Finite Group Databasebeta

← #11865852

Subgroups — #11865852

Order 1152.

1166 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5716C211yesmaximal, characteristic, complemented
2C2×#63925C212yesmaximal, complemented
3C4×D3×D12C211yesmaximal, characteristic, complemented
4#10592771C211yesmaximal, characteristic, complemented
5C3×#5072C211yesmaximal, characteristic, complemented
6#10591432C212yesmaximal, complemented
7#10591431C212yesmaximal, complemented
8#10591414C212yesmaximal, complemented
9#10590379C211yesmaximal, characteristic, complemented
10#10590363C211yesmaximal, characteristic, complemented
11#10590336C211yesmaximal, characteristic, complemented
12#83297—31nomaximal
13#80332—31nomaximal
14C2×C3×#237C2212yes—
15C2×D3×D12C2211yescharacteristic
16C2×C4×D32—21no—
17C32×#127C2211yescharacteristic, complemented
18C2×#3753C2212yes—
19C4×C4⋊(C3⋊D3)C2211yescharacteristic
20C3×#938—21no—
21C3×#938—21no—
22C3×#936C2212yescomplemented
23C3×D4×Dic3—21no—
24C3×D4×Dic3—21no—
25C2×C3×#213C2212yes—
26C3×C4×D12C2211yescharacteristic
27C3×C42×D3C2211yescharacteristic
28C3×#869C2211yescharacteristic, complemented
29#63925—28no—
30D3×(C2×D3)⋊C4—21no—
31C4×#626—21no—
32#63894C2211yescharacteristic
33Dic3×D12C2211yescharacteristic
34D3×C4⋊Dic3C2211yescharacteristic
35#63880—21no—
36C2×#3717C2212yes—
37C2×#3715C2212yes—
38C2×#3711C2212yes—
39#63659C2212yescomplemented
40C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
41C3×#832C2212yescomplemented
42C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
43Dic3⋊(C3⋊C8)C2212yescomplemented
44#63566C2212yescomplemented
45#63565C2212yescomplemented
46C3×#532—31no—
47C2×#949—32no—
48C2×#911—32no—
49#5716D311yescharacteristic, complemented
50C4×D3×D4—31no—
51C2×C4×D12—31no—
52C3×#459—31no—
53#5232—32no—
54#5153—32no—
55#5083—31no—
56#5083—31no—
57#5072—31no—
58#5072—31no—
59#5072—61no—
60#5072D311yescharacteristic, complemented
61#5057—32no—
62#5057—32no—
63#5019—31no—
64#4924—31no—
65#4841—31no—
66#4837—31no—
67#4787—32no—
68#4771—32no—
69C22×D32—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4C2312yes—
73C2×C4⋊(C3⋊D3)C2311yescharacteristic
74C2×C4×C3⋊D3—21no—
75C2×C3×#67—22no—
76C22×C3×Dic3—22no—
77C3×#237—24no—
78C3×#237—24no—
79C2×C3×D12C2311yescharacteristic
80C2×C3×C4×D3—21no—
81C2×C3×C4×D3D411yescharacteristic
82C2×C3×C4×D3D411yescharacteristic
83C2×C3×C4×D3C2311yescharacteristic
84C2×C3×Dic6C2312yes—
85C2×#626—21no—
86C2×#624—21no—
87C2×D3×Dic3—21no—
88C2×D3×Dic3—21no—
89D3×D12—22no—
90D3×D12D414yescomplemented
91C4×D32—44no—
