Finite Group Databasebeta

← #13175943

Subgroups — #13175943

Order 1728.

2580 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89039C211yesmaximal, characteristic
2C22×#6289C211yesmaximal, characteristic, complemented
3C22⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C3×#64095C211yesmaximal, characteristic, complemented
5C32×#886C211yesmaximal, characteristic, complemented
6#11723605C211yesmaximal, characteristic, complemented
7#11723602C211yesmaximal, characteristic, complemented
8#11723539C212yesmaximal, complemented
9#11723504C212yesmaximal, complemented
10#11723503C212yesmaximal, complemented
11#11723422C212yesmaximal, complemented
12C3⋊D3×#123—31nomaximal
13#10594295—34nomaximal
14C23×C3⋊(C3⋊D3)C2211yescharacteristic
15C22×D3×C3⋊D3C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2212yes—
17C2×C3×#638C2211yescharacteristic
18C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
19C2×C32×#67C2211yescharacteristic
20C22×C32×Dic3C2211yescharacteristic, complemented
21#89039—24no—
22C2×#6292C2211yescharacteristic
23C2×#6291C2211yescharacteristic
24C2×#6289—24no—
25C2×#6289C2212yes—
26C2×#6289C2211yescharacteristic
27C2×#6289C2212yescomplemented
28C33×C22⋊C4C2211yescharacteristic, complemented
29#88864C2211yescharacteristic, complemented
30#88859C2212yescomplemented
31C3×#3752C2212yescomplemented
32C3×C4⋊(C3⋊Dic3)C2212yescomplemented
33C32×(C2×D3)⋊C4C2212yescomplemented
34C32×C4⋊Dic3C2212yescomplemented
35#88806C2212yescomplemented
36#88806C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×D4×C3⋊D3—31no—
40C22×C4×C3⋊D3—31no—
41C2×#3811—34no—
42C22×#624—34no—
43C32×#123—31no—
44#64144—31no—
45#64138—31no—
46#64103—32no—
47C4⋊C4×C3⋊D3—32no—
48#64095D311yescharacteristic, complemented
49#64092—32no—
50C22⋊C4×C3⋊D3—31no—
51C22⋊C4×C3⋊D3—32no—
52C22⋊C4×C3⋊D3—34no—
53C22⋊C4×C3⋊D3—64no—
54C3×#886—34no—
55C3×#886D314yescomplemented
56#63961—34no—
57#63961—34no—
58#63921—38no—
59#63900—38no—
60#63854—38no—
61#63854—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)—24no—
64C22×C3⋊(C3⋊D3)D414yes—
65C22×C3⋊(C3⋊D3)C2311yescharacteristic
66C22×C3⋊(C3⋊D3)C2311yescharacteristic
67C22×C3×C3⋊D3C2311yescharacteristic
68C22×C32×D3C2311yescharacteristic
69C2×D3×C3⋊D3—22no—
70C2×D3×C3⋊D3—24no—
71C2×C33×C4C2312yes—
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C4×C3⋊(C3⋊D3)—24no—
74C3×#638—22no—
75C2×C3×C3⋊Dic3—22no—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C2×C3×C3⋊Dic3C2312yes—
78C32×#67—22no—
79C2×C32×Dic3—22no—
80C2×C32×Dic3C2312yes—
81C2×C32×Dic3C2311yescharacteristic
82#6292—22no—
83#6291—22no—
84#6289—24no—
85#6289—24no—
86#6289—22no—
87D3×#123—94no—
88D3×#123—91no—
89C23×C3⋊D3—31no—
90C23×C3⋊D3—34no—
91C23×C3⋊D3—31no—
92C23×C3⋊D3—64no—
93C22×D32—34no—
94C2×C32×D4—31no—
95C22×C32×C4—31no—
