Finite Group Databasebeta

← #11860322

Subgroups — #11860322

Order 1152.

1245 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64117C211yesmaximal, characteristic
2#10594954C211yesmaximal, characteristic, complemented
3C2×#63894C211yesmaximal, characteristic, complemented
4C3×#5121C211yesmaximal, characteristic, complemented
5C3×Dic3×M16C211yesmaximal, characteristic, complemented
6#10591106C211yesmaximal, characteristic, complemented
7#10591096C211yesmaximal, characteristic, complemented
8#10590773C212yesmaximal, complemented
9#10590768C212yesmaximal, complemented
10#10590763C212yesmaximal, complemented
11#10590758C212yesmaximal, complemented
12#79199—31nomaximal
13C4×#908—31nomaximal
14C2×#3828C2211yescharacteristic
15C22×C4⋊(C3⋊D3)C2211yescharacteristic
16C22×#624—21no—
17C2×C32×M16C2211yescharacteristic
18#64117C418yescomplemented
19C2×#3745C2212yes—
20C2×#3744C2212yes—
21C3×#926C2211yescharacteristic
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25C2×#3722—21no—
26#63952—21no—
27#63952—21no—
28#63894—21no—
29#63894—21no—
30#63894C2212yes—
31#63894C2212yes—
32#63894C2211yescharacteristic, complemented
33#63894C2211yescharacteristic, complemented
34C4×#624—21no—
35#63874—21no—
36#63844C2211yescharacteristic, complemented
37#63844C2211yescharacteristic, complemented
38C3×#818C2212yescomplemented
39C3×#817C2212yescomplemented
40C3×#815C2212yescomplemented
41C3×C8×Dic3C2212yescomplemented
42#63567C2212yescomplemented
43#63564C2212yes—
44#63564C2212yes—
45#63564C2212yescomplemented
46C2×#908—31no—
47C2×#908—31no—
48C2×#908—61no—
49#5522—31no—
50C2×#893—31no—
51C4×#235—31no—
52C2×C4×D12—31no—
53C3×#428—31no—
54C3×C4×M16—31no—
55#5133—31no—
56#5131—31no—
57#5121D311yescharacteristic, complemented
58Dic3×M16D311yescharacteristic, complemented
59#5001—31no—
60#4966—31no—
61#4889—32no—
62#4872—32no—
63#4869—32no—
64#4852—32no—
65#4711—32no—
66#4710—32no—
67C4×D24—32no—
68C4×#204—32no—
69C23×C3⋊D3—21no—
70C22×C32×C4C2311yescharacteristic
71C2×#638—21no—
72#3828—22no—
73#3828C2×C414yescomplemented
74C2×C4⋊(C3⋊D3)—21no—
75C2×C4⋊(C3⋊D3)—22no—
76C2×C4⋊(C3⋊D3)C2×C414yescomplemented
77C2×C4⋊(C3⋊D3)C2312yes—
78C2×C4⋊(C3⋊D3)C2311yescharacteristic
79C2×C4×C3⋊D3—21no—
80C2×#634C2311yescharacteristic
81C22×C3×Dic3—21no—
82C22×C3×Dic3D411yescharacteristic
83C22×C3×Dic3D411yescharacteristic
84C2×#624—22no—
85C2×#624—22no—
86C2×#624—21no—
87C2×#624—21no—
88C2×#624—44no—
89C32×M16C2×C414yescomplemented
90C2×C32×C8C2312yes—
91#3745C2×C418yescomplemented
92#3744C2×C418yescomplemented
