Finite Group Databasebeta

← C3⋊D3×#951 #13178001

Subgroups — C3⋊D3×#951 #13178001

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×#41×C3⋊D3C211yesmaximal, characteristic
2C2×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
3#11723992C211yesmaximal, characteristic, complemented
4C32×#951C211yesmaximal, characteristic, complemented
5#11723576C212yesmaximal, complemented
6C3⋊D3×#216C212yesmaximal, complemented
7#11723552C212yesmaximal, complemented
8C3⋊D3×#214C212yesmaximal, complemented
9#11723388C211yesmaximal, characteristic, complemented
10#11723375C211yesmaximal, characteristic, complemented
11C3⋊D3×#208C211yesmaximal, characteristic, complemented
12C3⋊D3×#137—31nomaximal
13D3×#951—34nomaximal
14C33×#41C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C3×#3832C2211yescharacteristic
17C3×Q8×C3⋊D3C2211yescharacteristic
18C3×#3830—21no—
19C3×#3830C2211yescharacteristic
20C3×D4×C3⋊D3—21no—
21C3×D4×C3⋊D3C2211yescharacteristic
22C3×#3828—21no—
23C3×#3828C2211yescharacteristic
24C2×C3×C4×C3⋊D3—21no—
25C2×C3×C4×C3⋊D3C2211yescharacteristic
26C2×C32×Dic6C2211yescharacteristic, complemented
27C2×#6293—21no—
28C2×Dic3×C3⋊D3—21no—
29C3⋊D3×Dic6—21no—
30C3⋊D3×Dic6—21no—
31C3⋊D3×Dic6C2212yes—
32C3⋊D3×Dic6C2212yescomplemented
33#88863C2212yescomplemented
34#88861C2212yescomplemented
35#88855C2211yescharacteristic, complemented
36C32×#216C2212yescomplemented
37C32×#214C2212yescomplemented
38C32×#208C2211yescharacteristic, complemented
39#88800C2212yescomplemented
40#88798C2212yescomplemented
41#88796C2212yescomplemented
42#88793C2212yescomplemented
43#88787C2212yescomplemented
44C3⋊D3×C3⋊C8C2212yes—
45#41×C3⋊D3D311yescharacteristic
46C2×Q8×C3⋊D3—31no—
47C3×D3×#41—34no—
48C2×D3×Dic6—34no—
49C32×#137—31no—
50#64160—31no—
51#64160—34no—
52#64160—64no—
53#64151—31no—
54#64127—32no—
55Q16×C3⋊D3—32no—
56#64124—32no—
57SD16×C3⋊D3—32no—
58#64118—31no—
59M16×C3⋊D3—31no—
60C3×#951D314yescomplemented
61#63943—38no—
62D3×#216—38no—
63#63929—38no—
64D3×#214—38no—
65#63831—34no—
66#63827—34no—
67D3×#208—34no—
68C22×C3×C3⋊D3—21no—
69C22×C3×C3⋊D3D411yescharacteristic
70C33×Q8C2311yescharacteristic
71C33×D4—21no—
72C33×D4C2311yescharacteristic
73C2×C33×C4—21no—
74C2×C33×C4C2311yescharacteristic
75C2×C3⋊(C3⋊Dic3)—21no—
76#6308—21no—
77#6308C2312yes—
78C3×#638—21no—
79C3×#638—21no—
80C3×#638—21no—
81C2×C3×C3⋊Dic3—21no—
82C2×C3×C3⋊Dic3D411yescharacteristic
83C3×C4⋊(C3⋊D3)—21no—
