Finite Group Databasebeta

← #13176646

Subgroups — #13176646

Order 1728.

1844 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724578C212yesmaximal, complemented
2C2×#88850C211yesmaximal, characteristic, complemented
3C3×#64113C211yesmaximal, characteristic, complemented
4C32×#904C211yesmaximal, characteristic, complemented
5#11723363C212yesmaximal, complemented
6#11723346C212yesmaximal, complemented
7#11723344C212yesmaximal, complemented
8#11723324C212yesmaximal, complemented
9#11723323C212yesmaximal, complemented
10#10596239—31nomaximal
11#10594685—34nomaximal
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C3×#3828C2212yescomplemented
14C32×#235C2212yescomplemented
15#89039—22no—
16#89029—22no—
17#89026C2212yescomplemented
18C4×D3×C3⋊D3—22no—
19#89020—22no—
20#89019C2212yescomplemented
21#89017C2212yes—
22#89015C2212yes—
23C2×C33×C8C2211yescharacteristic, complemented
24C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
25#88850C2212yescomplemented
26#88850C2212yescomplemented
27C3×#3746C2211yescharacteristic, complemented
28C3×#3745C2211yescharacteristic, complemented
29C3×#3744C2212yescomplemented
30C32×Dic12C2211yescharacteristic, complemented
31C32×D24C2211yescharacteristic, complemented
32C32×#204C2212yescomplemented
33#88798C2212yescomplemented
34#88793C2212yescomplemented
35#88792C2212yescomplemented
36#88789C2212yescomplemented
37#41×C3⋊D3—32no—
38#64306—38no—
39C32×#135—31no—
40#64159—32no—
41#64127—32no—
42#64124—32no—
43#64123—32no—
44#64120—32no—
45#64113D311yescharacteristic, complemented
46C2×#3743—31no—
47C2×#3743—34no—
48C2×#3743—64no—
49C3×#904—34no—
50C3×#904D314yescomplemented
51#63822—38no—
52#63811—38no—
53#63811—38no—
54#63798—38no—
55#63798—38no—
56C22×C3⋊(C3⋊D3)D411yescharacteristic
57C2×D3×C3⋊D3—22no—
58C2×C33×C4C2311yescharacteristic
59C2×C3⋊(C3⋊Dic3)D411yescharacteristic
60C4×C3⋊(C3⋊D3)D412yes—
61C4×C3⋊(C3⋊D3)C2312yes—
62C3×#638—22no—
63C3×C4⋊(C3⋊D3)C2312yes—
64C3×C4×C3⋊D3—22no—
65C3×#634C2312yes—
66C32×#67—22no—
67C32×D12C2312yes—
68C32×C4×D3—22no—
69C32×Dic6C2312yes—
70#6293—22no—
71#6292—22no—
72#6291—22no—
73#6290—22no—
74#6289—22no—
75Dic3×C3⋊D3—22no—
76D3×C3⋊Dic3—22no—
77C33×C8C2311yescharacteristic
78C33×C8C2311yescharacteristic
79C3⋊(C3⋊(C3⋊C8))C2312yes—
80#5768—94no—
81#5768—91no—
82C32×#41—32no—
83#3832—32no—
84Q8×C3⋊D3—32no—
85#3830—32no—
86#3830—62no—
87D4×C3⋊D3—32no—
88D4×C3⋊D3—62no—
89#3828—62no—
90#3828C2×D312yescomplemented
91C2×C4×C3⋊D3—31no—
92C2×C4×C3⋊D3—34no—
93C2×C4×C3⋊D3—62no—
94C2×C4×C3⋊D3—64no—
95C3×#235—38no—
96C3×#235C2×D318yescomplemented
97#3811—68no—
98#3805—68no—
99#3802—38no—
100C4×D32—68no—
101#3798—68no—
102#3797—38no—
103#3796—38no—
104#3796—38no—
105C32×Q16—31no—
106C32×SD16—32no—
107C32×D8—31no—
108C2×C32×C8—24no—
109C2×C32×C8C2×D314yescomplemented
110C2×C32×C8C2×D311yescharacteristic, complemented
111#3756—32no—
112#3755—32no—
113#3754—32no—
114#3753—32no—
115C2×C3⋊(C3⋊C8)—31no—
116C2×C3⋊(C3⋊C8)—34no—
117C2×C3⋊(C3⋊C8)—64no—
118#3746C2×D311yescharacteristic, complemented
