Finite Group Databasebeta

← #13171732

Subgroups — #13171732

Order 1728.

1538 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3805C211yesmaximal, characteristic
2C2×#88813C211yesmaximal, characteristic, complemented
3#67×C3⋊Dic3C212yesmaximal, complemented
4#11723595C211yesmaximal, characteristic, complemented
5C3×#67⋊Dic3C212yesmaximal, complemented
6#11722940C212yesmaximal, complemented
7#11722924C212yesmaximal, complemented
8#11722914C212yesmaximal, complemented
9D3×Dic32C212yesmaximal, complemented
10Dic3×#237—32nomaximal
11#10594050—31nomaximal
12C22×C3×C3⋊Dic3C2212yes—
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C2×C32×#67C2212yes—
15C2×D3×C3⋊Dic3C2212yes—
16C2×C3×#627C2211yescharacteristic
17C2×C3×#625C2211yescharacteristic
18C3×#3805C418yescomplemented
19C2×C3×D3×Dic3C2212yes—
20C2×C3×D3×Dic3C2212yes—
21#88864C2211yescharacteristic
22C32×#217C2211yescharacteristic
23#88813C2211yescharacteristic
24#88813C2211yescharacteristic
25#88813C2212yes—
26#88807C2212yes—
27Dic3⋊(C3⋊Dic3)C2212yes—
28(C2×D3)⋊(C3⋊Dic3)C2212yes—
29Dic3×C3⋊Dic3C2212yes—
30Dic3×C3⋊Dic3C2212yes—
31C3×Dic3⋊Dic3C2212yes—
32C3×(C2×D3)⋊Dic3C2212yes—
33C3×Dic32C2212yes—
34C2×C3×#237—32no—
35C2×#3805D311yescharacteristic
36D4×C3⋊Dic3—32no—
37#64086—31no—
38C3×D4×Dic3—32no—
39C3×#67⋊C4—32no—
40#67⋊Dic3—32no—
41#67⋊Dic3D312yescomplemented
42Dic3×#67—32no—
43Dic3×#67—32no—
44Dic3×#67—32no—
45Dic3×#67—62no—
46#63955—32no—
47#63955—32no—
48#63955—61no—
49C2×Dic32—31no—
50C2×Dic32—32no—
51C2×Dic32—32no—
52#63879—32no—
53#63878—32no—
54C4×D3×Dic3—32no—
55C4×D3×Dic3—32no—
56#63843—32no—
57Dic3×Dic6—32no—
58#63841—32no—
59#63837—32no—
60C23×C33C2311yescharacteristic
61C22×C32×D3C2312yes—
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C2×C3×C3⋊Dic3—24no—
64C2×C3×C3⋊Dic3C2×C414yescomplemented
65C2×C3×C3⋊Dic3C2312yes—
66C2×C3×C3⋊Dic3C2311yescharacteristic
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2312yes—
69C32×#67C2×C418yescomplemented
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2312yes—
72D3×C3⋊Dic3—24no—
73C3×#627C2×C414yescomplemented
74C3×#625C2×C414yescomplemented
75C3×D3×Dic3—24no—
76C3×D3×Dic3C2×C418yescomplemented
77Dic3×#41—91no—
78C4×#237—92no—
79C2×C32×D4—32no—
80C22×C3⋊Dic3—32no—
81C22×C3⋊Dic3C2×D312yes—
82C22×C3⋊Dic3C2×D311yescharacteristic
83C2×C3×#67—22no—
84C2×C3×#67—32no—
85C2×C3×#67C2×D312yes—
86C2×C3×#67C2×D312yes—
87C22×C3×Dic3—32no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—32no—
90C22×C3×Dic3—32no—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—61no—
93C22×C3×Dic3—62no—
94C3×#237—316no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—32no—
97C2×C3×Dic6—32no—
98C2×#627C2×D311yescharacteristic
