Finite Group Databasebeta

← #13170493

Subgroups — #13170493

Order 1728.

1276 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724618C211yesmaximal, characteristic
2C3⋊D3×#208C211yesmaximal, characteristic, complemented
3#11723352C212yesmaximal, complemented
4C3×#63817C212yesmaximal, complemented
5C2×C3×#3710C211yesmaximal, characteristic, complemented
6#11722826C214yesmaximal, complemented
7#11722820C212yesmaximal, complemented
8#11722814C212yesmaximal, complemented
9#10594707—31nomaximal
10D3×#901—32nomaximal
11C3×#3828C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13#89049C412yes—
14#89046C412yes—
15C4⋊#1113C411yescharacteristic
16C4×#1113C2212yes—
17#89042C2212yes—
18#89041C412yes—
19#89040C411yescharacteristic
20#88855C2211yescharacteristic, complemented
21C32×#208C2211yescharacteristic, complemented
22C2×C32×C3⋊C8C2212yescomplemented
23#88787C2214yescomplemented
24#88787C2212yescomplemented
25C3⋊D3×C3⋊C8C2212yes—
26C3⋊D3×C3⋊C8C2214yescomplemented
27C3×#3712C2214yescomplemented
28C3×#3710C2211yescharacteristic
29C3×#3710C2211yescharacteristic
30C3×#3710C2214yescomplemented
31C3×#3710C2212yescomplemented
32#64306—31no—
33D3×#235—32no—
34M16×C3⋊D3—31no—
35#64110—32no—
36C3×#907—32no—
37C3×#901—32no—
38C2×C3×D3×C8—32no—
39#63817D312yescomplemented
40C2×#3710D311yescharacteristic, complemented
41#63814—32no—
42#63814—32no—
43#63814—62no—
44D3×#208—32no—
45D3×#208—61no—
46#63804—34no—
47#63804—34no—
48#63803—34no—
49#63791—32no—
50D3×ΓL(1,𝔽25)—34no—
51D32×C8—32no—
52D32×C8—34no—
53C22×C3×C3⋊D3C2×C411yescharacteristic
54C2×#1113C2×C412yes—
55C2×C33×C4C2311yescharacteristic
56C3×#638C2×C414yes—
57C2×C3×C3⋊Dic3C2×C411yescharacteristic
58C3×C4⋊(C3⋊D3)C2×C412yes—
59C3×C4×C3⋊D3C2×C411yescharacteristic
60C3×C4×C3⋊D3C2×C411yescharacteristic
61C3×C4×C3⋊D3C2311yescharacteristic
62C3×C4×C3⋊D3C2311yescharacteristic
63C3×C4×C3⋊D3C2314yes—
64C3×#634C2×C412yes—
65#6296C2×C412yes—
66#6295C2×C414yes—
67#6295C2×C412yes—
68#6294C2×C412yes—
69#6294C2×C414yes—
70C3⋊(C3⋊(C3⋊C8))C2312yes—
71C32×C3⋊C8C2312yes—
72C32×C3⋊C8C2312yes—
73C32×C3⋊C8C2312yes—
74D3×#131—92no—
75C2×#901—91no—
76#3828C2×D312yes—
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C3×#235—32no—
79C3×#235—32no—
80C3×#235—62no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—61no—
83#3811—32no—
84D3×#67—34no—
85#3805—32no—
86#3804—34no—
87#3802—31no—
88D3×D12—32no—
89C4×D32—34no—
90C4×D32—32no—
91#3799—32no—
92#3798—34no—
93#3798—32no—
94#3797—31no—
95#3796—32no—
96#3795—32no—