93C3×#217—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2312yes—
96C3×C4⋊Dic3C2311yescharacteristic
97C3×C4⋊Dic3C2311yescharacteristic
98C3×C4×Dic3—21no—
99C3×C4×Dic3D411yescharacteristic
100C3×C4×Dic3D411yescharacteristic
101C3×C4×Dic3C2311yescharacteristic
102C3×C4×Dic3C2311yescharacteristic
103C3×#208—22no—
104C2×C3×C3⋊C8C2312yes—
105#3724—21no—
106#3724—21no—
107#3722—21no—
108#3722—21no—
109#3722—22no—
110#3722—22no—
111Dic32—21no—
112Dic32—21no—
113C4×AGL(1,ℤ/8)—91noSylow
114C2×#235—31no—
115C2×#235—31no—
116C2×#235—61no—
117C22×D12—31no—
118C22×D12—31no—
119C22×D12—61no—
120C22×C4×D3—61no—
121C22×C4×D3—61no—
122C2×C3×M16—21no—
123C2×C3×M16—31no—
124C2×C3×M16C2×D311yescharacteristic
125C2×C3×M16C2×D311yescharacteristic
126C3×#117—31no—
127C2×C3×C4⋊C4—31no—
128C2×C3×C42—31no—
129C4×#67—61no—
130C4×#67—61no—
131C2×(C2×D3)⋊C4—61no—
132C2×(C2×D3)⋊C4—61no—
133#926C2×D311yescharacteristic
134C2×C4⋊Dic3C2×D311yescharacteristic
135C2×C4×Dic3C2×D311yescharacteristic
136C2×#208C2×D311yescharacteristic
137#908—38no—
138#908—38no—
139#908—68no—
140C2×D24—32no—
141C2×D24—32no—
142C2×D24—62no—
143C2×#204—32no—
144C2×#204—32no—
145C2×#204—62no—
146#893—31no—
147#893—32no—
148#893—31no—
149#893—32no—
150#893—61no—
151#893—61no—
152#887—31no—
153#887—31no—
154#67⋊C4—61no—
155#67⋊C4—61no—
156C4×D12—32no—
157C4×D12—31no—
158C4×D12—32no—
159C4×D12—31no—
160C4×D12—61no—
161C4×D12—61no—
162#872—61no—
163#872—61no—
164C42×D3—61no—
165C42×D3—61no—
166C4×Dic6—31no—
167C4×Dic6—31no—
168C3×#107—32no—
169C3×#106—32no—
170C3×#97—32no—
171C3×C4×C8—32no—
172#821—32no—
173#821—32no—
174#821—32no—
175#818C2×D312yescomplemented
176#817C2×D312yescomplemented
177#816—32no—
178#815C2×D312yescomplemented
179C8×Dic3C2×D312yescomplemented
180#810—32no—
181#809—32no—
182#809—32no—
183#809—32no—
184C23×C32C2×D411yescharacteristic
185C22×C3⋊D3—24no—
186C22×C3⋊D3—22no—
187C22×C3⋊D3—21no—
188C22×C3⋊D3—42no—
189C22×C3⋊D3—42no—
190C2×C32×C4C22×C412yes—
191C2×C32×C4C2×D411yescharacteristic
192C2×C32×C4#4112yes—
193C2×C32×C4C2411yescharacteristic
194#638—24no—
195C2×C3⋊Dic3—21no—
196C4⋊(C3⋊D3)—24no—
197C4⋊(C3⋊D3)—22no—
198C4⋊(C3⋊D3)—21no—
199C4⋊(C3⋊D3)C22×C414yes—
200C4⋊(C3⋊D3)C22×C412yescomplemented
201C4×C3⋊D3—24no—
202#634—21no—
203#634C22×C412yescomplemented
204C2×C3×Dic3—22no—
205C2×C3×Dic3—21no—
206C2×C3×Dic3—21no—
207C2×C3×Dic3—22no—
208C2×C3×Dic3—41no—
209C2×C3×Dic3—42no—
210C2×C3×Dic3C2×D411yescharacteristic
211C2×C3×Dic3C2×D411yescharacteristic
212C2×C3×Dic3C2×D411yescharacteristic
213C2×C3×Dic3C2×D411yescharacteristic
214#624—42no—
