Finite Group Databasebeta

← #13155754

Subgroups — #13155754

Order 1728.

572 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723660C211yesmaximal, characteristic
2#11723352C211yesmaximal, characteristic, complemented
3C3×#63817C211yesmaximal, characteristic, complemented
4#11722060C211yesmaximal, characteristic, complemented
5#11722046C211yesmaximal, characteristic, complemented
6C3×#63570C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590976—31nomaximal
9#10590924—31nomaximal
10#10590342—31nomaximal
11C3×#3828C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13#88855C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C32×#208C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88812C411yescharacteristic, complemented
19#88811C411yescharacteristic, complemented
20#88811C411yescharacteristic, complemented
21#88810C411yescharacteristic, complemented
22#88787—21no—
23C3⋊D3×C3⋊C8—21no—
24C3×#3712—21no—
25C3×#3710—21no—
26#64110—31no—
27C3×#907—31no—
28C3×#901—31no—
29#63829—31no—
30#63829—31no—
31#63822—31no—
32#63817D311yescharacteristic, complemented
33#63814—31no—
34#63814—31no—
35#63814—61no—
36#63649—31no—
37#63632—31no—
38C3×#825—31no—
39C3×#815—31no—
40C3×#805—31no—
41C3×#803—31no—
42#63577—31no—
43#63577—31no—
44#63577—61no—
45#63570D311yescharacteristic, complemented
46#63569—31no—
47#63569—31no—
48#63569—61no—
49#63554D311yescharacteristic, complemented
50C2×C33×C4C2311yescharacteristic
51C3×#638—21no—
52C3×#638—21no—
53C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
54C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C3×#634C2×C411yescharacteristic
58C3×#634C2×C411yescharacteristic
59C2×C32×Dic3—21no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
62C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
63C32×C3⋊C8—21no—
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66#5145—91no—
67#4823—91no—
68#4717—91no—
69#3828C2×D311yescharacteristic
70#3828C2×D311yescharacteristic
71C3×#235—31no—
72C3×#235—31no—
73C3×#235—31no—
74C3×#235—31no—
75C3×#235—61no—
76C3×#235—61no—
77C32×M16—31no—
78C2×C32×C8—31no—
79C32×C42—31no—
80#3748—31no—
81#3748—31no—
82#3748—31no—
83#3748—61no—
84#3748—61no—
85#3748—61no—
86#3748—121no—
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88#3743—61no—
89C8×C3⋊D3—61no—
90C3×C4×Dic3—21no—
91C3×C4×Dic3C2×D311yescharacteristic
92C3×C4×Dic3C2×D311yescharacteristic
93C3×#208—21no—
94C3×#208C2×D311yescharacteristic
95C3×#208C2×D311yescharacteristic
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—61no—
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C3×ΓL(1,𝔽25)—61no—
103C3×ΓL(1,𝔽25)—61no—
104C3×D3×C8—61no—
105C3×D3×C8—61no—
106#3719—31no—
107#3719—31no—
108#3718—31no—
109#3717—31no—
110#3717—31no—
111#3716—31no—
112#3716—31no—
113#3715—31no—
114#3715—31no—
115#3714—31no—
116#3714—31no—
117#3713—31no—
118#3712—21no—
119#3711—62no—
120#3711—61no—
121#3711—61no—
122#3710—21no—
123D3×C3⋊C8—62no—
124D3×C3⋊C8—61no—
125D3×C3⋊C8—61no—
126C22×C33#4111yescharacteristic
127C2×C3×C3⋊D3—21no—
128C2×C3×C3⋊D3—21no—
129C33×C4C22×C411yescharacteristic
130C33×C4C2×D411yescharacteristic
131C3×C3⋊Dic3—21no—
132C3×C3⋊Dic3—21no—
133C32×Dic3—41no—
134C3×#131—91no—
135#950—91no—
136#950—91no—
137#948—91no—
138#907—31no—
139#904—91no—
140#901—31no—
141#901—91no—
142#901—91no—
143#901—181no—
144C3×#104—91no—
145C3×#97—91no—
146#837—91no—
147#825—31no—
148#822—91no—
149#822—91no—
150#822—181no—
151#819—91no—
152#815—31no—
153#805—31no—
154#805—91no—
155#805—91no—
156#805—181no—
157#803—31no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—41no—
162C2×C32×C4—61no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166#638—21no—
167#638—21no—
168C4⋊(C3⋊D3)C4×D311yescharacteristic
