Finite Group Databasebeta

← C2×#69526 #10822643

Subgroups — C2×#69526 #10822643

Order 768.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#924C211yesmaximal, characteristic
2C2×C3×#380C211yesmaximal, characteristic
3C22×#830C211yesmaximal, characteristic
4C22×#810C211yesmaximal, characteristic, complemented
5C22×#810C211yesmaximal, characteristic, complemented
6C22×#808C211yesmaximal, characteristic, complemented
7C2×#4529C211yesmaximal, characteristic, complemented
8#69526C218yesmaximal, direct product factor
9C2×#2035—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C23×Dic6C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4—21no—
15C22×#217—21no—
16C22×C4⋊Dic3C2211yescharacteristic
17C2×#924—22no—
18C2×#924—24no—
19C2×#924—24no—
20C2×#924C2214yes—
21C22×Dic3⋊C4—21no—
22C23×C3⋊C8C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C3×#380—22no—
25C3×#380C2214yes—
26C2×C3×#95—21no—
27C2×#830—24no—
28C2×#830C2214yes—
29C2×#810—24no—
30C2×#810—24no—
31C2×#810C2214yescomplemented
32C2×#810C2214yescomplemented
33C2×#808—24no—
34C2×#808C2214yescomplemented
35#4529C2214yescomplemented
36C22×C22⋊Q8—31no—
37C22×#106—31no—
38C22×#103—31no—
39C22×#103—31no—
40C22×C22⋊C8—31no—
41C2×#380D311yescharacteristic
42#2035—38no—
43C2×#340—31no—
44C25×C3D411yescharacteristic
45C23×C3×C4—22no—
46C23×C3×C4—22no—
47C23×C3×C4—22no—
48C23×C3×C4—22no—
49C23×C3×C4—22no—
50C23×C3×C4—22no—
51C23×C3×C4—21no—
52C23×C3×C4—21no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D412yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3—21no—
58C23×Dic3—21no—
59C22×Dic6—22no—
60C22×Dic6—22no—
61C22×Dic6—22no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—21no—
65C22×Dic6D414yes—
66C22×Dic6C2312yes—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—44no—
77C2×#217—28no—
78C2×#217—24no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3D414yes—
82C2×C4⋊Dic3C2312yes—
83#924—216no—
84#924—28no—
85#924—28no—
86C2×Dic3⋊C4—28no—
87C2×Dic3⋊C4—24no—
88C22×C3⋊C8—22no—
89C22×C3⋊C8—22no—
90C22×C3⋊C8—22no—
91C22×C3⋊C8—22no—
92C22×C3⋊C8—22no—
93C22×C3⋊C8D414yes—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2312yes—
96C3×#95—24no—
97#830—28no—
98#810—28no—
99#810—28no—
100#808—28no—
101C23×Q8—31no—
102C24×C4C2×D311yescharacteristic
103C23×C8—31no—
104C23×C8—31no—
105C2×C22⋊Q8—34no—
106C2×C22⋊Q8—64no—
107C2×C22⋊Q8—64no—
108C2×C22⋊Q8—62no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—61no—
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—61no—
115C2×#106—34no—
116C2×#106—64no—
117C2×#103—34no—
118C2×#103—34no—
119C2×#103—64no—
120C2×#103—64no—
121C2×C22⋊C8—34no—
122C2×C22⋊C8—64no—
123#380—22no—
124#380C2×D314yes—
125C2×#95—21no—
126#340—34no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3SD1614yes—
134C24×C3C2×D412yes—
135C24×C3C2×D411yescharacteristic
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—24no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—44no—
151C22×C3×C4—44no—
152C22×C3×C4—44no—
153C22×C3×C4—44no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4112yes—
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—24no—
170C22×Dic3—24no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—42no—
174C22×Dic3—42no—
175C22×Dic3—42no—
176C22×Dic3—42no—
177C2×Dic6—28no—
178C2×Dic6—28no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—24no—
