Finite Group Databasebeta

← #13178223

Subgroups — #13178223

Order 1728.

1282 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C22⋊C4×C3⋊D3C211yesmaximal, characteristic
2C3×#64087C212yesmaximal, complemented
3#11723723C211yesmaximal, characteristic
4#11723721C212yesmaximal, complemented
5C2×#88814C211yesmaximal, characteristic
6#11723692C212yesmaximal, complemented
7#11723688C212yesmaximal, complemented
8#11723674C212yesmaximal, complemented
9#11723671C212yesmaximal, complemented
10#10594180—31nomaximal
11#10594170—32nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×C3×C3⋊Dic3C2212yes—
14C2×C3×C4×C3⋊D3C2212yes—
15C2×C3×#634C2212yescomplemented
16C2×#6296C2212yescomplemented
17C2×#6294C2212yes—
18C33×C22⋊C4C2211yescharacteristic
19C3×#3757C2212yes—
20C3×#3757C2211yescharacteristic
21C3×#3752C2212yes—
22C3×C4⋊(C3⋊Dic3)C2212yescomplemented
23C3×#3750C2212yescomplemented
24C3×#3750C2212yescomplemented
25#88815C2212yescomplemented
26#88815C2212yescomplemented
27#88815C2212yescomplemented
28#88814C2211yescharacteristic
29#88814C2212yes—
30#88814C2211yescharacteristic
31#88814C2211yescharacteristic
32#88814C2211yescharacteristic
33C22⋊C4×C3⋊D3D311yescharacteristic
34#64087D312yescomplemented
35C3×D3×C22⋊C4—32no—
36C3×D3×C22⋊C4—61no—
37C3×#878—32no—
38C3×#878—32no—
39C3×#878—62no—
40#63982—32no—
41#63974—31no—
42D3×#217—32no—
43#63964—32no—
44#63964—32no—
45C2×#3722—31no—
46C2×(C2×D3)⋊Dic3—32no—
47#63899—32no—
48#63896—32no—
49#63895—32no—
50#63893—32no—
51#63888—32no—
52#63887—32no—
53#63871—32no—
54#63865—32no—
55#63857—32no—
56#63856—32no—
57#63851—32no—
58#63851—32no—
59C23×C33C2311yescharacteristic
60C22×C3×C3⋊D3—42no—
61C22×C3×C3⋊D3Q814yescomplemented
62C22×C3×C3⋊D3C2311yescharacteristic
63C22×C3×C3⋊D3C2311yescharacteristic
64C2×C33×C4C2312yes—
65C2×C3×C3⋊Dic3—42no—
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C3×C4×C3⋊D3—42no—
72C3×#634—42no—
73#6296—42no—
74#6294—42no—
75#5634—92no—
76#5483—91no—
77C23×C3⋊D3C2×D311yescharacteristic
78C23×C3×D3—32no—
79C23×C3×D3—61no—
80C22×C3⋊Dic3C2×D312yes—
81C2×C4×C3⋊D3C2×D312yes—
82C2×#634C2×D312yescomplemented
83C22×C3×Dic3—32no—
84C22×C3×Dic3—32no—
85C22×C3×Dic3—62no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—62no—
89C2×C3×Dic6—32no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—62no—
92C2×#627—32no—
93C2×#627—32no—
94C2×#627—32no—
95C2×#624—32no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98C32×C22⋊C4—22no—
99C32×C22⋊C4—21no—
100C32×C22⋊C4—41no—
101C32×C22⋊C4C2×D311yescharacteristic
102C32×C22⋊C4C2×D312yes—
103#3757C2×D312yes—
