Finite Group Databasebeta

← #13177325

Subgroups — #13177325

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C211yesmaximal, characteristic, complemented
2C2×#89026C211yesmaximal, characteristic, complemented
3C4×#6312C211yesmaximal, characteristic, complemented
4C3×#64139C211yesmaximal, characteristic, complemented
5#11723643C212yesmaximal, complemented
6#11723615C212yesmaximal, complemented
7D3×#3752C212yesmaximal, complemented
8#11723513C212yesmaximal, complemented
9#11723499C211yesmaximal, characteristic, complemented
10#11723456C211yesmaximal, characteristic, complemented
11#11723436C211yesmaximal, characteristic, complemented
12#10596300—31nomaximal
13#10595070—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C22×C33×C4C2211yescharacteristic, complemented
16C2×#6312C2211yescharacteristic
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×#638C2212yescomplemented
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C32×#67C2212yescomplemented
21C32×#235—24no—
22C2×C32×D12C2211yescharacteristic
23C2×C32×C4×D3C2212yescomplemented
24C2×C32×Dic6C2211yescharacteristic, complemented
25C2×#6290C2212yes—
26C2×D3×C3⋊Dic3C2212yescomplemented
27#89026—24no—
28#88864C2211yescharacteristic, complemented
29#88859C2211yescharacteristic
30#88857C2211yescharacteristic, complemented
31C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
32C3×#3752C2212yescomplemented
33C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
34#88807C2212yescomplemented
35#88804C2212yes—
36#88803C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×C32×#41—31no—
40C2×D4×C3⋊D3—31no—
41C2×C3×#235D314yescomplemented
42C2×#3802—34no—
43#64156—31no—
44#64155—31no—
45#64148—32no—
46#64146—32no—
47D4×C3⋊Dic3—31no—
48#64139D311yescharacteristic, complemented
49#64138—32no—
50C4×#638—31no—
51C4×#638—34no—
52C4×#638—64no—
53C2×#3752—32no—
54C3×#930—34no—
55#63983—38no—
56#63970—38no—
57D3×(C2×D3)⋊C4—38no—
58#63905—38no—
59#63898—34no—
60#63871—34no—
61#63863—34no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3C2312yes—
65C22×C32×D3C2312yes—
66C2×D3×C3⋊D3—24no—
67C2×D3×C3⋊D3—24no—
68C2×D3×C3⋊D3—42no—
69C2×C33×C4—22no—
70C2×C33×C4C2311yescharacteristic
71C2×C33×C4C2311yescharacteristic
72#6312—22no—
73C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
76C4×C3⋊(C3⋊D3)—22no—
77C3×#638—42no—
78C2×C3×C3⋊Dic3C2312yes—
79C3×C4⋊(C3⋊D3)—22no—
80C32×#67—24no—
81C2×C32×Dic3C2312yes—
82C32×D12D414yescomplemented
83C32×C4×D3—24no—
84C32×Dic6—22no—
85#6290—24no—
86D3×C3⋊Dic3—42no—
87#5829—94no—
88#5612—91no—
89C23×C3⋊D3—32no—
90C22×D32—38no—
91C32×#41—64no—
92C2×C32×Q8—31no—
93C2×C32×D4—32no—
94C2×C32×D4—31no—
95C22×C32×C4—24no—
96C22×C32×C4—32no—
97C22×C32×C4C2×D314yescomplemented
98C22×C32×C4C2×D311yescharacteristic, complemented
99C2×#638—31no—
100C2×#638—34no—
101C2×#638—32no—
102C2×#638—64no—
103C2×#638C2×D312yescomplemented
