—31no—
43C2×#837—31no—
44C2×#837—31no—
45C2×#837—31no—
46C2×#837—61no—
47#5197—34no—
48#5196—34no—
49#5145—34no—
50#5144—34no—
51C2×#825—31no—
52#5125—31no—
53#5124—31no—
54Dic3×M16—31no—
55Dic3×M16—31no—
56C32×#41—21no—
57C32×#41—21no—
58C32×#41C2×C412yes—
59C32×#41C2312yes—
60C2×C32×Q8C2×C411yescharacteristic
61C2×C32×D4—21no—
62C2×C32×D4C2×C411yescharacteristic
63C22×C32×C4—21no—
64C22×C32×C4C2311yescharacteristic
65C22×C3⋊Dic3—21no—
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
74C4×C3⋊Dic3—21no—
75C4×C3⋊Dic3C2311yescharacteristic
76C4×C3⋊Dic3C2311yescharacteristic
77#3748—21no—
78#3748C2312yes—
79C2×C3⋊(C3⋊C8)—21no—
80C3×#216C2×C414yescomplemented
81C3×#215C2×C414yescomplemented
82C3×#214C2×C414yescomplemented
83C3×#213C2×C414yescomplemented
84C3×#208—21no—
85C3×#208—24no—
86C3×#208C2×C412yes—
87C3×#208C2×C412yes—
88C3×#208D412yes—
89C3×#208D412yes—
90C3×#208C2312yes—
91C2×C3×C3⋊C8—21no—
92C2×C3×C3⋊C8—24no—
93C2×C3×C3⋊C8C2312yes—
94C2×C3×C3⋊C8C2312yes—
95#3711—24no—
96D3×C3⋊C8—24no—
97#2956—91noSylow
98C2×C3×#41—21no—
99C2×C3×#41—31no—
100C2×C3×#41C2×D311yescharacteristic
101C2×C3×#41C2×D311yescharacteristic
102C2×#235C2×D311yescharacteristic
103C3×#137—32no—
104C3×#137—32no—
105C3×AGL(1,ℤ/8)—32no—
106C3×AGL(1,ℤ/8)—32no—
107C3×#135—34no—
108C3×#131—34no—
109C3×#131—62no—
110C2×C3×M16—31no—
111C2×C3×M16—31no—
112C2×C3×M16—31no—
113C2×C3×M16—62no—
114#951Dic312yescomplemented
115#950C2×D314yes—
116#949Dic312yescomplemented
117#948—22no—
118#948—34no—
119#948—62no—
120#948C2×D314yescomplemented
121#942Dic312yescomplemented
122#932Dic312yescomplemented
123C2×C4×Dic3—31no—
124C2×C4×Dic3—31no—
125C2×C4×Dic3—61no—
126C2×#208—22no—
127C2×#208—31no—
128C2×#208—31no—
129C2×#208—61no—
130C2×#208—62no—
131C2×#208C2×D311yescharacteristic
132C2×#208C2×D311yescharacteristic
133C2×#208C2×D311yescharacteristic
134#907—62no—
135D3×M16—62no—
136#901—34no—
137#837—32no—
138#837—32no—
139#837—32no—
140#837—32no—
141#837—32no—
142#837—32no—
143#837—62no—
144#837—62no—
145#825—32no—
146#825—32no—
147#822—34no—
148#819—34no—
149#815—32no—
150#815—32no—
151C8×Dic3—32no—
152C8×Dic3—32no—
153C23×C32—21no—
154C23×C32#4111yescharacteristic
155C22×C3×D3—21no—
156C32×Q8—21no—
157C32×Q8C22×C412yes—
158C32×D4—22no—
159C32×D4—22no—
160C32×D4—21no—
161C32×D4C22×C412yes—
162C2×C32×C4—21no—
163C2×C32×C4—22no—
164C2×C32×C4—22no—
165C2×C32×C4C22×C411yescharacteristic
166C2×C32×C4C22×C411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4#4111yescharacteristic
170C2×C32×C4C2411yescharacteristic
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—42no—
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—
183C3⋊(C3⋊C8)—22no—
184C3×C3⋊C8—22no—
185C3×C3⋊C8C22×C414yes—
186C3×C3⋊C8C2×D414yes—
187#575—31no—
188#568—31no—
189#568—91no—
190#444—94no—
191#443—94no—
192#430—91no—
193C2×#104—91no—
194C2×#104—91no—
195C4×M16—91no—
196C3×#41—21no—
197C3×#41—21no—