92C32×C42C2311yescharacteristic
93#3753—24no—
94#3752—21no—
95C4⋊(C3⋊Dic3)C2311yescharacteristic
96C2×C3⋊(C3⋊C8)C2312yes—
97C3×#217—22no—
98C3×#217—22no—
99C3×#213—24no—
100C3×(C2×D3)⋊C4—21no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3—21no—
103C3×C4⋊Dic3C2311yescharacteristic
104C3×C4×Dic3D411yescharacteristic
105C3×C4×Dic3D411yescharacteristic
106C3×C4×Dic3C2311yescharacteristic
107C2×C3×C3⋊C8C2312yes—
108Dic3⋊Dic3—21no—
109#3722—21no—
110(C2×D3)⋊Dic3—21no—
111#3717—24no—
112#3715—24no—
113#3711—24no—
114#3036—91noSylow
115C2×C3×#41—32no—
116C2×#237C2×D312yes—
117C2×D3×D4—31no—
118C22×D12—31no—
119C22×C4×D3—61no—
120C22×C4×D3—61no—
121C2×C3×D8—32no—
122C3×C4⋊Q8—31no—
123C3×#127—21no—
124C3×#127C2×D311yescharacteristic, complemented
125C3×#127C2×D311yescharacteristic, complemented
126C3×#124—32no—
127C3×#121—61no—
128C3×#121—61no—
129C3×C4×D4—31no—
130C3×C4×D4—61no—
131C3×C4×D4—61no—
132C2×C3×C42—31no—
133#949—68no—
134C2×#215—32no—
135C2×#215—32no—
136#938—21no—
137#938—21no—
138#936C2×D312yescomplemented
139D4×Dic3—21no—
140D4×Dic3—21no—
141D4×Dic3—31no—
142C2×#214—32no—
143C2×#213—32no—
144C2×#213—32no—
145C2×#213—62no—
146C2×#213C2×D312yes—
147C4×#67—61no—
148C2×(C2×D3)⋊C4—61no—
149C2×C4⋊Dic3—31no—
150C2×#208—32no—
151#911—68no—
152C2×#204—32no—
153C2×ΓL(1,𝔽25)—32no—
154#893—31no—
155D3×C4⋊C4—31no—
156#67⋊C4—61no—
157D3×C22⋊C4—61no—
158C4×D12—31no—
159C4×D12—31no—
160C4×D12—31no—
161C4×D12—31no—
162C4×D12—61no—
163C4×D12—61no—
164C4×D12—61no—
165C4×D12C2×D311yescharacteristic
166C42×D3C2×D311yescharacteristic
167#869C2×D311yescharacteristic, complemented
168C3×C4⋊C8—31no—
169C3×#102—32no—
170#835—32no—
171#835—32no—
172#832—32no—
173#832—32no—
174#832—62no—
175#832C2×D312yescomplemented
176#816—32no—
177Dic3⋊C8—32no—
178#810—32no—
179C4⋊(C3⋊C8)—31no—
180C4⋊(C3⋊C8)—31no—
181C4⋊(C3⋊C8)—32no—
182C4⋊(C3⋊C8)—61no—
183C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
184C23×C32—22no—
185C22×C3⋊D3—21no—
186C22×C3×D3—21no—
187C22×C3×D3C2×D411yescharacteristic
188C2×D32—24no—
189C2×D32—42no—
190C2×D32—42no—
191C32×D4—22no—
192C32×D4—22no—
193C32×D4—42no—
194C32×D4—42no—
195C2×C32×C4C2×D411yescharacteristic
196C2×C32×C4C2×D411yescharacteristic
197C2×C32×C4C2411yescharacteristic
198C2×C3⋊Dic3—21no—
199C4⋊(C3⋊D3)—21no—
200C4⋊(C3⋊D3)C2×D412yes—
201C4×C3⋊D3—42no—
202C3×#67—44no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—22no—
205C2×C3×Dic3—44no—
206C2×C3×Dic3C2×D411yescharacteristic
207C2×C3×Dic3C2×D411yescharacteristic
208C2×C3×Dic3C2×D411yescharacteristic
