Finite Group Databasebeta

← #13176618

Subgroups — #13176618

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#3828C211yesmaximal, characteristic
2#11723942C211yesmaximal, characteristic, complemented
3C3×#64110C211yesmaximal, characteristic, complemented
4C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
5#11723356C211yesmaximal, characteristic, complemented
6#11723352C211yesmaximal, characteristic, complemented
7D3×#3748C211yesmaximal, characteristic, complemented
8#11723334C212yesmaximal, complemented
9#11723333C212yesmaximal, complemented
10#11723312C212yesmaximal, complemented
11C3⋊D3×ΓL(1,𝔽25)C212yesmaximal, complemented
12#10596216—31nomaximal
13#10594668—34nomaximal
14#89076C2211yescharacteristic
15C3×#3828C2211yescharacteristic
16C2×C32×C4×D3C2211yescharacteristic
17D3×#638C412yes—
18#89031C412yes—
19D3×C4⋊(C3⋊D3)C411yescharacteristic
20C4×D3×C3⋊D3C2212yes—
21#89022C411yescharacteristic
22#89021C411yescharacteristic
23#89020C2212yes—
24D3×#634C411yescharacteristic
25C2×C33×C8C2211yescharacteristic, complemented
26#88855C2211yescharacteristic, complemented
27#88850C2212yescomplemented
28C8×C3⋊(C3⋊D3)C2212yescomplemented
29C3×#3748C2211yescharacteristic, complemented
30C3×#3743C2212yescomplemented
31C3×C8×C3⋊D3C2212yescomplemented
32C2×C32×C3⋊C8C2211yescharacteristic, complemented
33C32×ΓL(1,𝔽25)C2211yescharacteristic
34C32×ΓL(1,𝔽25)C2211yescharacteristic
35C32×ΓL(1,𝔽25)C2211yescharacteristic, complemented
36C32×ΓL(1,𝔽25)C2211yescharacteristic, complemented
37#88788C2212yescomplemented
38#88787C2212yescomplemented
39#88786C2212yes—
40#88785C2212yescomplemented
41C3⋊D3×C3⋊C8C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43C2×#3828—31no—
44D3×#235—34no—
45C2×C32×M16—31no—
46C2×#3748—31no—
47#64116—32no—
48#64116—32no—
49M16×C3⋊D3—32no—
50M16×C3⋊D3—32no—
51#64110—31no—
52#64110—31no—
53#64110—31no—
54#64110—34no—
55#64110—64no—
56#64110D311yescharacteristic, complemented
57C3×#901—34no—
58C2×C3×ΓL(1,𝔽25)D314yescomplemented
59#63817—34no—
60#63814—34no—
61D3×#208—34no—
62#63805—38no—
63#63804—38no—
64#63791—38no—
65D3×ΓL(1,𝔽25)—38no—
66C22×C32×D3C2×C411yescharacteristic
67C2×D3×C3⋊D3C2×C412yes—
68C2×C33×C4C2311yescharacteristic
69#6312C2×C412yes—
70C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
71C4×C3⋊(C3⋊D3)C2312yes—
72#6308C2×C411yescharacteristic
73C3×#638C2×C412yes—
74C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
75C3×C4×C3⋊D3C2312yes—
76C3×#634C2×C411yescharacteristic
77C2×C32×Dic3C2×C411yescharacteristic
78C32×C4×D3C2×C411yescharacteristic
79C32×C4×D3C2×C411yescharacteristic
80C32×C4×D3C2311yescharacteristic
81C32×C4×D3C2311yescharacteristic
82#6293C2×C412yes—
83#6292C2×C412yes—
84#6291C2×C412yes—
85#6290C2×C412yes—
86#6289C2×C412yes—
87Dic3×C3⋊D3C2×C412yes—
88D3×C3⋊Dic3C2×C412yes—
89C33×C8C2311yescharacteristic
90C33×C8C2311yescharacteristic
91C3⋊(C3⋊(C3⋊C8))C2312yes—
