Finite Group Databasebeta

← #13162041

Subgroups — #13162041

Order 1728.

646 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64160C211yesmaximal, characteristic, complemented
2C3×#64158C211yesmaximal, characteristic
3C3×#64157C211yesmaximal, characteristic, complemented
4C2×#88811C211yesmaximal, characteristic, complemented
5#11722081C211yesmaximal, characteristic, complemented
6#11722074C211yesmaximal, characteristic, complemented
7#11722073C211yesmaximal, characteristic, complemented
8#10591681—31nomaximal
9#10591680—31nomaximal
10#10591679—31nomaximal
11C33×#41C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C3×#3756—21no—
15C3×#3755—21no—
16C3×#3754—21no—
17C3×#3753—21no—
18C3×#3748—21no—
19C3×#3748C2211yescharacteristic
20C3×#3748C2211yescharacteristic
21C3×#3748C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)—21no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24#88811—21no—
25#88811—21no—
26#64160D311yescharacteristic, complemented
27#64158D311yescharacteristic
28#64157D311yescharacteristic, complemented
29C3×#951—31no—
30C3×#951—31no—
31C3×#951—61no—
32C3×#949—31no—
33C3×#949—31no—
34C3×#949—61no—
35C3×#948—31no—
36C3×#948—31no—
37C3×#948—61no—
38C2×#3717—31no—
39C2×#3716—31no—
40C2×#3715—31no—
41#63578—31no—
42#63577—31no—
43#63577—31no—
44#63561—31no—
45#63560—31no—
46#63560—31no—
47#63559—31no—
48#63558—31no—
49#63558—31no—
50C22×C3×C3⋊D3—21no—
51C33×Q8D411yescharacteristic
52C33×D4—21no—
53C33×D4D411yescharacteristic
54C2×C33×C4—21no—
55C2×C33×C4C2311yescharacteristic
56C2×C3×C3⋊Dic3—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×#634—21no—
60C3×#634—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×C3⋊(C3⋊C8)D411yescharacteristic
65C3×C3⋊(C3⋊C8)D411yescharacteristic
66#5203—91no—
67#5200—91no—
68#5141—91no—
69C32×#41—21no—
70C32×#41—21no—
71C32×#41—21no—
72C32×#41—41no—
73C32×#41C2×D311yescharacteristic
74C32×#41C2×D311yescharacteristic
75C32×#41C2×D311yescharacteristic
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×#634C2×D311yescharacteristic
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—61no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84#3756—21no—
85#3755—21no—
86#3754—21no—
87#3753—21no—
88#3748—21no—
89#3748C2×D311yescharacteristic
90#3748C2×D311yescharacteristic
91#3748C2×D311yescharacteristic
92C2×C3⋊(C3⋊C8)—21no—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94C3×#216—62no—
95C3×#216—61no—
96C3×#216—61no—
97C3×#215—62no—
98C3×#215—61no—
99C3×#215—61no—
100C3×#214—62no—
101C3×#214—61no—
102C3×#214—61no—
103C3×#213—62no—
104C3×#213—61no—
105C3×#213—61no—
106C3×#208—31no—
107C3×#208—31no—
108C3×#208—31no—
109C3×#208—31no—
110C3×#208—31no—
111C3×#208—31no—
112C3×#208—61no—
113C3×#208—61no—
114C3×#208—61no—
115C3×#208—61no—
116C3×#208—61no—
117C3×#208—62no—
118C2×C3×C3⋊C8—31no—
119C2×C3×C3⋊C8—31no—
120C2×C3×C3⋊C8—61no—
121C2×C3×C3⋊C8—61no—
122C2×C3×C3⋊C8—61no—
123C2×C3×C3⋊C8—62no—