119#3745C2×D311yescharacteristic, complemented
120#3744C2×D312yescomplemented
121#3743—32no—
122#3743—32no—
123#3743—38no—
124#3743—38no—
125#3743—68no—
126#3743—68no—
127C3×Dic12—34no—
128C3×Dic12C2×D314yescomplemented
129C3×D24—34no—
130C3×D24C2×D314yescomplemented
131C3×#204—38no—
132C3×#204C2×D318yescomplemented
133#3718—38no—
134#3715—38no—
135#3715—38no—
136#3713—38no—
137C22×C33C2×D411yescharacteristic
138C2×C3⋊(C3⋊D3)—41no—
139C2×C3⋊(C3⋊D3)C2×D412yes—
140C2×C3×C3⋊D3—22no—
141C2×C32×D3—22no—
142D3×C3⋊D3—44no—
143C33×C4C2×D411yescharacteristic
144C33×C4C2411yescharacteristic
145C3⋊(C3⋊Dic3)C2×D412yes—
146C3×C3⋊Dic3—22no—
147C32×Dic3—22no—
148D3×#41—98no—
149D3×#41—92no—
150C3×#135—34no—
151C3×#135—91no—
152#950—98no—
153#950—92no—
154#918—98no—
155#918—92no—
156#915—98no—
157#915—92no—
158#914—98no—
159#914—92no—
160#911—98no—
161#911—92no—
162#904—34no—
163#904C3⋊D311yescharacteristic, complemented
164C2×ΓL(1,𝔽25)—94no—
165C2×ΓL(1,𝔽25)—91no—
166C2×ΓL(1,𝔽25)—184no—
167C22×C3⋊D3—31no—
168C22×C3⋊D3—34no—
169C22×C3⋊D3—64no—
170C22×C3⋊D3—62no—
171C2×D32—68no—
172C32×Q8—32no—
173C32×D4—32no—
174C32×D4—62no—
175C2×C32×C4—24no—
176C2×C32×C4—62no—
177C2×C32×C4C22×D314yes—
178C2×C32×C4C22×D311yescharacteristic
179#638—22no—
180#638—62no—
181#638—62no—
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—34no—
184C2×C3⋊Dic3—64no—
185C2×C3⋊Dic3—62no—
186C4⋊(C3⋊D3)—62no—
187C4⋊(C3⋊D3)C22×D312yes—
188C4×C3⋊D3—22no—
189C4×C3⋊D3—32no—
190C4×C3⋊D3—32no—
191C4×C3⋊D3—38no—
192C4×C3⋊D3—38no—
193C4×C3⋊D3—68no—
194C4×C3⋊D3—68no—
195C4×C3⋊D3—62no—
196C4×C3⋊D3—62no—
197#634—62no—
198#634C22×D312yes—
199C3×#67—28no—
200C3×#67—68no—
201C3×D12—38no—
202C3×D12C22×D318yes—
203C3×C4×D3—28no—
204C3×C4×D3—68no—
205C3×Dic6—38no—
206C3×Dic6C22×D318yes—
207#627—68no—
208#626—68no—
209#626—68no—
210#625—68no—
211#624—68no—
212D3×Dic3—68no—
213D3×Dic3—68no—
214C32×C8—24no—
215C32×C8—24no—
216C32×C8C22×D314yes—
217C32×C8C22×D314yes—
218C32×C8C22×D311yescharacteristic
219C32×C8C22×D311yescharacteristic
220C3⋊(C3⋊C8)—32no—
221C3⋊(C3⋊C8)—38no—
222C3⋊(C3⋊C8)—68no—
223#575—271noSylow
224C2×C33—21no—
225C2×C33C22×D411yescharacteristic
226C3⋊(C3⋊D3)—22no—
227C3×C3⋊D3—42no—
228C32×D3—42no—
229C3×#41—38no—
230C3×#41—92no—
231#239—98no—
232#239—92no—
233D3×Q8—98no—
234D3×Q8—92no—
235#237—98no—
236#237—92no—
237#237—182no—
238#237—188no—
239D3×D4—98no—
240D3×D4—92no—
241D3×D4—182no—
242D3×D4—188no—
243#235—38no—
244#235—182no—
245#235—188no—
246#235C2×C3⋊D312yescomplemented
247C2×C4×D3—94no—
248C2×C4×D3—91no—
249C2×C4×D3—184no—
250C2×C4×D3—182no—
251C2×C4×D3—188no—
252C3×Q16—34no—
253C3×Q16—91no—
254C3×SD16—38no—
255C3×SD16—92no—
256C3×D8—34no—
257C3×D8—91no—
258C2×C3×C8—24no—
259C2×C3×C8D3214yescomplemented
260C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
261#216—98no—
262#216—92no—
263#215—98no—
264#215—92no—
265#214—98no—
266#214—92no—
267#213—98no—
268#213—92no—
269C2×C3⋊C8—94no—
270C2×C3⋊C8—91no—
271C2×C3⋊C8—184no—
272Dic12—34no—
273Dic12C2×C3⋊D311yescharacteristic, complemented
