Finite Group Databasebeta

← #13155753

Subgroups — #13155753

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#11723369C211yesmaximal, characteristic
2#11723356C211yesmaximal, characteristic, complemented
3C3×#63817C211yesmaximal, characteristic, complemented
4#11722077C211yesmaximal, characteristic, complemented
5#11722047C211yesmaximal, characteristic, complemented
6C3×#63577C211yesmaximal, characteristic, complemented
7C3×#63555C211yesmaximal, characteristic, complemented
8#10590994—31nomaximal
9#10590924—31nomaximal
10#10590918—31nomaximal
11#89076C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C32×#208C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88800C411yescharacteristic, complemented
19#88797C411yescharacteristic, complemented
20#88796C411yescharacteristic, complemented
21#88791C411yescharacteristic, complemented
22#88788—21no—
23#88785—21no—
24C3×#3712—21no—
25C3×#3710—21no—
26#64113—31no—
27#64110—31no—
28C3×#907—31no—
29C3×#901—31no—
30#63829—31no—
31#63829—31no—
32#63829—61no—
33#63817—31no—
34#63817—31no—
35#63817—61no—
36#63817D311yescharacteristic, complemented
37#63652—31no—
38C3×#822—31no—
39C3×#819—31no—
40C3×#815—31no—
41C3×#815—31no—
42#63577D311yescharacteristic, complemented
43#63570—31no—
44#63570—31no—
45#63570—31no—
46#63570—61no—
47#63569—31no—
48#63568—31no—
49#63555D311yescharacteristic, complemented
50C2×C33×C4C2311yescharacteristic
51#6312—21no—
52C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
53C4×C3⋊(C3⋊D3)—21no—
54#6308C2×C411yescharacteristic
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—
61C32×C3⋊C8—21no—
62C32×C3⋊C8C2×C411yescharacteristic
63C32×C3⋊C8C2×C411yescharacteristic
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66#5145—91no—
67#4894—91no—
68#4717—91no—
69#3828—31no—
70#3828—31no—
71#3828—31no—
72#3828—61no—
73#3828—61no—
74#3828—61no—
75#3828—121no—
76#3828C2×D311yescharacteristic
77C3×#235—31no—
78C3×#235—31no—
79C3×#235—61no—
80C32×M16—31no—
81C2×C32×C8—31no—
82C2×C32×C8—31no—
83C4×C3⋊Dic3C2×D311yescharacteristic
84#3748C2×D311yescharacteristic
85#3746—31no—
86#3745—31no—
87#3744—31no—
88#3744—31no—
89#3743—61no—
90C8×C3⋊D3—61no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—61no—
94C3×#208—21no—
95C3×#208—31no—
96C3×#208—31no—
97C3×#208—61no—
98C3×#208C2×D311yescharacteristic
99C3×#208C2×D311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C3×ΓL(1,𝔽25)—61no—
107C3×ΓL(1,𝔽25)—61no—
108C3×D3×C8—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—21no—
123#3712—62no—
124#3712—61no—
125#3712—61no—
126#3710—21no—
127#3710—62no—
128#3710—61no—
129#3710—61no—
130C22×C33#4111yescharacteristic
131C2×C3⋊(C3⋊D3)—21no—
132C2×C3×C3⋊D3—21no—
133C33×C4C22×C411yescharacteristic
134C33×C4C2×D411yescharacteristic
135C3⋊(C3⋊Dic3)—21no—
136C3×C3⋊Dic3—21no—
137C3×C3⋊Dic3—41no—
138C3×#131—91no—
139#950—91no—
140#907—31no—
141#907—91no—
142#904—91no—
143#904—91no—
144#904—181no—
145#901—31no—
146#901—91no—
147#901—91no—
148#901—181no—
149C3×#108—91no—
150C3×#97—91no—
151#837—91no—
152#825—91no—
153#825—91no—
154#825—91no—
155#825—181no—
156#822—31no—
157#819—31no—
158#815—31no—
159#815—31no—
160#805—91no—
161#805—91no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169#638—21no—
170#638—61no—
171#638—62no—
172#638—61no—
173#638—61no—
174#638—121no—
175#638—121no—
176#638—122no—
177C2×C3⋊Dic3—21no—
178C4⋊(C3⋊D3)—31no—
179C4⋊(C3⋊D3)—31no—
180C4⋊(C3⋊D3)—31no—
181C4⋊(C3⋊D3)—61no—
182C4⋊(C3⋊D3)—61no—
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)—121no—
