Finite Group Databasebeta

← #13155752

Subgroups — #13155752

Order 1728.

624 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
1#11723377C211yesmaximal, characteristic
2#11723356C211yesmaximal, characteristic, complemented
3C3×#63814C211yesmaximal, characteristic, complemented
4#11722081C211yesmaximal, characteristic, complemented
5#11722048C211yesmaximal, characteristic, complemented
6C3×#63569C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10591663—31nomaximal
9#10590924—31nomaximal
10#10590342—31nomaximal
11#89076C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C3×#3748C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88799C411yescharacteristic, complemented
19#88795C411yescharacteristic, complemented
20#88794C411yescharacteristic, complemented
21#88790C411yescharacteristic, complemented
22#88788—21no—
23#88785—21no—
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26#64159—31no—
27#64110—31no—
28C3×#948—31no—
29C3×#901—31no—
30#63829—31no—
31#63829—31no—
32#63829—61no—
33#63817—31no—
34#63817—31no—
35#63817—61no—
36#63814D311yescharacteristic, complemented
37#63632—31no—
38C3×#837—31no—
39C3×#815—31no—
40C3×#805—31no—
41C3×#803—31no—
42#63578—31no—
43#63577—31no—
44#63577—31no—
45#63570—31no—
46#63570—31no—
47#63570—61no—
48#63569D311yescharacteristic, complemented
49#63554D311yescharacteristic, complemented
50C2×C33×C4C2311yescharacteristic
51#6312—21no—
52C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
53C4×C3⋊(C3⋊D3)—21no—
54#6308C2×C411yescharacteristic
55C32×#67—21no—
56C2×C32×Dic3—21no—
57C32×D12C2×C411yescharacteristic
58C32×C4×D3—21no—
59C32×Dic6C2×C411yescharacteristic
60C3×C3⋊(C3⋊C8)—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
63C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66#5145—91no—
67#4823—91no—
68#4717—91no—
69C32×#41—31no—
70#3828—31no—
71#3828—31no—
72#3828—31no—
73#3828—61no—
74#3828—61no—
75#3828—61no—
76#3828—121no—
77C3×#235—21no—
78C3×#235C2×D311yescharacteristic
79C3×#235C2×D311yescharacteristic
80C2×C32×C8—31no—
81C32×C42—31no—
82#3756—31no—
83#3755—31no—
84#3754—31no—
85#3753—31no—
86#3748C2×D311yescharacteristic
87#3748C2×D311yescharacteristic
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89#3743—61no—
90C8×C3⋊D3—61no—
91C3×C4×Dic3—21no—
92C3×C4×Dic3C2×D311yescharacteristic
93C3×C4×Dic3C2×D311yescharacteristic
94C3×#208—31no—
95C3×#208—31no—
96C3×#208—31no—
97C3×#208—31no—
98C3×#208—61no—
99C3×#208—61no—
100C3×#208—61no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C3×ΓL(1,𝔽25)—61no—
109C3×D3×C8—61no—
110#3719—31no—
111#3719—31no—
112#3719—61no—
113#3717—31no—
114#3717—31no—
115#3717—61no—
116#3716—31no—
117#3716—31no—
118#3716—61no—
119#3714—31no—
120#3714—31no—
121#3714—61no—
122#3712—62no—
123#3712—61no—
124#3712—61no—
125#3711—21no—
126#3710—62no—
127#3710—61no—
128#3710—61no—
129D3×C3⋊C8—21no—
130C22×C33#4111yescharacteristic
131C2×C3⋊(C3⋊D3)—21no—