84C3×C4⋊(C3⋊D3)C2311yescharacteristic
85C3×C4×C3⋊D3—21no—
86C3×C4×C3⋊D3—21no—
87C3×C4×C3⋊D3—21no—
88C3×C4×C3⋊D3D411yescharacteristic
89C3×C4×C3⋊D3D411yescharacteristic
90C3×C4×C3⋊D3C2311yescharacteristic
91C3×C4×C3⋊D3C2311yescharacteristic
92C3×#634—21no—
93C3×#634C2311yescharacteristic
94C2×C32×Dic3—21no—
95C32×Dic6—21no—
96C32×Dic6C2312yes—
97#6293—22no—
98#6293—22no—
99Dic3×C3⋊D3—22no—
100Dic3×C3⋊D3—22no—
101C3⋊(C3⋊(C3⋊C8))C2312yes—
102C32×C3⋊C8C2312yes—
103C2×#951—91no—
104D3×#137—94no—
105C32×#41—24no—
106C32×#41C2×D314yes—
107C32×#41C2×D311yescharacteristic
108C2×C32×Q8—31no—
109#3832C2×D311yescharacteristic
110Q8×C3⋊D3—32no—
111Q8×C3⋊D3—32no—
112Q8×C3⋊D3—61no—
113Q8×C3⋊D3—61no—
114Q8×C3⋊D3C2×D311yescharacteristic
115#3830—21no—
116#3830C2×D311yescharacteristic
117D4×C3⋊D3—21no—
118D4×C3⋊D3C2×D311yescharacteristic
119#3828—21no—
120#3828C2×D311yescharacteristic
121C2×C4×C3⋊D3—21no—
122C2×C4×C3⋊D3—61no—
123C2×C4×C3⋊D3C2×D311yescharacteristic
124C2×#634—31no—
125C2×#634—34no—
126C2×#634—61no—
127C2×#634—64no—
128C3×#239—34no—
129C3×D3×Q8—34no—
130C3×#237—34no—
131C3×#237—64no—
132C3×D3×D4—34no—
133C3×D3×D4—64no—
134C3×#235—34no—
135C3×#235—64no—
136C2×C3×C4×D3—34no—
137C2×C3×C4×D3—64no—
138C2×C3×Dic6C2×D314yescomplemented
139C2×#627—64no—
140C2×D3×Dic3—64no—
141D3×Dic6—38no—
142D3×Dic6—38no—
143D3×Dic6—64no—
144D3×Dic6—64no—
145C32×Q16—32no—
146C32×SD16—32no—
147C32×M16—31no—
148#3756—32no—
149#3756—38no—
150#3756—32no—
151#3756—68no—
152#3755—32no—
153#3754—32no—
154#3754—38no—
155#3754—68no—
156#3748—31no—
157#3748—34no—
158#3748—64no—
159#3746—32no—
160#3744—32no—
161#3743—32no—
162C8×C3⋊D3—32no—
163C3×#216C2×D318yescomplemented
164C3×#214C2×D318yescomplemented
165C3×#208C2×D314yescomplemented
166#3719—38no—
167#3718—38no—
168#3716—38no—
169#3715—38no—
170#3711—38no—
171D3×C3⋊C8—38no—
172C22×C33—21no—
173C22×C33C2×D411yescharacteristic
174C2×C3×C3⋊D3—22no—
175C2×C3×C3⋊D3—21no—
176C2×C3×C3⋊D3—42no—
177C2×C3×C3⋊D3C2×D411yescharacteristic
178C2×C3×C3⋊D3C2×D411yescharacteristic
179C33×C4—21no—
180C33×C4C2×D411yescharacteristic
181C33×C4C2411yescharacteristic
182C3⋊(C3⋊Dic3)—22no—
183C3×C3⋊Dic3—21no—
184C3×C3⋊Dic3C2×D411yescharacteristic
185C3×C3⋊Dic3C2×D411yescharacteristic
186C32×Dic3—22no—
187C2×C3×#41—91no—
188D3×#41—34no—
189C2×D3×Q8—94no—
190C22×Dic6—91no—
191C3×#137—34no—
192#951—94no—
193#951—91no—
194#951—91no—
195#951—92no—
196#951—92no—