185C4⋊(C3⋊D3)C4×D311yescharacteristic
186C4×C3⋊D3—21no—
187C4×C3⋊D3—61no—
188C4×C3⋊D3—62no—
189C4×C3⋊D3—61no—
190C4×C3⋊D3—61no—
191C4×C3⋊D3—121no—
192C4×C3⋊D3—121no—
193C4×C3⋊D3—122no—
194#634—31no—
195#634—31no—
196#634—31no—
197#634—61no—
198#634—61no—
199#634—61no—
200#634—121no—
201#634C4×D311yescharacteristic
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—
217C32×C8—31no—
218C32×C8—31no—
219C32×C8—31no—
220C32×C8—31no—
221C32×C8—61no—
222C3⋊(C3⋊C8)—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—21no—
229C3×C3⋊C8—41no—
230C3×C3⋊C8—62no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8C4×D311yescharacteristic
234C3×C3⋊C8C4×D311yescharacteristic
235C3×C3⋊C8C4×D311yescharacteristic
236C3×C3⋊C8C4×D311yescharacteristic
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—
245C3×C3⋊D3—41no—
246C3×#41—91no—
247#235—31no—
248#235—31no—
249#235—61no—
250#235—91no—
251#235—91no—
252#235—91no—
253#235—181no—
254#235—181no—
255#235—181no—
256#235—361no—
257C3×M16—31no—
258C3×M16—31no—
259C3×M16—61no—
260C3×M16—91no—
261C3×M16—181no—
262C2×C3×C8—31no—
263C2×C3×C8—31no—
264C2×C3×C8—31no—
265C2×C3×C8—31no—
266C2×C3×C8—61no—
267C2×C3×C8—61no—
268C2×C3×C8—181no—
269C3×C42—91no—
270#216—91no—
271#215—91no—
272#214—91no—
273#213—91no—
274C4×Dic3—31no—
275C4×Dic3—31no—
276C4×Dic3—61no—
277#208—31no—
278#208—31no—
279#208—61no—
280#208D3211yescharacteristic
281C2×C3⋊C8D3211yescharacteristic
282C2×C3⋊C8D3211yescharacteristic
283Dic12—91no—
284Dic12—91no—
285Dic12—181no—
286D24—91no—
287D24—91no—
288D24—181no—
289#204—91no—
290#204—91no—
291#204—91no—
292#204—91no—
293#204—181no—
294#204—181no—
295ΓL(1,𝔽25)—61no—
296ΓL(1,𝔽25)—61no—
297ΓL(1,𝔽25)—181no—
298ΓL(1,𝔽25)—181no—
299ΓL(1,𝔽25)—182no—
300ΓL(1,𝔽25)—181no—
301D3×C8—61no—
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#23511yescharacteristic
312C22×C32#23511yescharacteristic
313C22×C32#23911yescharacteristic
314C2×C3⋊D3—21no—
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—61no—
323C2×C3×D3—61no—
324C2×C3×D3—121no—
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×C4C2×C4×D311yescharacteristic
334C32×C4C2×C4×D311yescharacteristic
335C32×C4C2×C4×D311yescharacteristic
336C32×C4C2×D1211yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C3⋊Dic3—21no—
340C3⋊Dic3—41no—
341C3⋊Dic3—61no—
342C3⋊Dic3—62no—
343C3⋊Dic3—61no—
344C3⋊Dic3—61no—
345C3⋊Dic3—121no—
346C3⋊Dic3—121no—
347C3⋊Dic3—122no—
348C3×Dic3—61no—
349C3×Dic3—61no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C3×Dic3—121no—
353C3×Dic3—122no—
354#135—271no—
355#131—91no—
356#131—271no—
357#108—91no—
358#104—271no—
359#104—271no—
360#97—91no—
361C33#44411yescharacteristic, complemented, Sylow
362C3×Q8—91no—
363C3×D4—91no—
364C3×D4—181no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—41no—
369C2×C3×C4—181no—
370C2×C3×C4—181no—
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3211yescharacteristic
373C2×C3×C4C2×D3211yescharacteristic
374#67—61no—
375#67—61no—
376#67—121no—
377#67—182no—
378#67—181no—
379#67—181no—
380#67—181no—
381#67—361no—
382#67—361no—
383#67—362no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—121no—
387D12—31no—
388D12—31no—
389D12—61no—
390D12—91no—
391D12—91no—
392D12—91no—
393D12—181no—
394D12—181no—
395D12—181no—
396D12—361no—
397C4×D3—61no—
398C4×D3—61no—
399C4×D3—121no—
400C4×D3—182no—
401C4×D3—181no—
402C4×D3—181no—
403C4×D3—181no—
404C4×D3—361no—
405C4×D3—361no—
406C4×D3—362no—
407Dic6—31no—
408Dic6—31no—
409Dic6—61no—
