Finite Group Databasebeta

← #10797136

Subgroups — #10797136

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×C3×#568C211yesmaximal, characteristic, complemented
2C2×#5822C211yesmaximal, characteristic
3C2×#5232C211yesmaximal, characteristic, complemented
4C2×#5133C211yesmaximal, characteristic, complemented
5C2×#5132C211yesmaximal, characteristic, complemented
6#72019C214yesmaximal, complemented
7#72018C214yesmaximal, complemented
8#71998C211yesmaximal, characteristic, complemented
9C2×#4554C211yesmaximal, characteristic, complemented
10#21004—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C3×#568—24no—
14C3×#568—24no—
15C2×C3×#131—22no—
16C2×C3×#131C2214yes—
17C22×C3×M16—22no—
18C22×C3×M16C2211yescharacteristic
19C22×C3×M16C2211yescharacteristic
20C2×#951C2212yes—
21#5822—24no—
22#5822C418yescomplemented
23C2×#950—22no—
24C2×#949C2212yes—
25C2×#942—21no—
26C2×#932—21no—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×#208C2211yescharacteristic
29C22×#208C2211yescharacteristic
30#5232—22no—
31#5232C2214yes—
32C2×#835—21no—
33C2×#832—21no—
34C2×#824C2212yescomplemented
35#5133—22no—
36#5133—22no—
37#5132—24no—
38C2×#823C2212yescomplemented
39C2×#821C2212yescomplemented
40C2×#816C2212yescomplemented
41#4554C2214yescomplemented
42#4552C2214yescomplemented
43C2×#575—31no—
44C2×#568D311yescharacteristic, complemented
45C2×#417—31no—
46C2×#417—31no—
47C2×#413—31no—
48#1908—34no—
49#1907—34no—
50C2×#343—31no—
51#1738—31no—
52C2×C3×#41—21no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—21no—
56C2×C3×#41C2×C414yes—
57C2×C3×#41D412yes—
58C2×C3×#41D414yes—
59C2×C3×#41C2312yes—
60C22×C3×Q8D411yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4D411yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C22×#67—21no—
66C23×Dic3—21no—
67C2×#235—22no—
68C2×#235—21no—
69C2×#235—21no—
70C2×#235—21no—
71C2×#235—21no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235C2×C414yes—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×Dic6C2311yescharacteristic
78C3×#131—216no—
79C3×#131—28no—
80C3×#131—28no—
81C2×C3×M16—216no—
82C2×C3×M16—21no—
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C2×C3×M16—22no—
86C2×C3×M16—24no—
87C2×C3×M16D414yes—
88C2×C3×M16D412yes—
89C2×C3×M16D412yes—
90C2×C3×M16C2312yes—
91C22×C3×C8—21no—
92C22×C3×C8—24no—
93C22×C3×C8C2312yes—
94#951—24no—
95#951C2×C418yescomplemented
96#950—216no—
97#949—24no—
98#949C2×C418yescomplemented
99C2×#216—22no—
100#942—28no—
101C2×#215—22no—
102C2×#214—22no—
103#932—28no—
104C2×#213—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2312yes—
111C2×#208—21no—
112C2×#208—22no—
113C2×#208—21no—
114C2×#208—22no—
115C2×#208—22no—
116C2×#208C2×C414yes—
117C2×#208C2312yes—
118C2×#208C2312yes—
119C22×C3⋊C8—21no—
120C22×C3⋊C8—21no—
121#835—24no—
122#832—24no—
123#824—24no—
124#823—24no—
125#821—24no—
126#816—24no—
127C22×#41—31no—
128C22×#41C2×D311yescharacteristic
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—24no—
137#568—24no—
138C2×#131—22no—
139C2×#131C2×D314yes—
140C22×M16—22no—
141C22×M16—31no—
142C22×M16—31no—
143C22×M16C2×D311yescharacteristic
144C22×M16C2×D311yescharacteristic
145C22×C4⋊C4—31no—