92C3×C3⋊(C3⋊C8)C2312yes—
93C32×C3⋊C8C2311yescharacteristic
94C32×C3⋊C8C2311yescharacteristic
95#5750—91no—
96#5745—94no—
97C22×C32×C4—31no—
98C2×#638—32no—
99#3828—31no—
100#3828—34no—
101#3828—32no—
102#3828—31no—
103#3828—32no—
104#3828—31no—
105#3828—64no—
106#3828C2×D311yescharacteristic
107C2×C4⋊(C3⋊D3)—31no—
108C2×C4×C3⋊D3—32no—
109C2×#634—31no—
110C3×#235—34no—
111C2×C3×C4×D3C2×D314yes—
112D3×#67—38no—
113#3804—38no—
114D3×D12—34no—
115C4×D32—38no—
116#3799—34no—
117#3798—38no—
118#3795—34no—
119D3×Dic6—34no—
120C32×M16—31no—
121C32×M16—31no—
122C32×M16—31no—
123C32×M16—31no—
124C2×C32×C8—24no—
125C2×C32×C8—31no—
126C2×C32×C8C2×D314yescomplemented
127C2×C32×C8C2×D311yescharacteristic, complemented
128#3748—31no—
129#3748—34no—
130#3748—32no—
131#3748—64no—
132#3748C2×D311yescharacteristic, complemented
133C2×C3⋊(C3⋊C8)—32no—
134#3743—32no—
135#3743—32no—
136#3743—38no—
137#3743—32no—
138#3743—68no—
139#3743C2×D312yescomplemented
140C8×C3⋊D3—32no—
141C8×C3⋊D3—32no—
142C8×C3⋊D3—38no—
143C8×C3⋊D3—32no—
144C8×C3⋊D3—68no—
145C8×C3⋊D3C2×D312yescomplemented
146C3×#208—34no—
147C2×C3×C3⋊C8C2×D314yescomplemented
148C3×ΓL(1,𝔽25)—38no—
149C3×ΓL(1,𝔽25)C2×D314yes—
150C3×ΓL(1,𝔽25)C2×D314yes—
151C3×ΓL(1,𝔽25)C2×D314yescomplemented
152C3×ΓL(1,𝔽25)C2×D314yescomplemented
153C3×D3×C8—38no—
154#3712—38no—
155#3711—38no—
156#3711—38no—
157#3710—38no—
158D3×C3⋊C8—38no—
159D3×C3⋊C8—38no—
160C22×C33C22×C411yescharacteristic
161C2×C3⋊(C3⋊D3)C22×C412yes—
162C2×C3×C3⋊D3C22×C412yes—
163C2×C32×D3—41no—
164C2×C32×D3C22×C411yescharacteristic
165C2×C32×D3C22×C411yescharacteristic
166D3×C3⋊D3—42no—
167C33×C4C22×C411yescharacteristic
168C33×C4C2411yescharacteristic
169C3⋊(C3⋊Dic3)C22×C412yes—
170C3×C3⋊Dic3C22×C412yes—
171C32×Dic3C22×C411yescharacteristic
172C32×Dic3C22×C411yescharacteristic
173D3×#41—91no—
174C2×#235—94no—
175C3×#131—91no—
176C2×C3×M16—34no—
177#948—91no—
178C2×#208—94no—
179#907—98no—
180#907—98no—
181#907—91no—
182#907—92no—
183D3×M16—98no—
184D3×M16—98no—
185D3×M16—91no—
186D3×M16—92no—
187#901—34no—
188#901—94no—
189#901—94no—
190#901—94no—
191#901—91no—
192#901—92no—
193#901—184no—
194C2×ΓL(1,𝔽25)—92no—
195C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic, complemented
196C23×C32—31no—
197C22×C3⋊D3—32no—
198C22×C3×D3C4×D314yes—
199C2×D32—38no—
200C2×C32×C4—24no—
201C2×C32×C4—31no—
202C2×C32×C4—31no—
203C2×C32×C4—31no—
204C2×C32×C4—31no—
205C2×C32×C4—31no—
206C2×C32×C4C22×D314yes—
207C2×C32×C4C22×D311yescharacteristic
208#638—32no—
209#638—32no—
210#638—32no—
211#638—38no—
212#638—68no—
213#638C4×D312yes—
214C2×C3⋊Dic3—32no—
215C4⋊(C3⋊D3)—32no—
216C4⋊(C3⋊D3)—31no—
217C4⋊(C3⋊D3)—34no—
218C4⋊(C3⋊D3)—64no—
219C4⋊(C3⋊D3)C4×D311yescharacteristic
220C4×C3⋊D3—32no—
221C4×C3⋊D3—32no—
222C4×C3⋊D3—32no—
223C4×C3⋊D3—38no—
224C4×C3⋊D3—68no—
225C4×C3⋊D3C22×D312yes—
226#634—31no—
227#634—32no—
228#634—34no—