215#624—44no—
216#624—42no—
217#624—44no—
218C32×C8C22×C414yescomplemented
219C3×C3⋊C8—22no—
220C2×AGL(1,ℤ/8)—91no—
221C4×#41—91no—
222C2×C4×D4—91no—
223#442—92no—
224#440—92no—
225C4×SD16—92no—
226C4×D8—92no—
227#428—31no—
228#418—91no—
229#417—91no—
230C4×M16—31no—
231C23×D3—61no—
232C23×D3—61no—
233C23×D3—121no—
234C22×C3×C4—21no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—61no—
238C22×C3×C4C22×D311yescharacteristic
239C22×C3×C4C22×D311yescharacteristic
240C2×#67—61no—
241C2×#67—61no—
242C2×#67—121no—
243C22×Dic3—21no—
244C22×Dic3D1211yescharacteristic
245C22×Dic3D1211yescharacteristic
246#235—34no—
247#235—34no—
248#235—64no—
249#235—62no—
250#235—62no—
251#235—122no—
252C2×D12—34no—
253C2×D12—34no—
254C2×D12—32no—
255C2×D12—31no—
256C2×D12—32no—
257C2×D12—31no—
258C2×D12—62no—
259C2×D12—61no—
260C2×D12—64no—
261C2×D12—61no—
262C2×D12—62no—
263C2×D12—61no—
264C2×D12—62no—
265C2×D12—122no—
266C2×D12—121no—
267C2×C4×D3—61no—
268C2×C4×D3—61no—
269C2×C4×D3—61no—
270C2×C4×D3—61no—
271C2×C4×D3—62no—
272C2×C4×D3—62no—
273C2×C4×D3—61no—
274C2×C4×D3—61no—
275C2×C4×D3—62no—
276C2×C4×D3—62no—
277C2×C4×D3—121no—
278C2×C4×D3—124no—
279C2×C4×D3—124no—
280C2×Dic6—31no—
281C2×Dic6—31no—
282C2×Dic6—61no—
283C3×M16—24no—
284C3×M16—62no—
285C3×M16C4×D314yescomplemented
286C3×M16C4×D314yescomplemented
287C2×C3×C8—22no—
288C2×C3×C8—32no—
289C2×C3×C8C22×D312yes—
290C2×C3×C8C22×D312yes—
291C3×C4⋊C4—31no—
292C3×C4⋊C4—31no—
293C3×C4⋊C4—32no—
294C3×C4⋊C4—61no—
295C3×C22⋊C4—61no—
296C3×C42—31no—
297C3×C42—31no—
298C3×C42—31no—
299C3×C42—31no—
300C3×C42—61no—
301#217—21no—
302(C2×D3)⋊C4—61no—
303(C2×D3)⋊C4—61no—
304(C2×D3)⋊C4—62no—
305(C2×D3)⋊C4—62no—
306(C2×D3)⋊C4—61no—
307(C2×D3)⋊C4—61no—
308(C2×D3)⋊C4—62no—
309(C2×D3)⋊C4—62no—
310C4⋊Dic3—21no—
311C4⋊Dic3C22×D312yes—
312C4⋊Dic3C22×D311yescharacteristic
313C4⋊Dic3C22×D311yescharacteristic
314Dic3⋊C4—61no—
315Dic3⋊C4—61no—
316Dic3⋊C4—61no—
317Dic3⋊C4—61no—
318C4×Dic3—21no—
319C4×Dic3—61no—
320C4×Dic3—61no—
321C4×Dic3—61no—
322C4×Dic3—61no—
323C4×Dic3D1211yescharacteristic
324C4×Dic3D1211yescharacteristic
325C4×Dic3C22×D311yescharacteristic
326C4×Dic3C22×D311yescharacteristic
327#208—22no—
328C2×C3⋊C8C22×D312yes—
329D24—38no—
330D24—38no—
331D24—68no—
332#204—38no—
333#204—38no—
334#204—68no—
335C22×C32—21no—
336C22×C32—21no—
337C22×C32C4×D412yes—
338C22×C32C22×D411yescharacteristic
339C2×C3⋊D3—22no—
340C2×C3⋊D3—24no—
341C2×C3⋊D3—48no—
342C2×C3⋊D3—42no—
343C2×C3⋊D3—41no—
344C2×C3⋊D3—41no—
345C2×C3⋊D3—42no—
346C32×C4C4×D412yes—
347C32×C4C23×C411yescharacteristic
