Finite Group Databasebeta

← #13176685

Subgroups — #13176685

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724578C211yesmaximal, characteristic
2M16×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#64116C211yesmaximal, characteristic, complemented
4C32×#907C211yesmaximal, characteristic, complemented
5C2×#88788C211yesmaximal, characteristic, complemented
6#11723352C211yesmaximal, characteristic, complemented
7#11723349C211yesmaximal, characteristic, complemented
8#11723334C212yesmaximal, complemented
9#11723331C212yesmaximal, complemented
10C3⋊D3×ΓL(1,𝔽25)C212yesmaximal, complemented
11D3×#3743C212yesmaximal, complemented
12#10596221—31nomaximal
13#10594708—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C3×#3828C2211yescharacteristic
16C32×#235C2211yescharacteristic
17#89039C412yes—
18#89029C412yes—
19#89026C411yescharacteristic
20C4×D3×C3⋊D3C2212yes—
21#89020C2212yes—
22#89019C411yescharacteristic
23#89017C411yescharacteristic
24#89015C411yescharacteristic
25C33×M16C2211yescharacteristic, complemented
26#88855C2211yescharacteristic, complemented
27#88850C2212yes—
28C8×C3⋊(C3⋊D3)C2212yescomplemented
29C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
30C3×#3743C2212yescomplemented
31C3×C8×C3⋊D3C2212yescomplemented
32C2×C32×C3⋊C8C2211yescharacteristic, complemented
33C32×ΓL(1,𝔽25)C2212yescomplemented
34C32×D3×C8C2212yescomplemented
35#88788C2211yescharacteristic
36#88788C2211yescharacteristic
37#88788C2211yescharacteristic, complemented
38#88788C2211yescharacteristic, complemented
39#88787C2212yescomplemented
40#88786C2212yescomplemented
41C3⋊D3×C3⋊C8C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43#41×C3⋊D3—31no—
44#64306—34no—
45C32×#131—31no—
46#64157—31no—
47#64116—32no—
48#64116D311yescharacteristic, complemented
49M16×C3⋊D3—31no—
50M16×C3⋊D3—32no—
51M16×C3⋊D3—34no—
52M16×C3⋊D3—64no—
53#64110—31no—
54#64110—32no—
55C2×#3743—31no—
56C2×#3743—32no—
57C3×#907—34no—
58C3×#907D314yescomplemented
59C2×#3712—34no—
60#63814—34no—
61#63814—34no—
62#63805—38no—
63#63803—38no—
64D3×ΓL(1,𝔽25)—38no—
65D3×ΓL(1,𝔽25)—38no—
66C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
67C2×D3×C3⋊D3C2×C412yes—
68C2×C33×C4C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
70C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
71C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
72C4×C3⋊(C3⋊D3)C2311yescharacteristic
73C4×C3⋊(C3⋊D3)C2311yescharacteristic
74C3×#638C2×C412yes—
75C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
76C3×C4×C3⋊D3C2312yes—
77C3×#634C2×C411yescharacteristic
78C32×#67C2×C412yes—
79C32×D12C2×C411yescharacteristic
80C32×C4×D3C2312yes—
81C32×Dic6C2×C411yescharacteristic
82#6293C2×C412yes—
83#6292C2×C412yes—
84#6291C2×C412yes—
85#6290C2×C412yes—
86#6289C2×C412yes—
87Dic3×C3⋊D3C2×C412yes—
88D3×C3⋊Dic3C2×C412yes—
89C33×C8C2312yes—
