Finite Group Databasebeta

← C2×#70274 #10822641

Subgroups — C2×#70274 #10822641

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×#946C211yesmaximal, characteristic
2C22×C3×#103C211yesmaximal, characteristic, complemented
3C22×#835C211yesmaximal, characteristic, complemented
4C22×(C2×D3)⋊C8C211yesmaximal, characteristic
5C2×#4987C211yesmaximal, characteristic
6C22×#808C211yesmaximal, characteristic, complemented
7C2×#4551C211yesmaximal, characteristic, complemented
8#70274C218yesmaximal, direct product factor
9C2×#2035—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C23×C4×D3C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C2×#946—24no—
15C2×#946—22no—
16C2×#946—24no—
17C2×#946C2214yes—
18C22×(C2×D3)⋊C4—21no—
19C22×(C2×D3)⋊C4—21no—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4—21no—
23C23×C3⋊C8C2211yescharacteristic
24C2×C3×#103—24no—
25C2×C3×#103C2214yescomplemented
26C2×#835—24no—
27C2×#835C2214yescomplemented
28C2×C3⋊#95—21no—
29C2×(C2×D3)⋊C8—24no—
30C2×(C2×D3)⋊C8C2214yes—
31#4987—22no—
32#4987C2214yes—
33C2×#808—24no—
34C2×#808C2214yescomplemented
35#4551C2214yescomplemented
36C22×C22⋊Q8—31no—
37C22×#106—31no—
38C22×#103—31no—
39C22×#103D311yescharacteristic, complemented
40C22×C22⋊C8—31no—
41C2×#380—31no—
42#2035—38no—
43C2×#340—31no—
44C24×D3D411yescharacteristic
45C22×C3×Q8—22no—
46C22×C3×Q8—21no—
47C22×C3×Q8—22no—
48C22×C3×Q8—22no—
49C22×C3×Q8—22no—
50C22×C3×Q8—22no—
51C22×C3×Q8D414yes—
52C22×C3×Q8C2312yes—
53C23×C3×C4—21no—
54C23×C3×C4—21no—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3—21no—
57C23×Dic3—21no—
58C23×Dic3D411yescharacteristic
59C22×C4×D3—22no—
60C22×C4×D3—22no—
61C22×C4×D3—22no—
62C22×C4×D3—22no—
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3D412yes—
66C22×C4×D3C2312yes—
67C22×C3×C8—22no—
68C22×C3×C8—22no—
69C22×C3×C8—22no—
70C22×C3×C8C2312yes—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75#946—216no—
76#946—28no—
77#946—28no—
78C2×(C2×D3)⋊C4—28no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—44no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3D414yes—
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2312yes—
90C2×Dic3⋊C4—28no—
91C2×Dic3⋊C4—24no—
92C22×C3⋊C8—22no—
93C22×C3⋊C8—22no—
94C22×C3⋊C8D414yes—
95C22×C3⋊C8C2312yes—
96C3×#103—28no—
97#835—28no—
98C3⋊#95—24no—
99(C2×D3)⋊C8—28no—
100#808—28no—
101C23×Q8C2×D311yescharacteristic
102C24×C4—31no—
103C23×C8—31no—
104C23×C8C2×D311yescharacteristic
105C2×C22⋊Q8—34no—
106C2×C22⋊Q8—64no—
107C2×C22⋊Q8—62no—
108C2×C22⋊Q8—64no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—61no—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C22⋊C4—61no—
114C22×C22⋊C4—61no—
115C2×#106—34no—
116C2×#106—64no—
117C2×#103—24no—
118C2×#103—34no—
119C2×#103—64no—
120C2×#103C2×D314yescomplemented
121C2×C22⋊C8—34no—
122C2×C22⋊C8—64no—
123#380—34no—
124#380—62no—
125C2×#95—61no—
126#340—34no—
127C24×C3C2×D411yescharacteristic
128C23×D3—22no—
129C23×D3—22no—
130C23×D3—22no—
131C23×D3—22no—
132C23×D3—22no—
133C23×D3—22no—
134C23×D3SD1614yes—
135C23×D3C2×D412yes—
136C2×C3×Q8—24no—
137C2×C3×Q8—22no—
138C2×C3×Q8—28no—
139C2×C3×Q8—28no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—42no—
145C2×C3×Q8—44no—
146C2×C3×Q8—42no—
147C2×C3×Q8—44no—
148C2×C3×Q8C2×D414yes—
149C22×C3×C4—24no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4#4112yes—
162C22×C3×C4#4112yes—
163C22×C3×C4#4112yes—
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—22no—
166C22×Dic3—24no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—42no—
172C22×Dic3—42no—
173C22×Dic3—42no—
174C22×Dic3—42no—
175C22×Dic3—42no—
176C22×Dic3C2×D412yes—
177C2×C4×D3—24no—
