Finite Group Databasebeta

← C3⋊D3×#909 #13176687

Subgroups — C3⋊D3×#909 #13176687

Order 1728.

1788 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3⋊D3×#235C211yesmaximal, characteristic
2C2×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
3#11723950C211yesmaximal, characteristic, complemented
4C3×M16×C3⋊D3C211yesmaximal, characteristic, complemented
5C32×#909C211yesmaximal, characteristic, complemented
6#11723381C211yesmaximal, characteristic, complemented
7#11723379C211yesmaximal, characteristic, complemented
8#11723340C212yesmaximal, complemented
9#11723329C212yesmaximal, complemented
10C3⋊D3×Dic12C212yesmaximal, complemented
11C3⋊D3×#204C212yesmaximal, complemented
12C3⋊D3×#137—31nomaximal
13D3×#909—34nomaximal
14#89076C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C32×#235C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic, complemented
19C3⋊D3×#67—21no—
20C2×#6293—21no—
21C2×Dic3×C3⋊D3—21no—
22#89028—21no—
23C3⋊D3×D12C2211yescharacteristic
24C4×D3×C3⋊D3—21no—
25#89020—21no—
26#89018C2211yescharacteristic
27C3⋊D3×Dic6—21no—
28C3⋊D3×Dic6—21no—
29C3⋊D3×Dic6C2212yes—
30C3⋊D3×Dic6C2211yescharacteristic
31C3⋊D3×Dic6C2212yescomplemented
32#89014C2211yescharacteristic
33C33×M16C2211yescharacteristic, complemented
34#88853C2212yescomplemented
35#88851C2212yescomplemented
36C3×#3748C2211yescharacteristic, complemented
37C3×#3743C2212yescomplemented
38C3×C8×C3⋊D3C2212yes—
39C32×Dic12C2212yescomplemented
40C32×#204C2212yescomplemented
41#88799C2212yescomplemented
42#88799C2212yescomplemented
43#88795C2212yescomplemented
44#88794C2212yescomplemented
45#41×C3⋊D3—31no—
46C2×Q8×C3⋊D3—31no—
47D3×#235—34no—
48C2×D3×Dic6—34no—
49C32×#137—31no—
50#64160—31no—
51#64151—31no—
52#64127—32no—
53Q16×C3⋊D3—32no—
54#64124—32no—
55SD16×C3⋊D3—32no—
56#64118—31no—
57#64118—34no—
58#64118—64no—
59M16×C3⋊D3D311yescharacteristic, complemented
60C3×#909D314yescomplemented
61C3×D3×M16—34no—
62#63833—34no—
63#63831—34no—
64#63801—38no—
65#63800—38no—
66D3×Dic12—38no—
67D3×#204—38no—
68C22×C3×C3⋊D3D411yescharacteristic
69C2×D3×C3⋊D3—21no—
70C2×C33×C4C2311yescharacteristic
71#6312—21no—
72C2×C3⋊(C3⋊Dic3)—21no—
73C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
74C4×C3⋊(C3⋊D3)—21no—
75#6308—21no—
76#6308C2312yes—
77#6308C2311yescharacteristic
78C2×C3×C3⋊Dic3D411yescharacteristic
79C3×C4×C3⋊D3D411yescharacteristic
80C3×C4×C3⋊D3D411yescharacteristic
81C3×C4×C3⋊D3C2311yescharacteristic