104#3757C2×D311yescharacteristic
105#3752C2×D312yes—
106C4⋊(C3⋊Dic3)C2×D312yescomplemented
107#3750C2×D312yescomplemented
108#3750C2×D312yescomplemented
109C3×#217—32no—
110C3×#217—32no—
111C3×#217—32no—
112C3×#217—62no—
113C3×#217—61no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—62no—
117C3×C4⋊Dic3—32no—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—62no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—62no—
125C3×Dic3⋊C4—62no—
126#3724—32no—
127#3724—32no—
128#3724—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135#3722—31no—
136#3722—32no—
137#3722—31no—
138#3722—31no—
139#3722—31no—
140(C2×D3)⋊Dic3—32no—
141(C2×D3)⋊Dic3—32no—
142(C2×D3)⋊Dic3—32no—
143(C2×D3)⋊Dic3—32no—
144(C2×D3)⋊Dic3—32no—
145(C2×D3)⋊Dic3—32no—
146C22×C33—22no—
147C22×C33C2×Q812yes—
148C22×C33C2411yescharacteristic
149C2×C3×C3⋊D3—22no—
150C2×C3×C3⋊D3—22no—
151C2×C3×C3⋊D3—44no—
152C2×C3×C3⋊D3C2×Q812yes—
153C2×C3×C3⋊D3C2×Q812yes—
154C33×C4—22no—
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3—22no—
157C3×C3⋊Dic3—22no—
158C3×C3⋊Dic3—22no—
159C3×C3⋊Dic3—22no—
160C3×C22⋊Q8—92no—
161C2×C3×C22⋊C4—91no—
162C2×#217—91no—
163C2×#217—91no—
164#946—92no—
165#946—92no—
166C2×(C2×D3)⋊C4—92no—
167#924—92no—
168#924—92no—
169#924—92no—
170#897—92no—
171#897—92no—
172#896—92no—
173#896—92no—
174#896—92no—
175#896—92no—
176D3×C22⋊C4—32no—
177D3×C22⋊C4—61no—
178D3×C22⋊C4—92no—
179#878—32no—
180#878—32no—
181#878—62no—
182#878—92no—
183#878—92no—
184#878—92no—
185#878—92no—
186C23×C32—21no—
187C23×C32—22no—
188C23×C32—41no—
189C23×C32C22×D311yescharacteristic
190C23×C32C22×D312yes—
191C22×C3⋊D3—42no—
192C22×C3⋊D3Dic614yescomplemented
193C22×C3⋊D3C22×D311yescharacteristic
194C22×C3⋊D3C22×D311yescharacteristic
195C22×C3×D3—32no—
196C22×C3×D3—32no—
197C22×C3×D3—38no—
198C22×C3×D3—61no—
199C22×C3×D3—61no—
200C22×C3×D3—64no—
201C22×C3×D3—122no—
202C22×C3×D3—122no—
203C22×C3×D3—124no—
204C2×C32×C4—22no—
205C2×C32×C4—22no—
206C2×C32×C4—22no—
207C2×C32×C4—42no—
208C2×C32×C4C22×D312yes—
209C2×C32×C4C22×D312yes—
210C2×C32×C4C22×D312yes—
211C2×C3⋊Dic3—42no—
212C2×C3⋊Dic3C22×D312yes—
213C2×C3⋊Dic3C22×D312yes—
214C2×C3⋊Dic3C22×D312yes—
215C2×C3⋊Dic3C22×D312yes—
216C2×C3⋊Dic3C22×D312yes—
217C4×C3⋊D3—42no—
218#634—42no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—62no—
230C2×C3×Dic3—62no—
231C2×C3×Dic3—62no—
232C2×C3×Dic3—62no—
233C2×C3×Dic3—62no—
234C2×C3×Dic3—122no—
235C2×C3×Dic3—122no—
236C2×C3×Dic3—124no—
237C3×C4×D3—122no—
238C3×C4×D3—122no—
239C3×C4×D3—124no—
240C3×Dic6—122no—
241C3×Dic6—122no—
242C3×Dic6—124no—
243#627—122no—