99C2×#625C2×D311yescharacteristic
100#3805Dic318yescomplemented
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—62no—
104C2×D3×Dic3C2×D312yes—
105C2×D3×Dic3C2×D312yes—
106C32×C22⋊C4—31no—
107#3757—31no—
108#3757—32no—
109#3757—32no—
110#3757—61no—
111#3757—62no—
112#3757—121no—
113C4⋊(C3⋊Dic3)—32no—
114#3750—32no—
115C4×C3⋊Dic3—32no—
116C4×C3⋊Dic3—32no—
117C3×#217—21no—
118C3×#217—32no—
119C3×#217C2×D312yes—
120C3×(C2×D3)⋊C4—32no—
121C3×C4⋊Dic3—32no—
122C3×Dic3⋊C4—32no—
123C3×C4×Dic3—32no—
124C3×C4×Dic3—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—62no—
130Dic3⋊Dic3—62no—
131Dic3⋊Dic3C2×D312yes—
132(C2×D3)⋊Dic3—32no—
133(C2×D3)⋊Dic3—32no—
134(C2×D3)⋊Dic3—62no—
135(C2×D3)⋊Dic3C2×D312yes—
136Dic32—32no—
137Dic32—32no—
138Dic32—32no—
139Dic32—32no—
140Dic32—31no—
141Dic32—32no—
142Dic32—31no—
143Dic32—32no—
144Dic32—32no—
145Dic32—32no—
146Dic32—32no—
147Dic32—62no—
148Dic32—62no—
149Dic32C2×D312yes—
150C22×C33—22no—
151C22×C33C22×C412yes—
152C22×C33C2411yescharacteristic
153C2×C32×D3—22no—
154C2×C32×D3—22no—
155C2×C32×D3C22×C414yes—
156C3⋊(C3⋊Dic3)—22no—
157C3×C3⋊Dic3—22no—
158C3×C3⋊Dic3C22×C412yes—
159C3×C3⋊Dic3C22×C412yes—
160C3×C3⋊Dic3#4114yescomplemented
161C32×Dic3—22no—
162C32×Dic3C22×C414yes—
163C2×C3×#41—91no—
164C2×#237—32no—
165C3×C4×D4—92no—
166Q8×Dic3—92no—
167D4×Dic3—32no—
168D4×Dic3—92no—
169D4×Dic3—92no—
170D4×Dic3—92no—
171D4×Dic3—182no—
172C4×#67—92no—
173C4×#67—92no—
174#926—92no—
175#926—91no—
176C2×C4×Dic3—92no—
177C2×C4×Dic3—92no—
178C2×C4×Dic3—92no—
179C2×C4×Dic3—91no—
180#892—92no—
181#887—92no—
182#67⋊C4—32no—
183#67⋊C4—92no—
184#877—92no—
185#877—92no—
186#877—181no—
187C42×D3—92no—
188C4×Dic6—92no—
189C23×C32—21no—
190C23×C32—22no—
191C23×C32—32no—
192C23×C32—41no—
193C23×C32C22×D312yes—
194C23×C32C22×D311yescharacteristic
195C22×C3×D3—22no—
196C22×C3×D3C22×D312yes—
197C22×C3×D3C22×D312yes—
198C32×D4—38no—
199C2×C32×C4—32no—
200C2×C32×C4—32no—
201C2×C3⋊Dic3—24no—
202C2×C3⋊Dic3—32no—
203C2×C3⋊Dic3—32no—
204C2×C3⋊Dic3—32no—
205C2×C3⋊Dic3—62no—
206C2×C3⋊Dic3—62no—
207C2×C3⋊Dic3—62no—
208C2×C3⋊Dic3—64no—
209C2×C3⋊Dic3—122no—
210C2×C3⋊Dic3C2×Dic314yescomplemented
211C2×C3⋊Dic3C22×D312yes—
212C2×C3⋊Dic3C22×D312yes—
213C2×C3⋊Dic3C22×D311yescharacteristic
214C2×C3⋊Dic3C22×D311yescharacteristic
215C3×#67—28no—
216C3×#67—38no—
217C3×#67C4×D318yescomplemented
218C3×#67C2×Dic318yescomplemented
219C2×C3×Dic3—22no—
220C2×C3×Dic3—22no—
221C2×C3×Dic3—38no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—38no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—61no—
230C2×C3×Dic3—62no—
231C2×C3×Dic3—61no—
232C2×C3×Dic3—64no—
233C2×C3×Dic3—62no—
234C2×C3×Dic3—64no—
235C2×C3×Dic3—64no—
236C2×C3×Dic3—64no—
237C2×C3×Dic3—124no—