82C3×C4×C3⋊D3C2311yescharacteristic
83C32×#67—21no—
84C2×C32×Dic3—21no—
85C32×D12C2311yescharacteristic
86C32×C4×D3—21no—
87C32×Dic6—21no—
88C32×Dic6C2312yes—
89C32×Dic6C2311yescharacteristic
90#6293—22no—
91#6293—22no—
92#6293—21no—
93#6292—21no—
94#6291—21no—
95#6290—21no—
96#6289—21no—
97Dic3×C3⋊D3—22no—
98Dic3×C3⋊D3—22no—
99Dic3×C3⋊D3—21no—
100D3×C3⋊Dic3—21no—
101C33×C8C2312yes—
102C3×C3⋊(C3⋊C8)C2312yes—
103D3×#137—94no—
104C2×#909—91no—
105C32×#41—31no—
106C2×C32×Q8—31no—
107#3832—31no—
108Q8×C3⋊D3—32no—
109Q8×C3⋊D3—32no—
110Q8×C3⋊D3—31no—
111Q8×C3⋊D3—61no—
112Q8×C3⋊D3—61no—
113#3830—31no—
114#3830—61no—
115D4×C3⋊D3—31no—
116D4×C3⋊D3—61no—
117#3828—31no—
118#3828—34no—
119#3828—61no—
120#3828—64no—
121C2×C4×C3⋊D3—61no—
122C2×C4×C3⋊D3—61no—
123C2×C4×C3⋊D3C2×D311yescharacteristic
124C2×#634—31no—
125C2×#634—34no—
126C2×#634—61no—
127C2×#634—64no—
128C3×#235C2×D314yes—
129C2×C3×C4×D3—34no—
130C2×C3×Dic6C2×D314yescomplemented
131D3×#67—64no—
132C2×#627—64no—
133#3804—64no—
134C2×D3×Dic3—64no—
135D3×D12—34no—
136C4×D32—64no—
137#3799—34no—
138#3798—64no—
139#3795—34no—
140D3×Dic6—38no—
141D3×Dic6—38no—
142D3×Dic6—34no—
143D3×Dic6—64no—
144D3×Dic6—64no—
145C32×Q16—32no—
146C32×SD16—32no—
147C32×M16—24no—
148C32×M16C2×D314yescomplemented
149C32×M16C2×D311yescharacteristic, complemented
150#3756—32no—
151#3756—32no—
152#3755—32no—
153#3754—32no—
154#3748C2×D311yescharacteristic, complemented
155#3746—32no—
156#3746—38no—
157#3746—68no—
158#3744—32no—
159#3744—38no—
160#3744—68no—
161#3743C2×D312yescomplemented
162C8×C3⋊D3C2×D312yes—
163C3×#208—34no—
164C3×Dic12C2×D318yescomplemented
165C3×#204C2×D318yescomplemented
166C3×ΓL(1,𝔽25)—38no—
167C3×D3×C8—38no—
168#3719—38no—
169#3719—38no—
170#3717—38no—
171#3716—38no—
172C22×C33C2×D411yescharacteristic
173C2×C3⋊(C3⋊D3)—21no—
174C2×C3×C3⋊D3—22no—
175C2×C3×C3⋊D3C2×D411yescharacteristic
176C2×C3×C3⋊D3C2×D411yescharacteristic
177C2×C32×D3—21no—
178D3×C3⋊D3—42no—
179C33×C4C2×D411yescharacteristic
180C33×C4C2411yescharacteristic
181C3⋊(C3⋊Dic3)—21no—
182C3⋊(C3⋊Dic3)—22no—
183C3×C3⋊Dic3C2×D411yescharacteristic
184C3×C3⋊Dic3C2×D411yescharacteristic
185C32×Dic3—22no—
186C32×Dic3—21no—
187D3×#41—94no—
188C2×D3×Q8—94no—
189C2×#235—91no—
190C22×Dic6—91no—
191C3×#137—34no—
192C2×C3×M16—91no—
193#951—94no—
194#942—94no—
195#918—98no—
196D3×Q16—98no—
197#915—98no—
198D3×SD16—98no—
199#909—91no—
200#909—91no—
201#909—94no—
202#909—91no—
203#909—92no—