198C3×#41—21no—
199C3×#41—21no—
200C3×#41—22no—
201C3×#41—22no—
202C3×#41—32no—
203C3×#41—32no—
204C3×#41—41no—
205C3×#41—41no—
206C3×#41—61no—
207C3×#41—61no—
208C3×#41C4×D312yes—
209C3×#41C2×Dic312yes—
210C3×#41C22×D312yes—
211C3×#41C22×D312yes—
212C2×C3×Q8—21no—
213C2×C3×Q8—31no—
214C2×C3×Q8C4×D311yescharacteristic
215C2×C3×Q8C2×Dic311yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—31no—
220C2×C3×D4—41no—
221C2×C3×D4—61no—
222C2×C3×D4C4×D311yescharacteristic
223C2×C3×D4C2×Dic311yescharacteristic
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—41no—
228C22×C3×C4—61no—
229C22×C3×C4C22×D311yescharacteristic
230C22×C3×C4C22×D311yescharacteristic
231C2×#67—21no—
232C22×Dic3—61no—
233C22×Dic3—61no—
234C22×Dic3—121no—
235#235—21no—
236#235—21no—
237#235C2×Dic312yes—
238#235C22×D312yes—
239C2×D12C2×Dic311yescharacteristic
240C2×C4×D3—21no—
241C2×Dic6C2×Dic311yescharacteristic
242C3×Q16—34no—
243C3×SD16—34no—
244C3×SD16—34no—
245C3×D8—34no—
246C3×M16—32no—
247C3×M16—32no—
248C3×M16—32no—
249C3×M16—32no—
250C3×M16—32no—
251C3×M16—64no—
252C3×M16—61no—
253C2×C3×C8—32no—
254C2×C3×C8—32no—
255C2×C3×C8—64no—
256C2×C3×C8—61no—
257#216C2×Dic314yescomplemented
258#215C2×Dic314yescomplemented
259#214C2×Dic314yescomplemented
260#213C2×Dic314yescomplemented
261C4×Dic3—31no—
262C4×Dic3—31no—
263C4×Dic3—31no—
264C4×Dic3—31no—
265C4×Dic3—61no—
266C4×Dic3—61no—
267C4×Dic3—61no—
268C4×Dic3—61no—
269C4×Dic3—121no—
270#208—21no—
271#208—24no—
272#208—32no—
273#208—32no—
274#208—62no—
275#208—61no—
276#208—61no—
277#208—64no—
278#208—121no—
279#208C2×Dic312yes—
280#208C2×Dic312yes—
281#208#6712yes—
282#208#6712yes—
283#208C22×D312yes—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—24no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8—61no—
288C2×C3⋊C8—64no—
289C2×C3⋊C8—121no—
290C2×C3⋊C8C22×D312yes—
291C2×C3⋊C8C22×D312yes—
292ΓL(1,𝔽25)—64no—
293D3×C8—64no—
294C22×C32—22no—
295C22×C32—41no—
296C22×C32—81no—
297C22×C32C4×D411yescharacteristic
298C22×C32C4×D411yescharacteristic
299C22×C32C2×#4111yescharacteristic
300C2×C3×D3—22no—
301C2×C3×D3—81no—
302C32×C4—22no—
303C32×C4C4×D411yescharacteristic
304C32×C4C4×D411yescharacteristic
305C32×C4C23×C411yescharacteristic
306C32×C4C22×D411yescharacteristic
307C3⋊Dic3—22no—
308C3⋊Dic3—41no—
309C3×Dic3—22no—
310C2×#41—31no—
311C2×#41D3211yescharacteristic
312#137—32no—
313#137—32no—
314AGL(1,ℤ/8)—32no—
315AGL(1,ℤ/8)—32no—
316#135—34no—
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—91no—
326C2×M16—182no—
327C2×C42—91no—
328#108—94no—
329#104—92no—
330#104—92no—
331#104—92no—
332#104—92no—
333#97—92no—
334C4×C8—92no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—41no—
339C23×C3—61no—
340C23×C3#23511yescharacteristic
341C23×C3#23711yescharacteristic
342C22×D3—21no—
343C3×Q8—21no—
344C3×Q8—21no—
345C3×Q8—22no—
346C3×Q8—32no—
347C3×Q8—41no—
348C3×Q8—61no—
349C3×Q8C2×C4×D312yes—