97D3×Dic6—32no—
98C32×M16—31no—
99C2×C32×C8—32no—
100#3748—32no—
101#3748—31no—
102#3748—61no—
103#3748—62no—
104#3748—121no—
105#3743—34no—
106#3743—32no—
107C8×C3⋊D3—32no—
108C8×C3⋊D3—34no—
109C3×#208—21no—
110C3×#208C2×D312yescomplemented
111C2×C3×C3⋊C8—22no—
112C2×C3×C3⋊C8C2×D312yescomplemented
113C2×C3×C3⋊C8C2×D312yescomplemented
114C3×ΓL(1,𝔽25)—34no—
115C3×ΓL(1,𝔽25)—34no—
116C3×D3×C8—32no—
117C3×D3×C8—32no—
118C3×D3×C8—32no—
119C3×D3×C8—32no—
120C3×D3×C8—34no—
121C3×D3×C8—34no—
122#3712C2×D314yescomplemented
123#3711—34no—
124#3711—34no—
125#3711—34no—
126#3711—62no—
127#3711—64no—
128#3710C2×D311yescharacteristic
129#3710C2×D311yescharacteristic
130#3710C2×D314yescomplemented
131#3710C2×D312yescomplemented
132D3×C3⋊C8—34no—
133D3×C3⋊C8—34no—
134D3×C3⋊C8—34no—
135D3×C3⋊C8—62no—
136D3×C3⋊C8—64no—
137C22×C33C22×C411yescharacteristic
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3C22×C414yes—
140C2×C3×C3⋊D3C22×C411yescharacteristic
141C2×C3×C3⋊D3C22×C411yescharacteristic
142#1113—24no—
143C33×C4C22×C411yescharacteristic
144C33×C4C2411yescharacteristic
145C3×C3⋊Dic3C22×C411yescharacteristic
146C3×C3⋊Dic3C22×C414yes—
147C3×C3⋊Dic3C22×C411yescharacteristic
148D3×#41—92no—
149C2×#235—91no—
150C3×#131—92no—
151C22×C3×C8—91no—
152#948—92no—
153C2×#208—91no—
154#907—32no—
155#907—94no—
156D3×M16—94no—
157D3×M16—92no—
158D3×M16—181no—
159#901—32no—
160#901—94no—
161#901—94no—
162#901—92no—
163#901—92no—
164#901—182no—
165C2×ΓL(1,𝔽25)—92no—
166C2×D3×C8—32no—
167C2×D3×C8—92no—
168C2×D3×C8—94no—
169C22×C3⋊D3C4×D311yescharacteristic
170C22×C3×D3—32no—
171C22×C3×D3—61no—
172C2×D32—34no—
173C2×D32—32no—
174C2×C32×C4—22no—
175C2×C32×C4—21no—
176C2×C32×C4—41no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D312yes—
179#638C4×D314yes—
180C2×C3⋊Dic3C4×D311yescharacteristic
181C4⋊(C3⋊D3)C4×D312yes—
182C4×C3⋊D3C4×D311yescharacteristic
183C4×C3⋊D3C4×D311yescharacteristic
184C4×C3⋊D3C22×D311yescharacteristic
185C4×C3⋊D3C22×D311yescharacteristic
186C4×C3⋊D3C22×D314yes—
187#634C4×D312yes—
188C3×#67—34no—
189C3×#67—34no—
190C3×#67—64no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—61no—
193C3×D12—32no—
194C3×D12—32no—
195C3×D12—62no—
196C3×C4×D3—32no—
197C3×C4×D3—32no—
198C3×C4×D3—32no—
199C3×C4×D3—34no—
200C3×C4×D3—34no—
201C3×C4×D3—32no—
202C3×C4×D3—61no—
203C3×C4×D3—64no—
204C3×C4×D3—61no—
205C3×C4×D3—61no—
206C3×C4×D3—61no—
207C3×Dic6—32no—
208C3×Dic6—32no—
209C3×Dic6—62no—
210#627—34no—
211#627—32no—
212#626—34no—
213#626—34no—