90C3⋊(C3⋊(C3⋊C8))C2312yes—
91C3×C3⋊(C3⋊C8)C2311yescharacteristic
92C3×C3⋊(C3⋊C8)C2311yescharacteristic
93C32×C3⋊C8C2311yescharacteristic
94C32×C3⋊C8C2311yescharacteristic
95#5750—94no—
96#5750—91no—
97C32×#41—31no—
98#3832—31no—
99Q8×C3⋊D3—31no—
100#3830—32no—
101#3830—31no—
102D4×C3⋊D3—32no—
103D4×C3⋊D3—31no—
104#3828—32no—
105#3828C2×D311yescharacteristic
106C2×C4×C3⋊D3—31no—
107C2×C4×C3⋊D3—34no—
108C2×C4×C3⋊D3—32no—
109C2×C4×C3⋊D3—64no—
110C3×#235—34no—
111C3×#235C2×D314yes—
112#3811—38no—
113#3805—38no—
114#3802—34no—
115C4×D32—38no—
116#3798—38no—
117#3797—34no—
118#3796—34no—
119#3796—34no—
120C32×M16—24no—
121C32×M16—32no—
122C32×M16C2×D314yescomplemented
123C32×M16C2×D311yescharacteristic, complemented
124C2×C32×C8—31no—
125C2×C32×C8—32no—
126#3748—31no—
127#3748—34no—
128#3748—32no—
129#3748—64no—
130C2×C3⋊(C3⋊C8)—32no—
131C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
132#3743—32no—
133#3743—32no—
134#3743—38no—
135#3743—31no—
136#3743—31no—
137#3743—31no—
138#3743—31no—
139#3743—68no—
140#3743C2×D312yescomplemented
141C8×C3⋊D3—32no—
142C8×C3⋊D3—32no—
143C8×C3⋊D3—38no—
144C8×C3⋊D3—68no—
145C8×C3⋊D3C2×D312yescomplemented
146C2×C3×C3⋊C8—34no—
147C2×C3×C3⋊C8C2×D314yescomplemented
148C3×ΓL(1,𝔽25)—38no—
149C3×ΓL(1,𝔽25)C2×D318yescomplemented
150C3×D3×C8—38no—
151C3×D3×C8C2×D318yescomplemented
152#3712—34no—
153#3712—34no—
154#3712—34no—
155#3712—34no—
156#3711—38no—
157#3711—38no—
158D3×C3⋊C8—38no—
159D3×C3⋊C8—38no—
160C22×C33C22×C411yescharacteristic
161C2×C3⋊(C3⋊D3)—41no—
162C2×C3⋊(C3⋊D3)C22×C411yescharacteristic
163C2×C3⋊(C3⋊D3)C22×C411yescharacteristic
164C2×C3×C3⋊D3C22×C412yes—
165C2×C32×D3C22×C412yes—
166D3×C3⋊D3—42no—
167C33×C4C22×C411yescharacteristic
168C33×C4C2411yescharacteristic
169C3⋊(C3⋊Dic3)C22×C411yescharacteristic
170C3⋊(C3⋊Dic3)C22×C411yescharacteristic
171C3×C3⋊Dic3C22×C412yes—
172C32×Dic3C22×C412yes—
173D3×#41—94no—
174D3×#41—91no—
175C3×#131—34no—
176C3×#131—91no—
177#948—94no—
178#948—91no—
179#907—34no—
180#907—98no—
181#907—92no—
182#907C3⋊D311yescharacteristic, complemented
183D3×M16—98no—
184D3×M16—94no—
185D3×M16—91no—
186D3×M16—92no—
187D3×M16—184no—
188#901—98no—
189#901—94no—
190#901—91no—
191#901—92no—
192C2×ΓL(1,𝔽25)—98no—
193C2×ΓL(1,𝔽25)—94no—
194C2×ΓL(1,𝔽25)—91no—
195C2×ΓL(1,𝔽25)—92no—
196C22×C3⋊D3—31no—
197C22×C3⋊D3—32no—
198C22×C3⋊D3—34no—
199C22×C3⋊D3—64no—
200C2×D32—38no—
201C32×Q8—31no—
202C32×D4—32no—
203C32×D4—31no—
204C2×C32×C4—24no—
205C2×C32×C4—32no—
206C2×C32×C4C22×D314yes—
207C2×C32×C4C22×D311yescharacteristic
208#638—32no—
209#638—32no—
210#638C4×D312yes—
211C2×C3⋊Dic3—31no—
212C2×C3⋊Dic3—32no—
213C2×C3⋊Dic3—34no—
214C2×C3⋊Dic3—64no—
215C4⋊(C3⋊D3)—32no—
216C4⋊(C3⋊D3)C4×D311yescharacteristic
217C4×C3⋊D3—32no—
218C4×C3⋊D3—31no—
219C4×C3⋊D3—31no—
220C4×C3⋊D3—31no—
221C4×C3⋊D3—31no—
222C4×C3⋊D3—32no—
223C4×C3⋊D3—34no—
224C4×C3⋊D3—34no—