197#951—91no—
198#951—184no—
199#951C3⋊D311yescharacteristic, direct product factor
200C2×#216—92no—
201#942—94no—
202C2×#214—92no—
203C2×#208—91no—
204#918—98no—
205D3×Q16—98no—
206#915—98no—
207D3×SD16—98no—
208#909—94no—
209D3×M16—94no—
210C22×C3⋊D3—21no—
211C22×C3⋊D3#6711yescharacteristic
212C22×C3×D3—34no—
213C22×C3×D3—64no—
214C32×Q8—24no—
215C32×Q8—32no—
216C32×Q8—61no—
217C32×Q8C22×D314yes—
218C32×Q8C22×D311yescharacteristic
219C32×D4—21no—
220C32×D4—24no—
221C32×D4—24no—
222C32×D4—44no—
223C32×D4C22×D314yes—
224C32×D4C22×D311yescharacteristic
225C2×C32×C4—21no—
226C2×C32×C4—24no—
227C2×C32×C4—24no—
228C2×C32×C4—44no—
229C2×C32×C4—61no—
230C2×C32×C4C22×D314yes—
231C2×C32×C4C22×D311yescharacteristic
232#638—21no—
233#638—21no—
234#638—21no—
235C2×C3⋊Dic3—21no—
236C2×C3⋊Dic3—61no—
237C2×C3⋊Dic3—64no—
238C2×C3⋊Dic3—124no—
239C2×C3⋊Dic3#6711yescharacteristic
240C4⋊(C3⋊D3)—21no—
241C4⋊(C3⋊D3)C22×D311yescharacteristic
242C4×C3⋊D3—21no—
243C4×C3⋊D3—21no—
244C4×C3⋊D3—21no—
245C4×C3⋊D3—62no—
246C4×C3⋊D3—62no—
247C4×C3⋊D3#6711yescharacteristic
248C4×C3⋊D3#6711yescharacteristic
249C4×C3⋊D3C22×D311yescharacteristic
250C4×C3⋊D3C22×D311yescharacteristic
251#634—21no—
252#634—32no—
253#634—38no—
254#634—68no—
255#634—61no—
256#634—62no—
257#634—62no—
258#634—64no—
259#634—124no—
260#634C22×D311yescharacteristic
261C3×#67—64no—
262C3×#67—64no—
263C3×#67—64no—
264C2×C3×Dic3—24no—
265C2×C3×Dic3—34no—
266C2×C3×Dic3—64no—
267C3×D12—34no—
268C3×D12—64no—
269C3×C4×D3—34no—
270C3×C4×D3—34no—
271C3×C4×D3—34no—
272C3×C4×D3—34no—
273C3×C4×D3—64no—
274C3×C4×D3—64no—
275C3×C4×D3—64no—
276C3×Dic6—24no—
277C3×Dic6—34no—
278C3×Dic6—64no—
279C3×Dic6C22×D318yes—
280#627—68no—
281#627—68no—
282D3×Dic3—68no—
283D3×Dic3—68no—
284C32×C8—32no—
285C3⋊(C3⋊C8)—32no—
286C3⋊(C3⋊C8)—38no—
287C3⋊(C3⋊C8)—68no—
288C3×C3⋊C8C22×D318yes—
289C2×#137—271noSylow
290C2×C33—21no—
291C2×C33—41no—
292C2×C33C22×D411yescharacteristic
293C3×C3⋊D3—21no—
294C3×C3⋊D3—41no—
295C3×C3⋊D3#13712yescomplemented
296C3×#41—24no—
297C3×#41—91no—
298C3×#41—91no—
299C3×#41—91no—
300C3×#41—181no—
301C3×#41—181no—
302C3×#41D3214yes—
303C3×#41C2×C3⋊D311yescharacteristic
304C2×C3×Q8—34no—
305C2×C3×Q8—91no—
306C2×C3×D4—91no—
307C2×C3×D4—181no—
308C22×C3×C4—91no—
309C22×C3×C4—181no—
310C22×Dic3—181no—
311#239—34no—
312D3×Q8—34no—
313D3×Q8—98no—
314D3×Q8—98no—
315D3×Q8—184no—
316D3×Q8—184no—
317#237—34no—
318#237—64no—
319D3×D4—34no—