96C2×#638—31no—
97C2×#638C2×D311yescharacteristic
98C22×C3⋊Dic3C2×D311yescharacteristic, complemented
99D4×C3⋊D3—64no—
100C2×C4⋊(C3⋊D3)—31no—
101C2×C4×C3⋊D3—32no—
102C2×C4×C3⋊D3—38no—
103C2×C4×C3⋊D3—32no—
104C2×C4×C3⋊D3—32no—
105C2×C4×C3⋊D3—31no—
106C2×C4×C3⋊D3—64no—
107C2×C4×C3⋊D3—68no—
108C2×C3×#67—34no—
109C2×C3×#67C2×D314yes—
110C22×C3×Dic3—34no—
111C22×C3×Dic3C2×D314yescomplemented
112#3811—616no—
113C2×#626—34no—
114C2×#626—34no—
115C2×#624—38no—
116C2×#624—38no—
117C2×#624—34no—
118C2×#624—616no—
119C32×C4⋊C4—32no—
120C32×C22⋊C4—24no—
121C32×C22⋊C4—32no—
122C32×C22⋊C4C2×D314yescomplemented
123C32×C22⋊C4C2×D311yescharacteristic, complemented
124#3757—31no—
125#3757—34no—
126#3757—32no—
127#3757—64no—
128#3752—32no—
129#3752—32no—
130#3752—38no—
131#3752—68no—
132#3752C2×D312yescomplemented
133C4⋊(C3⋊Dic3)C2×D312yescomplemented
134#3750—32no—
135#3750—32no—
136C3×(C2×D3)⋊C4—38no—
137C3×(C2×D3)⋊C4C2×D318yescomplemented
138C3×C4⋊Dic3—38no—
139C3×C4⋊Dic3C2×D318yescomplemented
140#3724—38no—
141#3724—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33—22no—
145C22×C33C2×D412yes—
146C22×C33C2411yescharacteristic
147C2×C3⋊(C3⋊D3)—22no—
148C2×C3⋊(C3⋊D3)—28no—
149C2×C3⋊(C3⋊D3)C2×D412yes—
150C2×C3⋊(C3⋊D3)C2×D412yes—
151C2×C3⋊(C3⋊D3)#4114yes—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—22no—
154C2×C32×D3—21no—
155C2×C32×D3—22no—
156D3×C3⋊D3—44no—
157C33×C4—22no—
158C3⋊(C3⋊Dic3)—22no—
159C3×C3⋊Dic3—22no—
160C3×C3⋊Dic3—21no—
161C32×Dic3—22no—
162C32×Dic3—21no—
163C2×D3×D4—94no—
164C2×D3×D4—91no—
165C22×C4×D3—94no—
166C22×C4×D3—91no—
167C3×#123—34no—
168C3×#123—91no—
169#935—94no—
170#935—91no—
171#929—94no—
172#929—91no—
173#894—98no—
174#894—92no—
175D3×C4⋊C4—98no—
176D3×C4⋊C4—92no—
177#886—34no—
178#886C3⋊D311yescharacteristic, complemented
179#883—98no—
180#883—92no—
181D3×C22⋊C4—98no—
182D3×C22⋊C4—94no—
183D3×C22⋊C4—91no—
184D3×C22⋊C4—92no—
185D3×C22⋊C4—184no—
186C23×C32—24no—
187C23×C32—31no—
188C23×C32C22×D314yes—
189C23×C32C22×D311yescharacteristic
190C22×C3⋊D3—31no—
191C22×C3⋊D3—34no—
192C22×C3⋊D3—31no—
193C22×C3⋊D3—34no—
194C22×C3⋊D3—316no—
195C22×C3⋊D3—34no—
196C22×C3⋊D3—64no—
197C22×C3⋊D3—64no—
198C22×C3⋊D3—616no—
199C22×C3⋊D3—64no—
200C22×C3⋊D3—62no—
201C22×C3⋊D3—64no—
202C22×C3⋊D3—616no—
203C22×C3⋊D3—1216no—
204C22×C3⋊D3C22×D311yescharacteristic
205C22×C3×D3—34no—
206C22×C3×D3C22×D314yes—
207C2×D32—616no—
208C2×D32—68no—
209C32×D4—62no—
210C2×C32×C4—28no—
211C2×C32×C4—31no—
212C2×C32×C4—32no—
213C2×C32×C4—62no—
214C2×C32×C4C22×D318yes—
215C2×C32×C4C22×D312yes—
216#638—22no—
217#638—62no—
218C2×C3⋊Dic3—22no—
219C2×C3⋊Dic3—32no—
220C2×C3⋊Dic3—38no—
221C2×C3⋊Dic3—68no—
222C2×C3⋊Dic3C22×D311yescharacteristic
223C2×C3⋊Dic3C22×D312yes—
224C4⋊(C3⋊D3)—62no—
225C4×C3⋊D3—64no—
226C4×C3⋊D3—64no—
227C4×C3⋊D3—62no—