204#909—92no—
205#909—184no—
206#909C3⋊D311yescharacteristic, direct product factor
207D3×M16—34no—
208C2×Dic12—92no—
209C2×#204—92no—
210C22×C3⋊D3—61no—
211C22×C3⋊D3D1211yescharacteristic
212C22×C3×D3—34no—
213C2×D32—64no—
214C32×Q8—32no—
215C32×Q8—31no—
216C32×Q8—61no—
217C32×D4—31no—
218C32×D4—61no—
219C2×C32×C4—24no—
220C2×C32×C4—61no—
221C2×C32×C4—61no—
222C2×C32×C4C22×D314yes—
223C2×C32×C4C22×D311yescharacteristic
224#638—61no—
225#638—61no—
226#638—61no—
227#638—64no—
228#638—124no—
229C2×C3⋊Dic3—61no—
230C2×C3⋊Dic3—61no—
231C2×C3⋊Dic3—64no—
232C2×C3⋊Dic3—124no—
233C2×C3⋊Dic3D1211yescharacteristic
234C4⋊(C3⋊D3)—31no—
235C4⋊(C3⋊D3)—34no—
236C4⋊(C3⋊D3)—64no—
237C4⋊(C3⋊D3)—61no—
238C4×C3⋊D3—61no—
239C4×C3⋊D3—62no—
240C4×C3⋊D3—62no—
241C4×C3⋊D3—61no—
242C4×C3⋊D3—61no—
243C4×C3⋊D3—64no—
244C4×C3⋊D3—124no—
245C4×C3⋊D3D1211yescharacteristic
246C4×C3⋊D3D1211yescharacteristic
247C4×C3⋊D3C22×D311yescharacteristic
248C4×C3⋊D3C22×D311yescharacteristic
249#634—32no—
250#634—31no—
251#634—38no—
252#634—34no—
253#634—68no—
254#634—64no—
255#634—61no—
256#634—62no—
257#634—62no—
258#634—61no—
259#634—64no—
260#634—124no—
261C3×#67—24no—
262C2×C3×Dic3—24no—
263C2×C3×Dic3—34no—
264C3×D12C22×D314yes—
265C3×C4×D3—24no—
266C3×C4×D3—34no—
267C3×C4×D3—34no—
268C3×C4×D3—34no—
269C3×C4×D3—34no—
270C3×Dic6—24no—
271C3×Dic6C22×D318yes—
272C3×Dic6C22×D314yes—
273#627—68no—
274#627—68no—
275#627—64no—
276#626—64no—
277#626—64no—
278#625—64no—
279#624—64no—
280D3×Dic3—68no—
281D3×Dic3—68no—
282D3×Dic3—64no—
283D3×Dic3—64no—
284C32×C8—28no—
285C32×C8C22×D318yes—
286C32×C8C22×D312yes—
287C3⋊(C3⋊C8)C22×D312yes—
288C3×C3⋊C8—38no—
289C2×#137—271noSylow
290C2×C33—21no—
291C2×C33C22×D411yescharacteristic
292C3⋊(C3⋊D3)—41no—
293C3×C3⋊D3—21no—
294C3×C3⋊D3#13712yescomplemented
295C32×D3—41no—
296C3×#41—34no—
297C2×C3×Q8—34no—
298C22×C3×C4—91no—
299C2×#67—181no—
300C22×Dic3—181no—
301#239—94no—
302D3×Q8—98no—
303D3×Q8—98no—
304D3×Q8—94no—
305D3×Q8—184no—
306D3×Q8—184no—
307#237—94no—
308#237—184no—
309D3×D4—94no—
310D3×D4—184no—
311#235—94no—
312#235—91no—
313#235—91no—
314#235—91no—
315#235—184no—
316#235—181no—
317#235—184no—
318#235—181no—
319#235C2×C3⋊D311yescharacteristic
320C2×D12—91no—
321C2×C4×D3—34no—
322C2×C4×D3—181no—
323C2×C4×D3—184no—
324C2×C4×D3—184no—
325C2×Dic6—94no—
326C2×Dic6—91no—
327C2×Dic6—92no—
328C2×Dic6—91no—
329C2×Dic6—92no—
330C2×Dic6—184no—
331C2×Dic6—181no—
332C2×Dic6—181no—
333C2×Dic6—181no—