320D3×D4—64no—
321#235—34no—
322#235—64no—
323C2×C4×D3—34no—
324C2×C4×D3—64no—
325C2×C4×D3—184no—
326C2×Dic6—94no—
327C2×Dic6—91no—
328C2×Dic6—92no—
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—34no—
339#216—98no—
340#216—98no—
341#216—92no—
342#216—92no—
343#216—92no—
344#216—188no—
345#216C2×C3⋊D312yescomplemented
346#215—98no—
347#214—98no—
348#214—92no—
349#214—92no—
350#214—92no—
351#214—188no—
352#214C2×C3⋊D312yescomplemented
353#208—94no—
354#208—91no—
355#208—92no—
356#208—184no—
357#208C2×C3⋊D311yescharacteristic, complemented
358C2×C3⋊C8—92no—
359Dic12—98no—
360#204—98no—
361ΓL(1,𝔽25)—98no—
362D3×C8—98no—
363C22×C32—21no—
364C22×C32—24no—
365C22×C32—24no—
366C22×C32—44no—
367C22×C32D3×D414yes—
368C22×C32C2×#6711yescharacteristic
369C2×C3⋊D3—22no—
370C2×C3⋊D3—21no—
371C2×C3⋊D3—42no—
372C2×C3⋊D3C2×#6711yescharacteristic
373C2×C3⋊D3C2×#6711yescharacteristic
374C2×C3×D3—34no—
375C2×C3×D3—34no—
376C2×C3×D3—68no—
377C2×C3×D3—64no—
378C2×C3×D3—128no—
379C32×C4—21no—
380C32×C4—24no—
381C32×C4—24no—
382C32×C4—24no—
383C32×C4—44no—
384C32×C4—62no—
385C32×C4D3×D414yes—
386C32×C4C2×#6711yescharacteristic
387C32×C4C23×D314yes—
388C32×C4C23×D311yescharacteristic
389C3⋊Dic3—21no—
390C3⋊Dic3—62no—
391C3⋊Dic3—68no—
392C3⋊Dic3—128no—
393C3⋊Dic3C2×#6711yescharacteristic
394C3⋊Dic3C2×#6711yescharacteristic
395C3×Dic3—28no—
396C3×Dic3—34no—
397C3×Dic3—34no—
398C3×Dic3—64no—
399C2×#41—91no—
400C22×Q8—271no—
401#137—31no—
402#137—271no—
403#137—272no—
404#137—272no—
405#137—271no—
406#137—271no—
407C2×Q16—272no—
408C2×SD16—272no—
409C2×M16—271no—
410C33C2×#13711yescharacteristic, complemented, Sylow
411C23×C3—91no—
412C23×C3—181no—
413C22×D3—34no—
414C22×D3—64no—
415C3×Q8—24no—
416C3×Q8—38no—
417C3×Q8—64no—
418C3×Q8—91no—
419C3×Q8—181no—
420C3×Q8C2×D3214yes—
421C3×Q8C22×C3⋊D311yescharacteristic
422C3×D4—24no—
423C3×D4—21no—
424C3×D4—24no—
425C3×D4—44no—
426C3×D4—91no—
427C3×D4—181no—
428C3×D4—181no—
429C3×D4—181no—
430C3×D4—181no—
431C3×D4C2×D3214yes—
432C3×D4C22×C3⋊D311yescharacteristic
433C2×C3×C4—24no—
434C2×C3×C4—21no—
435C2×C3×C4—24no—
436C2×C3×C4—44no—
437C2×C3×C4—64no—
438C2×C3×C4—91no—
439C2×C3×C4—91no—
440C2×C3×C4—91no—
441C2×C3×C4—91no—
442C2×C3×C4—91no—
443C2×C3×C4—181no—
444C2×C3×C4—181no—
445C2×C3×C4—181no—
446C2×C3×C4—181no—
447C2×C3×C4C2×D3214yes—
448C2×C3×C4C22×C3⋊D311yescharacteristic
449#67—64no—
450#67—64no—
451#67—64no—
452C2×Dic3—21no—
