Finite Group Databasebeta

← #13178278

Subgroups — #13178278

Order 1728.

1082 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64087C212yesmaximal
2C3×#64086C211yesmaximal, characteristic
3C2×#88815C211yesmaximal, characteristic
4#11723721C212yesmaximal
5#11723715C211yesmaximal, characteristic
6C4⋊#6296C211yesmaximal, characteristic, complemented
7#11723679C211yesmaximal, characteristic, complemented
8#11723679C211yesmaximal, characteristic, complemented
9#11723678C211yesmaximal, characteristic, complemented
10#11723669C212yesmaximal, complemented
11#11723666C212yesmaximal, complemented
12#10594035—32nomaximal
13#10594033—31nomaximal
14C22×C3×C3⋊Dic3C2212yes—
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×#634C2212yes—
17C2×#6296C2212yes—
18C33×C22⋊C4C2211yescharacteristic
19C3×#3757C2211yescharacteristic
20C3×#3757C2212yes—
21C3×C4⋊(C3⋊Dic3)C2212yes—
22C3×#3750C2211yescharacteristic
23C3×#3750C2211yescharacteristic
24C3×#3750C2212yes—
25C3×#3750C2212yes—
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C3×C4×C3⋊Dic3C2211yescharacteristic
28#88815C2212yes—
29#88815C2212yes—
30#88815C2212yes—
31#88815C2212yes—
32#88815C2211yescharacteristic
33#88815C2211yescharacteristic
34#88815C2211yescharacteristic
35#88815C2211yescharacteristic
36#88813C2212yes—
37#64087D312yes—
38#64086D311yescharacteristic
39C3×#878—32no—
40C3×#878—32no—
41C3×#878—62no—
42C3×#877—32no—
43C3×#877—61no—
44#63982—32no—
45C2×#3724—31no—
46C2×Dic3⋊Dic3—32no—
47#63964—32no—
48#63964—32no—
49#63957—31no—
50#63955—32no—
51#63919—31no—
52#63918—32no—
53#63873—31no—
54#63870—31no—
55#63870—31no—
56#63867—32no—
57#63867—32no—
58#63866—32no—
59#63847—32no—
60#63846—32no—
61#63845—32no—
62#63840—32no—
63#63839—32no—
64#63839—32no—
65C23×C33C2311yescharacteristic
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C3×C3⋊Dic3—42no—
69C2×C3×C3⋊Dic3Q814yescomplemented
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C3×#634—42no—
79#6296—42no—
80#5623—92no—
81#5519—91no—
82C22×C3⋊Dic3C2×D312yes—
83C22×C3⋊Dic3C2×D311yescharacteristic
84C2×#634C2×D312yes—
85C22×C3×Dic3—32no—
86C22×C3×Dic3—32no—
87C22×C3×Dic3—32no—
88C22×C3×Dic3—61no—
89C22×C3×Dic3—62no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—32no—
92C2×C3×Dic6—62no—
93C2×#627—32no—
94C2×#627—32no—
95C2×#627—32no—
96C32×C22⋊C4—21no—
97C32×C22⋊C4—22no—
98C32×C22⋊C4—41no—
99C32×C22⋊C4C2×D312yes—
100C32×C22⋊C4C2×D311yescharacteristic
101#3757C2×D311yescharacteristic
102#3757C2×D312yes—
103C4⋊(C3⋊Dic3)C2×D312yes—
104#3750C2×D311yescharacteristic
105#3750C2×D311yescharacteristic
106#3750C2×D312yes—
107#3750C2×D312yes—
108C4×C3⋊Dic3C2×D311yescharacteristic
109C4×C3⋊Dic3C2×D311yescharacteristic
110C3×#217—32no—
111C3×#217—32no—
112C3×#217—32no—
113C3×#217—61no—
114C3×#217—62no—
115C3×C4⋊Dic3—32no—
116C3×C4⋊Dic3—32no—
117C3×C4⋊Dic3—62no—
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×Dic3⋊C4—61no—
125C3×Dic3⋊C4—61no—
126C3×Dic3⋊C4—62no—
127C3×Dic3⋊C4—62no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—32no—
130C3×C4×Dic3—61no—
131C3×C4×Dic3—61no—
132#3724—32no—
133#3724—32no—
134#3724—32no—