104C22×C3⋊Dic3—32no—
105D4×C3⋊D3—64no—
106C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
107C2×C4×C3⋊D3—31no—
108C2×C4×C3⋊D3—34no—
109C2×C4×C3⋊D3—64no—
110C2×C3×#67—38no—
111C2×C3×#67C2×D318yescomplemented
112C3×#235—216no—
113C2×C3×D12—34no—
114C2×C3×D12C2×D314yes—
115C2×C3×C4×D3C2×D318yescomplemented
116C2×C3×Dic6C2×D314yescomplemented
117C2×#625—38no—
118C2×D3×Dic3—38no—
119#3802—616no—
120#3757—31no—
121#3757—34no—
122#3757—32no—
123#3757—64no—
124#3752—32no—
125#3752—32no—
126#3752—31no—
127#3752—34no—
128#3752—32no—
129#3752—64no—
130#3752C2×D312yescomplemented
131C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
132#3750—31no—
133#3750—34no—
134#3750—32no—
135#3750—64no—
136C4×C3⋊Dic3—31no—
137C4×C3⋊Dic3—34no—
138C4×C3⋊Dic3—64no—
139C3×(C2×D3)⋊C4—38no—
140C3×C4⋊Dic3—34no—
141Dic3⋊Dic3—38no—
142#3722—38no—
143(C2×D3)⋊Dic3—38no—
144(C2×D3)⋊Dic3—38no—
145(C2×D3)⋊Dic3—38no—
146C22×C33—21no—
147C22×C33—21no—
148C22×C33—21no—
149C22×C33C2411yescharacteristic
150C2×C3⋊(C3⋊D3)—21no—
151C2×C3⋊(C3⋊D3)—21no—
152C2×C3⋊(C3⋊D3)#4112yescomplemented
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3—22no—
156C2×C32×D3—22no—
157C2×C32×D3—22no—
158C2×C32×D3C2×D414yes—
159D3×C3⋊D3—48no—
160C33×C4—21no—
161C33×C4C2×D412yes—
162C3⋊(C3⋊Dic3)—21no—
163C3⋊(C3⋊Dic3)—21no—
164C3⋊(C3⋊Dic3)—21no—
165C3×C3⋊Dic3—22no—
166C32×Dic3—22no—
167C2×C3×#41—34no—
168C2×D3×D4—94no—
169C2×D3×D4—91no—
170C2×#235C3⋊D311yescharacteristic, complemented
171C3×#121—91no—
172#947—94no—
173#946—94no—
174#939—98no—
175#938—92no—
176#937—98no—
177#937—92no—
178#936—91no—
179D4×Dic3—94no—
180#930—34no—
181#929—98no—
182C4×#67—94no—
183C4×#67—91no—
184C4×#67—184no—
185C2×(C2×D3)⋊C4—98no—
186#897—91no—
187#893—91no—
188#883—92no—
189D3×C22⋊C4—92no—
190C23×C32—24no—
191C23×C32—32no—
192C23×C32C22×D314yes—
193C23×C32C22×D311yescharacteristic
194C22×C3⋊D3—31no—
195C22×C3⋊D3—34no—
196C22×C3⋊D3—64no—
197C22×C3⋊D3—64no—
198C22×C3⋊D3—64no—
199C22×C3⋊D3—122no—
200C22×C3⋊D3C22×D312yes—
201C22×C3×D3—38no—
202C22×C3×D3C22×D318yes—
203C2×D32—616no—
204C2×D32—616no—
205C2×D32—128no—
206C32×Q8—62no—
207C32×D4—34no—
208C32×D4—64no—
209C2×C32×C4—22no—
210C2×C32×C4—24no—
211C2×C32×C4—24no—
212C2×C32×C4—28no—
213C2×C32×C4—32no—
214C2×C32×C4—48no—
215C2×C32×C4—64no—
216C2×C32×C4C22×D314yes—
217C2×C32×C4C22×D314yes—
218C2×C32×C4C22×D311yescharacteristic
219C2×C32×C4C22×D311yescharacteristic
220#638—42no—
221#638—64no—
222#638—62no—
223#638—68no—
224#638—616no—
225C2×C3⋊Dic3—31no—
226C2×C3⋊Dic3—31no—
227C2×C3⋊Dic3—31no—
228C2×C3⋊Dic3—34no—
229C2×C3⋊Dic3—34no—
230C2×C3⋊Dic3—34no—
231C2×C3⋊Dic3—64no—
232C2×C3⋊Dic3—64no—
233C2×C3⋊Dic3—64no—
234C2×C3⋊Dic3—122no—
235C2×C3⋊Dic3C22×D312yes—
236C4⋊(C3⋊D3)—22no—
237C4×C3⋊D3—62no—
238C4×C3⋊D3—68no—
239C4×C3⋊D3—128no—
240C3×#67—216no—
241C3×#67—128no—
242C2×C3×Dic3—38no—
243C2×C3×Dic3C22×D318yes—
244C3×D12—68no—
245C3×D12#67116yescomplemented