334C2×Dic6—184no—
335C2×Dic6C2×C3⋊D311yescharacteristic, complemented
336C3×Q16—38no—
337C3×SD16—38no—
338C3×M16—24no—
339C3×M16—91no—
340C3×M16—92no—
341C3×M16D3214yescomplemented
342C3×M16C2×C3⋊D311yescharacteristic, complemented
343C2×C3×C8—92no—
344#216—98no—
345#216—98no—
346#215—98no—
347#214—98no—
348#208—34no—
349Dic12—92no—
350Dic12—92no—
351Dic12—98no—
352Dic12—92no—
353Dic12—188no—
354Dic12C2×C3⋊D312yescomplemented
355#204—92no—
356#204—92no—
357#204—98no—
358#204—92no—
359#204—188no—
360#204C2×C3⋊D312yescomplemented
361ΓL(1,𝔽25)—38no—
362D3×C8—38no—
363C22×C32—24no—
364C22×C32—61no—
365C22×C32C2×D1211yescharacteristic
366C22×C32D3×D414yes—
367C2×C3⋊D3—22no—
368C2×C3⋊D3—61no—
369C2×C3⋊D3—64no—
370C2×C3⋊D3—124no—
371C2×C3⋊D3—122no—
372C2×C3⋊D3C2×D1211yescharacteristic
373C2×C3⋊D3C2×D1211yescharacteristic
374C2×C3×D3—24no—
375C2×C3×D3—34no—
376C2×C3×D3—34no—
377C2×C3×D3—68no—
378D32—128no—
379C32×C4—24no—
380C32×C4—24no—
381C32×C4—62no—
382C32×C4—61no—
383C32×C4C2×D1211yescharacteristic
384C32×C4D3×D414yes—
385C32×C4C23×D314yes—
386C32×C4C23×D311yescharacteristic
387C3⋊Dic3—62no—
388C3⋊Dic3—61no—
389C3⋊Dic3—64no—
390C3⋊Dic3—68no—
391C3⋊Dic3—128no—
392C3⋊Dic3—124no—
393C3⋊Dic3C2×D1211yescharacteristic
394C3⋊Dic3C2×D1211yescharacteristic
395C3×Dic3—28no—
396C3×Dic3—24no—
397C3×Dic3—34no—
398C3×Dic3—34no—
399C2×#41—271no—
400C22×Q8—271no—
401#137—31no—
402#137—272no—
403#137—272no—
404#137—271no—
405#137—271no—
406#137—271no—
407C2×Q16—272no—
408C2×SD16—272no—
409C2×M16—91no—
410C33C2×#13711yescharacteristic, complemented, Sylow
411C23×C3—91no—
412C22×D3—34no—
413C22×D3—181no—
414C22×D3—184no—
415C3×Q8—38no—
416C3×Q8—34no—
417C3×Q8—64no—
418C3×D4—34no—
419C3×D4—64no—
420C2×C3×C4—24no—
421C2×C3×C4—64no—
422C2×C3×C4—64no—
423C2×C3×C4—91no—
424C2×C3×C4—91no—
425C2×C3×C4—91no—
426C2×C3×C4—91no—
427C2×C3×C4—91no—
428C2×C3×C4C2×D3214yes—
429C2×C3×C4C22×C3⋊D311yescharacteristic
430#67—21no—
431#67—181no—
432#67—184no—
433#67—181no—
434#67—184no—
435#67—184no—
436#67—181no—
437#67—364no—
438C2×Dic3—21no—
439C2×Dic3—34no—
440C2×Dic3—182no—
441C2×Dic3—182no—
442C2×Dic3—181no—
443C2×Dic3—184no—
444C2×Dic3—184no—
445C2×Dic3—181no—
446C2×Dic3—364no—
447D12—94no—
448D12—91no—
449D12—184no—
450D12—184no—
451D12—181no—
452D12C22×C3⋊D311yescharacteristic
453C4×D3—21no—
454C4×D3—34no—
455C4×D3—34no—
456C4×D3—34no—
457C4×D3—34no—
458C4×D3—181no—
459C4×D3—184no—
460C4×D3—184no—
461C4×D3—188no—