410Dic6—91no—
411Dic6—91no—
412Dic6—91no—
413Dic6—181no—
414Dic6—181no—
415Dic6—181no—
416Dic6—361no—
417C3×C8—31no—
418C3×C8—31no—
419C3×C8—31no—
420C3×C8—31no—
421C3×C8—31no—
422C3×C8—31no—
423C3×C8—31no—
424C3×C8—31no—
425C3×C8—61no—
426C3×C8—61no—
427C3×C8—61no—
428C3×C8—61no—
429C3×C8—61no—
430C3×C8—61no—
431C3×C8—121no—
432C3×C8—181no—
433C3⋊C8—21no—
434C3⋊C8—62no—
435C3⋊C8—61no—
436C3⋊C8—61no—
437C3⋊C8#62411yescharacteristic
438C3⋊C8#62411yescharacteristic
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×C32C4×D1211yescharacteristic
453C2×C32#89311yescharacteristic
454C2×C32C4×#6711yescharacteristic
455C3⋊D3—41no—
456C3⋊D3—121no—
457C3⋊D3—122no—
458C3⋊D3—121no—
459C3⋊D3—121no—
460C3⋊D3—241no—
461C3⋊D3—241no—
462C3⋊D3—242no—
463C3×D3—121no—
464C3×D3—121no—
465C3×D3—241no—
466#41—91no—
467#41—271no—
468Q16—271no—
469SD16—271no—
470SD16—271no—
471D8—271no—
472M16—31no—
473M16—91no—
474M16—181no—
475M16—541no—
476C2×C8—31no—
477C2×C8—31no—
478C2×C8—181no—
479C2×C8—541no—
480C42—91no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—41no—
485C22×C3—181no—
486C22×C3#379811yescharacteristic
487C22×C3#379911yescharacteristic
488C22×C3#379911yescharacteristic
489C2×D3—61no—
490C2×D3—61no—
491C2×D3—121no—
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—181no—
508C3×C4—361no—
509C3×C4C4×D3211yescharacteristic
510C3×C4C4×D3211yescharacteristic
511C3×C4D3×D1211yescharacteristic
512C3×C4C2×#62411yescharacteristic
513C3×C4C2×#62611yescharacteristic
514C3×C4D3×#6711yescharacteristic
515Dic3—61no—
516Dic3—61no—
517Dic3—121no—
518Dic3—121no—
519Dic3—121no—
520Dic3—122no—
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#489411yescharacteristic, complemented
534C32#514511yescharacteristic, complemented
535Q8—91no—
536Q8—271no—
537D4—91no—
538D4—181no—
539D4—271no—
540D4—541no—
541C2×C4—181no—
542C2×C4—181no—
543C2×C4—541no—
544C2×C4D3311yescharacteristic
545C8—31no—
546C8—31no—
547C8—31no—
548C8—31no—
549C8—61no—
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—361no—
563C2×C3#6389411yescharacteristic
564C2×C3C4×#62611yescharacteristic
565C2×C3#6390411yescharacteristic
566D3—121no—
567D3—121no—
568D3—241no—
569D3—362no—
570D3—361no—
571D3—361no—
572D3—361no—
573D3—721no—
574D3—721no—
575D3—722no—
576C22—181no—
577C22—541no—
578C22#8896511yescharacteristic
579C4—181no—
580C4—361no—
581C4—541no—
582C4D3×#62411yescharacteristic
583C4D3×#62611yescharacteristic
584C3—21no—
585C3—21no—
586C3—21no—
587C3—41no—
588C3#1059091811yescharacteristic, complemented
589C3#1059092411yescharacteristic, complemented
590C3#1059099411yescharacteristic, complemented
591C2—21no—
592C2—361no—
593C2—1081no—
594C2#1172293311yescharacteristic
subgroupG/H
centerC4D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radical#11723369C2
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×#63817C2semidirect
4. #11722077C2semidirect
5. #11722047C2semidirect
6. C3×#63577C2semidirect
7. C3×#63555C2semidirect
18. #88800C4semidirect
19. #88797C4semidirect
20. #88796C4semidirect
21. #88791C4semidirect
36. #63817D3semidirect
42. #63577D3semidirect
49. #63555D3semidirect
361. C33#444semidirect
532. C32#4717semidirect
533. C32#4894semidirect
534. C32#5145semidirect
588. C3#10590918semidirect
589. C3#10590924semidirect
590. C3#10590994semidirect

Maximal subgroups

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

subgroupindex
#117233692
#117233562
C3×#638172
#117220772
#117220472
C3×#635772
C3×#635552
#105909943
#105909243
#105909183