Finite Group Databasebeta

← #13169381

Subgroups — #13169381

Order 1728.

1138 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723665C211yesmaximal, characteristic, complemented
2C4×D3×C3⋊Dic3C211yesmaximal, characteristic
3#11723448C212yesmaximal, complemented
4C3×#63851C212yesmaximal
5C3×#63838C211yesmaximal, characteristic, complemented
6#11723035C211yesmaximal, characteristic, complemented
7#11723032C211yesmaximal, characteristic, complemented
8#11722981C211yesmaximal, characteristic
9#11722976C211yesmaximal, characteristic
10#11722940C212yesmaximal, complemented
11#11722916C212yesmaximal
12#10594810—31nomaximal
13#10594386—32nomaximal
14C2×C32×C4×D3C2211yescharacteristic
15C2×#6296C2211yescharacteristic
16C2×D3×C3⋊Dic3C2211yescharacteristic
17C2×D3×C3⋊Dic3C2211yescharacteristic
18C2×C3×#627C2212yes—
19C2×C3×#627C2211yescharacteristic
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3750C2212yes—
22C3×C4×C3⋊Dic3C2211yescharacteristic
23C32×Dic3⋊C4C2212yes—
24#88815C2211yescharacteristic
25#88813C2212yes—
26#88807C2212yes—
27#88806C2212yes—
28#88806C2212yes—
29Dic3⋊(C3⋊Dic3)C2212yes—
30Dic3×C3⋊Dic3C2211yescharacteristic
31Dic3×C3⋊Dic3C2211yescharacteristic
32C3×#3724C2211yescharacteristic
33C3×Dic3⋊Dic3C2212yes—
34C3×(C2×D3)⋊Dic3C2212yes—
35C3×(C2×D3)⋊Dic3C2212yes—
36C3×Dic32C2212yes—
37C2×C4×C3⋊Dic3—31no—
38#64098—32no—
39C3×#924—32no—
40C3×#896—32no—
41C3×#887—32no—
42#63982—31no—
43#63982—31no—
44#63957—32no—
45#63919—31no—
46#63893—32no—
47#63887—32no—
48#63877—32no—
49#63877—32no—
50C4×D3×Dic3—32no—
51C4×D3×Dic3—61no—
52#63873—31no—
53#63870—32no—
54#63870—32no—
55#63870—62no—
56#63851D312yes—
57#63843—32no—
58Dic3×Dic6—32no—
59#63839—32no—
60#63839—32no—
61#63838—31no—
62#63838D311yescharacteristic, complemented
63#63837—32no—
64#63837—32no—
65C22×C32×D3C2311yescharacteristic
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2311yescharacteristic
76C32×C4×D3—22no—
77#6296—22no—
78D3×C3⋊Dic3—22no—
79D3×C3⋊Dic3Q814yescomplemented
80C3×#627—22no—
81C3×#627—24no—
82#5622—92no—
83#5578—91no—
84C22×C32×C4—31no—
85C22×C3⋊Dic3—31no—
86C22×C3⋊Dic3—31no—
87C2×C3×C4×D3—21no—
88C2×C3×C4×D3C2×D312yes—
89C2×C3×Dic6—32no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—32no—
92C2×#627—32no—
93C2×#627—31no—
94C2×#627C2×D312yes—
95C2×#627C2×D311yescharacteristic
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—61no—
99C2×D3×Dic3—61no—
100C32×C4⋊C4—32no—
101C4⋊(C3⋊Dic3)—32no—
102#3750—32no—
103#3750—32no—
104#3750—32no—
105#3750C2×D312yes—