209C3×D12—21no—
210C3×D12C2×D412yes—
211C3×C4×D3—22no—
212C3×C4×D3—41no—
213C3×C4×D3—42no—
214C3×C4×D3C2×D412yes—
215C3×C4×D3#4112yes—
216C3×Dic6—22no—
217C3×Dic6—22no—
218#626—24no—
219#624—42no—
220D3×Dic3—42no—
221D3×Dic3—42no—
222C3⋊(C3⋊C8)—22no—
223C3×C3⋊C8—22no—
224C2×AGL(1,ℤ/8)—92no—
225#532—31no—
226C2×C4×D4—91no—
227#463—92no—
228#463—92no—
229#461—91no—
230#461—91no—
231#459—31no—
232#459—91no—
233#423—91no—
234#417—92no—
235C23×D3—61no—
236C23×D3—61no—
237C3×#41—64no—
238C3×#41—64no—
239C2×C3×Q8—32no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4—22no—
245C2×C3×D4—31no—
246C2×C3×D4—41no—
247C2×C3×D4—41no—
248C2×C3×D4—62no—
249C2×C3×D4C22×D312yes—
250C2×C3×D4C22×D312yes—
251C22×C3×C4—31no—
252C22×C3×C4—31no—
253C22×C3×C4—31no—
254C22×C3×C4—62no—
255C22×C3×C4—61no—
256C2×#67—22no—
257C2×#67—61no—
258C22×Dic3—22no—
259C22×Dic3—61no—
260C22×Dic3—61no—
261#237—24no—
262#237—24no—
263D3×D4—34no—
264D3×D4—62no—
265C2×D12—31no—
266C2×D12—31no—
267C2×D12—34no—
268C2×D12—61no—
269C2×D12—62no—
270C2×D12—61no—
271C2×D12C22×D311yescharacteristic
272C2×C4×D3—21no—
273C2×C4×D3—61no—
274C2×C4×D3—61no—
275C2×C4×D3—61no—
276C2×C4×D3—61no—
277C2×C4×D3—61no—
278C2×C4×D3—61no—
279C2×C4×D3—121no—
280C2×C4×D3—124no—
281C2×C4×D3—124no—
282C2×C4×D3#6711yescharacteristic
283C2×C4×D3#6711yescharacteristic
284C2×C4×D3C22×D311yescharacteristic
285C2×Dic6C22×D312yes—
286C3×D8—64no—
287C2×C3×C8—32no—
288C3×C4⋊C4—31no—
289C3×C4⋊C4—61no—
290C3×C4⋊C4—61no—
291C3×C22⋊C4—62no—
292C3×C22⋊C4—62no—
293C3×C22⋊C4—61no—
294C3×C42—21no—
295C3×C42—31no—
296C3×C42—31no—
297C3×C42—31no—
298C3×C42C22×D311yescharacteristic
299C3×C42C22×D311yescharacteristic
300#217—22no—
301#217—22no—
302#217—61no—
303#215—64no—
304#215—64no—
305#214—64no—
306#213—24no—
307#213—64no—
308#213—64no—
309#213—124no—
310(C2×D3)⋊C4—21no—
311(C2×D3)⋊C4—61no—
312(C2×D3)⋊C4—61no—
313(C2×D3)⋊C4—61no—
314(C2×D3)⋊C4—61no—
315(C2×D3)⋊C4—61no—
316(C2×D3)⋊C4—121no—
317C4⋊Dic3—21no—
318C4⋊Dic3—21no—
319C4⋊Dic3—31no—
320C4⋊Dic3—31no—
321C4⋊Dic3—61no—
322C4⋊Dic3—61no—
323C4⋊Dic3C22×D311yescharacteristic
324Dic3⋊C4—61no—
325C4×Dic3#6711yescharacteristic
326C4×Dic3#6711yescharacteristic
327C4×Dic3C22×D311yescharacteristic
328#208—64no—
329C2×C3⋊C8—32no—
330C2×C3⋊C8—32no—
331C2×C3⋊C8—62no—
332C2×C3⋊C8C22×D312yes—
333#204—64no—
334ΓL(1,𝔽25)—64no—
335C22×C32—42no—
336C22×C32—42no—
337C22×C32—42no—
338C22×C32C22×D411yescharacteristic
339C2×C3⋊D3—22no—
340C2×C3⋊D3—41no—
341C2×C3⋊D3—41no—
342C2×C3×D3—21no—
343C2×C3×D3—21no—
344C2×C3×D3—22no—