214#626—34no—
215#625—34no—
216#625—32no—
217#624—32no—
218#624—34no—
219D3×Dic3—34no—
220D3×Dic3—34no—
221D3×Dic3—34no—
222C32×C8—32no—
223C32×C8—32no—
224C32×C8—32no—
225C3⋊(C3⋊C8)—34no—
226C3⋊(C3⋊C8)—32no—
227C3⋊(C3⋊C8)—62no—
228C3⋊(C3⋊C8)—64no—
229C3⋊(C3⋊C8)—122no—
230C3×C3⋊C8—22no—
231C3×C3⋊C8—22no—
232C3×C3⋊C8—22no—
233C3×C3⋊C8C22×D312yes—
234C3×C3⋊C8C22×D312yes—
235C3×C3⋊C8C22×D314yes—
236C3×C3⋊C8C22×D312yes—
237C3×C3⋊C8C22×D312yes—
238C2×#131—271noSylow
239C2×C33—21no—
240C2×C33C23×C411yescharacteristic
241C3×C3⋊D3—24no—
242C3×C3⋊D3—21no—
243C3×C3⋊D3#13112yescomplemented
244C3×#41—92no—
245C22×C3×C4—91no—
246C2×#67—92no—
247#239—92no—
248D3×Q8—92no—
249#237—92no—
250#237—94no—
251D3×D4—92no—
252D3×D4—94no—
253#235—32no—
254#235—32no—
255#235—62no—
256#235—92no—
257#235—92no—
258#235—94no—
259#235—92no—
260C2×D12—91no—
261C2×C4×D3—32no—
262C2×C4×D3—61no—
263C2×C4×D3—92no—
264C2×C4×D3—94no—
265C2×Dic6—91no—
266C3×M16—32no—
267C3×M16—61no—
268C3×M16—94no—
269C2×C3×C8—32no—
270C2×C3×C8—32no—
271C2×C3×C8—62no—
272C2×C3×C8—91no—
273C2×C3×C8—94no—
274C2×C3×C8—91no—
275C2×C3×C8—92no—
276#208—91no—
277#208—92no—
278#208—92no—
279#208—94no—
280#208—182no—
281#208—181no—
282#208—361no—
283#208D3211yescharacteristic, complemented
284C2×C3⋊C8—92no—
285C2×C3⋊C8—94no—
286C2×C3⋊C8D3212yescomplemented
287ΓL(1,𝔽25)—34no—
288ΓL(1,𝔽25)—34no—
289ΓL(1,𝔽25)—94no—
290ΓL(1,𝔽25)—94no—
291ΓL(1,𝔽25)—94no—
292ΓL(1,𝔽25)—184no—
293ΓL(1,𝔽25)—182no—
294D3×C8—32no—
295D3×C8—32no—
296D3×C8—32no—
297D3×C8—32no—
298D3×C8—34no—
299D3×C8—34no—
300D3×C8—94no—
301D3×C8—94no—
302D3×C8—94no—
303D3×C8—184no—
304D3×C8—182no—
305C22×C32—21no—
306C22×C32—22no—
307C22×C32—41no—
308C22×C32C2×C4×D312yes—
309C22×C32C2×C4×D311yescharacteristic
310C2×C3⋊D3—22no—
311C2×C3⋊D3C2×C4×D314yes—
312C2×C3⋊D3C2×C4×D311yescharacteristic
313C2×C3⋊D3C2×C4×D311yescharacteristic
314C2×C3×D3—32no—
315C2×C3×D3—34no—
316C2×C3×D3—34no—
317C2×C3×D3—32no—
318C2×C3×D3—61no—
319C2×C3×D3—61no—
320C2×C3×D3—64no—
321C2×C3×D3—64no—
322C2×C3×D3—122no—
323D32—68no—
324D32—64no—
325C32×C4—22no—
326C32×C4—21no—
327C32×C4—22no—
328C32×C4—21no—
329C32×C4—41no—
330C32×C4—41no—
331C32×C4C2×C4×D311yescharacteristic
332C32×C4C2×C4×D312yes—
333C32×C4C23×D311yescharacteristic
334C32×C4C23×D312yes—
335C3⋊Dic3C2×C4×D311yescharacteristic
336C3⋊Dic3C2×C4×D314yes—
337C3⋊Dic3C2×C4×D311yescharacteristic
338C3×Dic3—32no—
339C3×Dic3—34no—
340C3×Dic3—34no—
341C3×Dic3—32no—
342C3×Dic3—64no—
343C3×Dic3—61no—
344C3×Dic3—61no—