106C4×C3⋊Dic3—32no—
107C4×C3⋊Dic3—31no—
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—31no—
110C4×C3⋊Dic3—62no—
111C4×C3⋊Dic3—61no—
112C4×C3⋊Dic3—121no—
113C4×C3⋊Dic3C2×D311yescharacteristic
114C3×#217—32no—
115C3×#217—32no—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—32no—
118C3×C4⋊Dic3—32no—
119C3×Dic3⋊C4—22no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—62no—
125C3×Dic3⋊C4C2×D312yes—
126C3×Dic3⋊C4C2×D312yes—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—32no—
130C3×C4×Dic3—61no—
131#3724—31no—
132#3724—32no—
133#3724—32no—
134#3724—32no—
135#3724—32no—
136#3724—62no—
137#3724—62no—
138#3724C2×D311yescharacteristic
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—62no—
145Dic3⋊Dic3—62no—
146Dic3⋊Dic3C2×D312yes—
147(C2×D3)⋊Dic3C2×D312yes—
148(C2×D3)⋊Dic3C2×D312yes—
149Dic32—32no—
150Dic32—32no—
151Dic32—32no—
152Dic32—32no—
153Dic32—32no—
154Dic32—61no—
155Dic32—61no—
156Dic32C2×D312yes—
157C22×C33C2411yescharacteristic
158C2×C32×D3—21no—
159C2×C32×D3—21no—
160C2×C32×D3C2×Q812yes—
161C33×C4—21no—
162C3⋊(C3⋊Dic3)—21no—
163C3⋊(C3⋊Dic3)C2×Q812yes—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3C2×Q812yes—
167C3×C3⋊Dic3#4112yes—
168C32×Dic3—22no—
169C32×Dic3—22no—
170C32×Dic3#4112yes—
171C3×C22⋊Q8—92no—
172C3×C4×Q8—91no—
173#946—91no—
174Q8×Dic3—92no—
175#944—91no—
176#944—92no—
177#924—32no—
178C2×C4×Dic3—92no—
179C2×C4×Dic3—181no—
180#897—91no—
181#896—32no—
182#889—92no—
183#889—92no—
184#889—92no—
185#889—182no—
186#888—92no—
187#888—91no—
188#887—32no—
189#887—92no—
190#887—92no—
191#878—92no—
192#878—92no—
193#877—92no—
194#877—92no—
195#872—92no—
196C42×D3—91no—
197C4×Dic6—91no—
198C4×Dic6—92no—
199C23×C32—31no—
200C22×C3×D3—21no—
201C22×C3×D3C22×D312yes—
202C2×C32×C4—21no—
203C2×C32×C4—22no—
204C2×C32×C4—31no—
205C2×C32×C4—32no—
206C2×C32×C4—32no—
207C2×C32×C4—41no—
208C2×C32×C4—62no—
209C2×C32×C4C22×D312yes—
210C2×C32×C4C22×D311yescharacteristic
211C2×C3⋊Dic3—32no—
212C2×C3⋊Dic3—31no—
213C2×C3⋊Dic3—34no—
214C2×C3⋊Dic3—32no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—61no—
217C2×C3⋊Dic3—61no—
218C2×C3⋊Dic3—62no—
219C2×C3⋊Dic3—62no—
220C2×C3⋊Dic3—62no—
221C2×C3⋊Dic3—121no—
222C2×C3⋊Dic3—121no—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D311yescharacteristic
226C2×C3⋊Dic3C22×D311yescharacteristic
227C2×C3×Dic3—21no—
228C2×C3×Dic3—22no—
229C2×C3×Dic3—22no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—32no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—32no—
235C2×C3×Dic3—32no—
236C2×C3×Dic3—61no—
