Finite Group Databasebeta

← #13172239

Subgroups — #13172239

Order 1728.

1198 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723715C211yesmaximal, characteristic
2#11723642C212yesmaximal
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4C3×#63964C212yesmaximal
5C3×#63957C211yesmaximal, characteristic
6#11723037C211yesmaximal, characteristic, complemented
7#11723032C211yesmaximal, characteristic, complemented
8#11723018C211yesmaximal, characteristic, complemented
9#11723015C211yesmaximal, characteristic, complemented
10#11722943C212yesmaximal, complemented
11#11722940C212yesmaximal, complemented
12#10594810—31nomaximal
13#10594029—32nomaximal
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C2×#6293C2212yes—
18C2×C3×#627C2212yes—
19C3×#3757C2212yes—
20C32×#217C2212yes—
21#88815C2211yescharacteristic
22#88815C2211yescharacteristic
23#88813C2212yes—
24#88807C2212yes—
25#88806C2212yes—
26Dic3⋊(C3⋊Dic3)C2212yes—
27Dic3×C3⋊Dic3C2211yescharacteristic
28Dic3×C3⋊Dic3C2211yescharacteristic
29Dic3×C3⋊Dic3C2211yescharacteristic
30Dic3×C3⋊Dic3C2211yescharacteristic
31C3×#3724C2211yescharacteristic
32C3×#3724C2211yescharacteristic
33C3×#3724C2212yes—
34C3×Dic3⋊Dic3C2212yes—
35C3×Dic3⋊Dic3C2212yes—
36C3×Dic32C2212yes—
37C2×C4×C3⋊Dic3—31no—
38#64087—32no—
39C3×#924—32no—
40C3×#878—32no—
41C3×#877—32no—
42#63982—32no—
43#63982—32no—
44#63982—62no—
45#63964D312yes—
46#63957—31no—
47#63957D311yescharacteristic
48#63955—32no—
49C2×Dic32—32no—
50C2×Dic32—61no—
51#63919—31no—
52#63873—31no—
53#63870—31no—
54#63870—31no—
55#63846—32no—
56#63845—32no—
57#63844—32no—
58#63843—32no—
59Dic3×Dic6—32no—
60#63840—32no—
61#63839—32no—
62#63838—32no—
63#63837—32no—
64#63837—32no—
65C23×C33C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)—22no—
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3×C3⋊Dic3Q814yescomplemented
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C32×Dic3—22no—
75C2×C32×Dic3C2312yes—
76C2×C32×Dic3C2312yes—
77C2×C32×Dic3C2311yescharacteristic
78C2×C32×Dic3C2311yescharacteristic
79#6293—24no—
80C3×#627—24no—
81#5622—92no—
82#5517—91no—
83C22×C32×C4—31no—
84C22×C3⋊Dic3—32no—
85C22×C3⋊Dic3—31no—
86C22×C3⋊Dic3—61no—
87C22×C3⋊Dic3—62no—
88C22×C3⋊Dic3—121no—
89C22×C3⋊Dic3C2×D311yescharacteristic
90C2×#634—32no—
91C22×C3×Dic3—21no—
92C22×C3×Dic3—32no—
93C22×C3×Dic3—61no—
94C22×C3×Dic3C2×D312yes—
95C2×C3×Dic6—32no—
96C2×C3×Dic6—32no—
97C2×#627—32no—
98C2×#627—32no—
99C2×#627—62no—
100C2×#627C2×D312yes—
101C32×C22⋊C4—32no—
102#3757C2×D312yes—
103C4⋊(C3⋊Dic3)—32no—
104#3750—32no—
105#3750—32no—