184C2×Dic6—22no—
185C2×Dic6—44no—
186C2×Dic6—42no—
187C2×Dic6—42no—
188C2×Dic6—44no—
189C2×Dic6C2×D414yes—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C22⋊C4—48no—
197#217—216no—
198C4⋊Dic3—24no—
199C4⋊Dic3—22no—
200C4⋊Dic3C2×D414yes—
201Dic3⋊C4—216no—
202C2×C3⋊C8—28no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—44no—
209C2×C3⋊C8C2×D414yes—
210C25#6711yescharacteristic
211C22×Q8—34no—
212C22×Q8—32no—
213C22×Q8—62no—
214C22×Q8—61no—
215C22×Q8—62no—
216C22×Q8—62no—
217C22×Q8—62no—
218C22×Q8—62no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—61no—
228C23×C4—61no—
229C23×C4#6712yes—
230C23×C4#6711yescharacteristic
231C23×C4C22×D312yes—
232C23×C4C22×D311yescharacteristic
233C22×C8—34no—
234C22×C8—32no—
235C22×C8—32no—
236C22×C8—62no—
237C22×C8—62no—
238C22×C8—62no—
239C22×C8—62no—
240C22×C8—62no—
241C22⋊Q8—68no—
242C22⋊Q8—68no—
243C22⋊Q8—616no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—68no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4C22×D312yes—
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—68no—
261C2×C22⋊C4—64no—
262#106—68no—
263#103—68no—
264#103—68no—
265C22⋊C8—68no—
266#95—24no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—28no—
274C23×C3—28no—
275C23×C3—28no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—44no—
279C23×C3—44no—
280C23×C3#12012yes—
281C23×C3#12112yes—
282C23×C3#12312yes—
283C23×C3C2×SD1614yes—
284C23×C3C2×SD1612yes—
285C23×C3#13512yes—
286C23×C3#13712yes—
287C23×C3#13712yes—
288C23×C3C22×D411yescharacteristic
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—44no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—48no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—84no—
305C2×C3×C4—84no—
306C2×C3×C4#12012yes—
307C2×C3×C4#12114yes—
308C2×C3×C4#12114yes—
309C2×C3×C4#12112yes—
310C2×C3×C4#12314yes—
311C2×C3×C4#12312yes—
312C2×C3×C4#12414yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—48no—
326C2×Dic3—48no—
327Dic6—24no—
328Dic6—24no—
329Dic6—42no—
330Dic6—42no—
331Dic6—48no—
332C3⋊C8—24no—
333C3⋊C8—42no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24#21414yes—
341C24C2×#6712yes—
342C24C2×#6711yescharacteristic
343C2×Q8—34no—
344C2×Q8—64no—
345C2×Q8—62no—
346C2×Q8—68no—
347C2×Q8—68no—
348C2×Q8—62no—
349C2×Q8—62no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—124no—
353C2×Q8—124no—
354C2×Q8—122no—
355C2×Q8—122no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—64no—
381C22×C4—64no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—122no—
385C22×C4—122no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4#23512yes—
389C22×C4#23512yes—
390C22×C4D3×D412yes—
391C22×C4D3×D412yes—
392C22×C4D3×D412yes—
393C22×C4#23712yes—
394C22×C4C2×#6712yes—
395C22×C4C2×#6711yescharacteristic
396C22×C4C23×D311yescharacteristic
397C2×C8—34no—
398C2×C8—68no—
399C2×C8—62no—
400C2×C8—62no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—124no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—34no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—616no—
413C4⋊C4—64no—
414C4⋊C4—62no—
415C22⋊C4—48no—
416C22⋊C4—616no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—416no—
425C22×C3#39414yes—
426C22×C3#44814yes—
427C22×C3#45214yes—
428C22×C3#46514yes—
429C22×C3#47014yes—
430C22×C3#48314yes—
431C22×C3#48514yes—
432C22×C3C2×#12011yescharacteristic
433C22×C3C2×#12111yescharacteristic