132C2×C32×D3—21no—
133C33×C4C22×C411yescharacteristic
134C33×C4C2×D411yescharacteristic
135C3⋊(C3⋊Dic3)—21no—
136C32×Dic3—21no—
137C32×Dic3—41no—
138C3×#131—91no—
139#950—91no—
140#950—91no—
141#950—181no—
142#948—31no—
143#907—91no—
144#904—91no—
145#901—31no—
146#901—91no—
147#901—91no—
148#901—181no—
149C3×#104—91no—
150C3×#97—91no—
151#837—31no—
152#825—91no—
153#822—91no—
154#822—91no—
155#819—91no—
156#815—31no—
157#805—31no—
158#805—91no—
159#805—91no—
160#805—181no—
161#803—31no—
162C32×Q8—31no—
163C32×D4—31no—
164C32×D4—61no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—41no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174#638—61no—
175#638—62no—
176#638—61no—
177#638—61no—
178#638—121no—
179#638—121no—
180#638—122no—
181C4⋊(C3⋊D3)—31no—
182C4⋊(C3⋊D3)—31no—
183C4⋊(C3⋊D3)—31no—
184C4⋊(C3⋊D3)—61no—
185C4⋊(C3⋊D3)—61no—
186C4⋊(C3⋊D3)—61no—
187C4⋊(C3⋊D3)—121no—
188C4×C3⋊D3—61no—
189C4×C3⋊D3—62no—
190C4×C3⋊D3—61no—
191C4×C3⋊D3—61no—
192C4×C3⋊D3—121no—
193C4×C3⋊D3—121no—
194C4×C3⋊D3—122no—
195#634—31no—
196#634—31no—
197#634—31no—
198#634—61no—
199#634—61no—
200#634—61no—
201#634—121no—
202C3×#67—21no—
203C3×#67—21no—
204C3×#67—41no—
205C2×C3×Dic3—21no—
206C2×C3×Dic3—21no—
207C2×C3×Dic3—41no—
208C3×D12—21no—
209C3×D12C4×D311yescharacteristic
210C3×D12C4×D311yescharacteristic
211C3×C4×D3—21no—
212C3×C4×D3—21no—
213C3×C4×D3—41no—
214C3×Dic6—21no—
215C3×Dic6C4×D311yescharacteristic
216C3×Dic6C4×D311yescharacteristic
217C32×C8—31no—
218C32×C8—31no—
219C3⋊(C3⋊C8)—21no—
220C3⋊(C3⋊C8)—21no—
221C3⋊(C3⋊C8)C4×D311yescharacteristic
222C3⋊(C3⋊C8)C4×D311yescharacteristic
223C3×C3⋊C8—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—31no—
226C3×C3⋊C8—31no—
227C3×C3⋊C8—31no—
228C3×C3⋊C8—31no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—62no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—61no—
235C3×C3⋊C8—61no—
236C3×C3⋊C8—121no—
237C3×C3⋊C8D1211yescharacteristic
238C3×C3⋊C8D1211yescharacteristic
239C3×C3⋊C8#6711yescharacteristic
240C3×C3⋊C8#6711yescharacteristic
241#444—271noSylow
242C2×C33—21no—
243C2×C33C4×D411yescharacteristic
244C3⋊(C3⋊D3)—41no—
245C32×D3—41no—
246C3×#41—31no—
247C3×#41—31no—
248C3×#41—61no—
249#235—91no—
250#235—91no—
251#235—91no—
252#235—181no—
253#235—181no—
254#235—181no—
255#235—361no—
256#235D3211yescharacteristic
257C3×M16—91no—
258C3×M16—91no—
259C3×M16—181no—
260C2×C3×C8—31no—
261C2×C3×C8—31no—
262C2×C3×C8—61no—
263C2×C3×C8—91no—
264C2×C3×C8—181no—
265C3×C42—31no—
266C3×C42—31no—
267C3×C42—61no—
268#216—91no—
269#216—91no—
270#216—181no—
271#215—91no—
272#215—91no—
273#215—181no—
274#214—91no—
275#214—91no—
276#214—181no—
277#213—91no—
278#213—91no—
279#213—181no—
280C4×Dic3D3211yescharacteristic
281#208—31no—
282#208—31no—
283#208—31no—
284#208—31no—
285#208—61no—
286#208—61no—
287#208—61no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
291C2×C3⋊C8—61no—
292C2×C3⋊C8D3211yescharacteristic
