Finite Group Databasebeta

← #13162038

Subgroups — #13162038

Order 1728.

632 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64159C211yesmaximal, characteristic
2C3×#64157C211yesmaximal, characteristic, complemented
3#11723657C211yesmaximal, characteristic, complemented
4C3×#63664C211yesmaximal, characteristic, complemented
5#11722081C211yesmaximal, characteristic, complemented
6#11722077C211yesmaximal, characteristic, complemented
7#11722070C211yesmaximal, characteristic, complemented
8#10591675—31nomaximal
9#10591673—31nomaximal
10#10591662—31nomaximal
11C33×#41C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C3×#3756C411yescharacteristic, complemented
14C3×#3755C411yescharacteristic, complemented
15C3×#3754C411yescharacteristic, complemented
16C3×#3753C411yescharacteristic, complemented
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C3×#3748—21no—
19C3×#3748C2211yescharacteristic
20C3×#3748C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)—21no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24#88809—21no—
25#88808—21no—
26#64159D311yescharacteristic
27#64157D311yescharacteristic, complemented
28C3×#950—31no—
29C3×#950—31no—
30C3×#950—61no—
31C3×#948—31no—
32C3×#948—31no—
33C3×#948—61no—
34#63817—31no—
35#63815—31no—
36#63814—31no—
37#63664D311yescharacteristic, complemented
38C3×#837—31no—
39C3×#837—31no—
40C3×#837—61no—
41#63578—31no—
42#63577—31no—
43#63577—31no—
44#63570—31no—
45#63569—31no—
46#63568—31no—
47#63553—31no—
48Dic3×C3⋊C8—31no—
49Dic3×C3⋊C8—31no—
50C33×Q8C2×C411yescharacteristic
51C33×D4—21no—
52C33×D4C2×C411yescharacteristic
53C2×C33×C4—21no—
54C2×C33×C4C2311yescharacteristic
55C3×#638—21no—
56C2×C3×C3⋊Dic3—21no—
57C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
58C3×C4×C3⋊D3—21no—
59C3×#634C2×C411yescharacteristic
60C3×C3⋊(C3⋊C8)—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
63C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
64C3×C3⋊(C3⋊C8)D411yescharacteristic
65C3×C3⋊(C3⋊C8)D411yescharacteristic
66#5196—91no—
67#5144—91no—
68#4824—91no—
69C32×#41—21no—
70C32×#41—21no—
71C32×#41—21no—
72C32×#41—41no—
73C32×#41C2×D311yescharacteristic
74C32×#41C2×D311yescharacteristic
75C32×#41C2×D311yescharacteristic
76#3828C2×D311yescharacteristic
77C3×#235—31no—
78C3×#235—31no—
79C3×#235—61no—
80#3756Dic311yescharacteristic, complemented
81#3755Dic311yescharacteristic, complemented
82#3754Dic311yescharacteristic, complemented
83#3753Dic311yescharacteristic, complemented
84C4×C3⋊Dic3C2×D311yescharacteristic
85#3748—21no—
86#3748C2×D311yescharacteristic
87#3748C2×D311yescharacteristic
88C2×C3⋊(C3⋊C8)—21no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C3×#216—31no—
92C3×#216—31no—
93C3×#216—61no—
94C3×#215—31no—
95C3×#215—31no—
96C3×#215—61no—
97C3×#214—31no—
98C3×#214—31no—
99C3×#214—61no—
100C3×#213—31no—
101C3×#213—31no—
102C3×#213—61no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—61no—
106C3×#208—31no—
107C3×#208—31no—
108C3×#208—31no—
