Finite Group Databasebeta

← #13168644

Subgroups — #13168644

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2#11723667C211yesmaximal, characteristic, complemented
3#11723503C212yesmaximal, complemented
4C3×#63849C211yesmaximal, characteristic, complemented
5C3×#63841C212yesmaximal, complemented
6#11722919C212yesmaximal, complemented
7#11722916C212yesmaximal, complemented
8#11722906C212yesmaximal
9#11722900C212yesmaximal, complemented
10#10594969—31nomaximal
11#10594602—32nomaximal
12C22×D3×C3⋊D3C2211yescharacteristic
13C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×C4×D3C2211yescharacteristic
16C2×#6289C2214yes—
17C2×#6289C2211yescharacteristic
18C2×Dic3×C3⋊D3C2211yescharacteristic
19C2×D3×C3⋊Dic3C2211yescharacteristic
20C4×D3×C3⋊D3C418yescomplemented
21C3×C4⋊(C3⋊Dic3)C2212yescomplemented
22C32×C4⋊Dic3C2212yescomplemented
23#88814C2212yes—
24#88813C2212yescomplemented
25#88806C2214yescomplemented
26C3×#3722C2212yes—
27C3×(C2×D3)⋊Dic3C2214yes—
28C3×Dic32C2214yescomplemented
29C3×Dic32C2212yescomplemented
30C22×C4×C3⋊D3—31no—
31C2×C4×D32—32no—
32C2×C4×D32—61no—
33C4⋊C4×C3⋊D3—32no—
34C3×#926—32no—
35C3×#892—32no—
36C3×#892—32no—
37#63974—32no—
38#63957—32no—
39#63921—34no—
40#63900—32no—
41#63900—32no—
42#63900—62no—
43C4×#624—34no—
44C4×#624—32no—
45#63877—34no—
46#63877—34no—
47#63874—32no—
48#63849—31no—
49#63849D311yescharacteristic, complemented
50#63841—32no—
51#63841D312yescomplemented
52C22×C3⋊(C3⋊D3)C2311yescharacteristic
53C22×C3×C3⋊D3C2311yescharacteristic
54C22×C32×D3C2311yescharacteristic
55C2×D3×C3⋊D3—24no—
56C2×D3×C3⋊D3C2×C414yes—
57C2×C33×C4C2311yescharacteristic
58C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
59C4×C3⋊(C3⋊D3)C2×C414yes—
60C2×C3×C3⋊Dic3C2314yes—
61C2×C3×C3⋊Dic3C2311yescharacteristic
62C3×C4×C3⋊D3C2×C414yescomplemented
63C2×C32×Dic3C2314yes—
64C2×C32×Dic3C2311yescharacteristic
65C32×C4×D3C2×C414yescomplemented
66#6289—28no—
67#6289C2×C414yes—
68Dic3×C3⋊D3C2×C414yescomplemented
69D3×C3⋊Dic3C2×C414yescomplemented
70D3×#117—91no—
71D3×#117—92no—
72C23×C3⋊D3—31no—
73C22×D32—32no—
74C22×D32—61no—
75C22×C32×C4—31no—
76C22×C3⋊Dic3—31no—
77C2×C4×C3⋊D3—31no—
78C2×C4×C3⋊D3—32no—
79C2×C4×C3⋊D3—31no—
80C2×C4×C3⋊D3—31no—
81C2×C4×C3⋊D3—38no—
82C2×C4×C3⋊D3—34no—
83C2×C4×C3⋊D3—61no—
84C2×C4×C3⋊D3—62no—
85C2×C4×C3⋊D3—121no—
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×C3×C4×D3—21no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—61no—
90C2×C3×C4×D3C2×D312yes—
91C2×#624—32no—
92C2×#624—34no—
93C2×#624—34no—
94C2×#624—61no—
95C2×#624—64no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—61no—
