Finite Group Databasebeta

← #10797137

Subgroups — #10797137

Order 768.

1686 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5748C211yesmaximal, characteristic
2C2×#5745C211yesmaximal, characteristic, complemented
3C2×#5132C211yesmaximal, characteristic, complemented
4C2×#4966C211yesmaximal, characteristic, complemented
5C2×#4966C211yesmaximal, characteristic, complemented
6C2×C3×#343C211yesmaximal, characteristic, complemented
7#71994C211yesmaximal, characteristic, complemented
8#69007C214yesmaximal, complemented
9#69006C214yesmaximal, complemented
10#21004—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C22×#208C2211yescharacteristic
18#5748—24no—
19#5748C418yescomplemented
20C2×#909—21no—
21C2×#909C2212yes—
22C2×#908—21no—
23C2×#908C2212yes—
24#5745—24no—
25#5745—24no—
26C2×#907C2214yes—
27C2×D3×M16—22no—
28C2×#904—22no—
29C2×#901—22no—
30C2×#824C2212yescomplemented
31#5132—24no—
32C2×#823C2212yescomplemented
33C2×#810—21no—
34C2×#810C2212yescomplemented
35#4966—22no—
36#4966—22no—
37#4966—22no—
38#4966C2214yes—
39C2×#809—21no—
40C2×#809C2212yescomplemented
41C3×#343C2214yescomplemented
42#4558C2214yescomplemented
43C2×#575—31no—
44C2×#568—31no—
45C2×#417—31no—
46C2×#417—31no—
47C2×#413—31no—
48#1908—34no—
49#1907—34no—
50C2×#343D311yescharacteristic, complemented
51#1738—31no—
52C23×C3×C4—21no—
53C23×C3×C4C2311yescharacteristic
54C22×#67—21no—
55C22×#67—21no—
56C2×#235—21no—
57C2×#235—22no—
58C2×#235—21no—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235—21no—
62C2×#235—21no—
63C2×#235—21no—
64C2×#235—21no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235C2×C414yes—
68C2×#235C2×C414yes—
69C2×#235D414yes—
70C2×#235D412yes—
71C2×#235C2312yes—
72C22×D12D411yescharacteristic
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C22×C4×D3—21no—
76C22×Dic6D411yescharacteristic
77C22×Dic6C2311yescharacteristic
78C2×C3×M16—21no—
79C2×C3×M16—22no—
80C2×C3×M16—21no—
81C2×C3×M16—22no—
82C2×C3×M16C2×C414yes—
83C2×C3×M16D414yes—
84C2×C3×M16C2312yes—
85C2×C3×M16C2312yes—
86C22×C3×C8—21no—
87C22×C3×C8—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2312yes—
94C2×#208—22no—
95C2×#208—22no—
96C2×#208—21no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208D412yes—
100C2×#208D412yes—
101C2×#208C2312yes—
102C22×C3⋊C8—21no—
103C22×C3⋊C8C2312yes—
104#909—24no—
105#909—28no—
106#909C2×C418yescomplemented
107#908—24no—
108#908—28no—
109#908C2×C418yescomplemented
110#907—28no—
111#907—28no—
112D3×M16—216no—
113C2×Dic12—22no—
114#904—216no—
115C2×D24—22no—
116C2×#204—22no—
117C2×#204—22no—
118#901—216no—
119C2×ΓL(1,𝔽25)—24no—
120C2×D3×C8—24no—
121#824—24no—
122#823—24no—
123#810—24no—
124#810—24no—
125#809—24no—
126#809—24no—
127C22×#41—31no—
128C22×#41—31no—
129#575—38no—
130#575—64no—
131C2×#137—32no—
132C2×#137—61no—
133C2×AGL(1,ℤ/8)—32no—
134C2×AGL(1,ℤ/8)—61no—
135C2×#135—62no—
136#568—64no—
137#568—64no—
138C2×#131—34no—
139C2×#131—62no—
140C22×M16—31no—
141C22×M16—31no—
142C22×M16—62no—
143C22×M16C2×D311yescharacteristic
144C22×M16C2×D311yescharacteristic
145C22×C4⋊C4C2×D311yescharacteristic
146#417—34no—
147#417—62no—
148#417—62no—
149#417—62no—
150C2×#103—32no—