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#343—34no—
158#341—34no—
159C24×C3—21no—
160C24×C3#4111yescharacteristic
161C23×D3—21no—
162C3×#41—24no—
163C3×#41—24no—
164C3×#41—24no—
165C3×#41—24no—
166C3×#41—24no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41C22⋊C418yes—
170C3×#41C22×C414yes—
171C3×#41C2×D414yes—
172C2×C3×Q8—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8C22⋊C414yes—
175C2×C3×Q8C2×D412yes—
176C2×C3×D4—24no—
177C2×C3×D4—22no—
178C2×C3×D4—24no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4C22⋊C414yes—
183C2×C3×D4C2×D412yes—
184C22×C3×C4—21no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4C22×C412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D412yes—
196C22×C3×C4#4111yescharacteristic
197C22×C3×C4#4112yes—
198C22×C3×C4C2411yescharacteristic
199C2×#67—24no—
200C2×#67—41no—
201C2×#67—41no—
202C2×#67—42no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3—41no—
206C22×Dic3—41no—
207C22×Dic3—82no—
208#235—24no—
209#235—24no—
210#235—24no—
211#235—24no—
212#235—42no—
213#235—42no—
214#235—42no—
215#235—42no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12C22×C412yes—
222C2×C4×D3—24no—
223C2×C4×D3—41no—
224C2×C4×D3—41no—
225C2×C4×D3—42no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6C22×C412yes—
232C3×M16—216no—
233C3×M16—24no—
234C3×M16—24no—
235C3×M16—24no—
236C3×M16—22no—
237C3×M16C2×D414yes—
238C2×C3×C8—216no—
239C2×C3×C8—24no—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—22no—
243C2×C3×C8C2×D414yes—
244#216—28no—
245#215—28no—
246#214—28no—
247#213—28no—
248C4⋊Dic3—22no—
249C4⋊Dic3—22no—
250C4⋊Dic3—41no—
251C4⋊Dic3—41no—
252#208—24no—
253#208—22no—
254#208—22no—
255#208—22no—
256#208—42no—
257#208C22×C414yes—
258C2×C3⋊C8—24no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—42no—
262C2×#41—22no—
263C2×#41—21no—
264C2×#41—21no—
265C2×#41—22no—
266C2×#41—34no—
267C2×#41—62no—
268C2×#41—61no—
269C2×#41—61no—
270C2×#41—62no—
271C2×#41—62no—
272C2×#41—61no—
273C2×#41—61no—
274C2×#41C4×D314yes—
275C2×#41D1212yes—
276C2×#41#6714yes—
277C2×#41C22×D312yes—
278C22×Q8—31no—
279C22×Q8D1211yescharacteristic
280C22×D4—21no—
281C22×D4—31no—
282C22×D4—61no—
283C22×D4D1211yescharacteristic
284C23×C4—21no—
285C23×C4—61no—
286C23×C4—61no—
287C23×C4C22×D311yescharacteristic
288#137—38no—
289#137—64no—
290#137—68no—
291AGL(1,ℤ/8)—38no—
292AGL(1,ℤ/8)—64no—
293AGL(1,ℤ/8)—68no—
294#135—616no—
295C2×Q16—62no—
296C2×SD16—62no—
297C2×SD16—62no—
298C2×D8—62no—
299#131—216no—
300#131—28no—
301#131—28no—
302C2×M16—216no—
303C2×M16—21no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16—22no—
307C2×M16—24no—
308C2×M16—34no—
309C2×M16—32no—
310C2×M16—32no—
311C2×M16—61no—
312C2×M16—62no—
313C2×M16—61no—
314C2×M16—62no—
315C2×M16—62no—
316C2×M16#6714yes—
317C2×M16#6712yes—
318C2×M16#6712yes—
319C2×M16C22×D312yes—
320C22×C8—21no—
321C22×C8—24no—
322C22×C8—61no—
323C22×C8—61no—
324C22×C8C22×D312yes—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331#103—64no—
332#103—64no—
333#102—64no—
334#102—64no—
335#101—64no—
336#100—64no—
337C23×C3—22no—
338C23×C3—24no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—81no—