274D24—34no—
275D24C2×C3⋊D311yescharacteristic, complemented
276#204—38no—
277#204C2×C3⋊D312yescomplemented
278ΓL(1,𝔽25)—98no—
279ΓL(1,𝔽25)—98no—
280ΓL(1,𝔽25)—92no—
281ΓL(1,𝔽25)—92no—
282ΓL(1,𝔽25)—188no—
283ΓL(1,𝔽25)—188no—
284C22×C32—24no—
285C22×C32—62no—
286C22×C32D3×D414yes—
287C22×C32D3×D411yescharacteristic
288C2×C3⋊D3—22no—
289C2×C3⋊D3—32no—
290C2×C3⋊D3—38no—
291C2×C3⋊D3—68no—
292C2×C3⋊D3—124no—
293C2×C3⋊D3—121no—
294C2×C3⋊D3—124no—
295C2×C3⋊D3—128no—
296C2×C3×D3—28no—
297C2×C3×D3—68no—
298D32—1216no—
299C32×C4—24no—
300C32×C4—24no—
301C32×C4—62no—
302C32×C4D3×D414yes—
303C32×C4D3×D411yescharacteristic
304C32×C4C23×D314yes—
305C32×C4C23×D311yescharacteristic
306C3⋊Dic3—22no—
307C3⋊Dic3—32no—
308C3⋊Dic3—38no—
309C3⋊Dic3—68no—
310C3×Dic3—28no—
311C3×Dic3—68no—
312C2×#41—272no—
313#137—272no—
314#137—272no—
315AGL(1,ℤ/8)—272no—
316AGL(1,ℤ/8)—272no—
317#135—31no—
318#135—91no—
319#135—272no—
320C2×M16—271no—
321C33#57511yescharacteristic, complemented, Sylow
322C22×D3—94no—
323C22×D3—91no—
324C22×D3—184no—
325C22×D3—182no—
326C22×D3—188no—
327C3×Q8—38no—
328C3×Q8—92no—
329C3×D4—38no—
330C3×D4—68no—
331C3×D4—92no—
332C3×D4—182no—
333C2×C3×C4—24no—
334C2×C3×C4—68no—
335C2×C3×C4—182no—
336C2×C3×C4C2×D3214yes—
337C2×C3×C4C22×C3⋊D311yescharacteristic
338#67—22no—
339#67—68no—
340#67—182no—
341#67—188no—
342#67—182no—
343#67—188no—
344C2×Dic3—94no—
345C2×Dic3—91no—
346C2×Dic3—184no—
347C2×Dic3—182no—
348C2×Dic3—188no—
349D12—38no—
350D12—188no—
351D12—182no—
352D12C22×C3⋊D312yes—
353C4×D3—22no—
354C4×D3—68no—
355C4×D3—98no—
356C4×D3—92no—
357C4×D3—98no—
358C4×D3—92no—
359C4×D3—188no—
360C4×D3—188no—
361C4×D3—182no—
362C4×D3—188no—
363C4×D3—188no—
364C4×D3—182no—
365Dic6—38no—
366Dic6—182no—
367Dic6—188no—
368Dic6C22×C3⋊D312yes—
369C3×C8—24no—
370C3×C8—24no—
371C3×C8C2×D3214yes—
372C3×C8C2×D3214yes—
373C3×C8C22×C3⋊D311yescharacteristic
374C3×C8C22×C3⋊D311yescharacteristic
375C3⋊C8—98no—
376C3⋊C8—92no—
377C3⋊C8—188no—
378C2×C32—28no—
379C2×C32—24no—
380C2×C32—21no—
381C2×C32—24no—
382C2×C32—122no—
383C2×C32C2×D3×D414yes—
384C2×C32C2×D3×D411yescharacteristic
385C3⋊D3—42no—
386C3⋊D3—62no—
387C3⋊D3—68no—
388C3⋊D3—128no—
389C3×D3—48no—
390C3×D3—128no—
391#41—32no—
392#41—92no—
393#41—272no—
394#41—272no—
395#41—542no—
396#41—542no—
397C2×Q8—272no—
398C2×D4—272no—
399C2×D4—542no—
400C22×C4—271no—
401C22×C4—542no—
402Q16—31no—
403Q16—91no—
404Q16—272no—
405SD16—32no—
406SD16—92no—
407SD16—272no—
408SD16—272no—
409D8—31no—
410D8—91no—
411D8—272no—
412M16—272no—
413M16—272no—
414C2×C8—271no—
415C2×C8D3×C3⋊D311yescharacteristic, complemented
416C22×C3—24no—
417C22×C3—68no—
418C22×C3—182no—
419C22×C3#380214yes—
420C22×C3D4×C3⋊D311yescharacteristic
421C2×D3—22no—
422C2×D3—68no—
423C2×D3—98no—
424C2×D3—92no—
425C2×D3—188no—
426C2×D3—364no—
427C2×D3—3616no—
428C2×D3—364no—
429C2×D3—361no—
430C2×D3—368no—
431C3×C4—24no—
432C3×C4—24no—
433C3×C4—68no—
434C3×C4—182no—
435C3×C4#380214yes—
436C3×C4D4×C3⋊D311yescharacteristic
437C3×C4C22×D3214yes—