345C2×C3×D3—41no—
346C2×C3×D3—41no—
347C2×C3×D3#12112yes—
348D32—48no—
349C32×C4—21no—
350C32×C4AGL(1,ℤ/8)12yescomplemented
351C32×C4C22×D411yescharacteristic
352C32×C4C2×#4111yescharacteristic
353C3⋊Dic3—41no—
354C3×Dic3—21no—
355C3×Dic3—21no—
356C3×Dic3—42no—
357C3×Dic3—41no—
358C3×Dic3#12112yes—
359C2×#41—32no—
360C22×D4—91no—
361C23×C4—181no—
362AGL(1,ℤ/8)—188no—
363C2×SD16—92no—
364C2×SD16—92no—
365C2×D8—32no—
366C2×D8—92no—
367C2×M16—92no—
368C4⋊Q8—31no—
369#127D3211yescharacteristic, complemented
370#124—32no—
371#121—61no—
372#121—61no—
373C4×D4—31no—
374C4×D4—61no—
375C4×D4—61no—
376C4×D4—91no—
377C4×D4—91no—
378C4×D4—91no—
379C4×D4—181no—
380C4×D4—181no—
381C2×C4⋊C4—91no—
382C2×C22⋊C4—181no—
383C2×C42—31no—
384C4⋊C8—31no—
385C4⋊C8—92no—
386C4⋊C8—91no—
387#103—92no—
388#103—92no—
389#102—32no—
390#102—92no—
391C23×C3—22no—
392C23×C3—22no—
393C23×C3—31no—
394C23×C3—42no—
395C23×C3—61no—
396C22×D3—21no—
397C22×D3—64no—
398C22×D3—61no—
399C22×D3—61no—
400C22×D3—64no—
401C22×D3—121no—
402C22×D3—122no—
403C22×D3—122no—
404C22×D3—122no—
405C22×D3—122no—
406C22×D3C2×#6711yescharacteristic
407C3×Q8—62no—
408C3×Q8—62no—
409C3×D4—22no—
410C3×D4—22no—
411C3×D4—22no—
412C3×D4—22no—
413C3×D4—32no—
414C3×D4—42no—
415C3×D4—42no—
416C3×D4—42no—
417C3×D4—42no—
418C3×D4—42no—
419C3×D4—42no—
420C3×D4—61no—
421C3×D4—82no—
422C3×D4—82no—
423C3×D4—124no—
424C2×C3×C4—21no—
425C2×C3×C4—21no—
426C2×C3×C4—21no—
427C2×C3×C4—31no—
428C2×C3×C4—32no—
429C2×C3×C4—32no—
430C2×C3×C4—31no—
431C2×C3×C4—31no—
432C2×C3×C4—62no—
433C2×C3×C4—62no—
434C2×C3×C4—61no—
435C2×C3×C4—121no—
436C2×C3×C4—124no—
437C2×C3×C4—122no—
438C2×C3×C4D3×D411yescharacteristic
439C2×C3×C4D3×D411yescharacteristic
440C2×C3×C4C2×#6711yescharacteristic
441C2×C3×C4C2×#6711yescharacteristic
442C2×C3×C4C23×D311yescharacteristic
443C2×C3×C4C23×D311yescharacteristic
444#67—44no—
445#67—64no—
446C2×Dic3—21no—
447C2×Dic3—22no—
448C2×Dic3—44no—
449C2×Dic3—61no—
450C2×Dic3—61no—
451C2×Dic3—121no—
452C2×Dic3—122no—
453C2×Dic3—122no—
454C2×Dic3C2×#6711yescharacteristic
455C2×Dic3C2×#6711yescharacteristic
456C2×Dic3C2×#6711yescharacteristic
457D12—21no—
458D12—32no—
459D12—32no—
460D12—62no—
461D12—64no—
462D12—61no—
463D12—61no—
464D12—121no—
465D12D3×D412yes—
466C4×D3—22no—
467C4×D3—41no—
468C4×D3—42no—
469C4×D3—122no—
470C4×D3—122no—
471C4×D3—122no—
472C4×D3—122no—
473C4×D3—122no—
474C4×D3—122no—
475C4×D3—242no—
476C4×D3D3×D412yes—
477C4×D3#23712yes—
478Dic6—22no—
479Dic6—22no—
480C3×C8—62no—
481C3⋊C8—22no—
482C3⋊C8—62no—
483C3⋊C8—62no—