178C2×C4×D3—24no—
179C2×C4×D3—24no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—44no—
187C2×C4×D3—44no—
188C2×C4×D3—44no—
189C2×C4×D3—44no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—44no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—42no—
197(C2×D3)⋊C4—216no—
198(C2×D3)⋊C4—48no—
199C4⋊Dic3—24no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—42no—
204C4⋊Dic3C2×D414yes—
205Dic3⋊C4—216no—
206C2×C3⋊C8—28no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8C2×D414yes—
210C25—31no—
211C22×Q8—22no—
212C22×Q8—21no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8#6714yes—
218C22×Q8C22×D312yes—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—32no—
224C23×C4—61no—
225C23×C4—62no—
226C23×C4—62no—
227C23×C4—62no—
228C23×C4—62no—
229C23×C4—62no—
230C23×C4—62no—
231C23×C4—61no—
232C23×C4C22×D311yescharacteristic
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—34no—
237C22×C8—32no—
238C22×C8—62no—
239C22×C8—62no—
240C22×C8C22×D312yes—
241C22⋊Q8—68no—
242C22⋊Q8—616no—
243C22⋊Q8—68no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—68no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—68no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—124no—
262#106—68no—
263#103—28no—
264#103—68no—
265C22⋊C8—68no—
266#95—64no—
267C23×C3#12012yes—
268C23×C3#12112yes—
269C23×C3#12312yes—
270C23×C3C2×SD1612yes—
271C23×C3#13512yes—
272C23×C3#13712yes—
273C23×C3#13712yes—
274C23×C3C22×D411yescharacteristic
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—22no—
279C22×D3—22no—
280C22×D3—22no—
281C22×D3—28no—
282C22×D3—28no—
283C22×D3—28no—
284C22×D3—44no—
285C22×D3—44no—
286C22×D3—44no—
287C22×D3—44no—
288C22×D3C2×SD1614yes—
289C3×Q8—24no—
290C3×Q8—24no—
291C3×Q8—42no—
292C3×Q8—42no—
293C3×Q8—48no—
294C2×C3×C4—24no—
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—84no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12114yes—
304C2×C3×C4#12112yes—
305C2×C3×C4#12114yes—
306C2×C3×C4#12312yes—
307C2×C3×C4#12314yes—
308C2×C3×C4#12414yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—44no—
322C2×Dic3—48no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—84no—
327C4×D3—42no—
328C4×D3—42no—
329C4×D3—42no—
330C4×D3—42no—
331C4×D3—48no—
332C3×C8—42no—
333C3⋊C8—24no—
334C24—34no—
335C24—32no—
336C24—62no—
337C24—62no—
338C24—62no—
339C24—62no—
340C24—62no—
341C24—62no—
342C24D3×D411yescharacteristic
343C2×Q8—24no—
344C2×Q8—22no—
345C2×Q8—28no—
346C2×Q8—28no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—42no—
352C2×Q8—44no—
353C2×Q8—42no—
354C2×Q8—44no—
355C2×Q8C2×#6714yes—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—32no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—64no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4—64no—
372C22×C4—64no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—122no—
381C22×C4—122no—
382C22×C4—124no—
383C22×C4—124no—
384C22×C4—124no—
385C22×C4—124no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—122no—
389C22×C4#23512yes—
390C22×C4#23512yes—
391C22×C4D3×D411yescharacteristic
392C22×C4D3×D412yes—
393C22×C4D3×D412yes—
394C22×C4#23712yes—
395C22×C4C2×#6712yes—
396C22×C4C23×D311yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—34no—
401C2×C8—44no—
402C2×C8—68no—
403C2×C8—62no—
404C2×C8—62no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—34no—
408C4⋊C4—42no—
409C4⋊C4—616no—
410C4⋊C4—64no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—122no—
415C22⋊C4—616no—
416C22⋊C4—128no—
417C22×C3#39414yes—
418C22×C3#44814yes—
419C22×C3#45214yes—
420C22×C3#46514yes—
421C22×C3#47014yes—
422C22×C3#48314yes—
423C22×C3#48514yes—
