Finite Group Databasebeta

← #13176636

Subgroups — #13176636

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#3828C212yesmaximal, complemented
2#11723945C211yesmaximal, characteristic, complemented
3C3×#64113C211yesmaximal, characteristic, complemented
4C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
5#11723369C212yesmaximal, complemented
6#11723327C212yesmaximal, complemented
7#11723326C212yesmaximal, complemented
8#11723316C212yesmaximal, complemented
9#11723315C212yesmaximal, complemented
10#10596219—31nomaximal
11#10594676—34nomaximal
12#89076C2212yescomplemented
13C3×#3828C2212yescomplemented
14C2×C32×C4×D3C2211yescharacteristic
15D3×#638—22no—
16#89031—22no—
17D3×C4⋊(C3⋊D3)C2212yescomplemented
18C4×D3×C3⋊D3—22no—
19#89022C2212yes—
20#89021C2212yes—
21#89020—22no—
22D3×#634C2212yescomplemented
23C2×C33×C8C2211yescharacteristic, complemented
24#88853C2211yescharacteristic, complemented
25#88852C2211yescharacteristic, complemented
26#88851C2212yescomplemented
27C3×#3746C2211yescharacteristic, complemented
28C3×#3745C2211yescharacteristic, complemented
29C3×#3744C2212yescomplemented
30C2×C32×C3⋊C8C2211yescharacteristic, complemented
31C32×ΓL(1,𝔽25)C2212yescomplemented
32C32×ΓL(1,𝔽25)C2212yescomplemented
33#88800C2212yescomplemented
34#88797C2212yescomplemented
35#88796C2212yescomplemented
36#88791C2212yescomplemented
37C2×#3828—32no—
38D3×#235—38no—
39C2×C32×M16—31no—
40#64118—32no—
41#64118—32no—
42#64117—32no—
43#64117—32no—
44#64113—32no—
45#64113—31no—
46#64113—34no—
47#64113—64no—
48#64113D311yescharacteristic, complemented
49C3×#904—34no—
50C2×C3×ΓL(1,𝔽25)D314yescomplemented
51#63829—38no—
52#63801—38no—
53#63800—38no—
54#63795—38no—
55#63794—38no—
56C22×C32×D3D411yescharacteristic
57C2×D3×C3⋊D3—22no—
58C2×C33×C4C2311yescharacteristic
59#6312—22no—
60C4⋊(C3⋊(C3⋊D3))C2312yes—
61C4×C3⋊(C3⋊D3)—22no—
62#6308C2312yes—
63C3×#638—22no—
64C3×C4⋊(C3⋊D3)C2312yes—
65C3×C4×C3⋊D3—22no—
66C3×#634C2312yes—
67C2×C32×Dic3D411yescharacteristic
68C32×C4×D3D412yes—
69C32×C4×D3C2312yes—
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
79C32×C3⋊C8C2312yes—
80#5768—91no—
81#5748—94no—
82C22×C32×C4—31no—
83C2×#638—62no—
84#3828—32no—
85#3828—38no—
86#3828—32no—
87#3828—32no—
88#3828—62no—
89#3828—62no—
90#3828—68no—
91#3828C2×D312yescomplemented
92C2×C4⋊(C3⋊D3)—32no—
93C2×C4×C3⋊D3—62no—
94C2×#634—32no—
95C3×#235—38no—
96C2×C3×C4×D3C2×D314yes—
97D3×#67—68no—
98#3804—68no—
99D3×D12—38no—
100C4×D32—68no—
101#3799—38no—
102#3798—68no—
103#3795—38no—
104D3×Dic6—38no—
105C32×M16—32no—
106C32×M16—32no—
107C2×C32×C8—24no—
108C2×C32×C8—31no—
109C2×C32×C8C2×D314yescomplemented
110C2×C32×C8C2×D311yescharacteristic, complemented
111#3746—32no—
112#3746—31no—
113#3746—34no—
114#3746—64no—
115#3746C2×D311yescharacteristic, complemented
116#3745—32no—
117#3745—31no—
118#3745—34no—
119#3745—64no—
120#3745C2×D311yescharacteristic, complemented
121#3744—32no—
122#3744—32no—
123#3744—32no—