99C2×D3×Dic3—61no—
100C4×D32—316no—
101C4×D32—68no—
102C32×C4⋊C4—32no—
103C4⋊(C3⋊Dic3)C2×D312yescomplemented
104#3750—34no—
105C3×#217—34no—
106C3×(C2×D3)⋊C4—34no—
107C3×(C2×D3)⋊C4—34no—
108C3×C4⋊Dic3—22no—
109C3×C4⋊Dic3—32no—
110C3×C4⋊Dic3—32no—
111C3×C4⋊Dic3—62no—
112C3×C4⋊Dic3C2×D312yescomplemented
113C3×C4⋊Dic3C2×D312yescomplemented
114C3×C4×Dic3—34no—
115C3×C4×Dic3—34no—
116C3×C4×Dic3—34no—
117#3724—34no—
118#3724—34no—
119#3724—64no—
120#3722—32no—
121#3722C2×D312yes—
122(C2×D3)⋊Dic3—34no—
123(C2×D3)⋊Dic3C2×D314yes—
124Dic32—34no—
125Dic32—32no—
126Dic32C2×D314yescomplemented
127Dic32C2×D312yescomplemented
128C22×C33C2411yescharacteristic
129C2×C3⋊(C3⋊D3)C22×C412yes—
130C2×C3⋊(C3⋊D3)#4114yes—
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3C22×C412yes—
133C2×C32×D3—22no—
134C2×C32×D3C22×C412yes—
135D3×C3⋊D3—28no—
136C33×C4C22×C412yes—
137C3⋊(C3⋊Dic3)C22×C412yes—
138C3×C3⋊Dic3—24no—
139C3×C3⋊Dic3C22×C412yes—
140C32×Dic3—24no—
141C32×Dic3C22×C412yes—
142C22×C4×D3—92no—
143C22×C4×D3—91no—
144C22×C4×D3—181no—
145C3×#117—91no—
146C3×#117—92no—
147#926—32no—
148#926—91no—
149#892—32no—
150#892—32no—
151#892—92no—
152D3×C4⋊C4—92no—
153D3×C4⋊C4—92no—
154D3×C4⋊C4—92no—
155D3×C4⋊C4—182no—
156D3×C22⋊C4—92no—
157D3×C22⋊C4—94no—
158#877—92no—
159#877—94no—
160#872—94no—
161#872—92no—
162C42×D3—94no—
163C42×D3—92no—
164C23×C32—31no—
165C22×C3⋊D3—32no—
166C22×C3⋊D3—31no—
167C22×C3⋊D3—34no—
168C22×C3⋊D3—61no—
169C22×C3⋊D3—62no—
170C22×C3⋊D3—64no—
171C22×C3⋊D3—121no—
172C22×C3⋊D3C22×D311yescharacteristic
173C22×C3×D3—21no—
174C22×C3×D3—32no—
175C22×C3×D3—61no—
176C22×C3×D3C22×D312yes—
177C2×D32—38no—
178C2×D32—64no—
179C2×D32—64no—
180C2×D32—64no—
181C2×D32—68no—
182C2×C32×C4—22no—
183C2×C32×C4—21no—
184C2×C32×C4—31no—
185C2×C32×C4—34no—
186C2×C32×C4—34no—
187C2×C32×C4—41no—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D312yes—
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—32no—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—62no—
194C2×C3⋊Dic3—61no—
195C2×C3⋊Dic3—121no—
196C2×C3⋊Dic3C22×D314yes—
197C2×C3⋊Dic3C22×D311yescharacteristic
198C4×C3⋊D3—34no—
199C4×C3⋊D3—38no—
200C4×C3⋊D3—34no—
201C4×C3⋊D3—34no—
202C4×C3⋊D3—64no—
203C4×C3⋊D3—68no—
204C4×C3⋊D3—68no—
205C4×C3⋊D3—124no—
206C4×C3⋊D3C4×D314yescomplemented
207C2×C3×Dic3—21no—
208C2×C3×Dic3—24no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—61no—
213C2×C3×Dic3—64no—
214C2×C3×Dic3C22×D314yes—
215C2×C3×Dic3C22×D314yes—
216C2×C3×Dic3C22×D312yes—
217C3×C4×D3—24no—
218C3×C4×D3—38no—
219C3×C4×D3—64no—
220C3×C4×D3C4×D318yescomplemented
221#624—38no—
222#624—64no—
223#624—68no—
224#624—68no—