151C2×#103—61no—
152C2×#102—32no—
153C2×#102—61no—
154#413—64no—
155C2×#101—32no—
156C2×#100—32no—
157#343C2×D314yescomplemented
158#341—34no—
159C24×C3#4111yescharacteristic
160C23×D3—21no—
161C23×D3—21no—
162C22×C3×C4—22no—
163C22×C3×C4—41no—
164C22×C3×C4—41no—
165C22×C3×C4—82no—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4#4111yescharacteristic
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—24no—
177C2×#67—22no—
178C2×#67—24no—
179C2×#67—22no—
180C2×#67—24no—
181C2×#67—41no—
182C2×#67—41no—
183C2×#67—42no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186#235—24no—
187#235—24no—
188#235—24no—
189#235—24no—
190#235—24no—
191#235—22no—
192#235—22no—
193#235—24no—
194#235—24no—
195#235—24no—
196#235—24no—
197#235—42no—
198#235—42no—
199#235—42no—
200#235—42no—
201#235C22⋊C418yes—
202#235C22×C414yes—
203#235C2×D414yes—
204C2×D12—21no—
205C2×D12—22no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12C22⋊C414yes—
212C2×D12C22×C412yes—
213C2×D12C2×D412yes—
214C2×C4×D3—24no—
215C2×C4×D3—24no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—24no—
219C2×C4×D3—41no—
220C2×C4×D3—41no—
221C2×C4×D3—42no—
222C2×Dic6—21no—
223C2×Dic6—22no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6C22⋊C414yes—
230C2×Dic6C22×C412yes—
231C2×Dic6C2×D412yes—
232C3×M16—24no—
233C3×M16—24no—
234C3×M16—22no—
235C3×M16—22no—
236C3×M16C22×C414yes—
237C3×M16C2×D414yes—
238C2×C3×C8—24no—
239C2×C3×C8—24no—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—41no—
246#208—24no—
247#208—24no—
248#208—22no—
249#208—22no—
250#208—42no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—42no—
255C2×C3⋊C8C2×D414yes—
256Dic12—28no—
257D24—28no—
258#204—28no—
259#204—28no—
260ΓL(1,𝔽25)—216no—
261D3×C8—216no—
262C2×#41—34no—
263C2×#41—34no—
264C2×#41—34no—
265C2×#41—32no—
266C2×#41—32no—
267C2×#41—62no—
268C2×#41—62no—
269C2×#41—61no—
270C2×#41—61no—
271C2×#41—62no—
272C2×#41—61no—
273C2×#41—61no—
274C2×#41—62no—
275C2×#41—61no—
276C2×#41—61no—
277C2×#41—62no—
278C22×Q8—31no—
279C22×Q8—31no—
280C22×D4—31no—
281C22×D4—31no—
282C22×D4—61no—
283C22×D4—61no—
284C23×C4—21no—
285C23×C4—61no—
286C23×C4—61no—
287C23×C4C22×D311yescharacteristic
288#137—38no—
289#137—68no—
290#137—64no—
291AGL(1,ℤ/8)—38no—
292AGL(1,ℤ/8)—68no—
293AGL(1,ℤ/8)—64no—
294#135—616no—
295C2×Q16—62no—
296C2×SD16—62no—
297C2×SD16—62no—
298C2×D8—62no—
299#131—616no—
300#131—68no—
301#131—68no—
302C2×M16—21no—
303C2×M16—22no—
304C2×M16—21no—
305C2×M16—22no—
306C2×M16—32no—
307C2×M16—32no—
308C2×M16—32no—
309C2×M16—64no—
310C2×M16—616no—
311C2×M16—62no—
312C2×M16—62no—
313C2×M16—61no—
314C2×M16—62no—
315C2×M16—62no—
316C2×M16C4×D314yes—
317C2×M16D1214yes—
318C2×M16C22×D312yes—
319C2×M16C22×D312yes—
320C22×C8—21no—
321C22×C8—21no—
322C22×C8—32no—
323C22×C8—64no—
324C22×C8—61no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C22×D312yes—
331#103—64no—
332#103—64no—
333#102—64no—
334#102—64no—
335#101—64no—
336#100—64no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3C4×D412yes—
340C23×C3#12111yescharacteristic
341C23×C3#12111yescharacteristic
342C23×C3#12312yes—
343C23×C3C2×#4111yescharacteristic
344C22×D3—22no—
345C22×D3—24no—