124#3744—38no—
125#3744—68no—
126#3744C2×D312yescomplemented
127C2×C3×C3⋊C8C2×D314yescomplemented
128C3×Dic12—34no—
129C3×D24—34no—
130C3×#204—38no—
131C3×ΓL(1,𝔽25)C2×D318yescomplemented
132C3×ΓL(1,𝔽25)C2×D318yescomplemented
133#3719—38no—
134#3717—38no—
135#3716—38no—
136#3714—38no—
137C22×C33C2×D411yescharacteristic
138C2×C3⋊(C3⋊D3)—22no—
139C2×C3×C3⋊D3—22no—
140C2×C32×D3—41no—
141C2×C32×D3C2×D412yes—
142D3×C3⋊D3—44no—
143C33×C4C2×D411yescharacteristic
144C33×C4C2411yescharacteristic
145C3⋊(C3⋊Dic3)—22no—
146C3×C3⋊Dic3—22no—
147C32×Dic3C2×D412yes—
148D3×#41—92no—
149C2×#235—98no—
150C3×#135—91no—
151C2×C3×M16—34no—
152#950—92no—
153#918—92no—
154#915—92no—
155#914—92no—
156#911—92no—
157#909—98no—
158#909—98no—
159#908—98no—
160#908—98no—
161#904—34no—
162#904—98no—
163#904—94no—
164#904—91no—
165#904—184no—
166C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic, complemented
167C23×C32—31no—
168C22×C3⋊D3—62no—
169C22×C3×D3D1214yes—
170C2×D32—68no—
171C2×C32×C4—24no—
172C2×C32×C4—31no—
173C2×C32×C4—32no—
174C2×C32×C4—32no—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D311yescharacteristic
177#638—22no—
178#638—62no—
179#638—62no—
180#638—62no—
181#638—68no—
182#638—128no—
183C2×C3⋊Dic3—62no—
184C4⋊(C3⋊D3)—32no—
185C4⋊(C3⋊D3)—38no—
186C4⋊(C3⋊D3)—68no—
187C4⋊(C3⋊D3)—62no—
188C4⋊(C3⋊D3)C22×D312yes—
189C4×C3⋊D3—22no—
190C4×C3⋊D3—62no—
191C4×C3⋊D3—62no—
192C4×C3⋊D3—62no—
193C4×C3⋊D3—68no—
194C4×C3⋊D3—128no—
195#634—32no—
196#634—38no—
197#634—68no—
198#634—62no—
199#634C22×D312yes—
200C3×#67—68no—
201C2×C3×Dic3D1214yes—
202C3×D12—38no—
203C3×C4×D3—68no—
204C3×C4×D3D1218yes—
205C3×C4×D3C22×D318yes—
206C3×Dic6—38no—
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×C8—32no—
217C32×C8C22×D314yes—
218C32×C8C22×D314yes—
219C32×C8C22×D311yescharacteristic
220C32×C8C22×D311yescharacteristic
221C3×C3⋊C8C22×D318yes—
222#575—271noSylow
223C2×C33—21no—
224C2×C33C22×D411yescharacteristic
225C3⋊(C3⋊D3)—42no—
226C3×C3⋊D3—42no—
227C32×D3—22no—
228C3×#41—92no—
229C22×C3×C4—34no—
230C2×#67—188no—
231#239—92no—
232D3×Q8—92no—
233#237—92no—
234#237—182no—
235D3×D4—92no—
236D3×D4—182no—
237#235—38no—
238#235—98no—
239#235—98no—
240#235—98no—
241#235—92no—
242#235—188no—
243#235—188no—
244#235—188no—
245#235—182no—
246C2×D12—98no—
247C2×C4×D3—188no—
248C2×C4×D3—182no—
249C2×C4×D3C2×C3⋊D311yescharacteristic
250C2×Dic6—98no—
251C3×Q16—91no—
252C3×SD16—92no—
253C3×D8—91no—
254C3×M16—38no—
255C3×M16—38no—
256C2×C3×C8—24no—
257C2×C3×C8—34no—
258C2×C3×C8D3214yescomplemented
259C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
260#216—92no—
261#215—92no—
262#214—92no—
263#213—92no—
264C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
265Dic12—34no—
266Dic12—98no—
267Dic12—94no—
268Dic12—91no—
269Dic12—184no—
270D24—34no—
271D24—98no—
272D24—94no—
273D24—91no—
274D24—184no—
275#204—38no—
276#204—98no—
277#204—98no—
278#204—98no—
279#204—92no—
280#204—188no—