225#624—128no—
226D3×Dic3—38no—
227D3×Dic3—38no—
228D3×Dic3—64no—
229D3×Dic3—64no—
230C2×#117—271noSylow
231C2×C33C23×C411yescharacteristic
232C2×C33C2×#4112yes—
233C3⋊(C3⋊D3)#11714yescomplemented
234C3×C3⋊D3—22no—
235C32×D3—22no—
236C23×D3—92no—
237C23×D3—91no—
238C23×D3—181no—
239C22×C3×C4—32no—
240C22×C3×C4—61no—
241C22×C3×C4—91no—
242C22×Dic3—92no—
243C22×Dic3—91no—
244C22×Dic3—181no—
245C2×C4×D3—32no—
246C2×C4×D3—61no—
247C2×C4×D3—916no—
248C2×C4×D3—92no—
249C2×C4×D3—92no—
250C2×C4×D3—91no—
251C2×C4×D3—92no—
252C2×C4×D3—91no—
253C2×C4×D3—91no—
254C2×C4×D3—98no—
255C2×C4×D3—94no—
256C2×C4×D3—94no—
257C2×C4×D3—94no—
258C2×C4×D3—188no—
259C2×C4×D3—181no—
260C2×C4×D3—182no—
261C2×C4×D3—181no—
262C2×C4×D3—181no—
263C2×C4×D3—184no—
264C2×C4×D3—361no—
265C2×C4×D3D3211yescharacteristic
266C3×C4⋊C4—32no—
267C3×C4⋊C4—32no—
268C3×C4⋊C4—62no—
269C3×C4⋊C4—92no—
270C3×C22⋊C4—92no—
271C3×C22⋊C4—94no—
272C3×C42—94no—
273C3×C42—92no—
274#217—34no—
275#217—92no—
276(C2×D3)⋊C4—34no—
277(C2×D3)⋊C4—34no—
278(C2×D3)⋊C4—94no—
279C4⋊Dic3—32no—
280C4⋊Dic3—32no—
281C4⋊Dic3—62no—
282C4⋊Dic3D3212yescomplemented
283Dic3⋊C4—94no—
284Dic3⋊C4—94no—
285Dic3⋊C4—94no—
286Dic3⋊C4—184no—
287C4×Dic3—34no—
288C4×Dic3—34no—
289C4×Dic3—34no—
290C4×Dic3—92no—
291C4×Dic3—94no—
292C22×C32—22no—
293C22×C32—21no—
294C22×C32—32no—
295C22×C32—41no—
296C22×C32—62no—
297C22×C32C23×D311yescharacteristic
298C22×C32C23×D312yes—
299C2×C3⋊D3—22no—
300C2×C3⋊D3—34no—
301C2×C3⋊D3—32no—
302C2×C3⋊D3—34no—
303C2×C3⋊D3—34no—
304C2×C3⋊D3—34no—
305C2×C3⋊D3—62no—
306C2×C3⋊D3—64no—
307C2×C3⋊D3—64no—
308C2×C3⋊D3—64no—
309C2×C3⋊D3—64no—
310C2×C3⋊D3—68no—
311C2×C3⋊D3—122no—
312C2×C3⋊D3—124no—
313C2×C3⋊D3C2×C4×D312yes—
314C2×C3×D3—24no—
315C2×C3×D3—22no—
316C2×C3×D3—22no—
317C2×C3×D3—22no—
318C2×C3×D3—34no—
319C2×C3×D3—62no—
320C2×C3×D3—64no—
321C2×C3×D3—62no—
322C2×C3×D3—62no—
323C2×C3×D3C2×C4×D314yes—
324D32—616no—
325D32—616no—
326C32×C4—24no—
327C32×C4—22no—
328C32×C4—32no—
329C32×C4—42no—
330C32×C4—64no—
331C32×C4C2×C4×D312yes—
332C32×C4C2×C4×D314yes—
333C3⋊Dic3—24no—
334C3⋊Dic3—34no—
335C3⋊Dic3—32no—
336C3⋊Dic3—62no—
337C3⋊Dic3—64no—
338C3⋊Dic3—122no—
339C3⋊Dic3C2×C4×D312yes—
340C3×Dic3—24no—
341C3×Dic3—24no—
342C3×Dic3—22no—
343C3×Dic3—34no—
344C3×Dic3—44no—
345C3×Dic3—62no—
346C3×Dic3—64no—
347C3×Dic3—64no—
348C3×Dic3—124no—
349C3×Dic3C2×C4×D314yes—
350C23×C4—271no—
351#117—91no—
352#117—92no—
353#117—272no—
354#117—272no—
355#117—271no—
356C2×C4⋊C4—272no—
357C2×C22⋊C4—272no—
358C2×C42—272no—
359C33C2×#11711yescharacteristic, complemented, Sylow
360C23×C3—32no—
361C23×C3—61no—
362C23×C3—91no—
363C22×D3—32no—