246C3×C4×D3—216no—
247C3×Dic6—28no—
248#625—616no—
249D3×Dic3—128no—
250#546—271noSylow
251C2×C33—41no—
252C2×C33C22×D411yescharacteristic
253C2×C33C2×#4111yescharacteristic
254C2×C33H(5,𝔽2)11yescharacteristic
255C3⋊(C3⋊D3)—22no—
256C3×C3⋊D3—42no—
257C32×D3—24no—
258C23×D3—98no—
259C23×D3—92no—
260C3×#41—616no—
261C2×C3×Q8—34no—
262C2×C3×D4—38no—
263C2×C3×D4—34no—
264C2×C3×D4—92no—
265C2×C3×D4—91no—
266C22×C3×C4—24no—
267C22×C3×C4—38no—
268C22×C3×C4D3214yescomplemented
269C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
270C2×#67—38no—
271C2×#67—98no—
272C2×#67—94no—
273C2×#67—91no—
274C2×#67—92no—
275C2×#67—184no—
276C2×#67C2×C3⋊D312yescomplemented
277C22×Dic3—98no—
278D3×D4—1816no—
279D3×D4—184no—
280#235—24no—
281C2×D12—34no—
282C2×D12C2×C3⋊D311yescharacteristic
283C2×C4×D3—94no—
284C2×C4×D3—91no—
285C2×C4×D3—92no—
286C2×C4×D3—184no—
287C2×C4×D3C2×C3⋊D312yescomplemented
288C2×Dic6C2×C3⋊D311yescharacteristic, complemented
289C3×C4⋊C4—91no—
290C3×C22⋊C4—92no—
291#217—98no—
292#217—94no—
293#217—91no—
294#217—92no—
295#217—92no—
296#217—184no—
297(C2×D3)⋊C4—38no—
298(C2×D3)⋊C4—98no—
299(C2×D3)⋊C4—98no—
300(C2×D3)⋊C4—98no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—94no—
303(C2×D3)⋊C4—91no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—184no—
306C4⋊Dic3—34no—
307C4⋊Dic3—92no—
308Dic3⋊C4—98no—
309Dic3⋊C4—94no—
310Dic3⋊C4—91no—
311Dic3⋊C4—184no—
312C4×Dic3—94no—
313C4×Dic3—91no—
314C4×Dic3—184no—
315C22×C32—28no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32—24no—
319C22×C32—21no—
320C22×C32—24no—
321C22×C32—24no—
322C22×C32—24no—
323C22×C32—34no—
324C22×C32—44no—
325C22×C32—44no—
326C22×C32—62no—
327C22×C32—62no—
328C22×C32C23×D314yes—
329C22×C32C23×D311yescharacteristic
330C2×C3⋊D3—22no—
331C2×C3⋊D3—22no—
332C2×C3⋊D3—22no—
333C2×C3⋊D3—32no—
334C2×C3⋊D3—38no—
335C2×C3⋊D3—68no—
336C2×C3⋊D3—61no—
337C2×C3⋊D3—61no—
338C2×C3⋊D3—64no—
339C2×C3⋊D3—64no—
340C2×C3⋊D3—124no—
341C2×C3⋊D3—124no—
342C2×C3⋊D3—128no—
343C2×C3×D3—28no—
344C2×C3×D3—28no—
345C2×C3×D3—68no—
346C2×C3×D3—68no—
347C2×C3×D3—68no—
348C2×C3×D3C2×#67116yes—
349D32—1232no—
350C32×C4—21no—
351C32×C4—28no—
352C32×C4—24no—
353C32×C4—44no—
354C32×C4—62no—
355C32×C4D3×D412yes—
356C32×C4C2×#6718yes—
357C3⋊Dic3—22no—
358C3⋊Dic3—68no—
359C3⋊Dic3—61no—
360C3⋊Dic3—61no—
361C3⋊Dic3—64no—
362C3⋊Dic3—61no—
363C3⋊Dic3—64no—
364C3⋊Dic3—64no—
365C3⋊Dic3—124no—
366C3⋊Dic3—124no—
367C3×Dic3—28no—
368C3×Dic3—68no—
369C2×#41—31no—
370C22×D4—271no—
371#124—271no—
372#123—272no—
373C22⋊Q8—271no—
374#121—91no—
375#121—272no—
376#120—272no—
377C4×D4—271no—
378C4×D4—271no—
379C2×C22⋊C4—272no—
380C33#54611yescharacteristic, complemented, Sylow
381C23×C3—24no—
382C23×C3—38no—
383C23×C3—92no—
384C23×C3C2×D3214yes—
385C23×C3C22×C3⋊D311yescharacteristic
386C22×D3—38no—
387C22×D3—94no—
388C22×D3—91no—
389C22×D3—184no—
390C22×D3—184no—