169C4⋊(C3⋊D3)C4×D311yescharacteristic
170C4×C3⋊D3—21no—
171C4×C3⋊D3—21no—
172#634C4×D311yescharacteristic
173#634C4×D311yescharacteristic
174C3×#67—62no—
175C3×#67—61no—
176C3×#67—61no—
177C3×#67—61no—
178C3×#67—61no—
179C3×#67—121no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—41no—
183C3×D12—31no—
184C3×D12—31no—
185C3×D12—31no—
186C3×D12—31no—
187C3×D12—61no—
188C3×D12—61no—
189C3×C4×D3—62no—
190C3×C4×D3—61no—
191C3×C4×D3—61no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—121no—
195C3×Dic6—31no—
196C3×Dic6—31no—
197C3×Dic6—31no—
198C3×Dic6—31no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C32×C8—31no—
202C32×C8—31no—
203C32×C8—61no—
204C3⋊(C3⋊C8)—61no—
205C3⋊(C3⋊C8)—62no—
206C3⋊(C3⋊C8)—61no—
207C3⋊(C3⋊C8)—61no—
208C3⋊(C3⋊C8)—121no—
209C3⋊(C3⋊C8)—121no—
210C3⋊(C3⋊C8)—122no—
211C3⋊(C3⋊C8)C4×D311yescharacteristic
212C3⋊(C3⋊C8)C4×D311yescharacteristic
213C3×C3⋊C8—21no—
214C3×C3⋊C8—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—31no—
218C3×C3⋊C8—31no—
219C3×C3⋊C8—31no—
220C3×C3⋊C8—31no—
221C3×C3⋊C8—41no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8D1211yescharacteristic
225C3×C3⋊C8D1211yescharacteristic
226C3×C3⋊C8#6711yescharacteristic
227C3×C3⋊C8#6711yescharacteristic
228#444—271noSylow
229C2×C33—21no—
230C2×C33C4×D411yescharacteristic
231C3×C3⋊D3—41no—
232C3×C3⋊D3—41no—
233C3×#41—91no—
234C3×#41—91no—
235#235—31no—
236#235—31no—
237#235—31no—
238#235—31no—
239#235—61no—
240#235—61no—
241C3×M16—31no—
242C3×M16—31no—
243C3×M16—61no—
244C3×M16—181no—
245C2×C3×C8—31no—
246C2×C3×C8—31no—
247C2×C3×C8—61no—
248C2×C3×C8—91no—
249C2×C3×C8—181no—
250C3×C42—31no—
251C3×C42—31no—
252C3×C42—61no—
253#216—91no—
254#216—91no—
255#215—91no—
256#215—91no—
257#214—91no—
258#214—91no—
259#213—91no—
260#213—91no—
261C4×Dic3D3211yescharacteristic
262#208—91no—
263#208—91no—
264#208—91no—
265#208—181no—
266#208—181no—
267#208—181no—
268#208—181no—
269#208—361no—
270#208D3211yescharacteristic
271C2×C3⋊C8—31no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—61no—
274C2×C3⋊C8—181no—
275C2×C3⋊C8D3211yescharacteristic
276Dic12—91no—
277D24—91no—
278#204—91no—
279#204—91no—
280ΓL(1,𝔽25)—61no—
281ΓL(1,𝔽25)—61no—
282ΓL(1,𝔽25)—181no—
283ΓL(1,𝔽25)—181no—
284ΓL(1,𝔽25)—182no—
285D3×C8—61no—
286D3×C8—61no—
287D3×C8—181no—
288D3×C8—181no—
289D3×C8—182no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32#23511yescharacteristic
295C22×C32#23511yescharacteristic
296C22×C32#23711yescharacteristic
297C2×C3⋊D3—21no—
298C2×C3⋊D3—21no—
299C2×C3×D3—62no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—121no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—121no—
314C32×C4C2×C4×D311yescharacteristic
315C32×C4C2×C4×D311yescharacteristic
316C32×C4C2×C4×D311yescharacteristic
317C32×C4C2×D1211yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4C2×#6711yescharacteristic
320C3⋊Dic3—21no—
321C3⋊Dic3—21no—
322C3×Dic3—41no—
323C3×Dic3—41no—
324C3×Dic3—42no—
325C3×Dic3—62no—
326C3×Dic3—61no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—61no—
330C3×Dic3—121no—
331#135—271no—
332#131—91no—
333#131—271no—
334#108—271no—
335#104—91no—
336#104—271no—
337#97—91no—
338C33#44411yescharacteristic, complemented, Sylow
339C3×Q8—91no—
340C3×Q8—91no—
341C3×D4—91no—
342C3×D4—91no—
343C3×D4—181no—
344C3×D4—181no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—121no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4C2×D3211yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357#67—62no—
358#67—61no—
359#67—61no—
360#67—61no—
361#67—61no—
362#67—121no—
363C2×Dic3—21no—
364D12—31no—
365D12—31no—
366D12—31no—
367D12—31no—
368D12—61no—
369D12—61no—
370C4×D3—62no—
371C4×D3—61no—
372C4×D3—61no—
373C4×D3—61no—