462C4×D3—188no—
463C4×D3—181no—
464C4×D3—184no—
465C4×D3—181no—
466C4×D3—364no—
467Dic6—21no—
468Dic6—98no—
469Dic6—94no—
470Dic6—92no—
471Dic6—91no—
472Dic6—188no—
473Dic6—184no—
474Dic6—182no—
475Dic6—182no—
476Dic6—181no—
477Dic6—188no—
478Dic6—188no—
479Dic6—184no—
480Dic6—184no—
481Dic6—181no—
482Dic6—364no—
483Dic6C22×C3⋊D312yes—
484Dic6C22×C3⋊D311yescharacteristic
485C3×C8—28no—
486C3×C8—92no—
487C3×C8C2×D3218yes—
488C3×C8C22×C3⋊D312yes—
489C3⋊C8—38no—
490C2×C32—21no—
491C2×C32—24no—
492C2×C32—24no—
493C2×C32—44no—
494C2×C32—121no—
495C2×C32C22×D1211yescharacteristic
496C2×C32C2×D3×D414yes—
497C3⋊D3—21no—
498C3⋊D3—121no—
499C3⋊D3—124no—
500C3⋊D3—244no—
501C3⋊D3#90912yesdirect product factor
502C3×D3—38no—
503C3×D3—44no—
504C3×D3—64no—
505#41—31no—
506#41—271no—
507#41—271no—
508#41—271no—
509#41—541no—
510#41—541no—
511C2×Q8—31no—
512C2×Q8—272no—
513C2×Q8—271no—
514C2×Q8—272no—
515C2×Q8—271no—
516C2×Q8—541no—
517C2×Q8—541no—
518C2×Q8—541no—
519C2×D4—271no—
520C2×D4—541no—
521C22×C4—91no—
522C22×C4—541no—
523C22×C4—541no—
524Q16—32no—
525Q16—272no—
526Q16—272no—
527Q16—272no—
528SD16—32no—
529SD16—272no—
530SD16—272no—
531SD16—272no—
532M16—91no—
533M16—92no—
534M16D3×C3⋊D311yescharacteristic, complemented
535C2×C8—92no—
536C22×C3—24no—
537C22×C3—64no—
538C22×C3—91no—
539C22×C3—91no—
540C22×C3—182no—
541C22×C3D3×D1214yes—
542C22×C3D4×C3⋊D311yescharacteristic
543C2×D3—21no—
544C2×D3—34no—
545C2×D3—34no—
546C2×D3—68no—
547C2×D3—184no—
548C2×D3—181no—
549C2×D3—364no—
550C2×D3—368no—
551C2×D3—362no—
552C3×C4—24no—
553C3×C4—24no—
554C3×C4—68no—
555C3×C4—64no—
556C3×C4—91no—
557C3×C4—91no—
558C3×C4D3×D1214yes—
559C3×C4D4×C3⋊D311yescharacteristic
560C3×C4C22×D3214yes—
561C3×C4C23×C3⋊D311yescharacteristic
562Dic3—22no—
563Dic3—21no—
564Dic3—34no—
565Dic3—34no—
566Dic3—188no—
567Dic3—184no—
568Dic3—182no—
569Dic3—181no—
570Dic3—368no—
571Dic3—364no—
572C32—24no—
573C32C2×#90911yescharacteristic, complemented
574C32D3×#13714yescomplemented
575C23—91no—
576C23—541no—
577Q8—32no—
578Q8—31no—
579Q8—61no—
580Q8—272no—
581Q8—271no—
582Q8—542no—
583Q8—542no—
584Q8—541no—
585Q8—541no—
586D4—31no—
587D4—61no—
588D4—271no—
589D4—541no—
590D4—541no—
591D4—541no—
592D4—541no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—91no—
598C2×C4—91no—
599C2×C4—91no—
600C2×C4—541no—
601C2×C4—541no—
602C2×C4—542no—
603C2×C4—542no—
604C2×C4—541no—
605C2×C4—541no—
606C2×C4—541no—
607C2×C4C2×D3×C3⋊D311yescharacteristic
