Finite Group Databasebeta

← C2×#72021 #10796727

Subgroups — C2×#72021 #10796727

Order 768.

1794 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#568C211yesmaximal, characteristic, complemented
2C2×#5832C211yesmaximal, characteristic
3C2×#5240C211yesmaximal, characteristic, complemented
4C2×#5139C212yesmaximal, complemented
5#72021C218yesmaximal, direct product factor
6C2×#4555C212yesmaximal, complemented
7C2×#1884—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C22×#235C2211yescharacteristic
10C3×#568—22no—
11C3×#568—24no—
12C3×#568C2214yescomplemented
13C2×C3×#131—24no—
14C22×C3×M16—22no—
15C22×C3×M16C2212yes—
16#5832—22no—
17#5832—22no—
18#5832C414yescomplemented
19#5832C2214yes—
20C2×#947C411yescharacteristic, complemented
21C2×#944C411yescharacteristic, complemented
22C2×#940C411yescharacteristic, complemented
23C2×#939—22no—
24C2×#936C411yescharacteristic, complemented
25C2×C4×#67—22no—
26C2×#926C2212yes—
27C22×C4×Dic3C2211yescharacteristic
28#5240—22no—
29#5240C2214yescomplemented
30C2×#834—22no—
31#5139—24no—
32#5139C2218yescomplemented
33C2×#825—24no—
34#4555C2218yescomplemented
35C2×#568D311yescharacteristic, complemented
36C2×#532—31no—
37C2×#421—32no—
38C2×#410—31no—
39#1884—38no—
40C2×#339—32no—
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—22no—
45C2×C3×#41C2×C412yes—
46C2×C3×#41D412yes—
47C2×C3×#41D412yes—
48C2×C3×#41C2312yes—
49C22×C3×Q8C2×C411yescharacteristic
50C22×C3×D4C2×C411yescharacteristic
51C22×C3×D4D412yes—
52C23×C3×C4D412yes—
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C22×#67—21no—
56C23×Dic3—21no—
57C2×#235—21no—
58C2×#235—21no—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235C2×C412yes—
64C2×#235C2312yes—
65C22×D12C2×C411yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6C2×C411yescharacteristic
68C3×#131—216no—
69C3×#131—216no—
70C2×C3×M16—24no—
71C2×C3×M16—28no—
72C2×C3×M16—22no—
73C2×C3×M16—28no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16D414yescomplemented
77C2×C3×M16D414yescomplemented
78C2×C3×M16C2314yes—
79C22×C3×C8—24no—
80C22×C3×C8—22no—
81C2×#217—22no—
82C2×#217—22no—
83#947—22no—
84#947C2×C414yescomplemented
85#944—22no—
86#944C2×C414yescomplemented
87#940—22no—
88#940C2×C414yescomplemented
89#939—28no—
90#939—28no—
91#936—22no—
92#936C2×C414yescomplemented
93C4×#67—28no—
94C4×#67—28no—
95C2×(C2×D3)⋊C4—22no—
96#926—22no—
97#926—22no—
98#926—22no—
99#926C2314yes—
100C2×C4⋊Dic3—22no—
101C2×Dic3⋊C4—22no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—22no—
106C2×C4×Dic3C2×C411yescharacteristic
107C2×C4×Dic3C2×C411yescharacteristic
108C2×C4×Dic3C2×C412yes—
109C2×C4×Dic3C2312yes—
110#834—28no—
111#825—216no—
112C22×#41—31no—
113C22×#41C2×D311yescharacteristic
114#568—22no—
115#568—24no—
116#568C2×D314yescomplemented
117C2×#131—24no—
118C22×M16—22no—
119C22×M16C2×D312yes—
120#532—34no—
121#532—34no—
122#532—62no—
123#532—62no—
124C2×C4⋊Q8—31no—
125C2×#127—31no—
126C2×#124—31no—
127C2×#124—31no—
128C2×#121—62no—
129C2×C4×D4—62no—
130C2×#117—32no—
131C22×C42—31no—
132#421—38no—
133#421—64no—
134C2×#104—64no—
135#410—34no—
136#410—62no—
137C2×#99—62no—
138#339—38no—
139C24×C3C22⋊C412yes—