106C4×C3⋊Dic3—31no—
107C4×C3⋊Dic3—31no—
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—31no—
110C3×#217—22no—
111C3×#217—32no—
112C3×#217—32no—
113C3×#217—62no—
114C3×#217C2×D312yes—
115C3×#217C2×D312yes—
116C3×C4⋊Dic3—32no—
117C3×C4⋊Dic3—32no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×C4×Dic3—32no—
125C3×C4×Dic3—32no—
126#3724—31no—
127#3724—31no—
128#3724—32no—
129#3724—32no—
130#3724—62no—
131#3724C2×D312yes—
132#3724C2×D311yescharacteristic
133#3724C2×D311yescharacteristic
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—62no—
141Dic3⋊Dic3—62no—
142Dic3⋊Dic3C2×D312yes—
143Dic3⋊Dic3C2×D312yes—
144Dic32—32no—
145Dic32—32no—
146Dic32—32no—
147Dic32—32no—
148Dic32—32no—
149Dic32—32no—
150Dic32—32no—
151Dic32—61no—
152Dic32—61no—
153Dic32—61no—
154Dic32—61no—
155Dic32C2×D312yes—
156C22×C33—21no—
157C22×C33—21no—
158C22×C33C2×Q812yes—
159C22×C33C2411yescharacteristic
160C3⋊(C3⋊Dic3)—21no—
161C3⋊(C3⋊Dic3)#4112yes—
162C3×C3⋊Dic3—22no—
163C3×C3⋊Dic3—22no—
164C3×C3⋊Dic3C2×Q812yes—
165C3×C3⋊Dic3C2×Q812yes—
166C32×Dic3—21no—
167C32×Dic3—22no—
168C32×Dic3—22no—
169C32×Dic3#4112yes—
170C3×C22⋊Q8—92no—
171C3×#117—91no—
172Q8×Dic3—92no—
173#944—92no—
174#926—91no—
175#924—32no—
176C2×C4×Dic3—92no—
177C2×C4×Dic3—91no—
178C2×C4×Dic3—181no—
179#890—91no—
180#890—91no—
181#889—92no—
182#889—92no—
183#888—91no—
184#888—92no—
185#888—91no—
186#887—92no—
187#887—92no—
188#887—92no—
189#878—32no—
190#878—92no—
191#878—92no—
192#878—92no—
193#878—182no—
194#877—32no—
195#877—92no—
196C4×Dic6—92no—
197C4×Dic6—92no—
198C23×C32—21no—
199C23×C32—22no—
200C23×C32—41no—
201C23×C32C22×D311yescharacteristic
202C23×C32C22×D312yes—
203C2×C32×C4—31no—
204C2×C32×C4—32no—
205C2×C32×C4—32no—
206C2×C32×C4—31no—
207C2×C32×C4—62no—
208C2×C3⋊Dic3—32no—
209C2×C3⋊Dic3—31no—
210C2×C3⋊Dic3—32no—
211C2×C3⋊Dic3—31no—
212C2×C3⋊Dic3—61no—
213C2×C3⋊Dic3—61no—
214C2×C3⋊Dic3—62no—
215C2×C3⋊Dic3—62no—
216C2×C3⋊Dic3—62no—
217C2×C3⋊Dic3—64no—
218C2×C3⋊Dic3—64no—
219C2×C3⋊Dic3—121no—
220C2×C3⋊Dic3—121no—
221C2×C3⋊Dic3—124no—
222C2×C3⋊Dic3—124no—
223C2×C3⋊Dic3Dic614yescomplemented
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D312yes—
226C2×C3⋊Dic3C22×D311yescharacteristic
227C2×C3⋊Dic3C22×D311yescharacteristic
228#634—64no—
229C2×C3×Dic3—24no—
230C2×C3×Dic3—24no—
231C2×C3×Dic3—21no—
232C2×C3×Dic3—21no—
233C2×C3×Dic3—22no—
234C2×C3×Dic3—22no—
235C2×C3×Dic3—38no—
236C2×C3×Dic3—32no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—32no—
239C2×C3×Dic3—32no—
240C2×C3×Dic3—32no—
241C2×C3×Dic3—32no—
242C2×C3×Dic3—61no—
243C2×C3×Dic3—61no—
