Finite Group Databasebeta

← C2×#70045 #10796725

Subgroups — C2×#70045 #10796725

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×#5832C211yesmaximal, characteristic
2C2×#5819C211yesmaximal, characteristic, complemented
3C2×C3×#410C211yesmaximal, characteristic, complemented
4C2×#5005C212yesmaximal, complemented
5C2×#4532C212yesmaximal, complemented
6#70045C218yesmaximal, direct product factor
7C2×#1884—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C22×#235C2211yescharacteristic
10C2×C3×#117C2212yes—
11#5832—22no—
12#5832—22no—
13#5832C414yescomplemented
14#5832C2214yes—
15#5819—22no—
16#5819—24no—
17#5819C2214yescomplemented
18C2×#948—24no—
19C2×#947C411yescharacteristic, complemented
20C2×#944C411yescharacteristic, complemented
21C2×#940C411yescharacteristic, complemented
22C2×#939—22no—
23C2×#936C411yescharacteristic, complemented
24C2×C4×#67—22no—
25C22×C4×Dic3C2211yescharacteristic
26C22×#208—22no—
27C22×#208C2212yes—
28C3×#410—22no—
29C3×#410C2214yescomplemented
30C2×C3×#99—22no—
31#5005—24no—
32#5005C2218yescomplemented
33C2×#805—24no—
34#4532C2218yescomplemented
35C2×#568—31no—
36C2×#532—31no—
37C2×#421—32no—
38C2×#410D311yescharacteristic, complemented
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×#117—22no—
69C3×#117—22no—
70C3×#117—22no—
71C3×#117C2314yes—
72C2×C3×C4⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42—22no—
75C2×#217—22no—
76#948—216no—
77#948—216no—
78#947—22no—
79#947C2×C414yescomplemented
80#944—22no—
81#944C2×C414yescomplemented
82#940—22no—
83#940C2×C414yescomplemented
84#939—28no—
85#939—28no—
86#936—22no—
87#936C2×C414yescomplemented
88C4×#67—28no—
89C4×#67—28no—
90C2×(C2×D3)⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3C2×C411yescharacteristic
96C2×C4×Dic3C2×C411yescharacteristic
97C2×C4×Dic3C2×C412yes—
98C2×C4×Dic3C2312yes—
99C2×#208—24no—
100C2×#208—28no—
101C2×#208—22no—
102C2×#208—28no—
103C2×#208—22no—
104C2×#208—22no—
105C2×#208D414yescomplemented
106C2×#208D414yescomplemented
107C2×#208C2314yes—
108C22×C3⋊C8—24no—
109C22×C3⋊C8—22no—
110C3×#99—28no—
111#805—216no—
112C22×#41—31no—
113C22×#41C2×D311yescharacteristic
114#568—34no—
115#568—64no—
116#568—62no—
117C2×#131—64no—
118C22×M16—32no—
119C22×M16—62no—
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×#117C2×D312yes—
131C22×C42—31no—
132#421—38no—
133#421—64no—
134C2×#104—64no—
135#410—22no—
136#410C2×D314yescomplemented
137C2×#99—22no—
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×C4—42no—
167C22×C3×C4C22⋊C412yes—
168C22×C3×C4C22⋊C414yes—
169C22×C3×C4C22×C411yescharacteristic
170C22×C3×C4C22×C411yescharacteristic
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4#4112yes—
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—28no—
181C2×#67—22no—
182C2×#67—22no—
183C2×#67—48no—
184C2×#67—42no—
185C2×#67—42no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—44no—
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×C4⋊C4—24no—
215C3×C4⋊C4—42no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—42no—
218C3×C42—24no—
219C3×C42—42no—
220#217—28no—
221(C2×D3)⋊C4—28no—
222Dic3⋊C4—28no—
223C4×Dic3—22no—
224C4×Dic3—21no—
225C4×Dic3—21no—
226C4×Dic3—22no—
227C4×Dic3C22×C412yes—
228C4×Dic3C22×C412yes—
229#208—216no—
230#208—24no—
231#208—24no—
232#208—24no—
233#208C2×D418yescomplemented
234C2×C3⋊C8—216no—
235C2×C3⋊C8—24no—
236C2×C3⋊C8—24no—
237C2×C3⋊C8—24no—
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×D4#6712yes—
261C23×C4—61no—
262C23×C4—61no—
263C23×C4D1212yes—
264C23×C4C22×D311yescharacteristic
265#131—616no—
266#131—616no—
267C2×M16—34no—
268C2×M16—34no—
269C2×M16—34no—
270C2×M16—64no—
271C2×M16—68no—
272C2×M16—62no—
273C2×M16—62no—
274C2×M16—68no—
275C2×M16—62no—
276C22×C8—64no—
277C22×C8—62no—
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—22no—
291#117—22no—
292#117—22no—
293#117C22×D314yes—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—62no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C42—22no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42—32no—
303C2×C42—32no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42—61no—
307#104—616no—
308#99—28no—
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×C4—44no—
336C2×C3×C4—84no—
337C2×C3×C4C2×C22⋊C414yes—
338C2×C3×C4C2×C22⋊C411yescharacteristic
339C2×C3×C4C2×C22⋊C411yescharacteristic
340C2×C3×C4C4×D412yes—
341C2×C3×C4C4×D412yes—