244#627—122no—
245#627—122no—
246#624—122no—
247D3×Dic3—122no—
248D3×Dic3—122no—
249#543—271noSylow
250C2×C33—22no—
251C2×C33C22×Q811yescharacteristic
252C2×C33H(5,𝔽2)12yes—
253C3×C3⋊D3—22no—
254C3×C3⋊D3—22no—
255C24×C3—91no—
256C23×D3—32no—
257C23×D3—61no—
258C2×C3×Q8—92no—
259C22×C3×C4—92no—
260C22×C3×C4—92no—
261C22×Dic3—32no—
262C22×Dic3—32no—
263C22×Dic3—62no—
264C22×Dic3—92no—
265C2×C4×D3—32no—
266C2×C4×D3—32no—
267C2×C4×D3—62no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×Dic6—32no—
271C2×Dic6—32no—
272C2×Dic6—62no—
273C2×Dic6—92no—
274C2×Dic6—92no—
275C2×Dic6—92no—
276C3×C4⋊C4—92no—
277C3×C4⋊C4—92no—
278C3×C4⋊C4—92no—
279C3×C22⋊C4—22no—
280C3×C22⋊C4—21no—
281C3×C22⋊C4—41no—
282C3×C22⋊C4—91no—
283C3×C22⋊C4—92no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4D3211yescharacteristic
286C3×C22⋊C4D3212yes—
287#217—32no—
288#217—32no—
289#217—32no—
290#217—62no—
291#217—61no—
292#217—91no—
293#217—91no—
294#217—91no—
295#217—92no—
296#217—91no—
297(C2×D3)⋊C4—32no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—62no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306C4⋊Dic3—32no—
307C4⋊Dic3—32no—
308C4⋊Dic3—62no—
309C4⋊Dic3—92no—
310C4⋊Dic3—92no—
311C4⋊Dic3—92no—
312Dic3⋊C4—32no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—62no—
317Dic3⋊C4—62no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324C22×C32—22no—
325C22×C32—22no—
326C22×C32—22no—
327C22×C32—22no—
328C22×C32—21no—
329C22×C32—24no—
330C22×C32—22no—
331C22×C32—41no—
332C22×C32—42no—
333C22×C32—42no—
334C22×C32—42no—
335C22×C32—42no—
336C22×C32—82no—
337C22×C32C2×Dic612yes—
338C22×C32D3×Q814yes—
339C22×C32C23×D311yescharacteristic
340C22×C32C23×D312yes—
341C2×C3⋊D3—22no—
342C2×C3⋊D3—22no—
343C2×C3⋊D3—44no—
344C2×C3⋊D3C2×Dic612yes—
345C2×C3⋊D3C2×Dic612yes—
346C2×C3×D3—34no—
347C2×C3×D3—34no—
348C2×C3×D3—62no—
349C2×C3×D3—62no—
350C2×C3×D3—64no—
351C2×C3×D3—62no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—62no—
355C2×C3×D3—122no—
356C2×C3×D3—128no—
357C2×C3×D3—128no—
358C32×C4—22no—
359C32×C4—22no—
360C32×C4—22no—
361C32×C4—42no—
362C32×C4—42no—
363C32×C4—42no—
364C32×C4—82no—
365C3⋊Dic3—22no—
366C3⋊Dic3—22no—
367C3⋊Dic3—22no—
368C3⋊Dic3—22no—
369C3⋊Dic3—22no—
370C3×Dic3—64no—
371C3×Dic3—62no—
372C3×Dic3—62no—
373C3×Dic3—62no—
374C3×Dic3—62no—
375C3×Dic3—62no—
376C3×Dic3—62no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C3×Dic3—62no—
380C3×Dic3—62no—
381C3×Dic3—122no—
382C3×Dic3—122no—
383C3×Dic3—122no—
384C3×Dic3—122no—
385C22⋊Q8—92no—
386C22⋊Q8—272no—
387C22⋊Q8—272no—
388C22⋊Q8—272no—
389C22⋊Q8—272no—