453C2×Dic3—34no—
454C2×Dic3—64no—
455C2×Dic3—182no—
456C2×Dic3—182no—
457C2×Dic3—184no—
458C2×Dic3—181no—
459C2×Dic3—364no—
460D12—34no—
461D12—64no—
462C4×D3—34no—
463C4×D3—34no—
464C4×D3—34no—
465C4×D3—34no—
466C4×D3—64no—
467C4×D3—64no—
468C4×D3—64no—
469C4×D3—188no—
470C4×D3—188no—
471Dic6—21no—
472Dic6—34no—
473Dic6—64no—
474Dic6—98no—
475Dic6—92no—
476Dic6—188no—
477Dic6—182no—
478Dic6—182no—
479Dic6—188no—
480Dic6—188no—
481Dic6—184no—
482Dic6—181no—
483Dic6—364no—
484Dic6C22×C3⋊D312yes—
485C3×C8—38no—
486C3⋊C8—98no—
487C3⋊C8—92no—
488C3⋊C8—188no—
489C3⋊C8C22×C3⋊D312yes—
490C2×C32—21no—
491C2×C32—24no—
492C2×C32—24no—
493C2×C32—44no—
494C2×C32—41no—
495C2×C32—44no—
496C2×C32—84no—
497C2×C32C2×D3×D414yes—
498C2×C32C22×#6711yescharacteristic
499C3⋊D3—21no—
500C3⋊D3—41no—
501C3⋊D3#95112yesdirect product factor
502C3×D3—38no—
503C3×D3—64no—
504C3×D3—124no—
505#41—91no—
506#41—91no—
507#41—91no—
508#41—181no—
509#41—181no—
510#41D3×C3⋊D311yescharacteristic
511C2×Q8—31no—
512C2×Q8—91no—
513C2×Q8—272no—
514C2×Q8—272no—
515C2×Q8—271no—
516C2×Q8—541no—
517C2×Q8—541no—
518C2×Q8—541no—
519C2×D4—91no—
520C2×D4—181no—
521C22×C4—91no—
522C22×C4—181no—
523C22×C4—541no—
524Q16—32no—
525Q16—272no—
526Q16—272no—
527Q16—272no—
528SD16—32no—
529SD16—272no—
530SD16—272no—
531SD16—272no—
532M16—31no—
533M16—272no—
534M16—271no—
535C2×C8—272no—
536C22×C3—24no—
537C22×C3—21no—
538C22×C3—24no—
539C22×C3—44no—
540C22×C3—91no—
541C22×C3—91no—
542C22×C3—182no—
543C22×C3—181no—
544C22×C3—362no—
545C22×C3D3×#6714yes—
546C22×C3D4×C3⋊D311yescharacteristic
547C2×D3—34no—
548C2×D3—34no—
549C2×D3—68no—
550C2×D3—64no—
551C2×D3—128no—
552C3×C4—24no—
553C3×C4—21no—
554C3×C4—24no—
555C3×C4—24no—
556C3×C4—44no—
557C3×C4—68no—
558C3×C4—91no—
559C3×C4—91no—
560C3×C4—181no—
561C3×C4D3×#6714yes—
562C3×C4D4×C3⋊D311yescharacteristic
563C3×C4C22×D3214yes—
564C3×C4C23×C3⋊D311yescharacteristic
565Dic3—22no—
566Dic3—34no—
567Dic3—34no—
568Dic3—64no—
569Dic3—188no—
570Dic3—182no—
571Dic3—368no—
572C32—24no—
573C32D3×#13714yescomplemented
574C32C2×#95111yescharacteristic, complemented
575C23—91no—
576C23—181no—
577Q8—32no—
578Q8—61no—
579Q8—91no—
580Q8—181no—
581Q8—272no—
582Q8—542no—
583Q8—542no—
584Q8—541no—
585Q8C2×D3×C3⋊D311yescharacteristic
586D4—21no—
587D4—91no—
588D4—181no—
589D4—181no—
590D4—181no—
591D4—181no—
592D4C2×D3×C3⋊D311yescharacteristic
593C2×C4—21no—