228C4×C3⋊D3—64no—
229C4×C3⋊D3—616no—
230C4×C3⋊D3—1216no—
231C3×#67—28no—
232C3×#67—68no—
233C2×C3×Dic3—28no—
234C2×C3×Dic3—38no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3—68no—
237C2×C3×Dic3C22×D318yes—
238C2×C3×Dic3C22×D314yes—
239#626—68no—
240#626—68no—
241#624—616no—
242#624—616no—
243#624—68no—
244C2×#123—271noSylow
245C2×C33—22no—
246C2×C33C22×D411yescharacteristic
247C2×C33C2×#4112yes—
248C3⋊(C3⋊D3)—22no—
249C3⋊(C3⋊D3)#12314yescomplemented
250C3×C3⋊D3—41no—
251C32×D3—41no—
252C23×D3—94no—
253C23×D3—94no—
254C23×D3—91no—
255C23×D3—91no—
256C23×D3—184no—
257C2×C3×D4—34no—
258C2×C3×D4—91no—
259C22×C3×C4—34no—
260C22×C3×C4—91no—
261C2×#67—34no—
262C2×#67—94no—
263C2×#67—91no—
264C2×#67C2×C3⋊D311yescharacteristic
265C22×Dic3—34no—
266C22×Dic3C2×C3⋊D311yescharacteristic, complemented
267D3×D4—184no—
268D3×D4—1816no—
269C2×D12—94no—
270C2×D12—91no—
271C2×C4×D3—98no—
272C2×C4×D3—98no—
273C2×C4×D3—94no—
274C2×C4×D3—98no—
275C2×C4×D3—92no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—91no—
279C2×C4×D3—188no—
280C2×C4×D3—184no—
281C2×C4×D3—1816no—
282C3×C4⋊C4—38no—
283C3×C4⋊C4—92no—
284C3×C22⋊C4—24no—
285C3×C22⋊C4—38no—
286C3×C22⋊C4—92no—
287C3×C22⋊C4D3214yescomplemented
288C3×C22⋊C4C2×C3⋊D311yescharacteristic, complemented
289#217—98no—
290#217—94no—
291#217—91no—
292#217—92no—
293#217—184no—
294(C2×D3)⋊C4—38no—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—98no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—188no—
300(C2×D3)⋊C4C2×C3⋊D312yescomplemented
301C4⋊Dic3—38no—
302C4⋊Dic3C2×C3⋊D312yescomplemented
303Dic3⋊C4—98no—
304Dic3⋊C4—98no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307C22×C32—22no—
308C22×C32—24no—
309C22×C32—28no—
310C22×C32—28no—
311C22×C32—48no—
312C22×C32—61no—
313C22×C32—62no—
314C22×C32D3×D418yes—
315C22×C32D3×D412yes—
316C22×C32C23×D314yes—
317C22×C32C23×D311yescharacteristic
318C2×C3⋊D3—21no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3—32no—
321C2×C3⋊D3—34no—
322C2×C3⋊D3—32no—
323C2×C3⋊D3—38no—
324C2×C3⋊D3—316no—
325C2×C3⋊D3—38no—
326C2×C3⋊D3—68no—
327C2×C3⋊D3—616no—
328C2×C3⋊D3—68no—
329C2×C3⋊D3—68no—
330C2×C3⋊D3—62no—
331C2×C3⋊D3—632no—
332C2×C3⋊D3—68no—
333C2×C3⋊D3—1232no—
334C2×C3⋊D3—128no—
335C2×C3⋊D3—124no—
336C2×C3×D3—24no—
337C2×C3×D3—28no—
338C2×C3×D3—64no—
339C2×C3×D3—68no—
340D32—1216no—
341C32×C4—22no—
342C32×C4—28no—
343C32×C4—48no—
344C32×C4—62no—
345C32×C4—61no—
346C3⋊Dic3—22no—
347C3⋊Dic3—21no—
348C3⋊Dic3—62no—
349C3⋊Dic3—68no—
350C3⋊Dic3—128no—
351C3×Dic3—28no—
352C3×Dic3—24no—
353C3×Dic3—68no—
354C3×Dic3—64no—
355C22×D4—271no—
356C23×C4—271no—
357#123—31no—
358#123—91no—
359#123—272no—
360#123—272no—
361#123—271no—
362#123—271no—
363C2×C4⋊C4—272no—
364C2×C22⋊C4—272no—
365C2×C22⋊C4—271no—
366C33C2×#12311yescharacteristic, complemented, Sylow