237C2×C3×Dic3—61no—
238C2×C3×Dic3—62no—
239C2×C3×Dic3—62no—
240C2×C3×Dic3C22×D312yes—
241C2×C3×Dic3C22×D312yes—
242C2×C3×Dic3C22×D312yes—
243C2×C3×Dic3C22×D312yes—
244C2×C3×Dic3C22×D312yes—
245C3×C4×D3—24no—
246C3×C4×D3—24no—
247C3×Dic6—64no—
248C3×Dic6—64no—
249C3×Dic6—64no—
250#627—22no—
251#627—24no—
252#627—64no—
253#627—62no—
254D3×Dic3—38no—
255D3×Dic3—64no—
256D3×Dic3—64no—
257D3×Dic3—64no—
258#533—271noSylow
259C2×C33C22×Q811yescharacteristic
260C2×C33C2×#4111yescharacteristic
261C2×C33C2×#4111yescharacteristic
262C32×D3—22no—
263C2×C3×Q8—92no—
264C2×C3×Q8—91no—
265C22×C3×C4—32no—
266C22×C3×C4—61no—
267C22×Dic3—92no—
268C22×Dic3—92no—
269C22×Dic3—181no—
270C22×Dic3—181no—
271C2×C4×D3—91no—
272C2×C4×D3—91no—
273C2×C4×D3D3211yescharacteristic
274C2×Dic6—32no—
275C2×Dic6—32no—
276C2×Dic6—32no—
277C2×Dic6—91no—
278C2×Dic6—92no—
279C3×C4⋊C4—32no—
280C3×C4⋊C4—32no—
281C3×C4⋊C4—62no—
282C3×C4⋊C4—92no—
283C3×C4⋊C4—91no—
284C3×C4⋊C4—92no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4—92no—
287C3×C42—92no—
288C3×C42—91no—
289#217—32no—
290#217—32no—
291(C2×D3)⋊C4—32no—
292(C2×D3)⋊C4—32no—
293C4⋊Dic3—32no—
294C4⋊Dic3—91no—
295C4⋊Dic3—92no—
296C4⋊Dic3—92no—
297C4⋊Dic3—92no—
298C4⋊Dic3—182no—
299Dic3⋊C4—32no—
300Dic3⋊C4—32no—
301Dic3⋊C4—32no—
302Dic3⋊C4—32no—
303Dic3⋊C4—62no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—92no—
311Dic3⋊C4—92no—
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314Dic3⋊C4—182no—
315Dic3⋊C4—182no—
316Dic3⋊C4—182no—
317Dic3⋊C4D3212yes—
318C4×Dic3—32no—
319C4×Dic3—32no—
320C4×Dic3—32no—
321C4×Dic3—61no—
322C4×Dic3—92no—
323C4×Dic3—92no—
324C4×Dic3—92no—
325C4×Dic3—92no—
326C4×Dic3—92no—
327C4×Dic3—91no—
328C4×Dic3—92no—
329C4×Dic3—91no—
330C4×Dic3—91no—
331C4×Dic3—181no—
332C4×Dic3—181no—
333C4×Dic3—182no—
334C4×Dic3—181no—
335C4×Dic3—361no—
336C22×C32—22no—
337C22×C32—21no—
338C22×C32—32no—
339C22×C32—41no—
340C22×C32—61no—
341C22×C32—61no—
342C22×C32C23×D311yescharacteristic
343C22×C32C23×D312yes—
344C2×C3×D3—22no—
345C2×C3×D3—22no—
346C2×C3×D3—22no—
347C2×C3×D3—22no—
348C2×C3×D3—22no—
349C2×C3×D3D3×Q814yes—
350C32×C4—21no—
351C32×C4—22no—
352C32×C4—22no—
353C32×C4—32no—
354C32×C4—42no—
355C32×C4—42no—
356C32×C4—62no—
357C32×C4—62no—
358C3⋊Dic3—22no—
359C3⋊Dic3—22no—
360C3⋊Dic3—34no—
361C3⋊Dic3—32no—
362C3⋊Dic3—62no—
363C3⋊Dic3—64no—
364C3⋊Dic3—61no—
365C3⋊Dic3—62no—
366C3⋊Dic3—62no—
367C3⋊Dic3—122no—
368C3⋊Dic3—122no—
369C3⋊Dic3—122no—
370C3⋊Dic3#23512yes—
371C3⋊Dic3D3×Q812yes—
372C3×Dic3—22no—
373C3×Dic3—22no—