238C2×C3×Dic3C22×D312yes—
239C2×C3×Dic3C22×D312yes—
240C2×C3×Dic3C22×D312yes—
241C2×C3×Dic3C22×D312yes—
242C3×C4×D3—38no—
243C3×C4×D3—64no—
244C3×Dic6—38no—
245#627C2×Dic314yescomplemented
246#625C2×Dic314yescomplemented
247D3×Dic3—24no—
248D3×Dic3—64no—
249D3×Dic3—64no—
250D3×Dic3—124no—
251D3×Dic3C2×Dic318yescomplemented
252C4×#41—271noSylow
253C2×C33—22no—
254C2×C33C23×C411yescharacteristic
255C2×C33C2×#4112yes—
256C32×D3—24no—
257C3×#41—98no—
258C2×C3×Q8—91no—
259C2×C3×D4—32no—
260C2×C3×D4—32no—
261C2×C3×D4—62no—
262C2×C3×D4—91no—
263C22×C3×C4—92no—
264C22×C3×C4—92no—
265C22×C3×C4—91no—
266C22×C3×C4—92no—
267C2×#67—32no—
268C2×#67D3212yes—
269C22×Dic3—32no—
270C22×Dic3—32no—
271C22×Dic3—32no—
272C22×Dic3—32no—
273C22×Dic3—32no—
274C22×Dic3—61no—
275C22×Dic3—62no—
276C22×Dic3—92no—
277C22×Dic3—92no—
278C22×Dic3—182no—
279#237—316no—
280C2×C4×D3—32no—
281C2×C4×D3—32no—
282C2×C4×D3—92no—
283C2×Dic6—32no—
284C3×C4⋊C4—92no—
285C3×C22⋊C4—32no—
286C3×C22⋊C4—61no—
287C3×C22⋊C4—92no—
288C3×C42—92no—
289#217—32no—
290#217—92no—
291#217—92no—
292#217—92no—
293#217—91no—
294#217—182no—
295#217—181no—
296#217—182no—
297#217—361no—
298#217D3211yescharacteristic
299(C2×D3)⋊C4—32no—
300(C2×D3)⋊C4—92no—
301C4⋊Dic3—32no—
302C4⋊Dic3—92no—
303C4⋊Dic3—92no—
304C4⋊Dic3—92no—
305C4⋊Dic3—182no—
306Dic3⋊C4—32no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—182no—
311C4×Dic3—32no—
312C4×Dic3—32no—
313C4×Dic3—92no—
314C4×Dic3—92no—
315C4×Dic3—92no—
316C4×Dic3—92no—
317C4×Dic3—91no—
318C4×Dic3—91no—
319C4×Dic3—92no—
320C4×Dic3—92no—
321C4×Dic3—92no—
322C4×Dic3—92no—
323C4×Dic3—92no—
324C4×Dic3—92no—
325C4×Dic3—92no—
326C4×Dic3—182no—
327C4×Dic3—182no—
328C22×C32—22no—
329C22×C32—22no—
330C22×C32—22no—
331C22×C32—21no—
332C22×C32—22no—
333C22×C32—22no—
334C22×C32—24no—
335C22×C32—34no—
336C22×C32—41no—
337C22×C32—42no—
338C22×C32—42no—
339C22×C32—42no—
340C22×C32—42no—
341C22×C32—62no—
342C22×C32—62no—
343C22×C32—82no—
344C22×C32C2×C4×D314yes—
345C22×C32C22×Dic312yes—
346C22×C32C23×D312yes—
347C22×C32C23×D311yescharacteristic
348C2×C3×D3—22no—
349C2×C3×D3—22no—
350C2×C3×D3—22no—
351C2×C3×D3—22no—
352C2×C3×D3—24no—
353C2×C3×D3—42no—
354C2×C3×D3—42no—
355C2×C3×D3C2×C4×D314yes—
356C2×C3×D3C22×Dic314yes—
357C32×C4—34no—
358C32×C4—62no—
359C3⋊Dic3—22no—
360C3⋊Dic3—62no—
361C3⋊Dic3—62no—
362C3⋊Dic3—62no—
363C3⋊Dic3—122no—
364C3⋊Dic3—122no—
365C3⋊Dic3—122no—
366C3⋊Dic3—242no—
367C3⋊Dic3#23714yescomplemented
368C3⋊Dic3C22×Dic312yes—
369C3⋊Dic3C22×Dic312yes—
370C3×Dic3—22no—
371C3×Dic3—22no—
372C3×Dic3—24no—
373C3×Dic3—34no—
374C3×Dic3—34no—
375C3×Dic3—34no—
376C3×Dic3—34no—
377C3×Dic3—42no—
378C3×Dic3—62no—
379C3×Dic3—62no—
380C3×Dic3—64no—
381C3×Dic3—62no—
382C3×Dic3—62no—