244C2×C3×Dic3—64no—
245C2×C3×Dic3—62no—
246C2×C3×Dic3—62no—
247C2×C3×Dic3C22×D312yes—
248C2×C3×Dic3C22×D312yes—
249C2×C3×Dic3C22×D312yes—
250C2×C3×Dic3C22×D312yes—
251C2×C3×Dic3C22×D312yes—
252C2×C3×Dic3C22×D312yes—
253C3×Dic6—64no—
254C3×Dic6—64no—
255#627—24no—
256#627—64no—
257#627—64no—
258#627—124no—
259#533—271noSylow
260C2×C33—22no—
261C2×C33C22×Q811yescharacteristic
262C2×C33C2×#4111yescharacteristic
263C2×C33C2×#4111yescharacteristic
264C2×C3×Q8—92no—
265C22×C3×C4—32no—
266C22×C3×C4—61no—
267C22×C3×C4—91no—
268C22×Dic3—32no—
269C22×Dic3—61no—
270C22×Dic3—91no—
271C22×Dic3—92no—
272C22×Dic3—181no—
273C22×Dic3—182no—
274C22×Dic3—361no—
275C22×Dic3D3211yescharacteristic
276C2×Dic6—32no—
277C2×Dic6—32no—
278C2×Dic6—92no—
279C2×Dic6—92no—
280C2×Dic6—92no—
281C2×Dic6—182no—
282C3×C4⋊C4—92no—
283C3×C4⋊C4—92no—
284C3×C4⋊C4—92no—
285C3×C4⋊C4—91no—
286C3×C4⋊C4—91no—
287C3×C22⋊C4—32no—
288C3×C22⋊C4—32no—
289C3×C22⋊C4—62no—
290C3×C22⋊C4—92no—
291C3×C42—92no—
292#217—32no—
293#217—32no—
294#217—62no—
295#217D3212yes—
296C4⋊Dic3—32no—
297C4⋊Dic3—32no—
298C4⋊Dic3—91no—
299C4⋊Dic3—91no—
300C4⋊Dic3—92no—
301C4⋊Dic3—92no—
302C4⋊Dic3—92no—
303C4⋊Dic3—182no—
304Dic3⋊C4—32no—
305Dic3⋊C4—32no—
306Dic3⋊C4—32no—
307Dic3⋊C4—32no—
308Dic3⋊C4—32no—
309Dic3⋊C4—32no—
310Dic3⋊C4—92no—
311Dic3⋊C4—92no—
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315Dic3⋊C4—92no—
316Dic3⋊C4—92no—
317Dic3⋊C4—92no—
318Dic3⋊C4—182no—
319Dic3⋊C4—182no—
320C4×Dic3—32no—
321C4×Dic3—32no—
322C4×Dic3—92no—
323C4×Dic3—92no—
324C4×Dic3—92no—
325C4×Dic3—92no—
326C4×Dic3—92no—
327C4×Dic3—92no—
328C4×Dic3—92no—
329C4×Dic3—91no—
330C4×Dic3—91no—
331C4×Dic3—91no—
332C4×Dic3—91no—
333C4×Dic3—181no—
334C4×Dic3—181no—
335C4×Dic3—181no—
336C4×Dic3—181no—
337C22×C32—21no—
338C22×C32—21no—
339C22×C32—22no—
340C22×C32—22no—
341C22×C32—22no—
342C22×C32—22no—
343C22×C32—22no—
344C22×C32—21no—
345C22×C32—24no—
346C22×C32—22no—
347C22×C32—42no—
348C22×C32—41no—
349C22×C32—42no—
350C22×C32—42no—
351C22×C32—42no—
352C22×C32—42no—
353C22×C32C2×Dic612yes—
354C22×C32D3×Q814yes—
355C22×C32C23×D311yescharacteristic
356C22×C32C23×D312yes—
357C32×C4—32no—
358C32×C4—61no—
359C32×C4—62no—
360C32×C4—62no—
361C3⋊Dic3—22no—
362C3⋊Dic3—22no—
363C3⋊Dic3—32no—
364C3⋊Dic3—34no—
365C3⋊Dic3—64no—
366C3⋊Dic3—62no—
367C3⋊Dic3—61no—
368C3⋊Dic3—62no—
369C3⋊Dic3—62no—
370C3⋊Dic3—122no—
371C3⋊Dic3—122no—
372C3⋊Dic3—122no—
373C3⋊Dic3C2×Dic612yes—
374C3⋊Dic3C2×Dic612yes—
375C3×Dic3—22no—
376C3×Dic3—22no—
377C3×Dic3—22no—
378C3×Dic3—22no—
379C3×Dic3—22no—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—34no—