484C3⋊C8—122no—
485C2×C32—82no—
486C2×C32C2×#12111yescharacteristic
487C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
488C2×C32#57511yescharacteristic
489C3⋊D3—42no—
490C3×D3—22no—
491C3×D3—42no—
492C24—181no—
493#41—64no—
494#41—64no—
495C2×Q8—32no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—31no—
499C2×D4—62no—
500C2×D4—94no—
501C2×D4—91no—
502C2×D4—182no—
503C2×D4—181no—
504C2×D4C2×D3212yes—
505C22×C4—31no—
506C22×C4—31no—
507C22×C4—31no—
508C22×C4—62no—
509C22×C4—61no—
510C22×C4—181no—
511C22×C4—181no—
512C22×C4—181no—
513C22×C4—181no—
514C22×C4—364no—
515SD16—184no—
516SD16—184no—
517D8—64no—
518D8—184no—
519M16—184no—
520C2×C8—32no—
521C2×C8—92no—
522C4⋊C4—31no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—91no—
526C4⋊C4—181no—
527C22⋊C4—62no—
528C22⋊C4—62no—
529C22⋊C4—61no—
530C22⋊C4—181no—
531C22⋊C4—181no—
532C22⋊C4—181no—
533C42—31no—
534C42—31no—
535C42—31no—
536C42C2×D3211yescharacteristic
537C22×C3—21no—
538C22×C3—32no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—61no—
546C22×C3—61no—
547C22×C3—62no—
548C22×C3—82no—
549C22×C3—82no—
550C22×C3—82no—
551C22×C3—121no—
552C22×C3—121no—
553C22×C3C2×D3×D411yescharacteristic
554C22×C3C22×#6711yescharacteristic
555C2×D3—21no—
556C2×D3—21no—
557C2×D3—22no—
558C2×D3—41no—
559C2×D3—41no—
560C2×D3—62no—
561C2×D3—62no—
562C2×D3—122no—
563C2×D3—128no—
564C2×D3—128no—
565C2×D3—121no—
566C2×D3—121no—
567C2×D3—121no—
568C2×D3—121no—
569C2×D3—241no—
570C2×D3—241no—
571C2×D3#93812yes—
572C3×C4—21no—
573C3×C4—21no—
574C3×C4—21no—
575C3×C4—21no—
576C3×C4—22no—
577C3×C4—32no—
578C3×C4—41no—
579C3×C4—61no—
580C3×C4—61no—
581C3×C4—121no—
582C3×C4—122no—
583C3×C4#91112yescomplemented
584C3×C4#94912yescomplemented
585C3×C4C2×D3×D411yescharacteristic
586C3×C4C2×D3×D411yescharacteristic
587C3×C4C2×#23711yescharacteristic
588C3×C4D3×#4111yescharacteristic
589Dic3—21no—
590Dic3—21no—
591Dic3—41no—
592Dic3—42no—
593Dic3—121no—
594Dic3—121no—
595Dic3—241no—
596Dic3#93812yes—
597C32#303611yescharacteristic, complemented, Sylow
598C23—22no—
599C23—31no—
600C23—61no—
601C23—184no—
602C23—181no—
603C23—362no—
604C23—362no—
605Q8—62no—
606Q8—62no—
607D4—22no—
608D4—22no—
609D4—32no—
610D4—42no—
611D4—42no—
612D4—61no—
613D4—92no—
614D4—124no—
615D4—184no—
616D4—181no—
617C2×C4—31no—
618C2×C4—32no—
619C2×C4—32no—
620C2×C4—31no—
621C2×C4—31no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—61no—
625C2×C4—121no—
626C2×C4—124no—
627C2×C4—122no—
628C2×C4—181no—
629C2×C4—362no—
630C2×C4—362no—
631C2×C4—362no—
632C2×C4—362no—