293Dic12—91no—
294D24—91no—
295#204—91no—
296#204—91no—
297ΓL(1,𝔽25)—61no—
298ΓL(1,𝔽25)—181no—
299ΓL(1,𝔽25)—181no—
300ΓL(1,𝔽25)—182no—
301ΓL(1,𝔽25)—181no—
302D3×C8—61no—
303D3×C8—181no—
304D3×C8—181no—
305D3×C8—182no—
306D3×C8—181no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—41no—
311C22×C32—61no—
312C22×C32#23511yescharacteristic
313C22×C32#23511yescharacteristic
314C22×C32#23711yescharacteristic
315C2×C3⋊D3—61no—
316C2×C3⋊D3—62no—
317C2×C3⋊D3—61no—
318C2×C3⋊D3—61no—
319C2×C3⋊D3—121no—
320C2×C3⋊D3—121no—
321C2×C3⋊D3—122no—
322C2×C3×D3—21no—
323C2×C3×D3—21no—
324C2×C3×D3—41no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4—61no—
334C32×C4—121no—
335C32×C4C2×C4×D311yescharacteristic
336C32×C4C2×C4×D311yescharacteristic
337C32×C4C2×C4×D311yescharacteristic
338C32×C4C2×D1211yescharacteristic
339C32×C4D3×D411yescharacteristic
340C32×C4C2×#6711yescharacteristic
341C3⋊Dic3—61no—
342C3⋊Dic3—62no—
343C3⋊Dic3—61no—
344C3⋊Dic3—61no—
345C3⋊Dic3—121no—
346C3⋊Dic3—121no—
347C3⋊Dic3—122no—
348C3×Dic3—21no—
349C3×Dic3—21no—
350C3×Dic3—41no—
351C3×Dic3—41no—
352C3×Dic3—41no—
353C3×Dic3—42no—
354#135—271no—
355#131—91no—
356#131—271no—
357#108—271no—
358#104—91no—
359#104—271no—
360#97—91no—
361C33#44411yescharacteristic, complemented, Sylow
362C3×Q8—31no—
363C3×Q8—31no—
364C3×Q8—61no—
365C3×D4—31no—
366C3×D4—31no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—121no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—121no—
380C2×C3×C4—121no—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384#67—21no—
385#67—182no—
386#67—181no—
387#67—181no—
388#67—181no—
389#67—361no—
390#67—361no—
391#67—362no—
392C2×Dic3—21no—
393D12—91no—
394D12—91no—
395D12—91no—
396D12—181no—
397D12—181no—
398D12—181no—
399D12—361no—
400D12#62411yescharacteristic
401C4×D3—21no—
402C4×D3—182no—
403C4×D3—181no—
404C4×D3—181no—
405C4×D3—181no—
406C4×D3—361no—
407C4×D3—361no—
408C4×D3—362no—
409Dic6—91no—
410Dic6—91no—
411Dic6—91no—
412Dic6—181no—
413Dic6—181no—
414Dic6—181no—
415Dic6—361no—
416Dic6#62411yescharacteristic
417C3×C8—31no—
418C3×C8—31no—
419C3×C8—31no—
420C3×C8—31no—
421C3×C8—61no—
422C3×C8—61no—
423C3×C8—91no—
424C3×C8—91no—
425C3×C8—181no—
426C3×C8—181no—
427C3⋊C8—31no—
428C3⋊C8—31no—
429C3⋊C8—31no—
430C3⋊C8—31no—
431C3⋊C8—61no—
432C3⋊C8—61no—
433C3⋊C8—62no—
434C3⋊C8—61no—
435C3⋊C8—61no—
436C3⋊C8—61no—
437C3⋊C8—61no—
438C3⋊C8—121no—
439C3⋊C8#62611yescharacteristic
440C3⋊C8#62611yescharacteristic
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—81no—
452C2×C32—121no—
453C2×C32C4×D1211yescharacteristic
454C2×C32#67⋊C411yescharacteristic
455C2×C32C4×#6711yescharacteristic
456C3⋊D3—121no—
457C3⋊D3—122no—
458C3⋊D3—121no—
459C3⋊D3—121no—
460C3⋊D3—241no—
461C3⋊D3—241no—
462C3⋊D3—242no—
463C3×D3—41no—
464C3×D3—41no—
465C3×D3—81no—
466#41—31no—
467#41—271no—
468Q16—271no—
469SD16—271no—
470SD16—271no—
471D8—271no—
472M16—91no—
473M16—91no—
474M16—181no—