229#634—64no—
230#634C4×D311yescharacteristic
231C3×#67—38no—
232C2×C3×Dic3C4×D314yes—
233C3×D12—34no—
234C3×C4×D3—38no—
235C3×C4×D3C4×D314yes—
236C3×C4×D3C4×D314yes—
237C3×C4×D3C22×D314yes—
238C3×C4×D3C22×D314yes—
239C3×Dic6—34no—
240#627—38no—
241#626—38no—
242#626—38no—
243#625—38no—
244#624—38no—
245D3×Dic3—38no—
246D3×Dic3—38no—
247C32×C8—24no—
248C32×C8—24no—
249C32×C8—31no—
250C32×C8—31no—
251C32×C8C22×D314yes—
252C32×C8C22×D314yes—
253C32×C8C22×D311yescharacteristic
254C32×C8C22×D311yescharacteristic
255C3⋊(C3⋊C8)—32no—
256C3⋊(C3⋊C8)—38no—
257C3⋊(C3⋊C8)—68no—
258C3⋊(C3⋊C8)C22×D312yes—
259C3×C3⋊C8—38no—
260C3×C3⋊C8C22×D314yes—
261C3×C3⋊C8C22×D314yes—
262#568—271noSylow
263C2×C33—21no—
264C2×C33C23×C411yescharacteristic
265C3⋊(C3⋊D3)—22no—
266C3×C3⋊D3—22no—
267C32×D3—21no—
268C32×D3—21no—
269C3×#41—91no—
270C22×C3×C4—34no—
271C2×#67—98no—
272#239—91no—
273D3×Q8—91no—
274#237—92no—
275#237—91no—
276D3×D4—92no—
277D3×D4—91no—
278#235—34no—
279#235—94no—
280#235—98no—
281#235—94no—
282#235—98no—
283#235—94no—
284#235—91no—
285#235—92no—
286#235—184no—
287C2×D12—94no—
288C2×C4×D3—98no—
289C2×C4×D3—92no—
290C2×C4×D3C2×C3⋊D311yescharacteristic
291C2×Dic6—94no—
292C3×M16—34no—
293C3×M16—34no—
294C3×M16—34no—
295C3×M16—34no—
296C3×M16—91no—
297C3×M16—92no—
298C2×C3×C8—24no—
299C2×C3×C8—34no—
300C2×C3×C8—92no—
301C2×C3×C8D3214yescomplemented
302C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
303#208—34no—
304#208—98no—
305#208—94no—
306#208—91no—
307#208—92no—
308#208—184no—
309C2×C3⋊C8—98no—
310C2×C3⋊C8—92no—
311C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
312ΓL(1,𝔽25)—38no—
313ΓL(1,𝔽25)—98no—
314ΓL(1,𝔽25)—98no—
315ΓL(1,𝔽25)—92no—
316ΓL(1,𝔽25)—98no—
317ΓL(1,𝔽25)—92no—
318ΓL(1,𝔽25)—92no—
319ΓL(1,𝔽25)—188no—
320ΓL(1,𝔽25)C2×C3⋊D311yescharacteristic
321ΓL(1,𝔽25)C2×C3⋊D311yescharacteristic
322ΓL(1,𝔽25)C2×C3⋊D311yescharacteristic, complemented
323ΓL(1,𝔽25)C2×C3⋊D311yescharacteristic, complemented
324D3×C8—38no—
325D3×C8—98no—
326D3×C8—98no—
327D3×C8—92no—
328D3×C8—98no—
329D3×C8—92no—
330D3×C8—92no—
331D3×C8—188no—
332C22×C32—24no—
333C22×C32—31no—
334C22×C32—31no—
335C22×C32—121no—
336C22×C32C2×C4×D314yes—
337C22×C32C2×C4×D311yescharacteristic
338C2×C3⋊D3—32no—
339C2×C3⋊D3—38no—
340C2×C3⋊D3—68no—
341C2×C3⋊D3—122no—
342C2×C3⋊D3C2×C4×D312yes—
343C2×C3×D3—38no—
344C2×C3×D3—44no—
345C2×C3×D3C2×C4×D314yes—
346C2×C3×D3C2×C4×D314yes—
347D32—128no—
348C32×C4—24no—
349C32×C4—24no—
350C32×C4—31no—
351C32×C4—31no—
352C32×C4C2×C4×D314yes—
353C32×C4C2×C4×D311yescharacteristic
354C32×C4C23×D314yes—
355C32×C4C23×D311yescharacteristic
356C3⋊Dic3—32no—
357C3⋊Dic3—38no—
358C3⋊Dic3—68no—
359C3⋊Dic3C2×C4×D312yes—
360C3×Dic3—38no—
361C3×Dic3C2×C4×D314yes—
362C3×Dic3C2×C4×D314yes—
363C2×#41—271no—
364#131—91no—
365#131—271no—
366#131—271no—
367#131—272no—
368#131—272no—
369#131—271no—
370C2×M16—31no—
371C2×M16—272no—