438C3×C4C23×C3⋊D311yescharacteristic
439Dic3—22no—
440Dic3—68no—
441Dic3—98no—
442Dic3—92no—
443Dic3—188no—
444C32—24no—
445C32#576814yescomplemented
446C32#576811yescharacteristic, complemented
447C23—271no—
448C23—542no—
449Q8—32no—
450Q8—92no—
451Q8—542no—
452D4—32no—
453D4—62no—
454D4—92no—
455D4—182no—
456D4—542no—
457D4—542no—
458D4—542no—
459C2×C4—62no—
460C2×C4—182no—
461C2×C4—271no—
462C2×C4—272no—
463C2×C4—272no—
464C2×C4—542no—
465C2×C4—542no—
466C2×C4—542no—
467C2×C4C2×D3×C3⋊D311yescharacteristic
468C8—272no—
469C8C2×D3×C3⋊D311yescharacteristic
470C8C2×D3×C3⋊D311yescharacteristic
471C2×C3—28no—
472C2×C3—24no—
473C2×C3—21no—
474C2×C3—24no—
475C2×C3—128no—
476C2×C3—362no—
477C2×C3C2×#380214yes—
478C2×C3C2×D4×C3⋊D311yescharacteristic
479D3—42no—
480D3—128no—
481D3—188no—
482D3—182no—
483D3—368no—
484C22—62no—
485C22—182no—
486C22—272no—
487C22—1081no—
488C22—1084no—
489C22#8902611yescharacteristic
490C4—62no—
491C4—182no—
492C4—272no—
493C4#8902611yescharacteristic
494C4C22×D3×C3⋊D311yescharacteristic
495C3—24no—
496C3#1059468514yescomplemented
497C3#1059623911yescharacteristic, complemented
498C2—21no—
499C2—122no—
500C2—362no—
501C2—542no—
502C2C2×#8902611yescharacteristic
subgroupG/H
centerC4#89026
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×C8C22
socleC2×C33C22×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#575
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11724578C2semidirect
2. C2×#88850C2semidirect
3. C3×#64113C2semidirect
4. C32×#904C2semidirect
5. #11723363C2semidirect
6. #11723346C2semidirect
7. #11723344C2semidirect
8. #11723324C2semidirect
9. #11723323C2semidirect
13. C3×#3828C22semidirect
14. C32×#235C22semidirect
17. #89026C22semidirect
20. #89019C22semidirect
23. C2×C33×C8C22semidirect
24. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
25. #88850C22semidirect
26. #88850C22semidirect
27. C3×#3746C22semidirect
28. C3×#3745C22semidirect
29. C3×#3744C22semidirect
30. C32×Dic12C22semidirect
31. C32×D24C22semidirect
32. C32×#204C22semidirect
33. #88798C22semidirect
34. #88793C22semidirect
35. #88792C22semidirect
36. #88789C22semidirect
45. #64113D3semidirect
50. C3×#904D3semidirect
90. #3828C2×D3semidirect
96. C3×#235C2×D3semidirect
109. C2×C32×C8C2×D3semidirect
110. C2×C32×C8C2×D3semidirect
118. #3746C2×D3semidirect
119. #3745C2×D3semidirect
120. #3744C2×D3semidirect
128. C3×Dic12C2×D3semidirect
130. C3×D24C2×D3semidirect
132. C3×#204C2×D3semidirect
163. #904C3⋊D3semidirect
246. #235C2×C3⋊D3semidirect
259. C2×C3×C8D32semidirect
260. C2×C3×C8C2×C3⋊D3semidirect
273. Dic12C2×C3⋊D3semidirect
275. D24C2×C3⋊D3semidirect
277. #204C2×C3⋊D3semidirect
321. C33#575semidirect
415. C2×C8D3×C3⋊D3semidirect
445. C32#5768semidirect
446. C32#5768semidirect
496. C3#10594685semidirect
497. C3#10596239semidirect

Maximal subgroups

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

subgroupindex
#117245782
C2×#888502
C3×#641132
C32×#9042
#117233632
#117233462
#117233442
#117233242
#117233232
#105962393
#105946853