383C3×Dic3—122no—
384C3×Dic3C2×C4×D314yes—
385C3×Dic3C22×Dic314yes—
386C2×#41—91no—
387C4×Q8—272no—
388C4×D4—92no—
389C4×D4—272no—
390C4×D4—272no—
391#117—271no—
392#117—272no—
393C2×C42—272no—
394C2×C42—271no—
395C33C4×#4111yescharacteristic, complemented, Sylow
396C23×C3—21no—
397C23×C3—22no—
398C23×C3—32no—
399C23×C3—32no—
400C23×C3—41no—
401C23×C3—62no—
402C23×C3C2×D3211yescharacteristic
403C23×C3C2×D3212yes—
404C22×D3C2×D3212yes—
405C3×Q8—94no—
406C3×D4—38no—
407C3×D4—38no—
408C3×D4—68no—
409C3×D4—94no—
410C2×C3×C4—32no—
411C2×C3×C4—32no—
412C2×C3×C4—32no—
413C2×C3×C4—32no—
414C2×C3×C4—62no—
415C2×C3×C4—62no—
416C2×C3×C4—98no—
417C2×C3×C4—91no—
418C2×C3×C4—92no—
419C2×C3×C4—91no—
420C2×C3×C4—94no—
421C2×C3×C4—92no—
422C2×C3×C4—184no—
423C2×C3×C4—184no—
424#67—38no—
425#67D3×Dic318yescomplemented
426C2×Dic3—32no—
427C2×Dic3—32no—
428C2×Dic3—32no—
429C2×Dic3—32no—
430C2×Dic3—38no—
431C2×Dic3—38no—
432C2×Dic3—32no—
433C2×Dic3—32no—
434C2×Dic3—64no—
435C2×Dic3—61no—
436C2×Dic3—62no—
437C2×Dic3—61no—
438C2×Dic3—62no—
439C2×Dic3—64no—
440C2×Dic3—64no—
441C2×Dic3—64no—
442C2×Dic3—92no—
443C2×Dic3—92no—
444C2×Dic3—92no—
445C2×Dic3—124no—
446C2×Dic3—182no—
447C2×Dic3—182no—
448C2×Dic3—182no—
449C2×Dic3—184no—
450C2×Dic3—184no—
451C2×Dic3—362no—
452C2×Dic3—364no—
453C2×Dic3C2×D3212yes—
454C2×Dic3C2×D3212yes—
455C4×D3—38no—
456C4×D3—64no—
457C4×D3—184no—
458Dic6—38no—
459C2×C32—24no—
460C2×C32—22no—
461C2×C32—21no—
462C2×C32—22no—
463C2×C32—22no—
464C2×C32—22no—
465C2×C32—22no—
466C2×C32—41no—
467C2×C32—42no—
468C2×C32—42no—
469C2×C32—44no—
470C2×C32—64no—
471C2×C32—82no—
472C2×C32C22×C4×D312yes—
473C2×C32C2×#23712yes—
474C2×C32D3×#4112yes—
475C2×C32D3×#4112yes—
476C2×C32C23×Dic311yescharacteristic
477C3×D3—24no—
478C3×D3—24no—
479C3×D3—44no—
480#41—98no—
481C2×Q8—91no—
482C2×D4—32no—
483C2×D4—91no—
484C22×C4—92no—
485C22×C4—92no—
486C22×C4—91no—
487C22×C4—92no—
488C22×C4—272no—
489C4⋊C4—92no—
490C4⋊C4—272no—
491C4⋊C4—272no—
492C22⋊C4—31no—
493C22⋊C4—92no—
494C22⋊C4—272no—
495C22⋊C4—271no—
496C42—92no—
497C42—272no—
498C42—272no—
499C42—271no—
500C42—271no—
501C42—272no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—21no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—24no—
509C22×C3—34no—
510C22×C3—34no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—41no—
515C22×C3—42no—
516C22×C3—64no—
517C22×C3—62no—
518C22×C3—62no—
519C22×C3—62no—
520C22×C3—62no—
521C22×C3—82no—
522C22×C3—122no—
523C22×C3—122no—
524C22×C3C4×D3212yes—
525C22×C3C2×D3×Dic314yes—
526C22×C3C22×D3211yescharacteristic
527C22×C3C22×D3212yes—
528C2×D3—22no—
529C2×D3—22no—
530C2×D3C2×D3×Dic314yes—
531C3×C4—34no—
532C3×C4—34no—
533C3×C4—64no—
534C3×C4—62no—
535C3×C4—62no—