374C3×Dic3—22no—
375C3×Dic3—22no—
376C3×Dic3—22no—
377C3×Dic3—34no—
378C3×Dic3—34no—
379C3×Dic3—42no—
380C3×Dic3—42no—
381C3×Dic3—62no—
382C3×Dic3—62no—
383C3×Dic3—62no—
384C3×Dic3—62no—
385C3×Dic3—62no—
386C3×Dic3—62no—
387C3×Dic3—122no—
388C3×Dic3—122no—
389C3×Dic3#23714yes—
390C4⋊Q8—271no—
391C4⋊Q8—271no—
392#125—272no—
393C22⋊Q8—92no—
394C22⋊Q8—271no—
395C22⋊Q8—271no—
396C4×Q8—91no—
397C4×Q8—271no—
398C4×Q8—272no—
399#117—272no—
400C2×C42—271no—
401C33#53311yescharacteristic, complemented, Sylow
402C23×C3—32no—
403C23×C3—61no—
404C22×D3C2×D3211yescharacteristic
405C3×Q8—184no—
406C3×Q8—182no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—32no—
410C2×C3×C4—32no—
411C2×C3×C4—32no—
412C2×C3×C4—32no—
413C2×C3×C4—32no—
414C2×C3×C4—41no—
415C2×C3×C4—61no—
416C2×C3×C4—62no—
417C2×C3×C4—62no—
418C2×C3×C4—64no—
419C2×C3×C4—64no—
420C2×C3×C4—91no—
421C2×C3×C4—91no—
422C2×C3×C4—92no—
423C2×C3×C4—92no—
424C2×C3×C4C2×D3212yes—
425C2×C3×C4C2×D3211yescharacteristic
426C2×Dic3—32no—
427C2×Dic3—32no—
428C2×Dic3—32no—
429C2×Dic3—32no—
430C2×Dic3—32no—
431C2×Dic3—32no—
432C2×Dic3—61no—
433C2×Dic3—61no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3—98no—
437C2×Dic3—91no—
438C2×Dic3—92no—
439C2×Dic3—91no—
440C2×Dic3—92no—
441C2×Dic3—184no—
442C2×Dic3—181no—
443C2×Dic3—181no—
444C2×Dic3—182no—
445C2×Dic3—182no—
446C2×Dic3—184no—
447C2×Dic3—184no—
448C2×Dic3—361no—
449C2×Dic3—361no—
450C2×Dic3C2×D3212yes—
451C2×Dic3C2×D3212yes—
452C2×Dic3C2×D3211yescharacteristic
453C4×D3—22no—
454C4×D3—94no—
455C4×D3—182no—
456Dic6—64no—
457Dic6—64no—
458Dic6—64no—
459Dic6—184no—
460Dic6—182no—
461C2×C32—21no—
462C2×C32—22no—
463C2×C32—21no—
464C2×C32—22no—
465C2×C32—22no—
466C2×C32—21no—
467C2×C32—41no—
468C2×C32—41no—
469C2×C32—41no—
470C2×C32—62no—
471C2×C32C2×#23511yescharacteristic
472C2×C32C2×#23712yes—
473C2×C32C2×D3×Q812yes—
474C2×C32C2×D3×Q811yescharacteristic
475C2×C32D3×#4112yes—
476C2×C32D3×#4111yescharacteristic
477C3×D3—24no—
478C3×D3—24no—
479C2×Q8—91no—
480C2×Q8—92no—
481C2×Q8—271no—
482C22×C4—31no—
483C22×C4—271no—
484C22×C4—271no—
485C4⋊C4—32no—
486C4⋊C4—91no—
487C4⋊C4—92no—
488C4⋊C4—92no—
489C4⋊C4—272no—
490C4⋊C4—272no—
491C4⋊C4—272no—
492C4⋊C4—272no—
493C4⋊C4—271no—
494C22⋊C4—92no—
495C22⋊C4—92no—
496C42—92no—
497C42—91no—
498C42—271no—
499C42—271no—
500C42—272no—
501C42—271no—
502C22×C3—22no—
503C22×C3—21no—
504C22×C3—34no—
505C22×C3—41no—
506C22×C3—62no—
507C22×C3—62no—
508C22×C3—62no—
509C22×C3—62no—
510C22×C3—62no—
511C22×C3C22×D3211yescharacteristic