367C23×C3—24no—
368C23×C3—34no—
369C23×C3—91no—
370C23×C3C2×D3214yes—
371C23×C3C22×C3⋊D311yescharacteristic
372C22×D3—34no—
373C22×D3—94no—
374C22×D3—91no—
375C22×D3—94no—
376C22×D3—91no—
377C22×D3—916no—
378C22×D3—94no—
379C22×D3—184no—
380C22×D3—184no—
381C22×D3—1816no—
382C22×D3—182no—
383C22×D3—184no—
384C22×D3—188no—
385C22×D3—1816no—
386C22×D3—184no—
387C22×D3—1816no—
388C22×D3—3616no—
389C22×D3C22×C3⋊D311yescharacteristic
390C3×D4—68no—
391C3×D4—182no—
392C2×C3×C4—28no—
393C2×C3×C4—34no—
394C2×C3×C4—38no—
395C2×C3×C4—68no—
396C2×C3×C4—91no—
397C2×C3×C4—92no—
398C2×C3×C4—182no—
399C2×C3×C4C2×D3218yes—
400C2×C3×C4C22×C3⋊D312yes—
401#67—22no—
402#67—68no—
403#67—182no—
404#67—188no—
405C2×Dic3—22no—
406C2×Dic3—34no—
407C2×Dic3—38no—
408C2×Dic3—68no—
409C2×Dic3—98no—
410C2×Dic3—92no—
411C2×Dic3—188no—
412C2×Dic3C22×C3⋊D311yescharacteristic
413C2×Dic3C22×C3⋊D312yes—
414D12—188no—
415D12—182no—
416C4×D3—184no—
417C4×D3—184no—
418C4×D3—182no—
419C4×D3—1816no—
420C4×D3—1816no—
421C4×D3—188no—
422C4×D3—184no—
423C4×D3—1816no—
424C4×D3—3616no—
425C2×C32—22no—
426C2×C32—24no—
427C2×C32—28no—
428C2×C32—28no—
429C2×C32—48no—
430C2×C32—121no—
431C2×C32C2×D3×D414yes—
432C2×C32C2×D3×D411yescharacteristic
433C2×C32D3×#4118yes—
434C2×C32D3×#4112yes—
435C3⋊D3—34no—
436C3⋊D3—316no—
437C3⋊D3—41no—
438C3⋊D3—616no—
439C3⋊D3—62no—
440C3⋊D3—68no—
441C3⋊D3—128no—
442C3×D3—44no—
443C3×D3—124no—
444C24—271no—
445C24—271no—
446C2×D4—31no—
447C2×D4—91no—
448C2×D4—271no—
449C2×D4—271no—
450C2×D4—544no—
451C22×C4—31no—
452C22×C4—91no—
453C22×C4—272no—
454C22×C4—271no—
455C22×C4—272no—
456C22×C4—272no—
457C22×C4—544no—
458C4⋊C4—32no—
459C4⋊C4—92no—
460C4⋊C4—272no—
461C4⋊C4—272no—
462C22⋊C4—32no—
463C22⋊C4—92no—
464C22⋊C4—272no—
465C22⋊C4—272no—
466C22⋊C4—271no—
467C22⋊C4—272no—
468C22⋊C4D3×C3⋊D311yescharacteristic, complemented
469C22×C3—28no—
470C22×C3—22no—
471C22×C3—24no—
472C22×C3—28no—
473C22×C3—48no—
474C22×C3—64no—
475C22×C3—68no—
476C22×C3—181no—
477C22×C3—182no—
478C22×C3#380218yes—
479C22×C3D4×C3⋊D312yes—
480C22×C3C22×D3214yes—
481C22×C3C23×C3⋊D311yescharacteristic
482C2×D3—21no—
483C2×D3—22no—
484C2×D3—64no—
485C2×D3—68no—
486C2×D3—98no—
487C2×D3—916no—
488C2×D3—92no—
489C2×D3—94no—
490C2×D3—98no—
491C2×D3—92no—
492C2×D3—1816no—
493C2×D3—188no—
494C2×D3—188no—
495C2×D3—188no—
496C2×D3—1832no—
497C2×D3—188no—
498C2×D3—182no—
499C2×D3—3632no—
500C2×D3—368no—
501C2×D3—364no—
502C2×D3—3616no—
503C3×C4—28no—
504C3×C4—22no—
505C3×C4—48no—
506C3×C4—68no—
507C3×C4—64no—
508C3×C4—182no—
509C3×C4—181no—
510Dic3—22no—
511Dic3—21no—
512Dic3—68no—
513Dic3—64no—
514Dic3—188no—
515Dic3—182no—
516Dic3—368no—
517C32—24no—
518C32D3×#12314yescomplemented
519C32D3×#12311yescharacteristic, complemented
520C23—31no—
521C23—91no—
522C23—271no—
523C23—271no—