140C24×C3C2×D411yescharacteristic
141C23×D3—21no—
142C3×#41—24no—
143C3×#41—24no—
144C3×#41—24no—
145C3×#41—24no—
146C3×#41—24no—
147C3×#41—24no—
148C3×#41—24no—
149C3×#41—24no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8C22⋊C412yes—
153C2×C3×Q8C22×C412yes—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4C22⋊C412yes—
159C2×C3×D4C22⋊C414yes—
160C2×C3×D4C22×C412yes—
161C2×C3×D4C2×D414yes—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C22⋊C412yes—
167C22×C3×C4C22⋊C414yes—
168C22×C3×C4C22×C411yescharacteristic
169C22×C3×C4C22×C411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4#4112yes—
177C22×C3×C4#4112yes—
178C22×C3×C4C2411yescharacteristic
179C2×#67—28no—
180C2×#67—22no—
181C2×#67—22no—
182C2×#67—48no—
183C2×#67—42no—
184C2×#67—42no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—44no—
191C22×Dic3—42no—
192#235—24no—
193#235—24no—
194#235—24no—
195#235—24no—
196#235—42no—
197#235—42no—
198#235—42no—
199#235—42no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—21no—
203C2×D12C22×C412yes—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—41no—
207C2×C4×D3—41no—
208C2×C4×D3—41no—
209C2×C4×D3—41no—
210C2×Dic6—22no—
211C2×Dic6—22no—
212C2×Dic6—21no—
213C2×Dic6C22×C412yes—
214C3×M16—216no—
215C3×M16—24no—
216C3×M16—24no—
217C3×M16—24no—
218C3×M16C2×D418yescomplemented
219C2×C3×C8—216no—
220C2×C3×C8—24no—
221C2×C3×C8—24no—
222C2×C3×C8—24no—
223#217—28no—
224#217—24no—
225#217—42no—
226(C2×D3)⋊C4—28no—
227C4⋊Dic3—24no—
228C4⋊Dic3—42no—
229Dic3⋊C4—28no—
230C4×Dic3—22no—
231C4×Dic3—21no—
232C4×Dic3—21no—
233C4×Dic3—22no—
234C4×Dic3—24no—
235C4×Dic3—42no—
236C4×Dic3C22×C412yes—
237C4×Dic3C22×C412yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—32no—
243C2×#41—32no—
244C2×#41—61no—
245C2×#41—61no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41C4×D312yes—
251C2×#41D1212yes—
252C2×#41#6712yes—
253C2×#41C22×D312yes—
254C22×Q8—31no—
255C22×Q8C4×D311yescharacteristic
256C22×D4—31no—
257C22×D4—62no—
258C22×D4—61no—
259C22×D4C4×D311yescharacteristic
260C22×D4D1212yes—
261C23×C4—61no—
262C23×C4—61no—
263C23×C4#6712yes—
264C23×C4C22×D311yescharacteristic
265#131—216no—
266#131—216no—
267C2×M16—24no—
268C2×M16—28no—
269C2×M16—22no—
270C2×M16—28no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16#6714yescomplemented
274C2×M16#6714yescomplemented
275C2×M16C22×D314yes—
276C22×C8—24no—
277C22×C8—22no—
278C4⋊Q8—34no—
279C4⋊Q8—62no—
280#127—34no—
281#127—62no—
282#124—34no—
283#124—34no—
284#124—62no—
285#124—62no—
286#121—68no—
287#121—68no—
288C4×D4—68no—
289C4×D4—68no—
290#117—34no—
291#117—62no—
292#117—62no—
293#117—62no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4—62no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—32no—
302C2×C42—32no—
303C2×C42—61no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42—62no—
307#104—616no—
308#99—68no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—42no—
312C23×C3—44no—
313C23×C3C2×C22⋊C411yescharacteristic
314C23×C3C2×C22⋊C414yes—
315C23×C3C2×C22⋊C411yescharacteristic
316C23×C3#12112yes—
317C23×C3#12112yes—
318C23×C3C22×D411yescharacteristic
319C22×D3—22no—
320C22×D3—44no—
321C22×D3—41no—
322C22×D3—41no—