281ΓL(1,𝔽25)C2×C3⋊D312yescomplemented
282ΓL(1,𝔽25)C2×C3⋊D312yescomplemented
283C22×C32—24no—
284C22×C32—32no—
285C22×C32—121no—
286C22×C32C2×D1214yes—
287C22×C32D3×D411yescharacteristic
288C2×C3⋊D3—22no—
289C2×C3⋊D3—62no—
290C2×C3⋊D3—68no—
291C2×C3⋊D3—128no—
292C2×C3⋊D3—124no—
293C2×C3×D3—44no—
294C2×C3×D3—68no—
295C2×C3×D3C2×D1218yes—
296D32—1216no—
297C32×C4—24no—
298C32×C4—24no—
299C32×C4—32no—
300C32×C4C2×D1214yes—
301C32×C4D3×D411yescharacteristic
302C32×C4C23×D314yes—
303C32×C4C23×D311yescharacteristic
304C3⋊Dic3—22no—
305C3⋊Dic3—62no—
306C3⋊Dic3—68no—
307C3⋊Dic3—128no—
308C3×Dic3—68no—
309C3×Dic3C2×D1218yes—
310C2×#41—272no—
311#137—272no—
312#137—272no—
313AGL(1,ℤ/8)—272no—
314AGL(1,ℤ/8)—272no—
315#135—91no—
316#135—272no—
317#135—271no—
318C2×M16—31no—
319C33#57511yescharacteristic, complemented, Sylow
320C23×C3—34no—
321C22×D3—188no—
322C22×D3—182no—
323C22×D3C4⋊(C3⋊D3)11yescharacteristic
324C3×Q8—92no—
325C3×D4—92no—
326C3×D4—182no—
327C2×C3×C4—24no—
328C2×C3×C4—34no—
329C2×C3×C4—38no—
330C2×C3×C4—38no—
331C2×C3×C4—182no—
332C2×C3×C4C2×D3214yes—
333C2×C3×C4C22×C3⋊D311yescharacteristic
334#67—68no—
335#67—188no—
336#67—188no—
337#67—188no—
338#67—182no—
339#67—182no—
340#67—182no—
341#67—368no—
342C2×Dic3—188no—
343C2×Dic3—182no—
344C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
345D12—38no—
346D12—98no—
347D12—92no—
348D12—188no—
349D12—188no—
350D12—182no—
351C4×D3—68no—
352C4×D3—188no—
353C4×D3—188no—
354C4×D3—182no—
355C4×D3—188no—
356C4×D3—182no—
357C4×D3—182no—
358C4×D3—368no—
359C4×D3C4⋊(C3⋊D3)12yes—
360C4×D3C22×C3⋊D312yes—
361Dic6—38no—
362Dic6—98no—
363Dic6—92no—
364Dic6—188no—
365Dic6—188no—
366Dic6—182no—
367C3×C8—24no—
368C3×C8—24no—
369C3×C8—38no—
370C3×C8C2×D3214yes—
371C3×C8C2×D3214yes—
372C3×C8C22×C3⋊D311yescharacteristic
373C3×C8C22×C3⋊D311yescharacteristic
374C3⋊C8C22×C3⋊D312yes—
375C2×C32—21no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—44no—
379C2×C32—62no—
380C2×C32C22×D1214yes—
381C2×C32C2×D3×D411yescharacteristic
382C3⋊D3—42no—
383C3⋊D3—122no—
384C3⋊D3—128no—
385C3⋊D3—248no—
386C3×D3—28no—
387C3×D3—128no—
388#41—92no—
389#41—272no—
390#41—272no—
391#41—272no—
392#41—542no—
393#41—542no—
394C2×Q8—272no—
395C2×D4—272no—
396C2×D4—542no—
397C22×C4—31no—
398C22×C4—542no—
399Q16—91no—
400Q16—272no—
401Q16—271no—
402SD16—92no—
403SD16—272no—
404SD16—272no—
405SD16—272no—
406D8—91no—
407D8—272no—
408D8—271no—
409M16—32no—
410M16—32no—
411C2×C8—31no—
412C2×C8D3×C3⋊D311yescharacteristic, complemented
413C22×C3—24no—
414C22×C3—38no—
415C22×C3—124no—
416C22×C3—182no—
417C22×C3D3×D1214yes—
418C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
419C2×D3—41no—
420C2×D3—68no—
421C2×D3—188no—
422C2×D3—182no—
423C2×D3—368no—
424C2×D3—364no—
425C2×D3—3616no—
426C2×D3C2×C4⋊(C3⋊D3)12yes—
427C3×C4—24no—
428C3×C4—24no—
429C3×C4—38no—
430C3×C4—182no—
431C3×C4D3×D1214yes—
432C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
433C3×C4C22×D3214yes—