364C22×D3—61no—
365C22×D3—98no—
366C22×D3—92no—
367C22×D3—91no—
368C22×D3—94no—
369C22×D3—184no—
370C22×D3—182no—
371C22×D3—181no—
372C22×D3—184no—
373C22×D3—184no—
374C22×D3—184no—
375C22×D3—188no—
376C22×D3—361no—
377C22×D3C2×D3211yescharacteristic
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—32no—
381C2×C3×C4—38no—
382C2×C3×C4—34no—
383C2×C3×C4—34no—
384C2×C3×C4—41no—
385C2×C3×C4—64no—
386C2×C3×C4—61no—
387C2×C3×C4—64no—
388C2×C3×C4—91no—
389C2×C3×C4—94no—
390C2×C3×C4—94no—
391C2×C3×C4C2×D3212yes—
392C2×C3×C4C2×D3211yescharacteristic
393C2×Dic3—32no—
394C2×Dic3—34no—
395C2×Dic3—34no—
396C2×Dic3—61no—
397C2×Dic3—64no—
398C2×Dic3—98no—
399C2×Dic3—92no—
400C2×Dic3—91no—
401C2×Dic3—94no—
402C2×Dic3—184no—
403C2×Dic3—182no—
404C2×Dic3—181no—
405C2×Dic3—361no—
406C2×Dic3C2×D3214yes—
407C2×Dic3C2×D3211yescharacteristic
408C4×D3—38no—
409C4×D3—64no—
410C4×D3—98no—
411C4×D3—98no—
412C4×D3—94no—
413C4×D3—98no—
414C4×D3—94no—
415C4×D3—94no—
416C4×D3—184no—
417C4×D3—188no—
418C4×D3—184no—
419C4×D3—184no—
420C4×D3—188no—
421C4×D3—188no—
422C4×D3—188no—
423C4×D3—364no—
424C4×D3—368no—
425C4×D3#62414yescomplemented
426C2×C32—22no—
427C2×C32—22no—
428C2×C32—22no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—41no—
432C2×C32—42no—
433C2×C32—62no—
434C2×C32C22×C4×D311yescharacteristic
435C2×C32C22×C4×D312yes—
436C2×C32D3×#4112yes—
437C2×C32D3×#4112yes—
438C2×C32D3×#4112yes—
439C3⋊D3—22no—
440C3⋊D3—38no—
441C3⋊D3—34no—
442C3⋊D3—64no—
443C3⋊D3—68no—
444C3⋊D3—124no—
445C3×D3—24no—
446C3×D3—24no—
447C3×D3—64no—
448C3×D3—64no—
449C24—271no—
450C22×C4—31no—
451C22×C4—91no—
452C22×C4—271no—
453C22×C4—278no—
454C22×C4—274no—
455C22×C4—271no—
456C22×C4—271no—
457C22×C4—271no—
458C4⋊C4—32no—
459C4⋊C4—92no—
460C4⋊C4—274no—
461C22⋊C4—92no—
462C22⋊C4—94no—
463C22⋊C4—272no—
464C42—94no—
465C42—92no—
466C42—272no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—34no—
470C22×C3—41no—
471C22×C3—62no—
472C22×C3—64no—
473C22×C3—62no—
474C22×C3—62no—
475C22×C3—92no—
476C22×C3—182no—
477C22×C3C22×D3212yes—
478C22×C3C22×D3211yescharacteristic
479C2×D3—22no—
480C2×D3—34no—
481C2×D3—62no—
482C2×D3—64no—
483C2×D3—62no—
484C2×D3—62no—
485C2×D3—94no—
486C2×D3—94no—
487C2×D3—94no—
488C2×D3—94no—
489C2×D3—92no—
490C2×D3—182no—
491C2×D3—184no—
492C2×D3—184no—
493C2×D3—184no—
494C2×D3—184no—
495C2×D3—188no—
496C2×D3—1816no—
497C2×D3—1816no—
498C2×D3—362no—
499C2×D3—364no—
500C2×D3C2×#62412yes—
501C3×C4—22no—
502C3×C4—24no—
503C3×C4—34no—
504C3×C4—42no—
505C3×C4—62no—
506C3×C4—64no—
507C3×C4—64no—
508C3×C4—92no—
509C3×C4—124no—
510C3×C4—184no—
511C3×C4C4×D3214yes—
512C3×C4C2×#62412yes—