124#3717—61no—
125#3717—61no—
126#3716—61no—
127#3716—61no—
128#3715—61no—
129#3715—61no—
130C22×C33—21no—
131C22×C33#4111yescharacteristic
132C2×C3×C3⋊D3—41no—
133C2×C3×C3⋊D3—81no—
134C33×C4—21no—
135C33×C4C2×D411yescharacteristic
136C33×C4C2×D411yescharacteristic
137C3×C3⋊Dic3—41no—
138C3×#137—91no—
139C3×AGL(1,ℤ/8)—91no—
140C3×#131—91no—
141#951—31no—
142#951—31no—
143#951—61no—
144#949—31no—
145#949—31no—
146#949—61no—
147#948—31no—
148#948—31no—
149#948—61no—
150C2×#215—91no—
151C2×#214—91no—
152C2×#204—91no—
153#836—91no—
154#833—91no—
155#824—91no—
156#824—91no—
157#823—91no—
158#823—91no—
159#822—91no—
160#822—91no—
161#819—91no—
162C22×C3⋊D3—21no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—121no—
166C32×Q8—21no—
167C32×Q8—21no—
168C32×Q8—21no—
169C32×Q8—41no—
170C32×Q8D1211yescharacteristic
171C32×Q8#6711yescharacteristic
172C32×Q8#6711yescharacteristic
173C32×D4—21no—
174C32×D4—22no—
175C32×D4—21no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—21no—
179C32×D4—21no—
180C32×D4—41no—
181C32×D4—41no—
182C32×D4—41no—
183C32×D4—42no—
184C32×D4D1211yescharacteristic
185C32×D4#6711yescharacteristic
186C32×D4#6711yescharacteristic
187C2×C32×C4—21no—
188C2×C32×C4—22no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—41no—
195C2×C32×C4—41no—
196C2×C32×C4—41no—
197C2×C32×C4—42no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C3⋊Dic3—21no—
202C4⋊(C3⋊D3)—21no—
203C4⋊(C3⋊D3)—21no—
204#634—21no—
205#634—21no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—121no—
209C3×D12—62no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—61no—
214C3×D12—121no—
215C3×Dic6—62no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—61no—
220C3×Dic6—121no—
221C3⋊(C3⋊C8)—21no—
222C3⋊(C3⋊C8)—21no—
223C3⋊(C3⋊C8)—21no—
224C3⋊(C3⋊C8)#6711yescharacteristic
225C3⋊(C3⋊C8)#6711yescharacteristic
226C3×C3⋊C8—31no—
227C3×C3⋊C8—31no—
228C3×C3⋊C8—31no—
229C3×C3⋊C8—31no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—62no—
233C3×C3⋊C8—62no—
234C3×C3⋊C8—62no—
235C3×C3⋊C8—61no—
236C3×C3⋊C8—61no—
237C3×C3⋊C8—61no—
238C3×C3⋊C8—61no—
239C3×C3⋊C8—61no—
240C3×C3⋊C8—61no—
241#471—271noSylow
242C2×C33—21no—
243C2×C33—41no—
244C2×C33#12111yescharacteristic
245C3×C3⋊D3—81no—
246C3×#41—21no—
247C3×#41—21no—
248C3×#41—21no—
249C3×#41—41no—
250C3×#41D3211yescharacteristic
251C3×#41D3211yescharacteristic
252C3×#41D3211yescharacteristic
253C2×C3×Q8—91no—
254C2×C3×D4—91no—
255C2×D12—31no—
256C2×D12—31no—
257C2×D12—61no—
258C2×Dic6—31no—
259C2×Dic6—31no—
260C2×Dic6—61no—
261C3×Q16—181no—
262C3×SD16—181no—
263C3×SD16—181no—
264C3×D8—181no—
265C3×M16—91no—
266C3×M16—91no—
267C3×M16—91no—
268C3×M16—181no—
269C2×C3×C8—91no—
270C2×C3×C8—181no—
271#216—62no—
272#216—61no—
273#216—61no—
274#215—62no—
275#215—61no—
276#215—61no—
277#215—181no—
278#215—181no—
279#214—62no—
280#214—61no—
281#214—61no—
282#214—181no—
283#214—181no—
284#213—62no—
285#213—61no—
286#213—61no—