342C23×C3—81no—
343C23×C3C4×D412yes—
344C23×C3#12111yescharacteristic
345C23×C3#12111yescharacteristic
346C23×C3#12312yes—
347C23×C3C2×#4111yescharacteristic
348C22×D3—22no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—81no—
352C22×D3—81no—
353C3×Q8—22no—
354C3×Q8—24no—
355C3×Q8C2×C22⋊C414yes—
356C3×D4—24no—
357C3×D4—24no—
358C3×D4—28no—
359C3×D4—22no—
360C3×D4—24no—
361C3×D4C2×C22⋊C414yes—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—28no—
367C2×C3×C4C2×C22⋊C412yes—
368C2×C3×C4C2×C22⋊C412yes—
369C2×C3×C4C4×D412yes—
370C2×C3×C4C4×D412yes—
371C2×C3×C4C4×D412yes—
372C2×C3×C4#12012yes—
373C2×C3×C4#12012yes—
374C2×C3×C4#12112yes—
375C2×C3×C4#12112yes—
376C2×C3×C4#12111yescharacteristic
377C2×C3×C4#12111yescharacteristic
378C2×C3×C4#12312yes—
379C2×C3×C4C23×C411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385#67—44no—
386#67—44no—
387C2×Dic3—22no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—42no—
391C2×Dic3—82no—
392C2×Dic3—82no—
393C2×Dic3—82no—
394D12—22no—
395D12—22no—
396D12—41no—
397D12—41no—
398C4×D3—44no—
399C4×D3—44no—
400Dic6—22no—
401Dic6—22no—
402Dic6—41no—
403Dic6—41no—
404C3×C8—24no—
405C3×C8—24no—
406C3⋊C8—24no—
407C3⋊C8—42no—
408C24—21no—
409C24—61no—
410C24#23511yescharacteristic
411#41—24no—
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—22no—
417#41—22no—
418#41—64no—
419#41—64no—
420#41—64no—
421#41—64no—
422#41—122no—
423#41—122no—
424#41—122no—
425#41—122no—
426#41(C2×D3)⋊C418yes—
427#41C2×C4×D314yes—
428#41C2×#6714yes—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8(C2×D3)⋊C414yes—
438C2×Q8C2×D1212yes—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—32no—
446C2×D4—64no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—122no—
453C2×D4—121no—
454C2×D4—121no—
455C2×D4(C2×D3)⋊C414yes—
456C2×D4C2×D1212yes—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—61no—
463C22×C4—64no—
464C22×C4—62no—
465C22×C4—122no—
466C22×C4—121no—
467C22×C4—121no—
468C22×C4—121no—
469C22×C4—121no—
470C22×C4—242no—
471C22×C4C2×C4×D312yes—
472C22×C4C2×D1211yescharacteristic
473C22×C4C2×D1211yescharacteristic
474C22×C4#23511yescharacteristic
475C22×C4#23512yes—
476C22×C4C2×#6712yes—
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C2×#6712yes—
480C22×C4C23×D311yescharacteristic
481Q16—68no—
482SD16—68no—
483SD16—68no—
484D8—68no—
485M16—216no—
486M16—24no—
487M16—24no—
488M16—24no—
489M16—22no—
490M16—34no—
491M16—64no—
492M16—62no—
493M16—62no—
494M16—62no—
495M16—122no—
496M16C2×#6714yes—
497C2×C8—216no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—64no—
503C2×C8—62no—
504C2×C8—62no—
505C2×C8—122no—
506C2×C8C2×#6714yes—
507C4⋊C4—62no—
508C4⋊C4—62no—
509C4⋊C4—121no—
510C4⋊C4—121no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—24no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—84no—
523C22×C3#38612yes—
524C22×C3#38612yes—
525C22×C3C2×C4×D411yescharacteristic
526C22×C3C2×#12111yescharacteristic
527C22×C3C2×#12311yescharacteristic
528C2×D3—42no—
529C2×D3—82no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—82no—
533C3×C4—24no—
534C3×C4#38612yes—
535C3×C4#38612yes—
536C3×C4C22×C22⋊C411yescharacteristic