524C23—274no—
525C23—544no—
526C23—542no—
527C23—544no—
528C23C2×D3×C3⋊D311yescharacteristic
529D4—62no—
530D4—182no—
531D4—542no—
532D4—542no—
533C2×C4—32no—
534C2×C4—31no—
535C2×C4—62no—
536C2×C4—92no—
537C2×C4—91no—
538C2×C4—182no—
539C2×C4—272no—
540C2×C4—544no—
541C2×C4—544no—
542C2×C4—544no—
543C2×C4—542no—
544C2×C4C2×D3×C3⋊D312yes—
545C2×C3—22no—
546C2×C3—28no—
547C2×C3—24no—
548C2×C3—28no—
549C2×C3—48no—
550C2×C3—124no—
551C2×C3—361no—
552C2×C3C2×#380214yes—
553C2×C3#6431218yes—
554C2×C3C2×D4×C3⋊D311yescharacteristic
555C2×C3#41×C3⋊D312yes—
556D3—41no—
557D3—94no—
558D3—916no—
559D3—124no—
560D3—1816no—
561D3—182no—
562D3—188no—
563D3—368no—
564C22—22no—
565C22—61no—
566C22—62no—
567C22—181no—
568C22—182no—
569C22—272no—
570C22—274no—
571C22—272no—
572C22—548no—
573C22—542no—
574C22—1084no—
575C22#8902612yes—
576C22C22×D3×C3⋊D311yescharacteristic
577C4—22no—
578C4—62no—
579C4—61no—
580C4—182no—
581C4—181no—
582C4—542no—
583C3—24no—
584C3#1059429514yescomplemented
585C3C3⋊D3×#12311yescharacteristic, complemented
586C2—22no—
587C2—121no—
588C2—274no—
589C2—361no—
590C2—542no—
591C2C2×#8902611yescharacteristic
592C2#1172458612yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×#89039C2
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#6289C2semidirect
3. C22⋊C4×C3⋊(C3⋊D3)C2semidirect
4. C3×#64095C2semidirect
5. C32×#886C2semidirect
6. #11723605C2semidirect
7. #11723602C2semidirect
8. #11723539C2semidirect
9. #11723504C2semidirect
10. #11723503C2semidirect
11. #11723422C2semidirect
18. C22×C3×C3⋊Dic3C22semidirect
20. C22×C32×Dic3C22semidirect
27. C2×#6289C22semidirect
28. C33×C22⋊C4C22semidirect
29. #88864C22semidirect
30. #88859C22semidirect
31. C3×#3752C22semidirect
32. C3×C4⋊(C3⋊Dic3)C22semidirect
33. C32×(C2×D3)⋊C4C22semidirect
34. C32×C4⋊Dic3C22semidirect
35. #88806C22semidirect
36. #88806C22semidirect
37. #88803C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
48. #64095D3semidirect
55. C3×#886D3semidirect
98. C22×C3⋊Dic3C2×D3semidirect
111. C22×C3×Dic3C2×D3semidirect
122. C32×C22⋊C4C2×D3semidirect
123. C32×C22⋊C4C2×D3semidirect
132. #3752C2×D3semidirect
133. C4⋊(C3⋊Dic3)C2×D3semidirect
137. C3×(C2×D3)⋊C4C2×D3semidirect
139. C3×C4⋊Dic3C2×D3semidirect
178. #886C3⋊D3semidirect
249. C3⋊(C3⋊D3)#123semidirect
266. C22×Dic3C2×C3⋊D3semidirect
287. C3×C22⋊C4D32semidirect
288. C3×C22⋊C4C2×C3⋊D3semidirect
300. (C2×D3)⋊C4C2×C3⋊D3semidirect
302. C4⋊Dic3C2×C3⋊D3semidirect
366. C33C2×#123semidirect
468. C22⋊C4D3×C3⋊D3semidirect
518. C32D3×#123semidirect
519. C32D3×#123semidirect
584. C3#10594295semidirect
585. C3C3⋊D3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×#890392
C22×#62892
C22⋊C4×C3⋊(C3⋊D3)2
C3×#640952
C32×#8862
#117236052
#117236022
#117235392
#117235042
#117235032
#117234222
C3⋊D3×#1233
#105942953