346C22×D3—22no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—81no—
350C22×D3—81no—
351C22×D3—81no—
352C22×D3—81no—
353C2×C3×C4—42no—
354C2×C3×C4—82no—
355C2×C3×C4—82no—
356C2×C3×C4—82no—
357C2×C3×C4C2×C22⋊C412yes—
358C2×C3×C4C2×C22⋊C412yes—
359C2×C3×C4C4×D412yes—
360C2×C3×C4C4×D412yes—
361C2×C3×C4C4×D412yes—
362C2×C3×C4#12012yes—
363C2×C3×C4#12012yes—
364C2×C3×C4#12111yescharacteristic
365C2×C3×C4#12111yescharacteristic
366C2×C3×C4#12112yes—
367C2×C3×C4#12112yes—
368C2×C3×C4#12312yes—
369C2×C3×C4C23×C411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375#67—24no—
376#67—24no—
377#67—28no—
378#67—44no—
379#67—44no—
380C2×Dic3—22no—
381C2×Dic3—24no—
382C2×Dic3—22no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385D12—22no—
386D12—22no—
387D12—22no—
388D12—24no—
389D12—41no—
390D12—41no—
391D12C2×C22⋊C414yes—
392C4×D3—24no—
393C4×D3—24no—
394C4×D3—28no—
395C4×D3—44no—
396C4×D3—44no—
397Dic6—22no—
398Dic6—24no—
399Dic6—22no—
400Dic6—22no—
401Dic6—41no—
402Dic6—41no—
403Dic6C2×C22⋊C414yes—
404C3×C8—24no—
405C3×C8—24no—
406C3⋊C8—24no—
407C3⋊C8—42no—
408C24—61no—
409C24—61no—
410C24#23511yescharacteristic
411#41—38no—
412#41—34no—
413#41—34no—
414#41—64no—
415#41—64no—
416#41—64no—
417#41—64no—
418#41—64no—
419#41—62no—
420#41—62no—
421#41—64no—
422#41—64no—
423#41—64no—
424#41—64no—
425#41—122no—
426#41—122no—
427#41—122no—
428#41—122no—
429C2×Q8—34no—
430C2×Q8—32no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—62no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×D4—34no—
440C2×D4—32no—
441C2×D4—32no—
442C2×D4—64no—
443C2×D4—62no—
444C2×D4—64no—
445C2×D4—62no—
446C2×D4—64no—
447C2×D4—61no—
448C2×D4—62no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—121no—
455C2×D4—121no—
456C2×D4—122no—
457C22×C4—22no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—61no—
461C22×C4—64no—
462C22×C4—62no—
463C22×C4—64no—
464C22×C4—62no—
465C22×C4—61no—
466C22×C4—64no—
467C22×C4—82no—
468C22×C4—121no—
469C22×C4—121no—
470C22×C4—122no—
471C22×C4C2×C4×D312yes—
472C22×C4C2×D1212yes—
473C22×C4#23511yescharacteristic
474C22×C4D3×D412yes—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4D3×D411yescharacteristic
478C22×C4D3×D411yescharacteristic
479C22×C4#23912yes—
480C22×C4C23×D311yescharacteristic
481Q16—68no—
482SD16—68no—
483SD16—68no—
484D8—68no—
485M16—24no—
486M16—24no—
487M16—22no—
488M16—22no—
489M16—616no—
490M16—64no—
491M16—64no—
492M16—62no—
493M16—62no—
494M16—122no—
495M16C2×C4×D314yes—
496M16C2×D1214yes—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—34no—
502C2×C8—616no—
503C2×C8—62no—
504C2×C8—62no—
505C2×C8—62no—
506C2×C8—122no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3#38612yes—
520C22×C3#38612yes—
521C22×C3C2×C4×D411yescharacteristic
522C22×C3C2×#12111yescharacteristic
523C22×C3C2×#12311yescharacteristic
524C2×D3—24no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—84no—
529C2×D3—82no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—82no—
533C3×C4—82no—
534C3×C4#38612yes—
535C3×C4#38612yes—
536C3×C4C22×C22⋊C411yescharacteristic
537C3×C4C2×C4×D411yescharacteristic
538C3×C4C2×#12011yescharacteristic
539C3×C4C2×#12111yescharacteristic