109C3×#208—31no—
110C3×#208—61no—
111C3×#208—61no—
112C3×#208—61no—
113C3×#208—62no—
114C3×#208—61no—
115C2×C3×C3⋊C8—31no—
116C2×C3×C3⋊C8—31no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—31no—
119C2×C3×C3⋊C8—61no—
120C2×C3×C3⋊C8—61no—
121C2×C3×C3⋊C8—61no—
122C2×C3×C3⋊C8—61no—
123C2×C3×C3⋊C8—62no—
124#3712—61no—
125#3711—61no—
126#3711—61no—
127#3710—61no—
128D3×C3⋊C8—61no—
129D3×C3⋊C8—61no—
130C22×C33—21no—
131C22×C33#4111yescharacteristic
132C2×C3×C3⋊D3—21no—
133C33×C4—21no—
134C33×C4C22×C411yescharacteristic
135C33×C4C2×D411yescharacteristic
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3—21no—
138C3×#135—91no—
139C3×#131—91no—
140#950—31no—
141#950—31no—
142#950—61no—
143#948—31no—
144#948—31no—
145#948—61no—
146#948—91no—
147#907—91no—
148#901—91no—
149C3×#104—91no—
150#837—31no—
151#837—31no—
152#837—61no—
153#837—91no—
154#825—91no—
155#822—91no—
156#822—91no—
157#819—91no—
158C8×Dic3—91no—
159C8×Dic3—91no—
160#805—91no—
161C4×C3⋊C8—91no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×Q8—21no—
165C32×Q8—41no—
166C32×Q8C4×D311yescharacteristic
167C32×Q8C4×D311yescharacteristic
168C32×Q8C2×Dic311yescharacteristic
169C32×D4—21no—
170C32×D4—22no—
171C32×D4—21no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—41no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—42no—
180C32×D4C4×D311yescharacteristic
181C32×D4C4×D311yescharacteristic
182C32×D4C2×Dic311yescharacteristic
183C2×C32×C4—21no—
184C2×C32×C4—22no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—41no—
191C2×C32×C4—41no—
192C2×C32×C4—41no—
193C2×C32×C4—42no—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197#638—21no—
198C2×C3⋊Dic3—21no—
199C4⋊(C3⋊D3)C2×Dic311yescharacteristic
200C4×C3⋊D3—21no—
201#634C2×Dic311yescharacteristic
202C3×#67—61no—
203C3×#67—61no—
204C3×#67—121no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—121no—
208C3×D12—31no—
209C3×D12—31no—
210C3×D12—61no—
211C3×C4×D3—61no—
212C3×C4×D3—61no—
213C3×C4×D3—121no—
214C3×Dic6—31no—
215C3×Dic6—31no—
216C3×Dic6—61no—
217C3⋊(C3⋊C8)—21no—
218C3⋊(C3⋊C8)—21no—
219C3⋊(C3⋊C8)C2×Dic311yescharacteristic
220C3⋊(C3⋊C8)C2×Dic311yescharacteristic
221C3⋊(C3⋊C8)#6711yescharacteristic
222C3⋊(C3⋊C8)#6711yescharacteristic
223C3×C3⋊C8—31no—
224C3×C3⋊C8—31no—
225C3×C3⋊C8—31no—
226C3×C3⋊C8—31no—
227C3×C3⋊C8—31no—
228C3×C3⋊C8—31no—
229C3×C3⋊C8—31no—
230C3×C3⋊C8—31no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—61no—
235C3×C3⋊C8—62no—
236C3×C3⋊C8—61no—
237C3×C3⋊C8—61no—
238C3×C3⋊C8—61no—
239C3×C3⋊C8—61no—
240C3×C3⋊C8—121no—
241#443—271noSylow
242C2×C33—21no—
243C2×C33—41no—
244C2×C33C4×D411yescharacteristic
245C3×C3⋊D3—41no—
246C3×#41—21no—
247C3×#41—21no—
248C3×#41—21no—
249C3×#41—41no—
250C3×#41—91no—
251C3×#41D3211yescharacteristic
252C3×#41D3211yescharacteristic
253C3×#41D3211yescharacteristic
254#235—31no—
255#235—31no—
256#235—61no—
257C3×Q16—91no—
258C3×SD16—91no—
259C3×SD16—91no—
260C3×D8—91no—
261C3×M16—91no—
262C3×M16—91no—