135#3724—32no—
136#3724—31no—
137#3724—31no—
138#3724—31no—
139#3724—31no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—32no—
148Dic3⋊Dic3—32no—
149Dic3⋊Dic3—32no—
150Dic3⋊Dic3—32no—
151Dic3⋊Dic3—32no—
152Dic32—32no—
153Dic32—32no—
154Dic32—32no—
155C22×C33—21no—
156C22×C33—21no—
157C22×C33C2×Q812yes—
158C22×C33C2411yescharacteristic
159C33×C4—21no—
160C33×C4—21no—
161C3×C3⋊Dic3—22no—
162C3×C3⋊Dic3—21no—
163C3×C3⋊Dic3—22no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—21no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3C2×Q812yes—
168C3×C3⋊Dic3C2×Q812yes—
169C3×C22⋊Q8—92no—
170C3×#117—91no—
171#944—92no—
172#944—92no—
173#926—91no—
174C2×C4⋊Dic3—91no—
175#924—92no—
176C2×Dic3⋊C4—92no—
177#890—92no—
178#889—92no—
179#889—92no—
180#889—92no—
181#889—92no—
182#888—92no—
183#888—92no—
184#888—92no—
185#878—32no—
186#878—32no—
187#878—62no—
188#878—92no—
189#878—92no—
190#877—32no—
191#877—61no—
192#877—92no—
193#870—91no—
194#870—91no—
195#869—91no—
196#869—91no—
197C23×C32—22no—
198C23×C32—21no—
199C23×C32—41no—
200C23×C32C22×D311yescharacteristic
201C23×C32C22×D312yes—
202C2×C32×C4—21no—
203C2×C32×C4—21no—
204C2×C32×C4—22no—
205C2×C32×C4—22no—
206C2×C32×C4—41no—
207C2×C32×C4—41no—
208C2×C32×C4C22×D312yes—
209C2×C32×C4C22×D312yes—
210C2×C32×C4C22×D311yescharacteristic
211C2×C32×C4C22×D311yescharacteristic
212C2×C3⋊Dic3—42no—
213C2×C3⋊Dic3Dic614yescomplemented
214C2×C3⋊Dic3C22×D312yes—
215C2×C3⋊Dic3C22×D312yes—
216C2×C3⋊Dic3C22×D312yes—
217C2×C3⋊Dic3C22×D311yescharacteristic
218C2×C3⋊Dic3C22×D312yes—
219C2×C3⋊Dic3C22×D311yescharacteristic
220C2×C3⋊Dic3C22×D311yescharacteristic
221C2×C3⋊Dic3C22×D311yescharacteristic
222#634—42no—
223C2×C3×Dic3—38no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
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—64no—
240C2×C3×Dic3—61no—
241C2×C3×Dic3—61no—
242C2×C3×Dic3—62no—
243C2×C3×Dic3—62no—
244C2×C3×Dic3—62no—
245C2×C3×Dic3—122no—
246C2×C3×Dic3—122no—
247C2×C3×Dic3—124no—
248C3×Dic6—122no—
249C3×Dic6—122no—
250C3×Dic6—124no—
251#627—122no—
252#627—122no—
253#627—122no—
254#544—271noSylow
255C2×C33—22no—
256C2×C33C22×Q811yescharacteristic
257C2×C33H(5,𝔽2)11yescharacteristic
258C2×C332−1+411yescharacteristic
259C2×C3×Q8—92no—
260C22×C3×C4—91no—
261C22×C3×C4—92no—
262C22×Dic3—32no—
263C22×Dic3—32no—
264C22×Dic3—32no—
265C22×Dic3—61no—
266C22×Dic3—62no—
267C2×Dic6—32no—
268C2×Dic6—32no—
269C2×Dic6—62no—
270C2×Dic6—92no—
271C2×Dic6—92no—
272C2×Dic6—92no—
273C3×C4⋊C4—91no—
274C3×C4⋊C4—92no—
275C3×C4⋊C4—92no—
276C3×C4⋊C4—92no—
277C3×C4⋊C4—91no—
278C3×C22⋊C4—21no—
279C3×C22⋊C4—22no—
280C3×C22⋊C4—41no—
281C3×C22⋊C4—92no—
282C3×C22⋊C4—91no—
283C3×C22⋊C4D3212yes—
284C3×C22⋊C4D3211yescharacteristic
285C3×C42—91no—
286C3×C42—91no—
287#217—32no—
288#217—32no—
289#217—32no—
290#217—61no—
291#217—62no—
292C4⋊Dic3—32no—
293C4⋊Dic3—32no—
294C4⋊Dic3—62no—
295C4⋊Dic3—91no—
296C4⋊Dic3—91no—
297C4⋊Dic3—91no—
298C4⋊Dic3—92no—
299C4⋊Dic3—91no—
300C4⋊Dic3—92no—
301C4⋊Dic3—92no—
302C4⋊Dic3—92no—
303Dic3⋊C4—32no—
304Dic3⋊C4—32no—
305Dic3⋊C4—32no—
306Dic3⋊C4—32no—
307Dic3⋊C4—32no—
308Dic3⋊C4—32no—