374C4×D3—61no—
375C4×D3—121no—
376Dic6—31no—
377Dic6—31no—
378Dic6—31no—
379Dic6—31no—
380Dic6—61no—
381Dic6—61no—
382C3×C8—31no—
383C3×C8—31no—
384C3×C8—31no—
385C3×C8—31no—
386C3×C8—61no—
387C3×C8—61no—
388C3×C8—61no—
389C3×C8—61no—
390C3×C8—91no—
391C3×C8—91no—
392C3×C8—121no—
393C3⋊C8—21no—
394C3⋊C8—31no—
395C3⋊C8—31no—
396C3⋊C8—31no—
397C3⋊C8—31no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8—182no—
401C3⋊C8—181no—
402C3⋊C8—181no—
403C3⋊C8—181no—
404C3⋊C8—361no—
405C3⋊C8—361no—
406C3⋊C8—362no—
407C3⋊C8#62611yescharacteristic
408C3⋊C8#62611yescharacteristic
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32—81no—
420C2×C32C4×D1211yescharacteristic
421C2×C32#67⋊C411yescharacteristic
422C2×C32C4×#6711yescharacteristic
423C3⋊D3—41no—
424C3⋊D3—41no—
425C3×D3—122no—
426C3×D3—121no—
427C3×D3—121no—
428C3×D3—121no—
429C3×D3—121no—
430C3×D3—241no—
431#41—91no—
432#41—91no—
433Q16—271no—
434SD16—271no—
435SD16—271no—
436D8—271no—
437M16—31no—
438M16—181no—
439M16—271no—
440M16—541no—
441C2×C8—31no—
442C2×C8—91no—
443C2×C8—181no—
444C2×C8—541no—
445C42—31no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—41no—
450C22×C3—181no—
451C22×C3—181no—
452C22×C3#379511yescharacteristic
453C22×C3#379811yescharacteristic
454C22×C3#380411yescharacteristic
455C2×D3—62no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—61no—
460C2×D3—121no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—122no—
472C3×C4—181no—
473C3×C4—181no—
474C3×C4C4×D3211yescharacteristic
475C3×C4C4×D3211yescharacteristic
476C3×C4C4×D3211yescharacteristic
477C3×C4D3×D1211yescharacteristic
478C3×C4C2×#62611yescharacteristic
479C3×C4D3×#6711yescharacteristic
480Dic3—41no—
481Dic3—62no—
482Dic3—61no—
483Dic3—61no—
484Dic3—61no—
485Dic3—61no—
486Dic3—121no—
487C32—21no—
488C32—21no—
489C32—21no—
490C32—41no—
491C32#471711yescharacteristic, complemented
492C32#482311yescharacteristic, complemented
493C32#514511yescharacteristic, complemented
494Q8—91no—
495Q8—91no—
496D4—91no—
497D4—91no—
498D4—181no—
499D4—181no—
500C2×C4—61no—
501C2×C4—181no—
502C2×C4—181no—
503C2×C4D3311yescharacteristic
504C8—31no—
505C8—31no—
506C8—61no—
507C8—91no—
508C8—91no—
509C8—541no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—41no—
520C2×C3—81no—
521C2×C3—361no—
522C2×C3—361no—
523C2×C3#6387911yescharacteristic
524C2×C3#6388011yescharacteristic
525C2×C3C4×#62611yescharacteristic
526D3—122no—
527D3—121no—
528D3—121no—
529D3—121no—
530D3—121no—
531D3—241no—
532C22—181no—
533C22—181no—
534C22#8896011yescharacteristic
535C4—121no—
536C4—181no—
537C4—181no—
538C4#8894911yescharacteristic
539C4D3×#62611yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059034211yescharacteristic, complemented
545C3#1059092411yescharacteristic, complemented
546C3#1059097611yescharacteristic, complemented
547C2—21no—
548C2—361no—
549C2—361no—
550C2#1172293411yescharacteristic
subgroupG/H
centerC4D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radical#11723660C2
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. #11723352C2semidirect
3. C3×#63817C2semidirect
4. #11722060C2semidirect
5. #11722046C2semidirect
6. C3×#63570C2semidirect
7. C3×#63554C2semidirect
18. #88812C4semidirect
19. #88811C4semidirect
20. #88811C4semidirect
21. #88810C4semidirect
32. #63817D3semidirect
45. #63570D3semidirect
49. #63554D3semidirect
338. C33#444semidirect
491. C32#4717semidirect
492. C32#4823semidirect
493. C32#5145semidirect
544. C3#10590342semidirect
545. C3#10590924semidirect
546. C3#10590976semidirect

Maximal subgroups

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

subgroupindex
#117236602
#117233522
C3×#638172
#117220602
#117220462
C3×#635702
C3×#635542
#105909763
#105909243
#105903423