225C4×C3⋊D3—34no—
226C4×C3⋊D3—34no—
227C4×C3⋊D3—64no—
228C4×C3⋊D3—64no—
229C4×C3⋊D3—64no—
230C4×C3⋊D3—64no—
231C4×C3⋊D3C22×D312yes—
232#634—32no—
233#634C4×D311yescharacteristic
234C3×#67—38no—
235C3×#67C4×D318yes—
236C3×D12—34no—
237C3×D12C4×D314yes—
238C3×C4×D3—38no—
239C3×C4×D3C22×D318yes—
240C3×Dic6—34no—
241C3×Dic6C4×D314yes—
242#627—38no—
243#626—38no—
244#626—38no—
245#625—38no—
246#624—38no—
247D3×Dic3—38no—
248D3×Dic3—38no—
249C32×C8—28no—
250C32×C8—31no—
251C32×C8—31no—
252C32×C8C22×D318yes—
253C32×C8C22×D312yes—
254C3⋊(C3⋊C8)—32no—
255C3⋊(C3⋊C8)—38no—
256C3⋊(C3⋊C8)—68no—
257C3⋊(C3⋊C8)C22×D311yescharacteristic
258C3⋊(C3⋊C8)C22×D311yescharacteristic
259C3×C3⋊C8—34no—
260C3×C3⋊C8—34no—
261C3×C3⋊C8C22×D314yes—
262C3×C3⋊C8C22×D314yes—
263#568—271noSylow
264C2×C33—21no—
265C2×C33C23×C411yescharacteristic
266C3⋊(C3⋊D3)—21no—
267C3⋊(C3⋊D3)—21no—
268C3×C3⋊D3—22no—
269C32×D3—22no—
270C3×#41—34no—
271C3×#41—91no—
272#239—94no—
273#239—91no—
274D3×Q8—94no—
275D3×Q8—91no—
276#237—94no—
277#237—98no—
278#237—92no—
279#237—91no—
280D3×D4—94no—
281D3×D4—98no—
282D3×D4—92no—
283D3×D4—91no—
284#235—34no—
285#235—98no—
286#235—92no—
287#235C2×C3⋊D311yescharacteristic
288C2×C4×D3—98no—
289C2×C4×D3—94no—
290C2×C4×D3—91no—
291C2×C4×D3—92no—
292C2×C4×D3—184no—
293C3×M16—24no—
294C3×M16—38no—
295C3×M16—92no—
296C3×M16D3214yescomplemented
297C3×M16C2×C3⋊D311yescharacteristic, complemented
298C2×C3×C8—34no—
299C2×C3×C8—38no—
300C2×C3×C8—91no—
301C2×C3×C8—92no—
302#208—98no—
303#208—94no—
304#208—91no—
305#208—92no—
306#208—184no—
307C2×C3⋊C8—34no—
308C2×C3⋊C8—98no—
309C2×C3⋊C8—92no—
310C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
311ΓL(1,𝔽25)—38no—
312ΓL(1,𝔽25)—94no—
313ΓL(1,𝔽25)—94no—
314ΓL(1,𝔽25)—94no—
315ΓL(1,𝔽25)—94no—
316ΓL(1,𝔽25)—98no—
317ΓL(1,𝔽25)—92no—
318ΓL(1,𝔽25)—98no—
319ΓL(1,𝔽25)—92no—
320ΓL(1,𝔽25)—91no—
321ΓL(1,𝔽25)—91no—
322ΓL(1,𝔽25)—91no—
323ΓL(1,𝔽25)—91no—
324ΓL(1,𝔽25)—188no—
325ΓL(1,𝔽25)C2×C3⋊D312yescomplemented
326D3×C8—38no—
327D3×C8—98no—
328D3×C8—92no—
329D3×C8—98no—
330D3×C8—92no—
331D3×C8—188no—
332D3×C8C2×C3⋊D312yescomplemented
333C22×C32—24no—
334C22×C32—32no—
335C22×C32C2×C4×D314yes—
336C22×C32C2×C4×D311yescharacteristic
337C2×C3⋊D3—31no—
338C2×C3⋊D3—31no—
339C2×C3⋊D3—34no—
340C2×C3⋊D3—34no—
341C2×C3⋊D3—64no—
342C2×C3⋊D3—64no—
343C2×C3⋊D3—122no—
344C2×C3⋊D3—121no—
345C2×C3⋊D3—124no—
346C2×C3⋊D3—128no—
347C2×C3⋊D3C2×C4×D312yes—
348C2×C3×D3—38no—
349C2×C3×D3C2×C4×D318yes—
350D32—128no—
351C32×C4—24no—
352C32×C4—24no—
353C32×C4—32no—
354C32×C4C2×C4×D314yes—
355C32×C4C2×C4×D311yescharacteristic
356C32×C4C23×D314yes—
357C32×C4C23×D311yescharacteristic
358C3⋊Dic3—31no—
359C3⋊Dic3—31no—
360C3⋊Dic3—34no—
361C3⋊Dic3—34no—
362C3⋊Dic3—64no—
363C3⋊Dic3—64no—
364C3⋊Dic3C2×C4×D312yes—
365C3×Dic3—38no—
366C3×Dic3C2×C4×D318yes—
367C2×#41—271no—
368#131—31no—
369#131—91no—