345C2×#41—271no—
346#131—92no—
347#131—272no—
348#131—274no—
349C2×M16—272no—
350C2×M16—271no—
351C22×C8—91no—
352C22×C8—272no—
353C33C2×#13111yescharacteristic, complemented, Sylow
354C23×C3—91no—
355C22×D3—32no—
356C22×D3—61no—
357C22×D3—92no—
358C22×D3—94no—
359C3×Q8—92no—
360C3×D4—94no—
361C3×D4—92no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4—94no—
368C2×C3×C4—91no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4C2×D3212yes—
372C2×C3×C4C2×D3211yescharacteristic
373#67—34no—
374#67—34no—
375#67—64no—
376#67—94no—
377#67—94no—
378#67—94no—
379C2×Dic3—32no—
380C2×Dic3—61no—
381C2×Dic3—92no—
382C2×Dic3—94no—
383D12—32no—
384D12—32no—
385D12—62no—
386D12—92no—
387D12—94no—
388C4×D3—32no—
389C4×D3—32no—
390C4×D3—34no—
391C4×D3—34no—
392C4×D3—32no—
393C4×D3—32no—
394C4×D3—61no—
395C4×D3—61no—
396C4×D3—61no—
397C4×D3—64no—
398C4×D3—61no—
399C4×D3—94no—
400C4×D3—94no—
401C4×D3—94no—
402Dic6—32no—
403Dic6—32no—
404Dic6—62no—
405Dic6—92no—
406Dic6—94no—
407C3×C8—32no—
408C3×C8—32no—
409C3×C8—32no—
410C3×C8—34no—
411C3×C8—32no—
412C3×C8—62no—
413C3×C8—62no—
414C3×C8—62no—
415C3⋊C8—92no—
416C3⋊C8—94no—
417C3⋊C8—184no—
418C3⋊C8—182no—
419C3⋊C8—362no—
420C3⋊C8C2×D3212yes—
421C3⋊C8C2×D3212yes—
422C3⋊C8C2×D3212yes—
423C2×C32—21no—
424C2×C32—22no—
425C2×C32—22no—
426C2×C32—21no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32—81no—
431C2×C32C22×C4×D311yescharacteristic
432C2×C32C22×C4×D312yes—
433C3⋊D3—21no—
434C3⋊D3—24no—
435C3⋊D3#90112yescomplemented
436C3×D3—34no—
437C3×D3—62no—
438C3×D3—64no—
439C3×D3—64no—
440C3×D3—62no—
441C3×D3—124no—
442C3×D3—121no—
443#41—92no—
444#41—272no—
445#41—272no—
446#41—272no—
447C2×Q8—271no—
448C2×D4—271no—
449C2×D4—272no—
450C22×C4—91no—
451C22×C4—272no—
452M16—31no—
453M16—94no—
454M16—274no—
455M16—272no—
456M16—271no—
457C2×C8—32no—
458C2×C8—91no—
459C2×C8—91no—
460C2×C8—92no—
461C2×C8—94no—
462C2×C8—272no—
463C2×C8—274no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—41no—
467C22×C3—91no—
468C22×C3—94no—
469C22×C3—91no—
470C22×C3—182no—
471C22×C3C4×D3212yes—
472C22×C3C4×D3211yescharacteristic
473C2×D3—32no—
474C2×D3—32no—
475C2×D3—34no—
476C2×D3—34no—
477C2×D3—61no—
478C2×D3—61no—
479C2×D3—64no—
480C2×D3—64no—
481C2×D3—122no—
482C2×D3—188no—
483C2×D3—184no—
484C3×C4—22no—
485C3×C4—21no—
486C3×C4—22no—
487C3×C4—21no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—91no—
491C3×C4—94no—
492C3×C4—91no—
493C3×C4C4×D3211yescharacteristic
494C3×C4C4×D3212yes—
495C3×C4C22×D3212yes—