390C22⋊Q8—272no—
391C2×C22⋊C4—91no—
392C2×C22⋊C4—271no—
393C2×C22⋊C4—271no—
394C33#54311yescharacteristic, complemented, Sylow
395C23×C3—22no—
396C23×C3—21no—
397C23×C3—41no—
398C23×C3—91no—
399C23×C3—91no—
400C23×C3—94no—
401C23×C3—362no—
402C23×C3C2×D3212yes—
403C23×C3C2×D3211yescharacteristic
404C22×D3—32no—
405C22×D3—32no—
406C22×D3—38no—
407C22×D3—61no—
408C22×D3—64no—
409C22×D3—61no—
410C22×D3—122no—
411C22×D3—122no—
412C22×D3—124no—
413C3×Q8—362no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—42no—
418C2×C3×C4—92no—
419C2×C3×C4—92no—
420C2×C3×C4—92no—
421C2×C3×C4—92no—
422C2×C3×C4—92no—
423C2×C3×C4—362no—
424C2×C3×C4—362no—
425C2×C3×C4C2×D3212yes—
426C2×C3×C4C2×D3212yes—
427C2×C3×C4C2×D3212yes—
428C2×Dic3—32no—
429C2×Dic3—32no—
430C2×Dic3—32no—
431C2×Dic3—32no—
432C2×Dic3—32no—
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—62no—
439C2×Dic3—62no—
440C2×Dic3—62no—
441C2×Dic3—62no—
442C2×Dic3—62no—
443C2×Dic3—122no—
444C2×Dic3—122no—
445C2×Dic3—124no—
446C2×Dic3—362no—
447C4×D3—122no—
448C4×D3—122no—
449C4×D3—124no—
450C4×D3—362no—
451C4×D3—362no—
452Dic6—122no—
453Dic6—122no—
454Dic6—124no—
455Dic6—362no—
456Dic6—362no—
457Dic6—362no—
458C2×C32—24no—
459C2×C32—22no—
460C2×C32—22no—
461C2×C32—21no—
462C2×C32—22no—
463C2×C32—22no—
464C2×C32—22no—
465C2×C32—41no—
466C2×C32—42no—
467C2×C32—42no—
468C2×C32—44no—
469C2×C32—82no—
470C2×C32C22×Dic611yescharacteristic
471C2×C32#100412yes—
472C2×C32#100412yes—
473C2×C32C2×D3×Q812yes—
474C2×C32#100912yes—
475C3⋊D3—22no—
476C3⋊D3—22no—
477C3×D3—64no—
478C3×D3—64no—
479C3×D3—64no—
480C3×D3—122no—
481C24—91no—
482C2×Q8—92no—
483C2×Q8—272no—
484C22×C4—92no—
485C22×C4—92no—
486C22×C4—272no—
487C4⋊C4—92no—
488C4⋊C4—92no—
489C4⋊C4—92no—
490C4⋊C4—272no—
491C4⋊C4—272no—
492C4⋊C4—272no—
493C22⋊C4—91no—
494C22⋊C4—92no—
495C22⋊C4—92no—
496C22⋊C4—271no—
497C22⋊C4—271no—
498C22⋊C4—271no—
499C22⋊C4—272no—
500C22⋊C4—271no—
501C22⋊C4#111311yescharacteristic
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—21no—
507C22×C3—22no—
508C22×C3—24no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—41no—
513C22×C3—42no—
514C22×C3—82no—
515C22×C3—92no—
516C22×C3—92no—
517C22×C3—182no—
518C22×C3—182no—
519C22×C3—364no—
520C22×C3D3×Dic614yes—
521C22×C3#379712yes—
522C22×C3C22×D3212yes—
523C22×C3C22×D3211yescharacteristic
524C2×D3—34no—
525C2×D3—34no—
526C2×D3—62no—
527C2×D3—62no—
528C2×D3—64no—
529C2×D3—62no—
530C2×D3—62no—
531C2×D3—62no—
532C2×D3—62no—
533C2×D3—122no—
534C2×D3—128no—
535C2×D3—128no—
536C3×C4—22no—
537C3×C4—22no—
538C3×C4—22no—
539C3×C4—42no—
540C3×C4—42no—