348C32×C4C2×#4111yescharacteristic
349C3⋊Dic3—22no—
350C3×Dic3—21no—
351C3×Dic3—21no—
352C3×Dic3—21no—
353C3×Dic3—42no—
354C3×Dic3—41no—
355C3×Dic3—41no—
356C3×Dic3AGL(1,ℤ/8)12yescomplemented
357C2×#41—91no—
358C22×D4—91no—
359C23×C4—181no—
360AGL(1,ℤ/8)—98no—
361C2×SD16—92no—
362C2×D8—92no—
363C2×M16—31no—
364C2×M16D3211yescharacteristic
365C4×Q8—91no—
366C4×Q8—91no—
367C4×D4—92no—
368C4×D4—92no—
369C4×D4—91no—
370C4×D4—91no—
371C4×D4—181no—
372C4×D4—181no—
373C4×D4—181no—
374C4×D4—181no—
375#117—31no—
376#117—181no—
377C2×C4⋊C4—31no—
378C2×C22⋊C4—181no—
379C2×C42—31no—
380C2×C42—181no—
381#107—32no—
382#106—32no—
383#103—92no—
384#102—92no—
385#102—92no—
386#102—92no—
387#97—32no—
388C4×C8—32no—
389C23×C3—21no—
390C23×C3C2×D1211yescharacteristic
391C23×C3D3×D411yescharacteristic
392C22×D3—64no—
393C22×D3—64no—
394C22×D3—61no—
395C22×D3—62no—
396C22×D3—61no—
397C22×D3—62no—
398C22×D3—121no—
399C22×D3—122no—
400C22×D3—124no—
401C22×D3—122no—
402C22×D3—122no—
403C22×D3—122no—
404C22×D3—122no—
405C22×D3—242no—
406C22×D3—242no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—31no—
412C2×C3×C4—31no—
413C2×C3×C4—31no—
414C2×C3×C4—31no—
415C2×C3×C4—62no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—62no—
419C2×C3×C4—121no—
420C2×C3×C4—122no—
421C2×C3×C4C2×C4×D312yes—
422C2×C3×C4C2×C4×D312yes—
423C2×C3×C4C2×D1211yescharacteristic
424C2×C3×C4#23512yes—
425C2×C3×C4D3×D411yescharacteristic
426C2×C3×C4#23912yes—
427C2×C3×C4C23×D311yescharacteristic
428C2×C3×C4C23×D311yescharacteristic
429#67—64no—
430#67—64no—
431#67—124no—
432C2×Dic3—22no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—22no—
436C2×Dic3—41no—
437C2×Dic3—42no—
438C2×Dic3—61no—
439C2×Dic3—61no—
440C2×Dic3—121no—
441C2×Dic3C2×D1211yescharacteristic
442C2×Dic3C2×D1211yescharacteristic
443C2×Dic3C2×D1211yescharacteristic
444C2×Dic3C2×D1211yescharacteristic
445D12—34no—
446D12—34no—
447D12—32no—
448D12—32no—
449D12—64no—
450D12—62no—
451D12—64no—
452D12—64no—
453D12—62no—
454D12—61no—
455D12—62no—
456D12—61no—
457D12—122no—
458D12—124no—
459D12—121no—
460C4×D3—64no—
461C4×D3—64no—
462C4×D3—124no—
463C4×D3—122no—
464C4×D3—124no—
465C4×D3—122no—
466C4×D3—124no—
467C4×D3—122no—
468C4×D3—124no—
469C4×D3—122no—
470C4×D3—124no—
471Dic6—32no—
472Dic6—32no—
473Dic6—62no—
474Dic6—61no—
475Dic6—61no—
476Dic6—121no—
477C3×C8—24no—
478C3×C8—62no—
479C3×C8C2×C4×D314yescomplemented
480C3×C8C2×C4×D314yescomplemented
481C3⋊C8—22no—
482C2×C32—22no—
483C2×C32C2×C4×D411yescharacteristic
484C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