608C8—92no—
609C8C2×D3×C3⋊D312yes—
610C2×C3—24no—
611C2×C3—21no—
612C2×C3—24no—
613C2×C3—44no—
614C2×C3—92no—
615C2×C3—124no—
616C2×C3—181no—
617C2×C3C2×D3×D1214yes—
618C2×C3C2×D4×C3⋊D311yescharacteristic
619D3—38no—
620D3—41no—
621D3—64no—
622D3—364no—
623D3—361no—
624D3—724no—
625C22—61no—
626C22—91no—
627C22—91no—
628C22—182no—
629C22—541no—
630C22—1082no—
631C22C3⋊D3×D1211yescharacteristic
632C4—61no—
633C4—62no—
634C4—91no—
635C4—91no—
636C4—541no—
637C4—542no—
638C4C3⋊D3×D1211yescharacteristic
639C4C22×D3×C3⋊D311yescharacteristic
640C3—24no—
641C3D3×#90914yescomplemented
642C3C3⋊D3×#13711yescharacteristic, complemented
643C2—21no—
644C2—92no—
645C2—121no—
646C2—181no—
647C2—1081no—
648C2C2×C3⋊D3×D1211yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×D12
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×M16C22
socleC2×C33C22×D4
prime radicalC3⋊D3×#235C2
hypercenterM16D3×C3⋊D3
hypocenterC33C2×#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊D3×Dic6C2semidirect
3. #11723950C2semidirect
4. C3×M16×C3⋊D3C2semidirect
5. C32×#909C2semidirect
6. #11723381C2semidirect
7. #11723379C2semidirect
8. #11723340C2semidirect
9. #11723329C2semidirect
10. C3⋊D3×Dic12C2semidirect
11. C3⋊D3×#204C2semidirect
15. C2×#6308C22semidirect
18. C2×C32×Dic6C22semidirect
31. C3⋊D3×Dic6C22semidirect
33. C33×M16C22semidirect
34. #88853C22semidirect
35. #88851C22semidirect
36. C3×#3748C22semidirect
37. C3×#3743C22semidirect
39. C32×Dic12C22semidirect
40. C32×#204C22semidirect
41. #88799C22semidirect
42. #88799C22semidirect
43. #88795C22semidirect
44. #88794C22semidirect
59. M16×C3⋊D3D3semidirect
60. C3×#909D3semidirect
130. C2×C3×Dic6C2×D3semidirect
148. C32×M16C2×D3semidirect
149. C32×M16C2×D3semidirect
154. #3748C2×D3semidirect
161. #3743C2×D3semidirect
164. C3×Dic12C2×D3semidirect
165. C3×#204C2×D3semidirect
206. #909C3⋊D3direct (N × H)
294. C3×C3⋊D3#137semidirect
335. C2×Dic6C2×C3⋊D3semidirect
341. C3×M16D32semidirect
342. C3×M16C2×C3⋊D3semidirect
354. Dic12C2×C3⋊D3semidirect
360. #204C2×C3⋊D3semidirect
410. C33C2×#137semidirect
501. C3⋊D3#909direct (N × H)
534. M16D3×C3⋊D3semidirect
573. C32C2×#909semidirect
574. C32D3×#137semidirect
641. C3D3×#909semidirect
642. C3C3⋊D3×#137semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2352
C2×C3⋊D3×Dic62
#117239502
C3×M16×C3⋊D32
C32×#9092
#117233812
#117233792
#117233402
#117233292
C3⋊D3×Dic122
C3⋊D3×#2042
C3⋊D3×#1373
D3×#9093