370#131—272no—
371#131—271no—
372#131—271no—
373#131—272no—
374C2×M16—271no—
375C2×M16—272no—
376C2×M16—272no—
377C2×M16—271no—
378C33#56811yescharacteristic, complemented, Sylow
379C22×D3—98no—
380C22×D3—94no—
381C22×D3—91no—
382C22×D3—92no—
383C22×D3—184no—
384C3×Q8—34no—
385C3×Q8—91no—
386C3×D4—38no—
387C3×D4—34no—
388C3×D4—92no—
389C3×D4—91no—
390C2×C3×C4—24no—
391C2×C3×C4—38no—
392C2×C3×C4—92no—
393C2×C3×C4C2×D3214yes—
394C2×C3×C4C22×C3⋊D311yescharacteristic
395#67—38no—
396#67—98no—
397#67—98no—
398#67—92no—
399#67—92no—
400#67C4×C3⋊D312yes—
401C2×Dic3—98no—
402C2×Dic3—94no—
403C2×Dic3—91no—
404C2×Dic3—92no—
405C2×Dic3—184no—
406D12—34no—
407D12—98no—
408D12—92no—
409D12C4×C3⋊D311yescharacteristic
410C4×D3—38no—
411C4×D3—98no—
412C4×D3—98no—
413C4×D3—92no—
414C4×D3—94no—
415C4×D3—94no—
416C4×D3—91no—
417C4×D3—91no—
418C4×D3—94no—
419C4×D3—94no—
420C4×D3—91no—
421C4×D3—91no—
422C4×D3—92no—
423C4×D3—184no—
424C4×D3—184no—
425C4×D3—184no—
426C4×D3—184no—
427C4×D3C22×C3⋊D312yes—
428Dic6—34no—
429Dic6—98no—
430Dic6—92no—
431Dic6C4×C3⋊D311yescharacteristic
432C3×C8—28no—
433C3×C8—34no—
434C3×C8—34no—
435C3×C8—91no—
436C3×C8—91no—
437C3×C8C2×D3218yes—
438C3×C8C22×C3⋊D312yes—
439C3⋊C8—34no—
440C3⋊C8—34no—
441C3⋊C8—98no—
442C3⋊C8—92no—
443C3⋊C8—188no—
444C3⋊C8C22×C3⋊D311yescharacteristic
445C3⋊C8C22×C3⋊D311yescharacteristic
446C2×C32—21no—
447C2×C32—24no—
448C2×C32—24no—
449C2×C32—44no—
450C2×C32—62no—
451C2×C32C22×C4×D314yes—
452C2×C32C22×C4×D311yescharacteristic
453C3⋊D3—22no—
454C3⋊D3—68no—
455C3⋊D3—64no—
456C3⋊D3—61no—
457C3⋊D3—61no—
458C3⋊D3—64no—
459C3⋊D3—124no—
460C3×D3—28no—
461C3×D3—68no—
462#41—31no—
463#41—91no—
464#41—272no—
465#41—272no—
466#41—271no—
467#41—271no—
468C2×Q8—271no—
469C2×D4—272no—
470C2×D4—271no—
471C22×C4—272no—
472C22×C4—271no—
473M16—32no—
474M16—92no—
475M16—271no—
476M16—271no—
477M16—271no—
478M16—271no—
479M16—272no—
480M16—272no—
481M16—271no—
482M16—272no—
483M16D3×C3⋊D311yescharacteristic, complemented
484C2×C8—31no—
485C2×C8—32no—
486C2×C8—91no—
487C2×C8—92no—
488C2×C8—272no—
489C2×C8—272no—
490C2×C8—272no—
491C22×C3—24no—
492C22×C3—38no—
493C22×C3—92no—
494C22×C3C4×D3214yes—
495C22×C3C2×C4×C3⋊D311yescharacteristic
496C2×D3—38no—
497C2×D3—94no—
498C2×D3—94no—
499C2×D3—91no—
500C2×D3—91no—
501C2×D3—184no—
502C2×D3—184no—
503C2×D3—362no—
504C2×D3—368no—
505C2×D3—361no—
506C2×D3—364no—
507C2×D3—368no—
508C2×D3C2×C4×C3⋊D312yes—
509C3×C4—24no—
510C3×C4—24no—
511C3×C4—38no—
512C3×C4—92no—
513C3×C4C4×D3214yes—
514C3×C4C2×C4×C3⋊D311yescharacteristic
515C3×C4C22×D3214yes—
516C3×C4C23×C3⋊D311yescharacteristic
517Dic3—38no—
518Dic3—94no—
519Dic3—94no—
520Dic3—91no—
521Dic3—91no—
522Dic3—184no—
523Dic3—184no—
524Dic3C2×C4×C3⋊D312yes—
525C32—24no—
526C32#575014yescomplemented
527C32#575011yescharacteristic, complemented
528C23—272no—
529C23—271no—
530Q8—31no—