540Dic3—24no—
541Dic3—42no—
542C23—41no—
543C23—41no—
544C23—62no—
545C23—64no—
546C23—62no—
547C23—121no—
548C23—121no—
549C23—241no—
550C23—241no—
551C23—241no—
552C23—241no—
553C23C4×D1212yes—
554C23#88411yescharacteristic
555C23#88411yescharacteristic
556C23#89412yes—
557C23C2×#23511yescharacteristic
558Q8—34no—
559Q8—64no—
560Q8—62no—
561Q8—62no—
562Q8—62no—
563Q8—121no—
564Q8—121no—
565D4—34no—
566D4—68no—
567D4—64no—
568D4—64no—
569D4—64no—
570D4—62no—
571D4—62no—
572D4—62no—
573D4—124no—
574D4—124no—
575D4—121no—
576D4—121no—
577C2×C4—42no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—68no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—62no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—124no—
590C2×C4—124no—
591C2×C4C4×D1212yes—
592C2×C4D3×C22⋊C412yes—
593C2×C4D3×C22⋊C412yes—
594C2×C4#88212yes—
595C2×C4#88212yes—
596C2×C4#88411yescharacteristic
597C2×C4#88411yescharacteristic
598C2×C4#89312yes—
599C2×C4#89312yes—
600C2×C4#89412yes—
601C2×C4#89512yes—
602C2×C4#89512yes—
603C2×C4C22×C4×D311yescharacteristic
604C2×C4C22×D1211yescharacteristic
605C2×C4C2×D3×D411yescharacteristic
606C2×C4C2×D3×D411yescharacteristic
607C2×C4C2×D3×D411yescharacteristic
608C2×C4C2×#23911yescharacteristic
609C8—24no—
610C8—24no—
611C8—64no—
612C8—122no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—22no—
616C2×C3—22no—
617C2×C3C2×#38611yescharacteristic
618C2×C3#306811yescharacteristic
619C2×C3#306911yescharacteristic
620D3—44no—
621D3—82no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—42no—
629C22—42no—
630C22—64no—
631C22—122no—
632C22—122no—
633C22—122no—
634C22—244no—
635C22—242no—
636C22—242no—
637C22—241no—
638C22—241no—
639C22#466512yes—
640C22#466512yes—
641C22C2×C4×D1211yescharacteristic
642C22C2×#88411yescharacteristic
643C22C2×#89411yescharacteristic
644C4—64no—
645C4—82no—
646C4—122no—
647C4#466512yes—
648C4#466512yes—
649C4C2×D3×C22⋊C411yescharacteristic
650C4C2×#88211yescharacteristic
651C4C2×#89311yescharacteristic
652C4C2×#89511yescharacteristic
653C3#2100411yescharacteristic, complemented, Sylow
654C2—22no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—124no—
659C2—242no—
660C2C2×#466511yescharacteristic
661C2#7870311yescharacteristic
662C2#7871211yescharacteristic
subgroupG/H
centerC22C2×#884
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#343C2
socleC22×C3C2×#121
prime radicalC2×#5748C2
cosocleC22×C3×C4C24
hypercenterC2×#343D3
hypocenterC3#21004
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#5745C2semidirect
3. C2×#5132C2semidirect
4. C2×#4966C2semidirect
5. C2×#4966C2semidirect
6. C2×C3×#343C2semidirect
7. #71994C2semidirect
8. #69007C2semidirect
9. #69006C2semidirect
19. #5748C4semidirect
30. C2×#824C22semidirect
32. C2×#823C22semidirect
34. C2×#810C22semidirect
40. C2×#809C22semidirect
41. C3×#343C22semidirect
42. #4558C22semidirect
50. C2×#343D3semidirect
106. #909C2×C4semidirect
109. #908C2×C4semidirect
157. #343C2×D3semidirect
653. C3#21004semidirect

Maximal subgroups

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

subgroupindex
C2×#57482
C2×#57452
C2×#51322
C2×#49662
C2×C3×#3432
#719942
#690072
#690062
#210043