391C22×D3—1816no—
392C22×D3—1816no—
393C22×D3—184no—
394C22×D3—362no—
395C22×D3—368no—
396C22×D3C22×C3⋊D312yes—
397C3×Q8—68no—
398C3×D4—316no—
399C3×D4—616no—
400C3×D4—182no—
401C3×D4—362no—
402C2×C3×C4—28no—
403C2×C3×C4—22no—
404C2×C3×C4—24no—
405C2×C3×C4—24no—
406C2×C3×C4—38no—
407C2×C3×C4—48no—
408C2×C3×C4—616no—
409C2×C3×C4—92no—
410C2×C3×C4C2×D3214yes—
411C2×C3×C4C2×D3214yes—
412C2×C3×C4C22×C3⋊D311yescharacteristic
413C2×C3×C4C22×C3⋊D311yescharacteristic
414#67—24no—
415#67—128no—
416#67—1816no—
417#67—188no—
418#67—182no—
419#67—1816no—
420#67—184no—
421C2×Dic3—38no—
422C2×Dic3—94no—
423C2×Dic3—94no—
424C2×Dic3—94no—
425C2×Dic3—91no—
426C2×Dic3—91no—
427C2×Dic3—91no—
428C2×Dic3—184no—
429C2×Dic3—184no—
430C2×Dic3—184no—
431C2×Dic3—368no—
432C2×Dic3C22×C3⋊D312yes—
433D12—68no—
434D12#63814yescomplemented
435C4×D3—24no—
436C4×D3—188no—
437C4×D3—182no—
438C4×D3—368no—
439C4×D3—362no—
440Dic6—22no—
441C2×C32—24no—
442C2×C32—24no—
443C2×C32—24no—
444C2×C32—48no—
445C2×C32—44no—
446C2×C32—41no—
447C2×C32—64no—
448C2×C32C2×D3×D411yescharacteristic
449C2×C32#100411yescharacteristic
450C2×C32D3×#4114yes—
451C2×C32D3×#4111yescharacteristic
452C2×C32#100914yes—
453C2×C32C22×#6714yes—
454C3⋊D3—42no—
455C3⋊D3—62no—
456C3⋊D3—68no—
457C3⋊D3—128no—
458C3×D3—216no—
459C3×D3—128no—
460C24—272no—
461#41—64no—
462C2×Q8—31no—
463C2×D4—32no—
464C2×D4—31no—
465C2×D4—92no—
466C2×D4—91no—
467C2×D4—272no—
468C2×D4—271no—
469C2×D4—544no—
470C22×C4—32no—
471C22×C4—272no—
472C22×C4—271no—
473C22×C4D3×C3⋊D311yescharacteristic, complemented
474C4⋊C4—91no—
475C4⋊C4—272no—
476C4⋊C4—271no—
477C22⋊C4—92no—
478C22⋊C4—272no—
479C22⋊C4—272no—
480C22⋊C4—272no—
481C22⋊C4—272no—
482C22⋊C4—271no—
483C22⋊C4—271no—
484C42—271no—
485C22×C3—28no—
486C22×C3—24no—
487C22×C3—24no—
488C22×C3—24no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—24no—
493C22×C3—316no—
494C22×C3—44no—
495C22×C3—44no—
496C22×C3—68no—
497C22×C3—68no—
498C22×C3—182no—
499C22×C3—182no—
500C22×C3—182no—
501C22×C3C22×D3214yes—
502C22×C3C23×C3⋊D311yescharacteristic
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—68no—
506C2×D3—68no—
507C2×D3—68no—
508C2×D3—98no—
509C2×D3—92no—
510C2×D3—188no—
511C2×D3—184no—
512C2×D3—184no—
513C2×D3—181no—
514C2×D3—181no—
515C2×D3—364no—
516C2×D3—364no—
517C2×D3—368no—
518C2×D3—3632no—
519C2×D3C2×#63814yes—
520C3×C4—24no—
521C3×C4—21no—
522C3×C4—28no—
523C3×C4—44no—
524C3×C4—68no—
525C3×C4—182no—
526C3×C4D3×#6718yes—
527C3×C4C2×#63812yes—
528Dic3—22no—
529Dic3—68no—
530Dic3—188no—
531Dic3—184no—
532Dic3—184no—
533Dic3—184no—
534Dic3—181no—
535Dic3—181no—
536Dic3—181no—
537Dic3—364no—
538Dic3—364no—
539C32—24no—
540C32#561211yescharacteristic, complemented
541C32#582914yescomplemented
542C23—32no—
543C23—92no—
544C23—271no—
545C23—544no—
546C23—544no—
547C23—1082no—
548C23C2×D3×C3⋊D311yescharacteristic
549Q8—62no—
550D4—34no—
551D4—64no—
552D4—182no—