531Q8—91no—
532Q8—272no—
533D4—32no—
534D4—31no—
535D4—92no—
536D4—91no—
537D4—272no—
538D4—272no—
539D4—272no—
540C2×C4—32no—
541C2×C4—92no—
542C2×C4—272no—
543C2×C4—272no—
544C2×C4—272no—
545C2×C4—271no—
546C2×C4—271no—
547C2×C4—271no—
548C2×C4—271no—
549C2×C4—271no—
550C2×C4C2×D3×C3⋊D311yescharacteristic
551C8—31no—
552C8—31no—
553C8—91no—
554C8—91no—
555C8—272no—
556C8C2×D3×C3⋊D312yes—
557C2×C3—21no—
558C2×C3—24no—
559C2×C3—24no—
560C2×C3—44no—
561C2×C3—68no—
562C2×C3—182no—
563C2×C3C2×C4×D3214yes—
564C2×C3C22×C4×C3⋊D311yescharacteristic
565D3—22no—
566D3—68no—
567D3—188no—
568D3—181no—
569D3—181no—
570D3—184no—
571D3—184no—
572D3—364no—
573C22—32no—
574C22—92no—
575C22—271no—
576C22—271no—
577C22—1081no—
578C22—1082no—
579C22C4×D3×C3⋊D311yescharacteristic
580C4—32no—
581C4—92no—
582C4—271no—
583C4—271no—
584C4C4×D3×C3⋊D311yescharacteristic
585C4C22×D3×C3⋊D311yescharacteristic
586C3—24no—
587C3#1059470814yescomplemented
588C3#1059622111yescharacteristic, complemented
589C2—21no—
590C2—62no—
591C2—182no—
592C2—541no—
593C2—541no—
594C2C2×C4×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC4C22×D3×C3⋊D3
commutator, socleC2×C33C23×C4
FittingC33×M16C22
prime radical#11724578C2
cosocleC33×C4C24
hypercenterM16D3×C3⋊D3
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. M16×C3⋊(C3⋊D3)C2semidirect
3. C3×#64116C2semidirect
4. C32×#907C2semidirect
5. C2×#88788C2semidirect
6. #11723352C2semidirect
7. #11723349C2semidirect
8. #11723334C2semidirect
9. #11723331C2semidirect
10. C3⋊D3×ΓL(1,𝔽25)C2semidirect
11. D3×#3743C2semidirect
25. C33×M16C22semidirect
26. #88855C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
29. C2×C3×C3⋊(C3⋊C8)C22semidirect
30. C3×#3743C22semidirect
31. C3×C8×C3⋊D3C22semidirect
32. C2×C32×C3⋊C8C22semidirect
33. C32×ΓL(1,𝔽25)C22semidirect
34. C32×D3×C8C22semidirect
37. #88788C22semidirect
38. #88788C22semidirect
39. #88787C22semidirect
40. #88786C22semidirect
41. C3⋊D3×C3⋊C8C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
48. #64116D3semidirect
58. C3×#907D3semidirect
122. C32×M16C2×D3semidirect
123. C32×M16C2×D3semidirect
131. C2×C3⋊(C3⋊C8)C2×D3semidirect
140. #3743C2×D3semidirect
145. C8×C3⋊D3C2×D3semidirect
147. C2×C3×C3⋊C8C2×D3semidirect
149. C3×ΓL(1,𝔽25)C2×D3semidirect
151. C3×D3×C8C2×D3semidirect
182. #907C3⋊D3semidirect
296. C3×M16D32semidirect
297. C3×M16C2×C3⋊D3semidirect
310. C2×C3⋊C8C2×C3⋊D3semidirect
325. ΓL(1,𝔽25)C2×C3⋊D3semidirect
332. D3×C8C2×C3⋊D3semidirect
378. C33#568semidirect
483. M16D3×C3⋊D3semidirect
526. C32#5750semidirect
527. C32#5750semidirect
587. C3#10594708semidirect
588. C3#10596221semidirect

Maximal subgroups

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

subgroupindex
#117245782
M16×C3⋊(C3⋊D3)2
C3×#641162
C32×#9072
C2×#887882
#117233522
#117233492
#117233342
#117233312
C3⋊D3×ΓL(1,𝔽25)2
D3×#37432
#105962213
#105947083