309Dic3⋊C4—61no—
310Dic3⋊C4—61no—
311Dic3⋊C4—62no—
312Dic3⋊C4—62no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315Dic3⋊C4—92no—
316Dic3⋊C4—92no—
317Dic3⋊C4—92no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325C4×Dic3—32no—
326C4×Dic3—32no—
327C4×Dic3—61no—
328C4×Dic3—61no—
329C4×Dic3—92no—
330C4×Dic3—92no—
331C4×Dic3—92no—
332C22×C32—22no—
333C22×C32—21no—
334C22×C32—21no—
335C22×C32—22no—
336C22×C32—22no—
337C22×C32—21no—
338C22×C32—24no—
339C22×C32—22no—
340C22×C32—41no—
341C22×C32—42no—
342C22×C32—42no—
343C22×C32—41no—
344C22×C32—42no—
345C22×C32—41no—
346C22×C32—81no—
347C22×C32—81no—
348C22×C32C2×Dic612yes—
349C22×C32D3×Q814yes—
350C22×C32C23×D311yescharacteristic
351C22×C32C23×D312yes—
352C32×C4—22no—
353C32×C4—21no—
354C32×C4—21no—
355C32×C4—22no—
356C32×C4—42no—
357C32×C4—41no—
358C32×C4—42no—
359C32×C4—41no—
360C32×C4—81no—
361C32×C4—81no—
362C3⋊Dic3—22no—
363C3⋊Dic3—22no—
364C3⋊Dic3—22no—
365C3⋊Dic3—21no—
366C3⋊Dic3—21no—
367C3⋊Dic3—22no—
368C3⋊Dic3C2×Dic612yes—
369C3⋊Dic3C2×Dic612yes—
370C3×Dic3—34no—
371C3×Dic3—34no—
372C3×Dic3—62no—
373C3×Dic3—62no—
374C3×Dic3—64no—
375C3×Dic3—62no—
376C3×Dic3—62no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C3×Dic3—62no—
380C3×Dic3—62no—
381C3×Dic3—62no—
382C3×Dic3—62no—
383C3×Dic3—62no—
384C3×Dic3—62no—
385C3×Dic3—122no—
386C3×Dic3—121no—
387C3×Dic3—122no—
388C3×Dic3—121no—
389C3×Dic3—122no—
390C4⋊Q8—271no—
391C4⋊Q8—271no—
392C4⋊Q8—272no—
393#125—271no—
394#125—271no—
395#125—272no—
396C22⋊Q8—92no—
397C22⋊Q8—272no—
398#117—91no—
399#117—271no—
400C2×C4⋊C4—271no—
401C33#54411yescharacteristic, complemented, Sylow
402C23×C3—21no—
403C23×C3—22no—
404C23×C3—41no—
405C23×C3C2×D3212yes—
406C23×C3C2×D3211yescharacteristic
407C3×Q8—362no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—91no—
415C2×C3×C4—91no—
416C2×C3×C4—94no—
417C2×C3×C4—92no—
418C2×C3×C4—92no—
419C2×C3×C4—92no—
420C2×C3×C4—91no—
421C2×C3×C4—92no—
422C2×C3×C4—91no—
423C2×C3×C4—362no—
424C2×C3×C4C2×D3212yes—
425C2×C3×C4C2×D3212yes—
426C2×C3×C4C2×D3211yescharacteristic
427C2×C3×C4C2×D3211yescharacteristic
428C2×Dic3—32no—
429C2×Dic3—32no—
430C2×Dic3—38no—
431C2×Dic3—32no—
432C2×Dic3—32no—
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—32no—
440C2×Dic3—32no—
441C2×Dic3—64no—
442C2×Dic3—61no—
443C2×Dic3—61no—
444C2×Dic3—62no—
445C2×Dic3—61no—
446C2×Dic3—61no—
447C2×Dic3—62no—
448C2×Dic3—62no—
449C2×Dic3—62no—
450C2×Dic3—122no—
451C2×Dic3—122no—
452C2×Dic3—124no—
453Dic6—122no—
454Dic6—122no—
455Dic6—124no—
456Dic6—362no—
457Dic6—362no—
458Dic6—362no—
459C2×C32—22no—
460C2×C32—24no—
461C2×C32—22no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—22no—
465C2×C32—21no—
466C2×C32—22no—
467C2×C32—41no—
468C2×C32—41no—
469C2×C32—41no—
470C2×C32—42no—
471C2×C32—44no—
472C2×C32—82no—
473C2×C32C22×Dic611yescharacteristic
474C2×C32#100412yes—
475C2×C32C2×D3×Q812yes—
476C2×C32#100911yescharacteristic
477C2×C32#101011yescharacteristic
478C2×C32#101012yes—
479C2×Q8—92no—
480C2×Q8—272no—
481C22×C4—91no—
482C22×C4—92no—
483C4⋊C4—91no—
484C4⋊C4—91no—
485C4⋊C4—92no—
486C4⋊C4—92no—
487C4⋊C4—92no—