424C22×C3C2×#12011yescharacteristic
425C22×C3C2×#12111yescharacteristic
426C22×C3C2×#12311yescharacteristic
427C22×C3C22×SD1611yescharacteristic
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×#13711yescharacteristic
430C22×C3C2×#13711yescharacteristic
431C2×D3—24no—
432C2×D3—24no—
433C2×D3—24no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—416no—
439C3×C4—44no—
440C3×C4—82no—
441C3×C4#39414yes—
442C3×C4C2×#12111yescharacteristic
443C3×C4C2×#12111yescharacteristic
444C3×C4C2×#12311yescharacteristic
445C3×C4C2×#12411yescharacteristic
446Dic3—42no—
447Dic3—44no—
448Dic3—82no—
449C23—34no—
450C23—68no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23—62no—
455C23—62no—
456C23—62no—
457C23—68no—
458C23—68no—
459C23—124no—
460C23—124no—
461C23—124no—
462C23—124no—
463C23#88312yes—
464C23#88412yes—
465C23D3×SD1612yes—
466C23#91512yes—
467C23#91812yes—
468C23#91912yes—
469C23#93712yes—
470C23C2×D3×D411yescharacteristic
471Q8—24no—
472Q8—24no—
473Q8—48no—
474Q8—42no—
475Q8—42no—
476C2×C4—24no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—64no—
486C2×C4—62no—
487C2×C4—84no—
488C2×C4—128no—
489C2×C4—124no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—128no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—244no—
501C2×C4#88312yes—
502C2×C4#88412yes—
503C2×C4#88414yes—
504C2×C4#88514yes—
505C2×C4#92914yes—
506C2×C4#93712yes—
507C2×C4#93914yes—
508C2×C4C2×#23511yescharacteristic
509C2×C4C2×#23511yescharacteristic
510C2×C4C2×D3×D411yescharacteristic
511C2×C4C2×D3×D411yescharacteristic
512C2×C4C2×#23711yescharacteristic
513C2×C4C22×#6711yescharacteristic
514C8—42no—
515C8—64no—
516C2×C3#203518yescomplemented
517C2×C3C2×#39411yescharacteristic
518C2×C3C2×#44811yescharacteristic
519C2×C3C2×#45211yescharacteristic
520C2×C3C2×#46511yescharacteristic
521C2×C3C2×#47011yescharacteristic
522C2×C3C2×#48311yescharacteristic
523C2×C3C2×#48511yescharacteristic
524D3—44no—
525C22—64no—
526C22—64no—
527C22—64no—
528C22—1216no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22#480514yes—
534C22#480714yes—
535C22#481414yes—
536C22#481614yes—
537C22#495914yes—
538C22#517014yes—
539C22#518714yes—
540C22C2×#88311yescharacteristic
541C22C2×#88411yescharacteristic
542C22C2×D3×SD1611yescharacteristic
543C22C2×#91511yescharacteristic
544C22C2×#91811yescharacteristic
545C22C2×#91911yescharacteristic
546C22C2×#93711yescharacteristic
547C4—44no—
548C4—82no—
549C4—124no—
550C4—122no—
551C4—242no—
552C4#495914yes—
553C4C2×#88411yescharacteristic
554C4C2×#88511yescharacteristic
555C4C2×#92911yescharacteristic
556C4C2×#93911yescharacteristic
557C3C2×#203511yescharacteristic, complemented, Sylow
558C2—124no—
559C2#7027418yesdirect product factor
560C2C2×#481411yescharacteristic
561C2C2×#480711yescharacteristic
562C2C2×#481611yescharacteristic
563C2C2×#480511yescharacteristic
564C2C2×#495911yescharacteristic
565C2C2×#517011yescharacteristic
566C2C2×#518711yescharacteristic
subgroupG/H
centerC24D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#103C2
socleC24×C3C2×D4
prime radicalC24×D3D4
hypercenterC22×#103D3
hypocenterC3C2×#2035
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C22×C3×#103C2semidirect
3. C22×#835C2semidirect
6. C22×#808C2semidirect
7. C2×#4551C2semidirect
8. #70274C2direct (N × H)
25. C2×C3×#103C22semidirect
27. C2×#835C22semidirect
34. C2×#808C22semidirect
35. #4551C22semidirect
39. C22×#103D3semidirect
120. C2×#103C2×D3semidirect
516. C2×C3#2035semidirect
557. C3C2×#2035semidirect
559. C2#70274direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9462
C22×C3×#1032
C22×#8352
C22×(C2×D3)⋊C82
C2×#49872
C22×#8082
C2×#45512
#702742
C2×#20353