372C2×M16—272no—
373C2×M16—271no—
374C33#56811yescharacteristic, complemented, Sylow
375C23×C3—34no—
376C22×D3—98no—
377C22×D3—92no—
378C22×D3C4×C3⋊D311yescharacteristic
379C3×Q8—91no—
380C3×D4—92no—
381C3×D4—91no—
382C2×C3×C4—24no—
383C2×C3×C4—34no—
384C2×C3×C4—34no—
385C2×C3×C4—34no—
386C2×C3×C4—34no—
387C2×C3×C4—34no—
388C2×C3×C4—92no—
389C2×C3×C4C2×D3214yes—
390C2×C3×C4C22×C3⋊D311yescharacteristic
391#67—38no—
392#67—98no—
393#67—98no—
394#67—98no—
395#67—92no—
396#67—92no—
397#67—92no—
398#67—188no—
399C2×Dic3—98no—
400C2×Dic3—92no—
401C2×Dic3C4×C3⋊D311yescharacteristic
402D12—34no—
403D12—98no—
404D12—92no—
405D12—94no—
406D12—91no—
407D12—184no—
408C4×D3—38no—
409C4×D3—98no—
410C4×D3—98no—
411C4×D3—92no—
412C4×D3—98no—
413C4×D3—92no—
414C4×D3—92no—
415C4×D3—188no—
416C4×D3C4×C3⋊D311yescharacteristic
417C4×D3C4×C3⋊D311yescharacteristic
418C4×D3C22×C3⋊D311yescharacteristic
419C4×D3C22×C3⋊D311yescharacteristic
420Dic6—34no—
421Dic6—98no—
422Dic6—94no—
423Dic6—91no—
424Dic6—92no—
425Dic6—184no—
426C3×C8—24no—
427C3×C8—24no—
428C3×C8—34no—
429C3×C8—34no—
430C3×C8—92no—
431C3×C8C2×D3214yes—
432C3×C8C2×D3214yes—
433C3×C8C22×C3⋊D311yescharacteristic
434C3×C8C22×C3⋊D311yescharacteristic
435C3⋊C8—38no—
436C3⋊C8—98no—
437C3⋊C8—92no—
438C3⋊C8—188no—
439C3⋊C8C22×C3⋊D311yescharacteristic
440C3⋊C8C22×C3⋊D311yescharacteristic
441C2×C32—21no—
442C2×C32—24no—
443C2×C32—24no—
444C2×C32—44no—
445C2×C32—61no—
446C2×C32—61no—
447C2×C32C22×C4×D314yes—
448C2×C32C22×C4×D311yescharacteristic
449C3⋊D3—22no—
450C3⋊D3—62no—
451C3⋊D3—68no—
452C3⋊D3—128no—
453C3×D3—24no—
454C3×D3—24no—
455C3×D3—68no—
456#41—91no—
457#41—272no—
458#41—272no—
459#41—271no—
460#41—271no—
461#41—271no—
462C2×Q8—271no—
463C2×D4—271no—
464C2×D4—272no—
465C22×C4—31no—
466C22×C4—272no—
467M16—31no—
468M16—31no—
469M16—31no—
470M16—31no—
471M16—91no—
472M16—92no—
473M16—272no—
474M16—272no—
475M16—272no—
476M16—271no—
477M16—272no—
478C2×C8—31no—
479C2×C8—92no—
480C2×C8—272no—
481C2×C8—272no—
482C2×C8—272no—
483C2×C8—272no—
484C2×C8D3×C3⋊D311yescharacteristic, complemented
485C22×C3—24no—
486C22×C3—34no—
487C22×C3—34no—
488C22×C3—92no—
489C22×C3—124no—
490C22×C3C4×D3214yes—
491C22×C3C2×C4×C3⋊D311yescharacteristic
492C2×D3—38no—
493C2×D3—41no—
494C2×D3—98no—
495C2×D3—92no—
496C2×D3—188no—
497C2×D3—362no—
498C2×D3—368no—
499C2×D3C2×C4×C3⋊D311yescharacteristic
500C2×D3C2×C4×C3⋊D311yescharacteristic
501C3×C4—24no—
502C3×C4—24no—
503C3×C4—34no—
504C3×C4—34no—
505C3×C4—92no—
506C3×C4C4×D3214yes—
507C3×C4C2×C4×C3⋊D311yescharacteristic
508C3×C4C22×D3214yes—
509C3×C4C23×C3⋊D311yescharacteristic
510Dic3—38no—
511Dic3—98no—
512Dic3—92no—
513Dic3—188no—
514Dic3C2×C4×C3⋊D311yescharacteristic
515Dic3C2×C4×C3⋊D311yescharacteristic
516C32—24no—
517C32#574514yescomplemented
518C32#575011yescharacteristic, complemented
519C23—31no—
520C23—272no—
521Q8—91no—
522Q8—272no—