488C4⋊C4—271no—
489C4⋊C4—271no—
490C4⋊C4—272no—
491C4⋊C4—271no—
492C4⋊C4—271no—
493C4⋊C4—272no—
494C4⋊C4—272no—
495C4⋊C4—272no—
496C22⋊C4—91no—
497C22⋊C4—92no—
498C22⋊C4#111311yescharacteristic
499C42—91no—
500C42—91no—
501C42—272no—
502C22×C3—21no—
503C22×C3—22no—
504C22×C3—21no—
505C22×C3—22no—
506C22×C3—21no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—24no—
510C22×C3—42no—
511C22×C3—41no—
512C22×C3—42no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—42no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3D3×Dic614yes—
519C22×C3#379712yes—
520C22×C3C22×D3212yes—
521C22×C3C22×D3211yescharacteristic
522C3×C4—21no—
523C3×C4—22no—
524C3×C4—21no—
525C3×C4—22no—
526C3×C4—42no—
527C3×C4—41no—
528C3×C4—42no—
529C3×C4—41no—
530C3×C4—81no—
531C3×C4—81no—
532C3×C4—92no—
533C3×C4—92no—
534C3×C4—181no—
535C3×C4—182no—
536C3×C4—181no—
537C3×C4—182no—
538C3×C4—182no—
539C3×C4—182no—
540Dic3—34no—
541Dic3—34no—
542Dic3—62no—
543Dic3—62no—
544Dic3—64no—
545Dic3—62no—
546Dic3—62no—
547Dic3—62no—
548Dic3—62no—
549Dic3—62no—
550Dic3—62no—
551Dic3—62no—
552Dic3—62no—
553Dic3—62no—
554Dic3—62no—
555Dic3—122no—
556Dic3—122no—
557Dic3—121no—
558Dic3—122no—
559Dic3—121no—
560C32—21no—
561C32—22no—
562C32—41no—
563C32#551911yescharacteristic, complemented
564C32#562312yescomplemented
565C23C2×#111311yescharacteristic
566Q8—362no—
567Q8—1082no—
568C2×C4—91no—
569C2×C4—91no—
570C2×C4—94no—
571C2×C4—91no—
572C2×C4—91no—
573C2×C4—92no—
574C2×C4—92no—
575C2×C4—92no—
576C2×C4—92no—
577C2×C4—362no—
578C2×C4C2×#111311yescharacteristic
579C2×C4C2×#111311yescharacteristic
580C2×C3—22no—
581C2×C3—24no—
582C2×C3—21no—
583C2×C3—22no—
584C2×C3—21no—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—42no—
589C2×C3—44no—
590C2×C3—41no—
591C2×C3—41no—
592C2×C3—41no—
593C2×C3—82no—
594C2×C3C2×D3×Dic612yes—
595C2×C3C2×#379711yescharacteristic
596C2×C3#6430911yescharacteristic
597C2×C3#6430912yes—
598C2×C3#6431311yescharacteristic
599C2×C3#6431412yes—
600C22—21no—
601C22—21no—
602C22#8904012yes—
603C22C22×#111311yescharacteristic
604C4—21no—
605C4—21no—
606C4—92no—
607C4—92no—
608C4—181no—
609C4—182no—
610C4—182no—
611C4—182no—
612C4—182no—
613C4—181no—
614C3—21no—
615C3—22no—
616C3—41no—
617C3#1059403311yescharacteristic, complemented
618C3#1059403512yescomplemented
619C2—22no—
620C2C2×#8904011yescharacteristic
621C2#1172462011yescharacteristic
622C2#1172462311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC23×C33C23
hypercenterC22⋊C4#1113
hypocenterC33#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
6. C4⋊#6296C2semidirect
7. #11723679C2semidirect
8. #11723679C2semidirect
9. #11723678C2semidirect
10. #11723669C2semidirect
11. #11723666C2semidirect
69. C2×C3×C3⋊Dic3Q8semidirect
213. C2×C3⋊Dic3Dic6semidirect
401. C33#544semidirect
563. C32#5519semidirect
564. C32#5623semidirect
617. C3#10594033semidirect
618. C3#10594035semidirect

Maximal subgroups

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

subgroupindex
C3×#640872
C3×#640862
C2×#888152
#117237212
#117237152
C4⋊#62962
#117236792
#117236782
#117236692
#117236662
#105940353
#105940333