350C3×Q8C22×Dic312yes—
351C3×D4—22no—
352C3×D4—22no—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4—21no—
356C3×D4—21no—
357C3×D4—22no—
358C3×D4—32no—
359C3×D4—41no—
360C3×D4—42no—
361C3×D4—42no—
362C3×D4—62no—
363C3×D4—62no—
364C3×D4—61no—
365C3×D4C2×C4×D312yes—
366C3×D4C22×Dic312yes—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—41no—
382C2×C3×C4—61no—
383C2×C3×C4—62no—
384C2×C3×C4—62no—
385C2×C3×C4C2×C4×D311yescharacteristic
386C2×C3×C4C2×C4×D311yescharacteristic
387C2×C3×C4#23511yescharacteristic
388C2×C3×C4D3×D411yescharacteristic
389C2×C3×C4D3×D411yescharacteristic
390C2×C3×C4#23711yescharacteristic
391C2×C3×C4C22×Dic311yescharacteristic
392C2×C3×C4C22×Dic311yescharacteristic
393C2×C3×C4C2×#6711yescharacteristic
394C2×C3×C4C2×#6711yescharacteristic
395C2×C3×C4C23×D311yescharacteristic
396C2×C3×C4C23×D311yescharacteristic
397#67—22no—
398#67—22no—
399C2×Dic3—21no—
400C2×Dic3—61no—
401C2×Dic3—61no—
402C2×Dic3—61no—
403C2×Dic3—61no—
404C2×Dic3—121no—
405C2×Dic3—121no—
406C2×Dic3—122no—
407C2×Dic3—122no—
408C2×Dic3—124no—
409D12—21no—
410D12C22×Dic312yes—
411C4×D3—22no—
412C4×D3—22no—
413Dic6—21no—
414Dic6C22×Dic312yes—
415C3×C8—34no—
416C3×C8—34no—
417C3×C8—62no—
418C3⋊C8—22no—
419C3⋊C8—62no—
420C3⋊C8—62no—
421C3⋊C8—122no—
422C3⋊C8C22×Dic314yes—
423C3⋊C8C2×#6714yes—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—42no—
428C2×C32C2×C4×D411yescharacteristic
429C3×D3—42no—
430#41—21no—
431#41—21no—
432#41—32no—
433#41—32no—
434#41—61no—
435#41—61no—
436#41D3×Dic312yes—
437#41C2×D3212yes—
438C2×Q8—31no—
439C2×Q8D3×Dic311yescharacteristic
440C2×D4—21no—
441C2×D4—31no—
442C2×D4—61no—
443C2×D4D3×Dic311yescharacteristic
444C22×C4—21no—
445C22×C4—61no—
446C22×C4—181no—
447C22×C4C2×D3211yescharacteristic
448Q16—34no—
449SD16—34no—
450SD16—34no—
451D8—34no—
452M16—32no—
453M16—32no—
454M16—32no—
455M16—32no—
456M16—32no—
457M16—64no—
458M16—61no—
459M16—92no—
460M16—184no—
461M16—181no—
462C2×C8—32no—
463C2×C8—32no—
464C2×C8—64no—
465C2×C8—61no—
466C2×C8—184no—
467C2×C8—181no—
468C42—91no—
469C42—91no—
470C42—181no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—42no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—42no—
480C22×C3—62no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—82no—
484C22×C3—241no—
485C22×C3C4×#6711yescharacteristic
486C22×C3C4×#6711yescharacteristic
487C22×C3D4×Dic311yescharacteristic
488C22×C3D4×Dic311yescharacteristic
489C22×C3C2×#23511yescharacteristic
490C22×C3C2×#23711yescharacteristic
491C2×D3—22no—
492C2×D3—81no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—42no—
500C3×C4—62no—
501C3×C4C4×#6711yescharacteristic
502C3×C4C4×#6711yescharacteristic
503C3×C4D4×Dic311yescharacteristic
504C3×C4D4×Dic311yescharacteristic
505C3×C4C22×C4×D311yescharacteristic
506C3×C4C2×D3×D411yescharacteristic
507C3×C4C23×Dic311yescharacteristic
508C3×C4C22×#6711yescharacteristic
509Dic3—22no—