434C3×C4C23×C3⋊D311yescharacteristic
435Dic3—68no—
436Dic3—188no—
437Dic3—182no—
438Dic3—368no—
439Dic3C2×C4⋊(C3⋊D3)12yes—
440C32—24no—
441C32#574814yescomplemented
442C32#576811yescharacteristic, complemented
443C23—31no—
444C23—542no—
445Q8—92no—
446Q8—272no—
447Q8—542no—
448D4—92no—
449D4—182no—
450D4—272no—
451D4—542no—
452D4—542no—
453D4—542no—
454D4—542no—
455C2×C4—31no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—182no—
459C2×C4—542no—
460C2×C4—542no—
461C2×C4—542no—
462C2×C4—542no—
463C2×C4C2×D3×C3⋊D311yescharacteristic
464C8—32no—
465C8C2×D3×C3⋊D311yescharacteristic
466C8C2×D3×C3⋊D311yescharacteristic
467C2×C3—24no—
468C2×C3—21no—
469C2×C3—24no—
470C2×C3—44no—
471C2×C3—68no—
472C2×C3—362no—
473C2×C3C2×D3×D1214yes—
474C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
475D3—22no—
476D3—128no—
477D3—368no—
478D3—362no—
479D3—728no—
480C22—32no—
481C22—121no—
482C22—182no—
483C22—542no—
484C22—1084no—
485C22D3×C4⋊(C3⋊D3)11yescharacteristic
486C4—32no—
487C4—182no—
488C4—542no—
489C4D3×C4⋊(C3⋊D3)11yescharacteristic
490C4C22×D3×C3⋊D311yescharacteristic
491C3—24no—
492C3#1059467614yescomplemented
493C3#1059621911yescharacteristic, complemented
494C2—21no—
495C2—62no—
496C2—362no—
497C2—1082no—
498C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC4D3×C4⋊(C3⋊D3)
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. D3×#3828C2semidirect
2. #11723945C2semidirect
3. C3×#64113C2semidirect
4. C2×C32×ΓL(1,𝔽25)C2semidirect
5. #11723369C2semidirect
6. #11723327C2semidirect
7. #11723326C2semidirect
8. #11723316C2semidirect
9. #11723315C2semidirect
12. #89076C22semidirect
13. C3×#3828C22semidirect
17. D3×C4⋊(C3⋊D3)C22semidirect
22. D3×#634C22semidirect
23. C2×C33×C8C22semidirect
24. #88853C22semidirect
25. #88852C22semidirect
26. #88851C22semidirect
27. C3×#3746C22semidirect
28. C3×#3745C22semidirect
29. C3×#3744C22semidirect
30. C2×C32×C3⋊C8C22semidirect
31. C32×ΓL(1,𝔽25)C22semidirect
32. C32×ΓL(1,𝔽25)C22semidirect
33. #88800C22semidirect
34. #88797C22semidirect
35. #88796C22semidirect
36. #88791C22semidirect
48. #64113D3semidirect
50. C2×C3×ΓL(1,𝔽25)D3semidirect
91. #3828C2×D3semidirect
109. C2×C32×C8C2×D3semidirect
110. C2×C32×C8C2×D3semidirect
115. #3746C2×D3semidirect
120. #3745C2×D3semidirect
126. #3744C2×D3semidirect
127. C2×C3×C3⋊C8C2×D3semidirect
131. C3×ΓL(1,𝔽25)C2×D3semidirect
132. C3×ΓL(1,𝔽25)C2×D3semidirect
166. C2×ΓL(1,𝔽25)C3⋊D3semidirect
258. C2×C3×C8D32semidirect
259. C2×C3×C8C2×C3⋊D3semidirect
264. C2×C3⋊C8C2×C3⋊D3semidirect
281. ΓL(1,𝔽25)C2×C3⋊D3semidirect
282. ΓL(1,𝔽25)C2×C3⋊D3semidirect
319. C33#575semidirect
412. C2×C8D3×C3⋊D3semidirect
441. C32#5748semidirect
442. C32#5768semidirect
492. C3#10594676semidirect
493. C3#10596219semidirect

Maximal subgroups

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

subgroupindex
D3×#38282
#117239452
C3×#641132
C2×C32×ΓL(1,𝔽25)2
#117233692
#117233272
#117233262
#117233162
#117233152
#105962193
#105946763