553D4—362no—
554D4—542no—
555D4—544no—
556C2×C4—22no—
557C2×C4—32no—
558C2×C4—64no—
559C2×C4—92no—
560C2×C4—271no—
561C2×C4—271no—
562C2×C4—271no—
563C2×C4—542no—
564C2×C4—1082no—
565C2×C4C2×D3×C3⋊D311yescharacteristic
566C2×C4C2×D3×C3⋊D311yescharacteristic
567C2×C3—24no—
568C2×C3—24no—
569C2×C3—24no—
570C2×C3—48no—
571C2×C3—41no—
572C2×C3—44no—
573C2×C3—616no—
574C2×C3—362no—
575C2×C3#6430614yes—
576C2×C3#6430714yes—
577C2×C3C2×D3×#6714yes—
578C2×C3#41×C3⋊D311yescharacteristic
579C2×C3#6436611yescharacteristic
580C2×C3C22×#63811yescharacteristic
581D3—24no—
582D3—128no—
583D3—182no—
584D3—188no—
585D3—368no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—34no—
590C22—62no—
591C22—62no—
592C22—182no—
593C22—182no—
594C22—182no—
595C22—272no—
596C22—541no—
597C22—541no—
598C22—1088no—
599C22C22×D3×C3⋊D311yescharacteristic
600C4—21no—
601C4—62no—
602C4—182no—
603C4—541no—
604C4—541no—
605C4—541no—
606C4D3×#63812yes—
607C3—24no—
608C3#1059507014yescomplemented
609C3#1059630011yescharacteristic, complemented
610C2—41no—
611C2—64no—
612C2—362no—
613C2—542no—
614C2#1172457811yescharacteristic
615C2#1172458011yescharacteristic
616C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#546
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#235C2semidirect
2. C2×#89026C2semidirect
3. C4×#6312C2semidirect
4. C3×#64139C2semidirect
5. #11723643C2semidirect
6. #11723615C2semidirect
7. D3×#3752C2semidirect
8. #11723513C2semidirect
9. #11723499C2semidirect
10. #11723456C2semidirect
11. #11723436C2semidirect
15. C22×C33×C4C22semidirect
18. C2×C3×#638C22semidirect
20. C2×C32×#67C22semidirect
23. C2×C32×C4×D3C22semidirect
24. C2×C32×Dic6C22semidirect
26. C2×D3×C3⋊Dic3C22semidirect
28. #88864C22semidirect
30. #88857C22semidirect
31. C4×C3⋊(C3⋊Dic3)C22semidirect
32. C3×#3752C22semidirect
33. C3×C4⋊(C3⋊Dic3)C22semidirect
34. #88807C22semidirect
36. #88803C22semidirect
37. #88803C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
41. C2×C3×#235D3semidirect
48. #64139D3semidirect
82. C32×D12D4semidirect
97. C22×C32×C4C2×D3semidirect
98. C22×C32×C4C2×D3semidirect
103. C2×#638C2×D3semidirect
111. C2×C3×#67C2×D3semidirect
115. C2×C3×C4×D3C2×D3semidirect
116. C2×C3×Dic6C2×D3semidirect
130. #3752C2×D3semidirect
131. C4⋊(C3⋊Dic3)C2×D3semidirect
152. C2×C3⋊(C3⋊D3)#41semidirect
170. C2×#235C3⋊D3semidirect
245. C3×D12#67semidirect
268. C22×C3×C4D32semidirect
269. C22×C3×C4C2×C3⋊D3semidirect
276. C2×#67C2×C3⋊D3semidirect
287. C2×C4×D3C2×C3⋊D3semidirect
288. C2×Dic6C2×C3⋊D3semidirect
380. C33#546semidirect
434. D12#638semidirect
473. C22×C4D3×C3⋊D3semidirect
540. C32#5612semidirect
541. C32#5829semidirect
608. C3#10595070semidirect
609. C3#10596300semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×#890262
C4×#63122
C3×#641392
#117236432
#117236152
D3×#37522
#117235132
#117234992
#117234562
#117234362
#105963003
#105950703