263C3×M16—181no—
264C2×C3×C8—91no—
265C2×C3×C8—91no—
266C2×C3×C8—181no—
267C3×C42—91no—
268#216—31no—
269#216—31no—
270#216—61no—
271#215—31no—
272#215—31no—
273#215—61no—
274#214—31no—
275#214—31no—
276#214—61no—
277#213—31no—
278#213—31no—
279#213—61no—
280C4×Dic3—31no—
281C4×Dic3—31no—
282C4×Dic3—61no—
283#208—31no—
284#208—31no—
285#208—31no—
286#208—31no—
287#208—61no—
288#208—61no—
289#208—61no—
290#208—61no—
291#208—62no—
292#208—181no—
293C2×C3⋊C8—31no—
294C2×C3⋊C8—31no—
295C2×C3⋊C8—31no—
296C2×C3⋊C8—31no—
297C2×C3⋊C8—61no—
298C2×C3⋊C8—61no—
299C2×C3⋊C8—61no—
300C2×C3⋊C8—61no—
301C2×C3⋊C8—62no—
302C2×C3⋊C8—181no—
303ΓL(1,𝔽25)—181no—
304ΓL(1,𝔽25)—181no—
305D3×C8—181no—
306D3×C8—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—21no—
322C2×C3×D3—61no—
323C2×C3×D3—61no—
324C2×C3×D3—121no—
325C32×C4—21no—
326C32×C4—22no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—41no—
336C32×C4—41no—
337C32×C4—41no—
338C32×C4—41no—
339C32×C4—42no—
340C32×C4C2×C4×D311yescharacteristic
341C32×C4C2×C4×D311yescharacteristic
342C32×C4D3×D411yescharacteristic
343C32×C4D3×D411yescharacteristic
344C32×C4C22×Dic311yescharacteristic
345C32×C4C2×#6711yescharacteristic
346C3⋊Dic3—22no—
347C3⋊Dic3—21no—
348C3×Dic3—62no—
349C3×Dic3—62no—
350C3×Dic3—61no—
351C3×Dic3—61no—
352C3×Dic3—121no—
353C3×Dic3—122no—
354#135—91no—
355#131—91no—
356#131—271no—
357#108—271no—
358#104—91no—
359#104—271no—
360C4×C8—271no—
361C33#44311yescharacteristic, complemented, Sylow
362C3×Q8—21no—
363C3×Q8—21no—
364C3×Q8—21no—
365C3×Q8—41no—
366C3×Q8—91no—
367C3×Q8D3×Dic311yescharacteristic
368C3×Q8D3×Dic311yescharacteristic
369C3×Q8#62411yescharacteristic
370C3×D4—22no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—21no—
374C3×D4—21no—
375C3×D4—21no—
376C3×D4—21no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—42no—
380C3×D4—41no—
381C3×D4—91no—
382C3×D4—181no—
383C3×D4D3×Dic311yescharacteristic
384C3×D4D3×Dic311yescharacteristic
385C3×D4#62411yescharacteristic
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—42no—
396C2×C3×C4—41no—
397C2×C3×C4—181no—
398C2×C3×C4—181no—
399C2×C3×C4C2×D3211yescharacteristic
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402#67—61no—
403#67—61no—
404#67—121no—
405C2×Dic3—61no—
406C2×Dic3—61no—
407C2×Dic3—121no—
408D12—31no—
409D12—31no—
410D12—61no—
411C4×D3—61no—
412C4×D3—61no—
413C4×D3—121no—
414Dic6—31no—
415Dic6—31no—
416Dic6—61no—
417C3×C8—91no—
418C3×C8—91no—
419C3×C8—91no—
420C3×C8—91no—
421C3×C8—181no—
422C3×C8—181no—
423C3⋊C8—31no—
424C3⋊C8—31no—
425C3⋊C8—31no—
426C3⋊C8—31no—
427C3⋊C8—31no—
428C3⋊C8—31no—
429C3⋊C8—31no—
430C3⋊C8—31no—
431C3⋊C8—61no—
432C3⋊C8—61no—
433C3⋊C8—61no—
434C3⋊C8—61no—
435C3⋊C8—62no—
436C3⋊C8—61no—
437C3⋊C8—61no—
438C3⋊C8—61no—
439C3⋊C8—61no—
440C3⋊C8—121no—
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#67⋊C411yescharacteristic
460C2×C32C4×#6711yescharacteristic
461C2×C32D4×Dic311yescharacteristic
462C3⋊D3—41no—
463C3×D3—121no—
464C3×D3—121no—