323C3×Q8—24no—
324C3×Q8—24no—
325C3×D4—28no—
326C3×D4—28no—
327C3×D4—24no—
328C3×D4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4C2×C22⋊C414yes—
336C2×C3×C4C2×C22⋊C411yescharacteristic
337C2×C3×C4C2×C22⋊C411yescharacteristic
338C2×C3×C4C4×D412yes—
339C2×C3×C4C4×D412yes—
340C2×C3×C4C4×D412yes—
341C2×C3×C4C4×D412yes—
342C2×C3×C4#12012yes—
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4#12011yescharacteristic
345C2×C3×C4#12112yes—
346C2×C3×C4#12112yes—
347C2×C3×C4#12311yescharacteristic
348C2×C3×C4#12311yescharacteristic
349C2×C3×C4#12312yes—
350C2×C3×C4C23×C411yescharacteristic
351C2×C3×C4C22×D412yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C2×#4112yes—
354#67—416no—
355#67—44no—
356#67—44no—
357C2×Dic3—24no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—44no—
362C2×Dic3—44no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—44no—
367C2×Dic3—84no—
368D12—24no—
369D12—42no—
370C4×D3—42no—
371C4×D3—42no—
372C4×D3—42no—
373C4×D3—42no—
374Dic6—24no—
375Dic6—42no—
376C3×C8—28no—
377C24—61no—
378C24(C2×D3)⋊C412yes—
379C24C2×D1211yescharacteristic
380#41—24no—
381#41—24no—
382#41—24no—
383#41—24no—
384#41—24no—
385#41—24no—
386#41—24no—
387#41—24no—
388#41—64no—
389#41—64no—
390#41—64no—
391#41—64no—
392#41—122no—
393#41—122no—
394#41—122no—
395#41—122no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—32no—
399C2×Q8—61no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×Q8(C2×D3)⋊C412yes—
403C2×Q8C2×C4×D312yes—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—32no—
409C2×D4—68no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—61no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—128no—
416C2×D4—122no—
417C2×D4—122no—
418C2×D4(C2×D3)⋊C414yes—
419C2×D4(C2×D3)⋊C412yes—
420C2×D4C2×C4×D312yes—
421C2×D4C2×D1214yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—124no—
434C22×C4—121no—
435C22×C4—121no—
436C22×C4—121no—
437C22×C4—121no—
438C22×C4—122no—
439C22×C4(C2×D3)⋊C412yes—
440C22×C4(C2×D3)⋊C414yes—
441C22×C4C2×C4×D311yescharacteristic
442C22×C4C2×C4×D311yescharacteristic
443C22×C4C2×D1211yescharacteristic
444C22×C4#23512yes—
445C22×C4#23512yes—
446C22×C4C2×#6714yes—
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6712yes—
450C22×C4C2×#6712yes—
451C22×C4C23×D311yescharacteristic
452M16—216no—
453M16—24no—
454M16—24no—
455M16—24no—
456M16C2×#6718yescomplemented
457C2×C8—216no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C4⋊C4—68no—
462C4⋊C4—64no—
463C4⋊C4—122no—
464C22⋊C4—68no—
465C22⋊C4—68no—
466C22⋊C4—64no—
467C22⋊C4—122no—
468C42—32no—
469C42—32no—
470C42—62no—
471C42—61no—
472C42—61no—
473C42—62no—
474C42—64no—
475C42—122no—
476C22×C3—22no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—44no—
483C22×C3—44no—
484C22×C3—82no—
485C22×C3#38611yescharacteristic
486C22×C3#38612yes—
487C22×C3#38611yescharacteristic
488C22×C3C22×C22⋊C411yescharacteristic
489C22×C3C2×#12112yes—
490C2×D3—44no—
491C2×D3—42no—
492C2×D3—82no—
493C2×D3—82no—
494C3×C4—24no—
495C3×C4#38612yes—
496C3×C4#38611yescharacteristic
497C3×C4#38611yescharacteristic
498C3×C4C2×C4×D412yes—
499C3×C4C2×#12011yescharacteristic
500C3×C4C2×#12311yescharacteristic