523Q8—271no—
524D4—92no—
525D4—91no—
526D4—272no—
527D4—272no—
528D4—272no—
529D4—272no—
530D4—271no—
531C2×C4—31no—
532C2×C4—31no—
533C2×C4—31no—
534C2×C4—31no—
535C2×C4—31no—
536C2×C4—92no—
537C2×C4—272no—
538C2×C4—272no—
539C2×C4—272no—
540C2×C4—272no—
541C2×C4C2×D3×C3⋊D311yescharacteristic
542C8—31no—
543C8—31no—
544C8—92no—
545C8—272no—
546C8C2×D3×C3⋊D311yescharacteristic
547C8C2×D3×C3⋊D311yescharacteristic
548C2×C3—24no—
549C2×C3—21no—
550C2×C3—24no—
551C2×C3—44no—
552C2×C3—64no—
553C2×C3—64no—
554C2×C3—182no—
555C2×C3C2×C4×D3214yes—
556C2×C3C22×C4×C3⋊D311yescharacteristic
557D3—21no—
558D3—21no—
559D3—68no—
560D3—188no—
561D3—182no—
562D3—368no—
563C22—31no—
564C22—31no—
565C22—92no—
566C22—121no—
567C22—272no—
568C22—1082no—
569C22C4×D3×C3⋊D311yescharacteristic
570C4—31no—
571C4—31no—
572C4—92no—
573C4—272no—
574C4C4×D3×C3⋊D311yescharacteristic
575C4C22×D3×C3⋊D311yescharacteristic
576C3—24no—
577C3#1059466814yescomplemented
578C3#1059621611yescharacteristic, complemented
579C2—21no—
580C2—61no—
581C2—61no—
582C2—182no—
583C2—542no—
584C2C2×C4×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC4C22×D3×C3⋊D3
commutator, socleC2×C33C23×C4
FittingC2×C33×C8C22
prime radicalD3×#3828C2
cosocleC33×C4C24
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11723942C2semidirect
3. C3×#64110C2semidirect
4. C2×C32×ΓL(1,𝔽25)C2semidirect
5. #11723356C2semidirect
6. #11723352C2semidirect
7. D3×#3748C2semidirect
8. #11723334C2semidirect
9. #11723333C2semidirect
10. #11723312C2semidirect
11. C3⋊D3×ΓL(1,𝔽25)C2semidirect
25. C2×C33×C8C22semidirect
26. #88855C22semidirect
27. #88850C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
29. C3×#3748C22semidirect
30. C3×#3743C22semidirect
31. C3×C8×C3⋊D3C22semidirect
32. C2×C32×C3⋊C8C22semidirect
35. C32×ΓL(1,𝔽25)C22semidirect
36. C32×ΓL(1,𝔽25)C22semidirect
37. #88788C22semidirect
38. #88787C22semidirect
40. #88785C22semidirect
41. C3⋊D3×C3⋊C8C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
56. #64110D3semidirect
58. C2×C3×ΓL(1,𝔽25)D3semidirect
126. C2×C32×C8C2×D3semidirect
127. C2×C32×C8C2×D3semidirect
132. #3748C2×D3semidirect
139. #3743C2×D3semidirect
145. C8×C3⋊D3C2×D3semidirect
147. C2×C3×C3⋊C8C2×D3semidirect
151. C3×ΓL(1,𝔽25)C2×D3semidirect
152. C3×ΓL(1,𝔽25)C2×D3semidirect
195. C2×ΓL(1,𝔽25)C3⋊D3semidirect
301. C2×C3×C8D32semidirect
302. C2×C3×C8C2×C3⋊D3semidirect
311. C2×C3⋊C8C2×C3⋊D3semidirect
322. ΓL(1,𝔽25)C2×C3⋊D3semidirect
323. ΓL(1,𝔽25)C2×C3⋊D3semidirect
374. C33#568semidirect
484. C2×C8D3×C3⋊D3semidirect
517. C32#5745semidirect
518. C32#5750semidirect
577. C3#10594668semidirect
578. C3#10596216semidirect

Maximal subgroups

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

subgroupindex
D3×#38282
#117239422
C3×#641102
C2×C32×ΓL(1,𝔽25)2
#117233562
#117233522
D3×#37482
#117233342
#117233332
#117233122
C3⋊D3×ΓL(1,𝔽25)2
#105962163
#105946683