287#208—31no—
288#208—31no—
289#208—31no—
290#208—31no—
291#208—31no—
292#208—31no—
293#208—61no—
294#208—61no—
295#208—61no—
296#208—61no—
297#208—61no—
298#208—62no—
299C2×C3⋊C8—31no—
300C2×C3⋊C8—31no—
301C2×C3⋊C8—61no—
302C2×C3⋊C8—61no—
303C2×C3⋊C8—61no—
304C2×C3⋊C8—62no—
305#204—181no—
306#204—181no—
307C22×C32—21no—
308C22×C32—22no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32—41no—
315C22×C32—41no—
316C22×C32—41no—
317C22×C32—42no—
318C22×C32#23511yescharacteristic
319C22×C32#23711yescharacteristic
320C22×C32#23711yescharacteristic
321C2×C3⋊D3—41no—
322C2×C3⋊D3—81no—
323C2×C3×D3—122no—
324C2×C3×D3—121no—
325C2×C3×D3—121no—
326C2×C3×D3—242no—
327C2×C3×D3—241no—
328C2×C3×D3—241no—
329C32×C4—21no—
330C32×C4—22no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—41no—
340C32×C4—41no—
341C32×C4—41no—
342C32×C4—41no—
343C32×C4—42no—
344C32×C4C2×D1211yescharacteristic
345C32×C4D3×D411yescharacteristic
346C32×C4D3×D411yescharacteristic
347C32×C4C2×#6711yescharacteristic
348C32×C4C2×#6711yescharacteristic
349C32×C4C2×#6711yescharacteristic
350C3⋊Dic3—41no—
351C3×Dic3—122no—
352C3×Dic3—121no—
353C3×Dic3—121no—
354#137—91no—
355AGL(1,ℤ/8)—91no—
356C2×SD16—271no—
357#131—91no—
358#108—271no—
359#101—271no—
360#100—271no—
361C33#47111yescharacteristic, complemented, Sylow
362C23×C3—181no—
363C22×D3—61no—
364C22×D3—61no—
365C22×D3—121no—
366C3×Q8—21no—
367C3×Q8—21no—
368C3×Q8—21no—
369C3×Q8—41no—
370C3×Q8—181no—
371C3×Q8—181no—
372C3×Q8#62511yescharacteristic
373C3×Q8#62611yescharacteristic
374C3×Q8#62611yescharacteristic
375C3×D4—22no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—21no—
379C3×D4—21no—
380C3×D4—21no—
381C3×D4—21no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—42no—
385C3×D4—41no—
386C3×D4—181no—
387C3×D4—181no—
388C3×D4#62511yescharacteristic
389C3×D4#62611yescharacteristic
390C3×D4#62611yescharacteristic
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—181no—
403C2×C3×C4C2×D3211yescharacteristic
404C2×C3×C4C2×D3211yescharacteristic
405C2×C3×C4C2×D3211yescharacteristic
406C2×Dic3—61no—
407C2×Dic3—61no—
408C2×Dic3—121no—
409D12—62no—
410D12—61no—
411D12—61no—
412D12—61no—
413D12—61no—
414D12—121no—
415Dic6—62no—
416Dic6—61no—
417Dic6—61no—
418Dic6—61no—
419Dic6—61no—
420Dic6—121no—
421C3×C8—91no—
422C3×C8—91no—
423C3×C8—181no—
424C3×C8—181no—
425C3×C8—181no—
426C3⋊C8—31no—
427C3⋊C8—31no—
428C3⋊C8—31no—
429C3⋊C8—31no—
430C3⋊C8—61no—
431C3⋊C8—61no—
432C3⋊C8—62no—
433C3⋊C8—62no—
434C3⋊C8—62no—
435C3⋊C8—61no—
436C3⋊C8—61no—
437C3⋊C8—61no—
438C3⋊C8—61no—
439C3⋊C8—61no—
440C3⋊C8—61no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32—41no—
450C2×C32—41no—
451C2×C32—41no—
452C2×C32—42no—
453C2×C32—41no—
454C2×C32—41no—
455C2×C32—81no—
456C2×C32—81no—
457C2×C32—81no—
458C2×C32—82no—
459C2×C32#93011yescharacteristic
460C2×C32#93911yescharacteristic
461C2×C32#93911yescharacteristic
462C3⋊D3—81no—
463C3×D3—242no—
464C3×D3—241no—