475M16—541no—
476C2×C8—31no—
477C2×C8—91no—
478C2×C8—181no—
479C2×C8—541no—
480C42—31no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—41no—
485C22×C3—61no—
486C22×C3—61no—
487C22×C3—121no—
488C22×C3#379811yescharacteristic
489C22×C3#380411yescharacteristic
490C22×C3#380411yescharacteristic
491C2×D3—21no—
492C2×D3—182no—
493C2×D3—181no—
494C2×D3—181no—
495C2×D3—181no—
496C2×D3—361no—
497C2×D3—361no—
498C2×D3—362no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—61no—
508C3×C4—61no—
509C3×C4—121no—
510C3×C4—121no—
511C3×C4—121no—
512C3×C4—122no—
513C3×C4C4×D3211yescharacteristic
514C3×C4C4×D3211yescharacteristic
515C3×C4D3×D1211yescharacteristic
516C3×C4C2×#62411yescharacteristic
517C3×C4C2×#62611yescharacteristic
518C3×C4D3×#6711yescharacteristic
519Dic3—21no—
520Dic3—41no—
521Dic3—182no—
522Dic3—181no—
523Dic3—181no—
524Dic3—181no—
525Dic3—361no—
526Dic3—361no—
527Dic3—362no—
528C32—21no—
529C32—21no—
530C32—21no—
531C32—41no—
532C32#471711yescharacteristic, complemented
533C32#482311yescharacteristic, complemented
534C32#514511yescharacteristic, complemented
535Q8—31no—
536Q8—271no—
537D4—31no—
538D4—61no—
539D4—271no—
540D4—541no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—541no—
544C2×C4D3311yescharacteristic
545C8—31no—
546C8—31no—
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—41no—
561C2×C3—81no—
562C2×C3—121no—
563C2×C3—121no—
564C2×C3—241no—
565C2×C3#6388011yescharacteristic
566C2×C3C4×#62611yescharacteristic
567C2×C3#6395211yescharacteristic
568D3—41no—
569D3—362no—
570D3—361no—
571D3—361no—
572D3—361no—
573D3—721no—
574D3—721no—
575D3—722no—
576C22—61no—
577C22—541no—
578C22#8895911yescharacteristic
579C4—61no—
580C4—121no—
581C4—541no—
582C4D3×#62411yescharacteristic
583C4D3×#62611yescharacteristic
584C3—21no—
585C3—21no—
586C3—21no—
587C3—41no—
588C3#1059034211yescharacteristic, complemented
589C3#1059092411yescharacteristic, complemented
590C3#1059166311yescharacteristic, complemented
591C2—21no—
592C2—121no—
593C2—1081no—
594C2#1172293211yescharacteristic
subgroupG/H
centerC4D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radical#11723377C2
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723356C2semidirect
3. C3×#63814C2semidirect
4. #11722081C2semidirect
5. #11722048C2semidirect
6. C3×#63569C2semidirect
7. C3×#63554C2semidirect
18. #88799C4semidirect
19. #88795C4semidirect
20. #88794C4semidirect
21. #88790C4semidirect
36. #63814D3semidirect
48. #63569D3semidirect
49. #63554D3semidirect
361. C33#444semidirect
532. C32#4717semidirect
533. C32#4823semidirect
534. C32#5145semidirect
588. C3#10590342semidirect
589. C3#10590924semidirect
590. C3#10591663semidirect

Maximal subgroups

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

subgroupindex
#117233772
#117233562
C3×#638142
#117220812
#117220482
C3×#635692
C3×#635542
#105916633
#105909243
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