594C2×C4—61no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—91no—
598C2×C4—91no—
599C2×C4—91no—
600C2×C4—181no—
601C2×C4—181no—
602C2×C4—181no—
603C2×C4—181no—
604C2×C4—542no—
605C2×C4—542no—
606C2×C4—541no—
607C2×C4C2×D3×C3⋊D311yescharacteristic
608C8—32no—
609C8—272no—
610C2×C3—24no—
611C2×C3—21no—
612C2×C3—24no—
613C2×C3—44no—
614C2×C3—44no—
615C2×C3—41no—
616C2×C3—84no—
617C2×C3—92no—
618C2×C3—181no—
619C2×C3—361no—
620C2×C3C2×D3×#6714yes—
621C2×C3C2×D4×C3⋊D311yescharacteristic
622D3—38no—
623D3—64no—
624D3—124no—
625C22—21no—
626C22—91no—
627C22—91no—
628C22—182no—
629C22—181no—
630C22—362no—
631C22C3⋊D3×#6711yescharacteristic
632C4—21no—
633C4—62no—
634C4—91no—
635C4—91no—
636C4—181no—
637C4—542no—
638C4C3⋊D3×#6711yescharacteristic
639C4C22×D3×C3⋊D311yescharacteristic
640C3—24no—
641C3D3×#95114yescomplemented
642C3C3⋊D3×#13711yescharacteristic, complemented
643C2—21no—
644C2—41no—
645C2—92no—
646C2—181no—
647C2—361no—
648C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×#67
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
prime radicalC3×#41×C3⋊D3C2
hypercenter#41D3×C3⋊D3
hypocenterC33C2×#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊D3×Dic6C2semidirect
3. #11723992C2semidirect
4. C32×#951C2semidirect
5. #11723576C2semidirect
6. C3⋊D3×#216C2semidirect
7. #11723552C2semidirect
8. C3⋊D3×#214C2semidirect
9. #11723388C2semidirect
10. #11723375C2semidirect
11. C3⋊D3×#208C2semidirect
15. C2×#6308C22semidirect
26. C2×C32×Dic6C22semidirect
32. C3⋊D3×Dic6C22semidirect
33. #88863C22semidirect
34. #88861C22semidirect
35. #88855C22semidirect
36. C32×#216C22semidirect
37. C32×#214C22semidirect
38. C32×#208C22semidirect
39. #88800C22semidirect
40. #88798C22semidirect
41. #88796C22semidirect
42. #88793C22semidirect
43. #88787C22semidirect
60. C3×#951D3semidirect
138. C2×C3×Dic6C2×D3semidirect
163. C3×#216C2×D3semidirect
164. C3×#214C2×D3semidirect
165. C3×#208C2×D3semidirect
199. #951C3⋊D3direct (N × H)
295. C3×C3⋊D3#137semidirect
335. C2×Dic6C2×C3⋊D3semidirect
345. #216C2×C3⋊D3semidirect
352. #214C2×C3⋊D3semidirect
357. #208C2×C3⋊D3semidirect
410. C33C2×#137semidirect
501. C3⋊D3#951direct (N × H)
573. C32D3×#137semidirect
574. C32C2×#951semidirect
641. C3D3×#951semidirect
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×#41×C3⋊D32
C2×C3⋊D3×Dic62
#117239922
C32×#9512
#117235762
C3⋊D3×#2162
#117235522
C3⋊D3×#2142
#117233882
#117233752
C3⋊D3×#2082
C3⋊D3×#1373
D3×#9513