465C3×D3—241no—
466#41#111311yescharacteristic
467C2×Q8—91no—
468C2×D4—91no—
469Q16—181no—
470SD16—181no—
471SD16—181no—
472SD16—541no—
473SD16—541no—
474D8—181no—
475M16—91no—
476M16—91no—
477M16—91no—
478M16—181no—
479C2×C8—91no—
480C2×C8—181no—
481C22×C3—22no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—42no—
491C22×C3—41no—
492C22×C3—361no—
493C22×C3—721no—
494C22×C3#380411yescharacteristic
495C22×C3#380411yescharacteristic
496C22×C3#380511yescharacteristic
497C2×D3—122no—
498C2×D3—121no—
499C2×D3—121no—
500C2×D3—242no—
501C2×D3—241no—
502C2×D3—241no—
503C3×C4—22no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—361no—
519C3×C4C2×#62511yescharacteristic
520C3×C4C2×#62611yescharacteristic
521C3×C4C2×#62611yescharacteristic
522C3×C4D3×#6711yescharacteristic
523C3×C4D3×#6711yescharacteristic
524C3×C4#381111yescharacteristic
525Dic3—122no—
526Dic3—121no—
527Dic3—121no—
528C32—21no—
529C32—21no—
530C32—21no—
531C32—41no—
532C32#514111yescharacteristic, complemented
533C32#520011yescharacteristic, complemented
534C32#520311yescharacteristic, complemented
535C23—181no—
536Q8—181no—
537Q8—181no—
538Q8#629511yescharacteristic
539D4—21no—
540D4—181no—
541D4—181no—
542D4#629511yescharacteristic
543C2×C4—21no—
544C2×C4—181no—
545C2×C4C2×#111311yescharacteristic
546C8—91no—
547C8—91no—
548C8—181no—
549C8—181no—
550C8—181no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—41no—
558C2×C3—41no—
559C2×C3—41no—
560C2×C3—42no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—81no—
566C2×C3—81no—
567C2×C3—81no—
568C2×C3—82no—
569C2×C3—721no—
570C2×C3#6397811yescharacteristic
571C2×C3#6397811yescharacteristic
572C2×C3#6398011yescharacteristic
573D3—242no—
574D3—241no—
575D3—241no—
576C22—21no—
577C22—361no—
578C22—721no—
579C22#8904611yescharacteristic
580C4—21no—
581C4—361no—
582C4C2×#629511yescharacteristic
583C4#8904911yescharacteristic
584C3—21no—
585C3—21no—
586C3—21no—
587C3—41no—
588C3#1059167911yescharacteristic, complemented
589C3#1059168011yescharacteristic, complemented
590C3#1059168111yescharacteristic, complemented
591C2—21no—
592C2—41no—
593C2—721no—
594C2#1172371711yescharacteristic
subgroupG/H
centerC2#11723717
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×#41C22
socleC2×C33#121
prime radicalC3×#64158C2
hypercenter#41#1113
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#64160C2semidirect
3. C3×#64157C2semidirect
4. C2×#88811C2semidirect
5. #11722081C2semidirect
6. #11722074C2semidirect
7. #11722073C2semidirect
26. #64160D3semidirect
28. #64157D3semidirect
361. C33#471semidirect
532. C32#5141semidirect
533. C32#5200semidirect
534. C32#5203semidirect
588. C3#10591679semidirect
589. C3#10591680semidirect
590. C3#10591681semidirect

Maximal subgroups

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

subgroupindex
C3×#641602
C3×#641582
C3×#641572
C2×#888112
#117220812
#117220742
#117220732
#105916813
#105916803
#105916793