512C22×C3C22×D3212yes—
513C2×D3—21no—
514C2×D3—21no—
515C2×D3#379712yes—
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—21no—
519C3×C4—34no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—62no—
523C3×C4—62no—
524C3×C4—62no—
525C3×C4—62no—
526C3×C4—62no—
527C3×C4—92no—
528C3×C4—92no—
529C3×C4—122no—
530C3×C4—122no—
531C3×C4—182no—
532C3×C4—182no—
533Dic3—22no—
534Dic3—22no—
535Dic3—34no—
536Dic3—34no—
537Dic3—62no—
538Dic3—62no—
539Dic3—62no—
540Dic3—62no—
541Dic3—62no—
542Dic3—62no—
543Dic3—92no—
544Dic3—94no—
545Dic3—122no—
546Dic3—122no—
547Dic3—182no—
548Dic3—184no—
549Dic3—182no—
550Dic3—182no—
551Dic3—181no—
552Dic3—362no—
553Dic3—362no—
554Dic3—362no—
555Dic3#380512yes—
556C32—22no—
557C32—21no—
558C32—41no—
559C32#557811yescharacteristic, complemented
560C32#562212yescomplemented
561C23—31no—
562Q8—184no—
563Q8—182no—
564Q8—542no—
565C2×C4—31no—
566C2×C4—32no—
567C2×C4—32no—
568C2×C4—62no—
569C2×C4—91no—
570C2×C4—91no—
571C2×C4—92no—
572C2×C4—92no—
573C2×C4—274no—
574C2×C4—271no—
575C2×C4—271no—
576C2×C4—542no—
577C2×C4D3311yescharacteristic
578C2×C3—21no—
579C2×C3—22no—
580C2×C3—21no—
581C2×C3—22no—
582C2×C3—22no—
583C2×C3—21no—
584C2×C3—41no—
585C2×C3—41no—
586C2×C3—41no—
587C2×C3—64no—
588C2×C3—64no—
589C2×C3C2×#379711yescharacteristic
590C2×C3D3×#23512yes—
591C2×C3#6430611yescharacteristic
592C2×C3D3×#23712yes—
593C2×C3D32×Q812yes—
594C2×C3C2×#380511yescharacteristic
595D3—22no—
596C22—32no—
597C22—61no—
598C22—61no—
599C22C2×D3311yescharacteristic
600C4—21no—
601C4—32no—
602C4—62no—
603C4—62no—
604C4—92no—
605C4—92no—
606C4—182no—
607C4—182no—
608C4—272no—
609C4—541no—
610C3—21no—
611C3—22no—
612C3—41no—
613C3#1059438612yescomplemented
614C3#1059481011yescharacteristic, complemented
615C2—62no—
616C2D3×#379711yescharacteristic
617C2#1172438711yescharacteristic
618C2D3×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C32×D3C23
hypercenterC2×C4D33
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723665C2semidirect
3. #11723448C2semidirect
5. C3×#63838C2semidirect
6. #11723035C2semidirect
7. #11723032C2semidirect
10. #11722940C2semidirect
62. #63838D3semidirect
79. D3×C3⋊Dic3Q8semidirect
401. C33#533semidirect
559. C32#5578semidirect
560. C32#5622semidirect
613. C3#10594386semidirect
614. C3#10594810semidirect

Maximal subgroups

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

subgroupindex
#117236652
C4×D3×C3⋊Dic32
#117234482
C3×#638512
C3×#638382
#117230352
#117230322
#117229812
#117229762
#117229402
#117229162
#105948103
#105943863