434C22×C3C2×#12311yescharacteristic
435C22×C3C22×SD1611yescharacteristic
436C22×C3C2×#13511yescharacteristic
437C22×C3C2×#13711yescharacteristic
438C22×C3C2×#13711yescharacteristic
439C3×C4—42no—
440C3×C4—82no—
441C3×C4—82no—
442C3×C4#39414yes—
443C3×C4C2×#12111yescharacteristic
444C3×C4C2×#12111yescharacteristic
445C3×C4C2×#12311yescharacteristic
446C3×C4C2×#12411yescharacteristic
447Dic3—44no—
448Dic3—44no—
449C23—28no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—28no—
454C23—22no—
455C23—28no—
456C23—22no—
457C23—22no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23#92912yes—
463C23C2×#21414yes—
464C23C2×#21412yes—
465C23#93712yes—
466C23#93912yes—
467C23#94212yes—
468C23#95012yes—
469C23#95112yes—
470C23C22×#6711yescharacteristic
471Q8—64no—
472Q8—64no—
473Q8—128no—
474Q8—122no—
475Q8—122no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—64no—
492C2×C4—64no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4—128no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—128no—
500C2×C4—122no—
501C2×C4#88314yes—
502C2×C4#88414yes—
503C2×C4#88414yes—
504C2×C4#88514yes—
505C2×C4#92912yes—
506C2×C4#93712yes—
507C2×C4#93912yes—
508C2×C4C2×#23511yescharacteristic
509C2×C4C2×#23511yescharacteristic
510C2×C4C2×D3×D411yescharacteristic
511C2×C4C2×D3×D411yescharacteristic
512C2×C4C2×D3×D411yescharacteristic
513C2×C4C2×#23711yescharacteristic
514C8—64no—
515C8—122no—
516C2×C3—44no—
517C2×C3#203518yescomplemented
518C2×C3C2×#39411yescharacteristic
519C2×C3C2×#44811yescharacteristic
520C2×C3C2×#45211yescharacteristic
521C2×C3C2×#46511yescharacteristic
522C2×C3C2×#47011yescharacteristic
523C2×C3C2×#48311yescharacteristic
524C2×C3C2×#48511yescharacteristic
525C22—24no—
526C22—24no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—416no—
532C22—42no—
533C22#495914yes—
534C22#497114yes—
535C22#499614yes—
536C22#504014yes—
537C22#504114yes—
538C22#504714yes—
539C22#505214yes—
540C22C2×#92911yescharacteristic
541C22C22×#21411yescharacteristic
542C22C2×#93711yescharacteristic
543C22C2×#93911yescharacteristic
544C22C2×#94211yescharacteristic
545C22C2×#95011yescharacteristic
546C22C2×#95111yescharacteristic
547C4—42no—
548C4—82no—
549C4—82no—
550C4—124no—
551C4—124no—
552C4#495914yes—
553C4C2×#88311yescharacteristic
554C4C2×#88411yescharacteristic
555C4C2×#88411yescharacteristic
556C4C2×#88511yescharacteristic
557C3C2×#203511yescharacteristic, complemented, Sylow
558C2—44no—
559C2#6952618yesdirect product factor
560C2C2×#495911yescharacteristic
561C2C2×#497111yescharacteristic
562C2C2×#499611yescharacteristic
563C2C2×#504111yescharacteristic
564C2C2×#504011yescharacteristic
565C2C2×#505211yescharacteristic
566C2C2×#504711yescharacteristic
subgroupG/H
centerC24C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#380C2
socleC24×C3C2×D4
prime radicalC25×C3D4
hypercenterC2×#380D3
hypocenterC3C2×#2035
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. C22×#810C2semidirect
5. C22×#810C2semidirect
6. C22×#808C2semidirect
7. C2×#4529C2semidirect
8. #69526C2direct (N × H)
31. C2×#810C22semidirect
32. C2×#810C22semidirect
34. C2×#808C22semidirect
35. #4529C22semidirect
517. C2×C3#2035semidirect
557. C3C2×#2035semidirect
559. C2#69526direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9242
C2×C3×#3802
C22×#8302
C22×#8102
C22×#8082
C2×#45292
#695262
C2×#20353