485C2×C32#57511yescharacteristic
486C3⋊D3—42no—
487C3⋊D3—44no—
488C24—181no—
489#41—94no—
490#41—182no—
491C2×Q8—91no—
492C2×D4—91no—
493C2×D4—92no—
494C2×D4—94no—
495C2×D4—181no—
496C2×D4—182no—
497C2×D4—181no—
498C22×C4—31no—
499C22×C4—31no—
500C22×C4—61no—
501C22×C4—181no—
502C22×C4—181no—
503C22×C4—182no—
504C22×C4—182no—
505C22×C4—181no—
506C22×C4—364no—
507C22×C4C2×D3211yescharacteristic
508SD16—98no—
509D8—98no—
510M16—62no—
511M16#62414yescomplemented
512C2×C8—32no—
513C2×C8C2×D3212yes—
514C4⋊C4—31no—
515C4⋊C4—31no—
516C4⋊C4—32no—
517C4⋊C4—61no—
518C4⋊C4—181no—
519C4⋊C4—181no—
520C22⋊C4—61no—
521C22⋊C4—182no—
522C22⋊C4—182no—
523C22⋊C4—181no—
524C22⋊C4—181no—
525C42—31no—
526C42—31no—
527C42—31no—
528C42—31no—
529C42—61no—
530C42—181no—
531C42—181no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—22no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3C4×D1212yes—
541C22×C3#89312yes—
542C22×C3C22×D1211yescharacteristic
543C22×C3C2×D3×D411yescharacteristic
544C2×D3—62no—
545C2×D3—64no—
546C2×D3—62no—
547C2×D3—64no—
548C2×D3—122no—
549C2×D3—124no—
550C2×D3—128no—
551C2×D3—128no—
552C2×D3—122no—
553C2×D3—121no—
554C2×D3—121no—
555C2×D3—122no—
556C2×D3—122no—
557C2×D3—121no—
558C2×D3—121no—
559C2×D3—122no—
560C2×D3—242no—
561C2×D3—242no—
562C2×D3—241no—
563C2×D3—241no—
564C2×D3—248no—
565C3×C4—21no—
566C3×C4—21no—
567C3×C4—22no—
568C3×C4—32no—
569C3×C4—61no—
570C3×C4—61no—
571C3×C4—61no—
572C3×C4—122no—
573C3×C4—121no—
574C3×C4—121no—
575C3×C4C4×D1212yes—
576C3×C4#89312yes—
577C3×C4C22×C4×D311yescharacteristic
578C3×C4C22×C4×D311yescharacteristic
579C3×C4C2×#23511yescharacteristic
580C3×C4C2×#23911yescharacteristic
581Dic3—21no—
582Dic3—21no—
583Dic3—21no—
584Dic3—42no—
585Dic3—41no—
586Dic3—41no—
587Dic3—62no—
588Dic3—62no—
589Dic3—122no—
590Dic3#90812yescomplemented
591C32C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
592C23—181no—
593C23—182no—
594C23—184no—
595C23—362no—
596C23—362no—
597C23D3×D1211yescharacteristic
598Q8—92no—
599Q8—181no—
600D4—92no—
601D4—94no—
602D4—184no—
603D4—182no—
604D4—181no—
605D4—184no—
606C2×C4—31no—
607C2×C4—31no—
608C2×C4—31no—
609C2×C4—31no—
610C2×C4—62no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—121no—
615C2×C4—122no—
616C2×C4—181no—
617C2×C4—184no—
618C2×C4—364no—
619C2×C4—362no—
620C2×C4—364no—
621C2×C4—362no—
622C2×C4#379912yes—
623C2×C4D3×D1211yescharacteristic
624C2×C4C2×#62412yes—
625C2×C4C22×D3211yescharacteristic
626C8—62no—
627C8C2×#62414yescomplemented
628C2×C3—22no—
629C2×C3—22no—
630C2×C3—21no—
631C2×C3—21no—