383C3×Dic3—34no—
384C3×Dic3—42no—
385C3×Dic3—42no—
386C3×Dic3—62no—
387C3×Dic3—62no—
388C3×Dic3—62no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—122no—
393C3×Dic3—122no—
394C3×Dic3#23714yes—
395C4⋊Q8—271no—
396C4⋊Q8—271no—
397#125—271no—
398#125—271no—
399C22⋊Q8—92no—
400C22⋊Q8—272no—
401C4×Q8—272no—
402C4×Q8—272no—
403#117—91no—
404#117—271no—
405C2×C42—271no—
406C33#53311yescharacteristic, complemented, Sylow
407C23×C3—21no—
408C23×C3—22no—
409C23×C3—41no—
410C23×C3C2×D3212yes—
411C23×C3C2×D3211yescharacteristic
412C3×Q8—184no—
413C2×C3×C4—32no—
414C2×C3×C4—32no—
415C2×C3×C4—32no—
416C2×C3×C4—32no—
417C2×C3×C4—32no—
418C2×C3×C4—32no—
419C2×C3×C4—61no—
420C2×C3×C4—61no—
421C2×C3×C4—62no—
422C2×C3×C4—62no—
423C2×C3×C4—64no—
424C2×C3×C4—64no—
425C2×C3×C4—91no—
426C2×C3×C4—91no—
427C2×C3×C4—94no—
428C2×C3×C4—92no—
429C2×C3×C4—92no—
430C2×Dic3—22no—
431C2×Dic3—32no—
432C2×Dic3—32no—
433C2×Dic3—38no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—64no—
439C2×Dic3—61no—
440C2×Dic3—61no—
441C2×Dic3—62no—
442C2×Dic3—62no—
443C2×Dic3—91no—
444C2×Dic3—92no—
445C2×Dic3—91no—
446C2×Dic3—92no—
447C2×Dic3—181no—
448C2×Dic3—181no—
449C2×Dic3—182no—
450C2×Dic3—182no—
451C2×Dic3—182no—
452C2×Dic3—184no—
453C2×Dic3—184no—
454C2×Dic3—361no—
455C2×Dic3—361no—
456C2×Dic3—364no—
457C2×Dic3—364no—
458C2×Dic3C2×D3212yes—
459C2×Dic3C2×D3212yes—
460C2×Dic3C2×D3211yescharacteristic
461C2×Dic3C2×D3211yescharacteristic
462Dic6—64no—
463Dic6—64no—
464Dic6—184no—
465Dic6—184no—
466Dic6—184no—
467Dic6—364no—
468C2×C32—22no—
469C2×C32—24no—
470C2×C32—24no—
471C2×C32—22no—
472C2×C32—21no—
473C2×C32—21no—
474C2×C32—22no—
475C2×C32—22no—
476C2×C32—21no—
477C2×C32—41no—
478C2×C32—41no—
479C2×C32—41no—
480C2×C32—44no—
481C2×C32—44no—
482C2×C32C22×Dic611yescharacteristic
483C2×C32C2×#23712yes—
484C2×C32C2×D3×Q812yes—
485C2×C32D3×#4112yes—
486C2×C32D3×#4111yescharacteristic
487C2×C32D3×#4111yescharacteristic
488C2×Q8—92no—
489C2×Q8—272no—
490C22×C4—31no—
491C22×C4—91no—
492C22×C4—271no—
493C4⋊C4—91no—
494C4⋊C4—91no—
495C4⋊C4—92no—
496C4⋊C4—92no—
497C4⋊C4—92no—
498C4⋊C4—272no—
499C4⋊C4—272no—
500C4⋊C4—272no—
501C4⋊C4—271no—
502C4⋊C4—271no—
503C22⋊C4—32no—
504C22⋊C4—92no—
505C42—92no—
506C42—271no—
507C42—271no—
508C42—271no—
509C42—271no—
510C42—272no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—21no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—21no—
517C22×C3—22no—
518C22×C3—21no—
519C22×C3—22no—
520C22×C3—24no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—41no—
526C22×C3—42no—
527C22×C3D3×Dic614yes—