465C3×D3—241no—
466#41—91no—
467#41#111311yescharacteristic
468Q16—91no—
469SD16—91no—
470SD16—91no—
471D8—91no—
472M16—91no—
473M16—91no—
474M16—181no—
475M16—541no—
476C2×C8—91no—
477C2×C8—91no—
478C2×C8—181no—
479C2×C8—541no—
480C42—91no—
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—181no—
493C22×C3#380411yescharacteristic
494C22×C3#380411yescharacteristic
495C22×C3#380511yescharacteristic
496C2×D3—61no—
497C2×D3—61no—
498C2×D3—121no—
499C3×C4—22no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—42no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—182no—
515C3×C4—181no—
516C3×C4C2×D3×Dic311yescharacteristic
517C3×C4C2×D3×Dic311yescharacteristic
518C3×C4C2×#62411yescharacteristic
519C3×C4D3×#6711yescharacteristic
520C3×C4D3×#6711yescharacteristic
521C3×C4#381111yescharacteristic
522Dic3—62no—
523Dic3—62no—
524Dic3—61no—
525Dic3—61no—
526Dic3—121no—
527Dic3—122no—
528C32—21no—
529C32—21no—
530C32—21no—
531C32—41no—
532C32#482411yescharacteristic, complemented
533C32#514411yescharacteristic, complemented
534C32#519611yescharacteristic, complemented
535Q8—91no—
536Q8#629411yescharacteristic
537D4—21no—
538D4—91no—
539D4—181no—
540D4#629411yescharacteristic
541C2×C4—21no—
542C2×C4—181no—
543C2×C4—181no—
544C2×C4C2×#111311yescharacteristic
545C8—91no—
546C8—91no—
547C8—91no—
548C8—91no—
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—361no—
570C2×C3#6395211yescharacteristic
571C2×C3Dic3×#6711yescharacteristic
572C2×C3#67⋊Dic311yescharacteristic
573D3—121no—
574D3—121no—
575D3—241no—
576C22—21no—
577C22—181no—
578C22#8904611yescharacteristic
579C4—21no—
580C4—182no—
581C4—181no—
582C4C2×#629411yescharacteristic
583C4#8904911yescharacteristic
584C3—21no—
585C3—21no—
586C3—21no—
587C3—41no—
588C3#1059166211yescharacteristic, complemented
589C3#1059167311yescharacteristic, complemented
590C3#1059167511yescharacteristic, complemented
591C2—21no—
592C2—41no—
593C2—361no—
594C2#1172372611yescharacteristic
subgroupG/H
centerC4#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC33×#41C22
socleC2×C33C4×D4
prime radicalC3×#64159C2
cosocleC2×C33×C4C23
hypercenter#41#1113
hypocenterC33#443
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C3×#64157C2semidirect
3. #11723657C2semidirect
4. C3×#63664C2semidirect
5. #11722081C2semidirect
6. #11722077C2semidirect
7. #11722070C2semidirect
13. C3×#3756C4semidirect
14. C3×#3755C4semidirect
15. C3×#3754C4semidirect
16. C3×#3753C4semidirect
27. #64157D3semidirect
37. #63664D3semidirect
80. #3756Dic3semidirect
81. #3755Dic3semidirect
82. #3754Dic3semidirect
83. #3753Dic3semidirect
361. C33#443semidirect
532. C32#4824semidirect
533. C32#5144semidirect
534. C32#5196semidirect
588. C3#10591662semidirect
589. C3#10591673semidirect
590. C3#10591675semidirect

Maximal subgroups

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

subgroupindex
C3×#641592
C3×#641572
#117236572
C3×#636642
#117220812
#117220772
#117220702
#105916753
#105916733
#105916623