633C2×C4D3×#6711yescharacteristic
634C2×C4D3×#6711yescharacteristic
635C2×C4C22×D3211yescharacteristic
636C8—62no—
637C8—182no—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—21no—
641C2×C3—62no—
642C2×C3—82no—
643C2×C3—82no—
644C2×C3—122no—
645C2×C3—162no—
646C2×C3D3×#12111yescharacteristic
647C2×C3C2×#91111yescharacteristic
648C2×C3#577311yescharacteristic
649C2×C3C2×#93811yescharacteristic
650C2×C3C2×#94911yescharacteristic
651C2×C3#582211yescharacteristic
652D3—22no—
653D3—42no—
654D3—122no—
655D3—122no—
656D3—242no—
657C22—32no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—61no—
662C22—61no—
663C22—62no—
664C22—121no—
665C22—121no—
666C22—182no—
667C22—368no—
668C22—361no—
669C22—361no—
670C22C2×D3×#6711yescharacteristic
671C4—21no—
672C4—32no—
673C4—61no—
674C4—61no—
675C4—121no—
676C4—122no—
677C4—361no—
678C4#6392512yescomplemented
679C4D32×D411yescharacteristic
680C4D3×#23711yescharacteristic
681C3—21no—
682C3#8033211yescharacteristic, complemented
683C3#8329711yescharacteristic, complemented
684C2—62no—
685C2—82no—
686C2—122no—
687C2—362no—
688C2C2×#6392511yescharacteristic
689C2#1059509511yescharacteristic
690C2D3×#93811yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#127C22
socleC22×C32C22×D4
hypercenter#127D32
hypocenterC32#3036
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C3×#5716C2semidirect
2. C2×#63925C2semidirect
3. C4×D3×D12C2semidirect
4. #10592771C2semidirect
5. C3×#5072C2semidirect
6. #10591432C2semidirect
7. #10591431C2semidirect
8. #10591414C2semidirect
9. #10590379C2semidirect
10. #10590363C2semidirect
11. #10590336C2semidirect
17. C32×#127C22semidirect
22. C3×#936C22semidirect
28. C3×#869C22semidirect
39. #63659C22semidirect
40. C4⋊(C3⋊(C3⋊C8))C22semidirect
41. C3×#832C22semidirect
42. C3×C4⋊(C3⋊C8)C22semidirect
43. Dic3⋊(C3⋊C8)C22semidirect
44. #63566C22semidirect
45. #63565C22semidirect
49. #5716D3semidirect
60. #5072D3semidirect
90. D3×D12D4semidirect
124. C3×#127C2×D3semidirect
125. C3×#127C2×D3semidirect
138. #936C2×D3semidirect
167. #869C2×D3semidirect
175. #832C2×D3semidirect
183. C4⋊(C3⋊C8)C2×D3semidirect
350. C32×C4AGL(1,ℤ/8)semidirect
369. #127D32semidirect
583. C3×C4#911semidirect
584. C3×C4#949semidirect
597. C32#3036semidirect
678. C4#63925semidirect
682. C3#80332semidirect
683. C3#83297semidirect

Maximal subgroups

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

subgroupindex
C3×#57162
C2×#639252
C4×D3×D122
#105927712
C3×#50722
#105914322
#105914312
#105914142
#105903792
#105903632
#105903362
#832973
#803323