528C22×C3#379712yes—
529C22×C3C22×D3212yes—
530C22×C3C22×D3211yescharacteristic
531C3×C4—34no—
532C3×C4—62no—
533C3×C4—62no—
534C3×C4—62no—
535C3×C4—62no—
536C3×C4—62no—
537C3×C4—62no—
538C3×C4—62no—
539C3×C4—92no—
540C3×C4—92no—
541C3×C4—122no—
542C3×C4—122no—
543C3×C4—182no—
544C3×C4—182no—
545Dic3—22no—
546Dic3—22no—
547Dic3—21no—
548Dic3—34no—
549Dic3—34no—
550Dic3—62no—
551Dic3—62no—
552Dic3—62no—
553Dic3—62no—
554Dic3—62no—
555Dic3—62no—
556Dic3—92no—
557Dic3—94no—
558Dic3—122no—
559Dic3—122no—
560Dic3—184no—
561Dic3—182no—
562Dic3—182no—
563Dic3—181no—
564Dic3—182no—
565Dic3—362no—
566Dic3—362no—
567Dic3—362no—
568Dic3#380512yes—
569C32—22no—
570C32—21no—
571C32—41no—
572C32#551711yescharacteristic, complemented
573C32#562212yescomplemented
574C23D3311yescharacteristic
575Q8—184no—
576Q8—544no—
577C2×C4—31no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—31no—
581C2×C4—62no—
582C2×C4—91no—
583C2×C4—91no—
584C2×C4—94no—
585C2×C4—92no—
586C2×C4—92no—
587C2×C4—271no—
588C2×C4—271no—
589C2×C4—542no—
590C2×C3—24no—
591C2×C3—24no—
592C2×C3—22no—
593C2×C3—22no—
594C2×C3—21no—
595C2×C3—22no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—22no—
599C2×C3—44no—
600C2×C3—44no—
601C2×C3—41no—
602C2×C3—41no—
603C2×C3—41no—
604C2×C3C2×D3×Dic612yes—
605C2×C3C2×#379711yescharacteristic
606C2×C3#6430611yescharacteristic
607C2×C3D3×#23712yes—
608C2×C3#6431212yes—
609C2×C3C2×#380511yescharacteristic
610C22—21no—
611C22—21no—
612C22#8895712yes—
613C22C2×D3311yescharacteristic
614C4—32no—
615C4—61no—
616C4—62no—
617C4—62no—
618C4—92no—
619C4—92no—
620C4—182no—
621C4—182no—
622C4—272no—
623C4—541no—
624C3—21no—
625C3—22no—
626C3—41no—
627C3#1059402912yescomplemented
628C3#1059481011yescharacteristic, complemented
629C2—22no—
630C2D3×#380511yescharacteristic
631C2#1172441111yescharacteristic
632C2C2×#8895711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC23×C33C23
hypercenterC23D33
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
6. #11723037C2semidirect
7. #11723032C2semidirect
8. #11723018C2semidirect
9. #11723015C2semidirect
10. #11722943C2semidirect
11. #11722940C2semidirect
69. C2×C3×C3⋊Dic3Q8semidirect
223. C2×C3⋊Dic3Dic6semidirect
406. C33#533semidirect
572. C32#5517semidirect
573. C32#5622semidirect
627. C3#10594029semidirect
628. C3#10594810semidirect

Maximal subgroups

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

subgroupindex
#117237152
#117236422
C2×Dic3×C3⋊Dic32
C3×#639642
C3×#639572
#117230372
#117230322
#117230182
#117230152
#117229432
#117229402
#105948103
#105940293