496C3×C4C22×D3211yescharacteristic
497Dic3—34no—
498Dic3—34no—
499Dic3—32no—
500Dic3—32no—
501Dic3—64no—
502Dic3—61no—
503Dic3—61no—
504C32—22no—
505C32—21no—
506C32—41no—
507C32C2×#90111yescharacteristic, complemented
508C32D3×#13112yescomplemented
509C23—91no—
510C23—272no—
511Q8—92no—
512Q8—272no—
513D4—94no—
514D4—92no—
515D4—272no—
516D4—274no—
517C2×C4—91no—
518C2×C4—91no—
519C2×C4—91no—
520C2×C4—94no—
521C2×C4—91no—
522C2×C4—91no—
523C2×C4—272no—
524C2×C4—274no—
525C2×C4D3311yescharacteristic
526C8—32no—
527C8—32no—
528C8—32no—
529C8—272no—
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—22no—
534C2×C3—41no—
535C2×C3—42no—
536C2×C3—41no—
537C2×C3—81no—
538C2×C3—92no—
539C2×C3—184no—
540C2×C3—181no—
541C2×C3C2×C4×D3212yes—
542C2×C3C2×C4×D3211yescharacteristic
543D3—34no—
544D3—62no—
545D3—62no—
546D3—64no—
547D3—64no—
548D3—121no—
549D3—124no—
550C22—91no—
551C22—91no—
552C22—94no—
553C22—182no—
554C22—544no—
555C22#8894911yescharacteristic
556C4—91no—
557C4—91no—
558C4—94no—
559C4#8894911yescharacteristic
560C4C2×D3311yescharacteristic
561C3—21no—
562C3—22no—
563C3—41no—
564C3D3×#90112yescomplemented
565C3#1059470711yescharacteristic, complemented
566C2—21no—
567C2—92no—
568C2—181no—
569C2—184no—
570C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC4C2×D33
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C23
prime radical#11724618C2
cosocleC33×C4C24
hypercenterC2×C4D33
hypocenterC33C2×#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊D3×#208C2semidirect
3. #11723352C2semidirect
4. C3×#63817C2semidirect
5. C2×C3×#3710C2semidirect
6. #11722826C2semidirect
7. #11722820C2semidirect
8. #11722814C2semidirect
20. #88855C22semidirect
21. C32×#208C22semidirect
22. C2×C32×C3⋊C8C22semidirect
23. #88787C22semidirect
24. #88787C22semidirect
26. C3⋊D3×C3⋊C8C22semidirect
27. C3×#3712C22semidirect
30. C3×#3710C22semidirect
31. C3×#3710C22semidirect
39. #63817D3semidirect
40. C2×#3710D3semidirect
110. C3×#208C2×D3semidirect
112. C2×C3×C3⋊C8C2×D3semidirect
113. C2×C3×C3⋊C8C2×D3semidirect
122. #3712C2×D3semidirect
130. #3710C2×D3semidirect
131. #3710C2×D3semidirect
243. C3×C3⋊D3#131semidirect
283. #208D32semidirect
286. C2×C3⋊C8D32semidirect
353. C33C2×#131semidirect
435. C3⋊D3#901semidirect
507. C32C2×#901semidirect
508. C32D3×#131semidirect
564. C3D3×#901semidirect
565. C3#10594707semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3⋊D3×#2082
#117233522
C3×#638172
C2×C3×#37102
#117228262
#117228202
#117228142
#105947073
D3×#9013