541C3×C4—42no—
542C3×C4—82no—
543C3×C4—182no—
544C3×C4—182no—
545C3×C4—182no—
546C3×C4—182no—
547C3×C4—182no—
548Dic3—64no—
549Dic3—62no—
550Dic3—62no—
551Dic3—62no—
552Dic3—62no—
553Dic3—62no—
554Dic3—62no—
555Dic3—62no—
556Dic3—62no—
557Dic3—62no—
558Dic3—62no—
559Dic3—122no—
560Dic3—122no—
561Dic3—122no—
562Dic3—122no—
563C32—22no—
564C32—21no—
565C32—41no—
566C32#548311yescharacteristic, complemented
567C32#563412yescomplemented
568C23—91no—
569C23—91no—
570C23—94no—
571C23—362no—
572C23C2×#111311yescharacteristic
573Q8—362no—
574Q8—1082no—
575C2×C4—92no—
576C2×C4—92no—
577C2×C4—92no—
578C2×C4—92no—
579C2×C4—92no—
580C2×C4—362no—
581C2×C4—362no—
582C2×C4—1082no—
583C2×C4C2×#111312yes—
584C2×C3—24no—
585C2×C3—22no—
586C2×C3—21no—
587C2×C3—22no—
588C2×C3—22no—
589C2×C3—22no—
590C2×C3—22no—
591C2×C3—42no—
592C2×C3—44no—
593C2×C3—41no—
594C2×C3—42no—
595C2×C3—82no—
596C2×C3—182no—
597C2×C3—182no—
598C2×C3C2×D3×Dic612yes—
599C2×C3C2×#379711yescharacteristic
600C2×C3#6431312yes—
601C2×C3#6431412yes—
602C2×C3#6431412yes—
603D3—64no—
604D3—64no—
605D3—64no—
606D3—122no—
607C22—22no—
608C22—92no—
609C22—92no—
610C22—182no—
611C22—182no—
612C22—364no—
613C22#8904012yes—
614C22C22×#111311yescharacteristic
615C4—22no—
616C4—182no—
617C4—182no—
618C4—182no—
619C4—182no—
620C4—182no—
621C3—21no—
622C3—22no—
623C3—41no—
624C3#1059417012yescomplemented
625C3#1059418011yescharacteristic, complemented
626C2—22no—
627C2—182no—
628C2—182no—
629C2C2×#8904011yescharacteristic
630C2#1172462312yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC23×C3×C3⋊D3C22
hypercenterC22⋊C4#1113
hypocenterC33#543
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64087C2semidirect
4. #11723721C2semidirect
6. #11723692C2semidirect
7. #11723688C2semidirect
8. #11723674C2semidirect
9. #11723671C2semidirect
15. C2×C3×#634C22semidirect
16. C2×#6296C22semidirect
22. C3×C4⋊(C3⋊Dic3)C22semidirect
23. C3×#3750C22semidirect
24. C3×#3750C22semidirect
25. #88815C22semidirect
26. #88815C22semidirect
27. #88815C22semidirect
34. #64087D3semidirect
61. C22×C3×C3⋊D3Q8semidirect
82. C2×#634C2×D3semidirect
106. C4⋊(C3⋊Dic3)C2×D3semidirect
107. #3750C2×D3semidirect
108. #3750C2×D3semidirect
192. C22×C3⋊D3Dic6semidirect
394. C33#543semidirect
566. C32#5483semidirect
567. C32#5634semidirect
624. C3#10594170semidirect
625. C3#10594180semidirect

Maximal subgroups

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

subgroupindex
C3×C22⋊C4×C3⋊D32
C3×#640872
#117237232
#117237212
C2×#888142
#117236922
#117236882
#117236742
#117236712
#105941803
#105941703