632C2×C3—21no—
633C2×C3—42no—
634C2×C3C2×C4×D1211yescharacteristic
635C2×C3C2×#89311yescharacteristic
636C2×C3C2×#90811yescharacteristic
637C2×C3#574811yescharacteristic
638C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
639C2×C3#577711yescharacteristic
640D3—124no—
641D3—122no—
642D3—124no—
643D3—122no—
644D3—244no—
645D3—242no—
646C22—21no—
647C22—21no—
648C22—182no—
649C22—184no—
650C22—368no—
651C22—362no—
652C22—361no—
653C22—361no—
654C22—362no—
655C22#6389412yes—
656C22C2×D3×D1211yescharacteristic
657C4—32no—
658C4—61no—
659C4—61no—
660C4—61no—
661C4—122no—
662C4—121no—
663C4—121no—
664C4—182no—
665C4#6389412yes—
666C4C2×#379911yescharacteristic
667C4C22×#62411yescharacteristic
668C3—21no—
669C3C4×#90811yescharacteristic, complemented
670C3#7919911yescharacteristic, complemented
671C2—22no—
672C2—364no—
673C2—362no—
674C2D3×#90811yescharacteristic
675C2#1059471111yescharacteristic
676C2C2×#6389411yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C22×D4
prime radicalC2×#64117C2
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32C4×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10594954C2semidirect
3. C2×#63894C2semidirect
4. C3×#5121C2semidirect
5. C3×Dic3×M16C2semidirect
6. #10591106C2semidirect
7. #10591096C2semidirect
8. #10590773C2semidirect
9. #10590768C2semidirect
10. #10590763C2semidirect
11. #10590758C2semidirect
18. #64117C4semidirect
32. #63894C22semidirect
33. #63894C22semidirect
36. #63844C22semidirect
37. #63844C22semidirect
38. C3×#818C22semidirect
39. C3×#817C22semidirect
40. C3×#815C22semidirect
41. C3×C8×Dic3C22semidirect
42. #63567C22semidirect
45. #63564C22semidirect
57. #5121D3semidirect
58. Dic3×M16D3semidirect
73. #3828C2×C4semidirect
76. C2×C4⋊(C3⋊D3)C2×C4semidirect
89. C32×M16C2×C4semidirect
91. #3745C2×C4semidirect
92. #3744C2×C4semidirect
175. #818C2×D3semidirect
176. #817C2×D3semidirect
178. #815C2×D3semidirect
179. C8×Dic3C2×D3semidirect
200. C4⋊(C3⋊D3)C22×C4semidirect
203. #634C22×C4semidirect
218. C32×C8C22×C4semidirect
285. C3×M16C4×D3semidirect
286. C3×M16C4×D3semidirect
356. C3×Dic3AGL(1,ℤ/8)semidirect
479. C3×C8C2×C4×D3semidirect
480. C3×C8C2×C4×D3semidirect
511. M16#624semidirect
590. Dic3#908semidirect
591. C32C4×AGL(1,ℤ/8)semidirect
627. C8C2×#624semidirect
669. C3C4×#908semidirect
670. C3#79199semidirect

Maximal subgroups

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

subgroupindex
C2×#641172
#105949542
C2×#638942
C3×#51212
C3×Dic3×M162
#105911062
#105910962
#105907732
#105907682
#105907632
#105907582
#791993
C4×#9083