510Dic3—62no—
511Dic3—62no—
512Dic3—122no—
513Dic3—121no—
514Dic3—121no—
515Dic3—122no—
516C32#295611yescharacteristic, complemented, Sylow
517C23—21no—
518C23—61no—
519C23#380411yescharacteristic
520Q8—21no—
521Q8—32no—
522Q8—61no—
523Q8C2×D3×Dic312yes—
524D4—22no—
525D4—22no—
526D4—21no—
527D4—32no—
528D4—62no—
529D4—62no—
530D4—61no—
531D4C2×D3×Dic312yes—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—61no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4—362no—
541C2×C4C2×D3×Dic311yescharacteristic
542C2×C4C2×D3×Dic311yescharacteristic
543C2×C4#380411yescharacteristic
544C2×C4D3×#6711yescharacteristic
545C2×C4D3×#6711yescharacteristic
546C2×C4C22×D3211yescharacteristic
547C8—34no—
548C8—34no—
549C8—62no—
550C8—182no—
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—42no—
562C2×C3—42no—
563C2×C3—82no—
564C2×C3—122no—
565C2×C3C2×C4×#6711yescharacteristic
566C2×C3C2×D4×Dic311yescharacteristic
567D3—42no—
568C22—22no—
569C22—41no—
570C22—62no—
571C22—81no—
572C22—241no—
573C22Dic3×#6711yescharacteristic
574C22Dic3×#6711yescharacteristic
575C22C2×#380411yescharacteristic
576C4—22no—
577C4—62no—
578C4—182no—
579C4—361no—
580C4Dic3×#6711yescharacteristic
581C4Dic3×#6711yescharacteristic
582C4C22×D3×Dic311yescharacteristic
583C4C2×D3×#6711yescharacteristic
584C3—21no—
585C3#8381111yescharacteristic, complemented
586C3#8400011yescharacteristic, complemented
587C2—21no—
588C2—21no—
589C2—21no—
590C2—42no—
591C2—122no—
592C2C2×Dic3×#6711yescharacteristic
subgroupG/H
centerC4C2×D3×#67
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC2×C32×#41C22
socleC2×C32C2×C4×D4
prime radicalC3×#5822C2
cosocleC2×C32×C4C24
hypercenterC2×#41D32
hypocenterC32#2956
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C3×#5819C2semidirect
3. #10594776C2semidirect
4. C2×#63664C2semidirect
5. #10591674C2semidirect
6. #10591673C2semidirect
7. #10591141C2semidirect
8. C2×#63568C2semidirect
9. #10591060C2semidirect
16. C3×#951C4semidirect
18. C3×#949C4semidirect
20. C3×#948C22semidirect
21. C3×#942C4semidirect
22. C3×#932C4semidirect
28. #63814C22semidirect
30. #63664C22semidirect
31. #63664C22semidirect
32. #63577C22semidirect
33. #63568C22semidirect
34. #63568C22semidirect
35. #63555C22semidirect
36. #63553C22semidirect
41. #5819D3semidirect
80. C3×#216C2×C4semidirect
81. C3×#215C2×C4semidirect
82. C3×#214C2×C4semidirect
83. C3×#213C2×C4semidirect
114. #951Dic3semidirect
116. #949Dic3semidirect
120. #948C2×D3semidirect
121. #942Dic3semidirect
122. #932Dic3semidirect
257. #216C2×Dic3semidirect
258. #215C2×Dic3semidirect
259. #214C2×Dic3semidirect
260. #213C2×Dic3semidirect
516. C32#2956semidirect
585. C3#83811semidirect
586. C3#84000semidirect

Maximal subgroups

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

subgroupindex
C3×#58222
C3×#58192
#105947762
C2×#636642
#105916742
#105916732
#105911412
C2×#635682
#105910602
#840003
#838113