537C3×C4C2×C4×D411yescharacteristic
538C3×C4C2×#12011yescharacteristic
539C3×C4C2×#12111yescharacteristic
540Dic3—42no—
541Dic3—82no—
542C23—22no—
543C23—24no—
544C23—41no—
545C23—41no—
546C23—62no—
547C23—81no—
548C23—81no—
549C23—121no—
550C23—121no—
551C23—241no—
552C23—241no—
553C23C4×#6712yes—
554C23#92912yes—
555C23#93011yescharacteristic
556C23#93011yescharacteristic
557C23C2×#23511yescharacteristic
558Q8—24no—
559Q8—22no—
560Q8—62no—
561Q8—62no—
562Q8—121no—
563Q8—121no—
564Q8C2×(C2×D3)⋊C414yes—
565D4—28no—
566D4—24no—
567D4—24no—
568D4—24no—
569D4—22no—
570D4—62no—
571D4—62no—
572D4—124no—
573D4—124no—
574D4—121no—
575D4—121no—
576D4C2×(C2×D3)⋊C414yes—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—62no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—124no—
586C2×C4—124no—
587C2×C4—122no—
588C2×C4—242no—
589C2×C4—242no—
590C2×C4—242no—
591C2×C4C2×(C2×D3)⋊C412yes—
592C2×C4C2×(C2×D3)⋊C412yes—
593C2×C4C4×#6712yes—
594C2×C4C4×#6712yes—
595C2×C4C4×#6712yes—
596C2×C4#92912yes—
597C2×C4#93011yescharacteristic
598C2×C4#93011yescharacteristic
599C2×C4#93012yes—
600C2×C4#93012yes—
601C2×C4#95312yes—
602C2×C4#95312yes—
603C2×C4C22×C4×D311yescharacteristic
604C2×C4C22×D1211yescharacteristic
605C2×C4C2×#23511yescharacteristic
606C2×C4C22×#6711yescharacteristic
607C2×C4C22×#6711yescharacteristic
608C2×C4C22×#6711yescharacteristic
609C8—24no—
610C8—24no—
611C8—64no—
612C8—122no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—22no—
616C2×C3—22no—
617C2×C3—44no—
618C2×C3C2×#38611yescharacteristic
619C2×C3#306811yescharacteristic
620C2×C3#306911yescharacteristic
621D3—82no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—84no—
634C22—122no—
635C22—242no—
636C22—242no—
637C22—241no—
638C22—241no—
639C22#521312yes—
640C22#521312yes—
641C22C2×C4×#6711yescharacteristic
642C22C2×#92911yescharacteristic
643C22C2×#93011yescharacteristic
644C4—24no—
645C4—122no—
646C4—242no—
647C4#521312yes—
648C4#521312yes—
649C4C22×(C2×D3)⋊C411yescharacteristic
650C4C2×C4×#6711yescharacteristic
651C4C2×#93011yescharacteristic
652C4C2×#95311yescharacteristic
653C3#2100411yescharacteristic, complemented, Sylow
654C2—22no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—44no—
659C2—242no—
660C2#8381411yescharacteristic
661C2#8381611yescharacteristic
662C2C2×#521311yescharacteristic
subgroupG/H
centerC22C2×#930
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#568C2
socleC22×C3C2×#121
prime radicalC2×#5822C2
cosocleC22×C3×C4C24
hypercenterC2×#568D3
hypocenterC3#21004
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
1. C2×C3×#568C2semidirect
3. C2×#5232C2semidirect
4. C2×#5133C2semidirect
5. C2×#5132C2semidirect
6. #72019C2semidirect
7. #72018C2semidirect
8. #71998C2semidirect
9. C2×#4554C2semidirect
22. #5822C4semidirect
34. C2×#824C22semidirect
38. C2×#823C22semidirect
39. C2×#821C22semidirect
40. C2×#816C22semidirect
41. #4554C22semidirect
42. #4552C22semidirect
44. C2×#568D3semidirect
95. #951C2×C4semidirect
98. #949C2×C4semidirect
653. C3#21004semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5682
C2×#58222
C2×#52322
C2×#51332
C2×#51322
#720192
#720182
#719982
C2×#45542
#210043