536C3×C4—92no—
537C3×C4—92no—
538C3×C4—94no—
539C3×C4—122no—
540C3×C4—182no—
541Dic3—22no—
542Dic3—34no—
543Dic3—34no—
544Dic3—34no—
545Dic3—34no—
546Dic3—62no—
547Dic3—62no—
548Dic3—64no—
549Dic3—62no—
550Dic3—62no—
551Dic3—122no—
552Dic3—182no—
553Dic3—182no—
554Dic3—182no—
555Dic3—362no—
556Dic3—362no—
557Dic3—362no—
558Dic3—722no—
559Dic3C2×D3×Dic314yes—
560C32—21no—
561C32—22no—
562C32—41no—
563C32C4×#23712yescomplemented
564C32Dic3×#4111yescharacteristic, complemented
565C23—32no—
566C23D3311yescharacteristic
567Q8—94no—
568D4—38no—
569D4—94no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—98no—
573C2×C4—92no—
574C2×C4—91no—
575C2×C4—91no—
576C2×C4—94no—
577C2×C4—92no—
578C2×C4—184no—
579C2×C4—184no—
580C2×C4—272no—
581C2×C4—544no—
582C2×C3—22no—
583C2×C3—24no—
584C2×C3—21no—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—22no—
588C2×C3—22no—
589C2×C3—44no—
590C2×C3—42no—
591C2×C3—41no—
592C2×C3—42no—
593C2×C3—64no—
594C2×C3—64no—
595C2×C3—82no—
596C2×C3—124no—
597C2×C3C2×C4×D3211yescharacteristic
598C2×C3D3×#23712yes—
599C2×C3D3×#23712yes—
600C2×C3#6431212yes—
601C2×C3C22×D3×Dic312yes—
602D3—24no—
603C22—22no—
604C22—34no—
605C22—62no—
606C22—62no—
607C22D32×Dic312yes—
608C22C2×D3311yescharacteristic
609C4—34no—
610C4—62no—
611C4—92no—
612C4—92no—
613C4—94no—
614C4—182no—
615C4—542no—
616C3—22no—
617C3—21no—
618C3—41no—
619C3#1059405011yescharacteristic, complemented
620C3Dic3×#23712yescomplemented
621C2—22no—
622C2—64no—
623C2C2×D32×Dic311yescharacteristic
624C2#1172440812yes—
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC23×C33C23
socle, cosocleC22×C33C24
prime radicalC2×C3×#3805C2
hypercenterC23D33
hypocenterC33C4×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88813C2semidirect
3. #67×C3⋊Dic3C2semidirect
4. #11723595C2semidirect
5. C3×#67⋊Dic3C2semidirect
6. #11722940C2semidirect
7. #11722924C2semidirect
8. #11722914C2semidirect
9. D3×Dic32C2semidirect
18. C3×#3805C4semidirect
41. #67⋊Dic3D3semidirect
64. C2×C3×C3⋊Dic3C2×C4semidirect
69. C32×#67C2×C4semidirect
73. C3×#627C2×C4semidirect
74. C3×#625C2×C4semidirect
76. C3×D3×Dic3C2×C4semidirect
100. #3805Dic3semidirect
160. C3×C3⋊Dic3#41semidirect
210. C2×C3⋊Dic3C2×Dic3semidirect
217. C3×#67C4×D3semidirect
218. C3×#67C2×Dic3semidirect
245. #627C2×Dic3semidirect
246. #625C2×Dic3semidirect
251. D3×Dic3C2×Dic3semidirect
367. C3⋊Dic3#237semidirect
395. C33C4×#41semidirect
425. #67D3×Dic3semidirect
563. C32C4×#237semidirect
564. C32Dic3×#41semidirect
619. C3#10594050semidirect
620. C3Dic3×#237semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38052
C2×#888132
#67×C3⋊Dic32
#117235952
C3×#67⋊Dic32
#117229402
#117229242
#117229142
D3×Dic322
Dic3×#2373
#105940503