513Dic3—24no—
514Dic3—34no—
515Dic3—62no—
516Dic3—64no—
517Dic3—64no—
518Dic3—94no—
519Dic3—92no—
520Dic3—124no—
521Dic3—182no—
522Dic3—184no—
523Dic3—362no—
524Dic3C2×#62412yes—
525C32—22no—
526C32—21no—
527C32—41no—
528C32D3×#11711yescharacteristic, complemented
529C32D3×#11712yescomplemented
530C23—31no—
531C23—91no—
532C23—274no—
533C23—271no—
534C23—544no—
535C2×C4—31no—
536C2×C4—34no—
537C2×C4—34no—
538C2×C4—91no—
539C2×C4—94no—
540C2×C4—94no—
541C2×C4—274no—
542C2×C4—274no—
543C2×C4—274no—
544C2×C4—271no—
545C2×C4—274no—
546C2×C4—548no—
547C2×C4D3311yescharacteristic
548C2×C3—22no—
549C2×C3—22no—
550C2×C3—22no—
551C2×C3—21no—
552C2×C3—22no—
553C2×C3—42no—
554C2×C3—41no—
555C2×C3—64no—
556C2×C3—64no—
557C2×C3—182no—
558C2×C3C2×C4×D3212yes—
559C2×C3#6430612yes—
560C2×C3#6431212yes—
561C2×C3#6431912yes—
562C2×C3C22×#62411yescharacteristic
563D3—22no—
564D3—64no—
565D3—64no—
566D3—98no—
567D3—94no—
568D3—184no—
569D3—188no—
570D3—364no—
571C22—32no—
572C22—62no—
573C22—92no—
574C22—182no—
575C22—274no—
576C22—272no—
577C22—548no—
578C22C2×D3311yescharacteristic
579C4—32no—
580C4—64no—
581C4—92no—
582C4—184no—
583C4—272no—
584C4D3×#62412yes—
585C3—21no—
586C3—22no—
587C3—41no—
588C3#1059460212yescomplemented
589C3#1059496911yescharacteristic, complemented
590C2—62no—
591C2—182no—
592C2—274no—
593C2#1172437612yes—
594C2C2×D3×#62411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC22×D3×C3⋊D3C22
hypercenterC2×C4D33
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723667C2semidirect
3. #11723503C2semidirect
4. C3×#63849C2semidirect
5. C3×#63841C2semidirect
6. #11722919C2semidirect
7. #11722916C2semidirect
9. #11722900C2semidirect
20. C4×D3×C3⋊D3C4semidirect
21. C3×C4⋊(C3⋊Dic3)C22semidirect
22. C32×C4⋊Dic3C22semidirect
24. #88813C22semidirect
25. #88806C22semidirect
28. C3×Dic32C22semidirect
29. C3×Dic32C22semidirect
49. #63849D3semidirect
51. #63841D3semidirect
62. C3×C4×C3⋊D3C2×C4semidirect
65. C32×C4×D3C2×C4semidirect
68. Dic3×C3⋊D3C2×C4semidirect
69. D3×C3⋊Dic3C2×C4semidirect
103. C4⋊(C3⋊Dic3)C2×D3semidirect
112. C3×C4⋊Dic3C2×D3semidirect
113. C3×C4⋊Dic3C2×D3semidirect
126. Dic32C2×D3semidirect
127. Dic32C2×D3semidirect
206. C4×C3⋊D3C4×D3semidirect
220. C3×C4×D3C4×D3semidirect
233. C3⋊(C3⋊D3)#117semidirect
282. C4⋊Dic3D32semidirect
359. C33C2×#117semidirect
425. C4×D3#624semidirect
528. C32D3×#117semidirect
529. C32D3×#117semidirect
588. C3#10594602semidirect
589. C3#10594969semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
#117236672
#117235032
C3×#638492
C3×#638412
#117229192
#117229162
#117229062
#117229002
#105949693
#105946023