501Dic3—24no—
502Dic3—42no—
503Dic3—42no—
504Dic3—84no—
505C23—22no—
506C23—22no—
507C23—42no—
508C23—44no—
509C23—62no—
510C23—124no—
511C23—121no—
512C23—121no—
513C23C2×(C2×D3)⋊C411yescharacteristic
514C23C2×(C2×D3)⋊C414yes—
515C23C2×(C2×D3)⋊C411yescharacteristic
516C23#93012yes—
517C23#93012yes—
518C23C22×D1211yescharacteristic
519Q8—24no—
520Q8—24no—
521Q8—64no—
522Q8—122no—
523D4—28no—
524D4—28no—
525D4—24no—
526D4—24no—
527D4—64no—
528D4—1216no—
529D4—124no—
530D4—124no—
531D4—122no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—64no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—124no—
543C2×C4—124no—
544C2×C4—122no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—124no—
552C2×C4—244no—
553C2×C4C2×(C2×D3)⋊C414yes—
554C2×C4C2×(C2×D3)⋊C411yescharacteristic
555C2×C4C2×(C2×D3)⋊C411yescharacteristic
556C2×C4C4×#6712yes—
557C2×C4C4×#6712yes—
558C2×C4C4×#6712yes—
559C2×C4C4×#6712yes—
560C2×C4#92911yescharacteristic
561C2×C4#92911yescharacteristic
562C2×C4#92912yes—
563C2×C4#93012yes—
564C2×C4#93012yes—
565C2×C4#95312yes—
566C2×C4#95311yescharacteristic
567C2×C4#95311yescharacteristic
568C2×C4C22×C4×D311yescharacteristic
569C2×C4C2×#23512yes—
570C2×C4C22×#6712yes—
571C2×C4C22×#6711yescharacteristic
572C8—28no—
573C2×C3—21no—
574C2×C3—22no—
575C2×C3—21no—
576C2×C3—22no—
577C2×C3—44no—
578C2×C3#188412yescomplemented
579C2×C3C2×#38611yescharacteristic
580D3—82no—
581C22—22no—
582C22—24no—
583C22—24no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—82no—
590C22—124no—
591C22—122no—
592C22—242no—
593C22—242no—
594C22#521311yescharacteristic
595C22#521312yes—
596C22#521311yescharacteristic
597C22C22×(C2×D3)⋊C411yescharacteristic
598C22C2×#93012yes—
599C4—24no—
600C4—64no—
601C4—122no—
602C4—122no—
603C4—244no—
604C4#521311yescharacteristic
605C4#521311yescharacteristic
606C4#521312yes—
607C4C2×C4×#6712yes—
608C4C2×#92911yescharacteristic
609C4C2×#95311yescharacteristic
610C3C2×#188411yescharacteristic, complemented, Sylow
611C2—21no—
612C2—22no—
613C2—21no—
614C2—22no—
615C2—44no—
616C2—242no—
617C2#7202112yesdirect product factor
618C2C2×#521311yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#568C2
socleC22×C3#386
prime radicalC2×#5832C2
cosocleC22×C3×C4C24
hypercenterC2×#568D3
hypocenterC3C2×#1884
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
1. C2×C3×#568C2semidirect
3. C2×#5240C2semidirect
4. C2×#5139C2semidirect
5. #72021C2direct (N × H)
6. C2×#4555C2semidirect
12. C3×#568C22semidirect
18. #5832C4semidirect
20. C2×#947C4semidirect
21. C2×#944C4semidirect
22. C2×#940C4semidirect
24. C2×#936C4semidirect
29. #5240C22semidirect
32. #5139C22semidirect
34. #4555C22semidirect
35. C2×#568D3semidirect
76. C2×C3×M16D4semidirect
77. C2×C3×M16D4semidirect
84. #947C2×C4semidirect
86. #944C2×C4semidirect
88. #940C2×C4semidirect
92. #936C2×C4semidirect
116. #568C2×D3semidirect
218. C3×M16C2×D4semidirect
273. C2×M16#67semidirect
274. C2×M16#67semidirect
456. M16C2×#67semidirect
578. C2×C3#1884semidirect
610. C3C2×#1884semidirect
617. C2#72021direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5682
C2×#58322
C2×#52402
C2×#51392
#720212
C2×#45552
C2×#18843