342C2×C3×C4C4×D412yes—
343C2×C3×C4C4×D412yes—
344C2×C3×C4#12012yes—
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4#12011yescharacteristic
347C2×C3×C4#12112yes—
348C2×C3×C4#12112yes—
349C2×C3×C4#12311yescharacteristic
350C2×C3×C4#12311yescharacteristic
351C2×C3×C4#12312yes—
352C2×C3×C4C23×C411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D412yes—
355C2×C3×C4C2×#4112yes—
356#67—416no—
357#67—44no—
358#67—44no—
359C2×Dic3—24no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—44no—
364C2×Dic3—44no—
365C2×Dic3—42no—
366C2×Dic3—41no—
367C2×Dic3—41no—
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×#6711yescharacteristic
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—62no—
413C2×D4—62no—
414C2×D4—61no—
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×#6714yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—124no—
435C22×C4—121no—
436C22×C4—121no—
437C22×C4—121no—
438C22×C4—121no—
439C22×C4(C2×D3)⋊C412yes—
440C22×C4(C2×D3)⋊C414yes—
441C22×C4C2×C4×D311yescharacteristic
442C22×C4C2×C4×D311yescharacteristic
443C22×C4C2×D1214yes—
444C22×C4C2×D1211yescharacteristic
445C22×C4C2×D1211yescharacteristic
446C22×C4D3×D412yes—
447C22×C4D3×D412yes—
448C22×C4#23712yes—
449C22×C4#23712yes—
450C22×C4C2×#6711yescharacteristic
451C22×C4C23×D311yescharacteristic
452M16—38no—
453M16—616no—
454M16—64no—
455M16—64no—
456M16—64no—
457C2×C8—616no—
458C2×C8—64no—
459C2×C8—64no—
460C2×C8—64no—
461C4⋊C4—24no—
462C4⋊C4—42no—
463C4⋊C4—68no—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—68no—
467C22⋊C4—68no—
468C42—24no—
469C42—32no—
470C42—32no—
471C42—42no—
472C42—62no—
473C42—61no—
474C42—61no—
475C42—62no—
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—84no—
496C3×C4#38612yes—
497C3×C4#38611yescharacteristic
498C3×C4#38611yescharacteristic
499C3×C4C2×C4×D412yes—
500C3×C4C2×#12011yescharacteristic
501C3×C4C2×#12311yescharacteristic
502Dic3—24no—
503Dic3—42no—
504Dic3—42no—
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#93912yes—
517C23#93912yes—
518C23C22×#6711yescharacteristic
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—44no—
539C2×C4—64no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—84no—
544C2×C4—124no—
545C2×C4—124no—
546C2×C4—122no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4#67⋊C412yes—
554C2×C4#67⋊C412yes—
555C2×C4#67⋊C412yes—
556C2×C4#67⋊C412yes—
557C2×C4#88212yes—
558C2×C4#88211yescharacteristic
559C2×C4#88211yescharacteristic
560C2×C4#88611yescharacteristic
561C2×C4#88611yescharacteristic
562C2×C4#88612yes—
563C2×C4C2×(C2×D3)⋊C414yes—
564C2×C4C2×(C2×D3)⋊C411yescharacteristic
565C2×C4C2×(C2×D3)⋊C411yescharacteristic
566C2×C4#93912yes—
567C2×C4#93912yes—
568C2×C4C22×C4×D311yescharacteristic
569C2×C4C22×D1211yescharacteristic
570C2×C4C2×D3×D412yes—
571C2×C4C2×#23712yes—
572C8—68no—
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#495811yescharacteristic
595C22#495812yes—
596C22#495811yescharacteristic
597C22C22×(C2×D3)⋊C411yescharacteristic
598C22C2×#93912yes—
599C4—24no—
600C4—64no—
601C4—84no—
602C4—122no—
603C4—122no—
604C4#495811yescharacteristic
605C4#495811yescharacteristic
606C4#495812yes—
607C4C2×#67⋊C412yes—
608C4C2×#88211yescharacteristic
609C4C2×#88611yescharacteristic
610C3C2×#188411yescharacteristic, complemented, Sylow
611C2—21no—
612C2—22no—
613C2—21no—
614C2—22no—
615C2—44no—
616C2—242no—
617C2#7004512yesdirect product factor
618C2C2×#495811yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#410C2
socleC22×C3#386
prime radicalC2×#5832C2
cosocleC22×C3×C4C24
hypercenterC2×#410D3
hypocenterC3C2×#1884
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#5819C2semidirect
3. C2×C3×#410C2semidirect
4. C2×#5005C2semidirect
5. C2×#4532C2semidirect
6. #70045C2direct (N × H)
13. #5832C4semidirect
17. #5819C22semidirect
19. C2×#947C4semidirect
20. C2×#944C4semidirect
21. C2×#940C4semidirect
23. C2×#936C4semidirect
29. C3×#410C22semidirect
32. #5005C22semidirect
34. #4532C22semidirect
38. C2×#410D3semidirect
79. #947C2×C4semidirect
81. #944C2×C4semidirect
83. #940C2×C4semidirect
87. #936C2×C4semidirect
105. C2×#208D4semidirect
106. C2×#208D4semidirect
136. #410C2×D3semidirect
233. #208C2×D4semidirect
578. C2×C3#1884semidirect
610. C3C2×#1884semidirect
617. C2#70045direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#58322
C2×#58192
C2×C3×#4102
C2×#50052
C2×#45322
#700452
C2×#18843