Finite Group Databasebeta

← C2×#82598 #11429962

Subgroups — C2×#82598 #11429962

Order 768.

4270 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#120C211yesmaximal, characteristic
2C22×#936C211yesmaximal, characteristic, complemented
3C2×#5247C212yesmaximal, complemented
4C2×#5222C212yesmaximal, complemented
5C2×#5220C212yesmaximal, complemented
6C2×#5220C211yesmaximal, characteristic, complemented
7#82598C2116yesmaximal, direct product factor
8C2×#4952C212yesmaximal, complemented
9C2×#4947C212yesmaximal, complemented
10C2×#4944C212yesmaximal, complemented
11C2×#2684—31nomaximal, Sylow
12C26×C3C2211yescharacteristic
13C23×C3×D4C2212yes—
14C23×C3×D4C2211yescharacteristic
15C24×Dic3C2211yescharacteristic, complemented
16C23×Dic6C2211yescharacteristic, complemented
17C2×C3×#120—28no—
18C2×C3×#120C2218yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×#217C2212yes—
22C22×#217C2212yes—
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C22×#217C2212yescomplemented
28C22×#217C2211yescharacteristic, complemented
29C2×#936—28no—
30C2×#936C2218yescomplemented
31C22×Dic3⋊C4C2212yescomplemented
32C22×C4×Dic3C2211yescharacteristic, complemented
33#5247C22116yescomplemented
34#5222C22116yescomplemented
35#5220C2218yescomplemented
36#5220C22116yescomplemented
37C2×C3⋊#95C2212yescomplemented
38C2×C3⋊#95C2212yescomplemented
39C2×C3⋊#95C2212yescomplemented
40C2×C3⋊#95C2212yescomplemented
41#4952C22116yescomplemented
42#4947C22116yescomplemented
43#4944C22116yescomplemented
44C22×#124—31no—
45C22×#120D311yescharacteristic
46#2684—316no—
47C2×#397—32no—
48C2×#394—32no—
49C2×#393—32no—
50C2×#388—32no—
51C2×#386—32no—
52C2×#386—31no—
53C2×#379—32no—
54C25×C3—24no—
55C25×C3—28no—
56C25×C3—44no—
57C25×C3—42no—
58C25×C3—42no—
59C25×C3C2311yescharacteristic
60C25×C3C2311yescharacteristic
61C25×C3C2312yes—
62C25×C3C2314yes—
63C22×C3×D4—28no—
64C22×C3×D4—28no—
65C22×C3×D4—42no—
66C22×C3×D4—42no—
67C22×C3×D4—44no—
68C22×C3×D4D418yes—
69C22×C3×D4C2318yes—
70C22×C3×D4C2314yes—
71C23×C3×C4C2312yes—
72C23×C3×C4C2311yescharacteristic
73C23×Dic3—42no—
74C23×Dic3—44no—
75C23×Dic3C2312yes—
76C23×Dic3C2312yes—
77C23×Dic3C2312yes—
78C23×Dic3C2312yes—
79C23×Dic3C2314yescomplemented
80C22×Dic6—24no—
81C22×Dic6—42no—
82C22×Dic6—42no—
83C22×Dic6C2314yescomplemented
84C3×#120—232no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—28no—
87C2×C3×C22⋊C4C2318yes—
88C2×C3×C22⋊C4C2314yes—
89C2×#217—28no—
90C2×#217—28no—
91C2×#217—28no—
92C2×#217—44no—
93C2×#217—44no—
94C2×#217—44no—
95C2×#217—44no—
96C2×#217—42no—
97C2×#217C2318yes—
98C2×#217C2318yes—
99C2×#217C2318yes—
100C2×#217C2318yes—
101C2×#217C2318yes—
102C2×#217C2318yes—
103C2×#217C2318yescomplemented
104C2×#217C2314yescomplemented
105#936—232no—
106C2×Dic3⋊C4—44no—
107C2×Dic3⋊C4C2318yescomplemented
108C2×C4×Dic3—24no—
109C2×C4×Dic3C2314yescomplemented
110C3⋊#95C2318yescomplemented
111C3⋊#95C2318yescomplemented
112C3⋊#95C2318yescomplemented
113C3⋊#95C2318yescomplemented
114C26C2×D311yescharacteristic
115C23×Q8—31no—
116C23×D4C2×D312yes—
117C23×D4C2×D311yescharacteristic
118C24×C4—31no—
119C2×#124—38no—
120C2×#124—68no—
121C2×#120—28no—
122C2×#120C2×D318yes—
123C22×C4⋊C4—32no—
124C22×C22⋊C4—32no—
125C22×C22⋊C4—32no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—31no—
130C22×C22⋊C4—32no—
131C22×C22⋊C4—32no—
132C22×C22⋊C4C2×D312yes—
133C22×C22⋊C4C2×D311yescharacteristic
134C22×C42—31no—
135#397—316no—
136#394—316no—
137#393—316no—
138#388—316no—
139#386—38no—
140#386—316no—
141#379—316no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C24×C3—24no—
147C24×C3—24no—
148C24×C3—24no—
149C24×C3—24no—
150C24×C3—24no—
151C24×C3—24no—
152C24×C3—24no—
153C24×C3—24no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—24no—
157C24×C3—22no—
158C24×C3—22no—
159C24×C3—24no—
160C24×C3—28no—
161C24×C3—216no—
162C24×C3—48no—
163C24×C3—44no—
164C24×C3—44no—
165C24×C3—432no—
166C24×C3—416no—
167C24×C3—416no—
168C24×C3—432no—
169C24×C3—416no—
170C24×C3C2×D414yes—
171C24×C3C2×D414yes—
172C24×C3#4118yes—
173C24×C3C2412yes—
174C24×C3C2412yes—
175C24×C3C2414yes—
176C24×C3C2411yescharacteristic
177C2×C3×D4—216no—
178C2×C3×D4—216no—
179C2×C3×D4—416no—
180C2×C3×D4—44no—
181C2×C3×D4—44no—
182C2×C3×D4—48no—
183C2×C3×D4—432no—
184C2×C3×D4C2×D4116yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2414yes—
192C22×C3×C4C2412yes—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—44no—
206C22×Dic3—48no—
207C22×Dic3—416no—
208C22×Dic3C2414yes—
209C22×Dic3C2414yes—
210C22×Dic3C2414yes—
211C22×Dic3C2414yes—
212C2×Dic6—28no—
213C2×Dic6—44no—
214C2×Dic6—44no—
215C2×Dic6—416no—
216C3×C22⋊C4—28no—
217C3×C22⋊C4—216no—
218#217—216no—
219#217—216no—
220#217—216no—
221#217—48no—
222#217—48no—
223#217—48no—
224#217—48no—
225#217—44no—
226Dic3⋊C4—48no—
227C4×Dic3—28no—
228C25—28no—
229C25—24no—
230C25—44no—
231C25—42no—
232C25—42no—
233C25C22×D312yes—
234C25C22×D311yescharacteristic
235C25C22×D311yescharacteristic
236C25C22×D314yes—
237C22×Q8—34no—
238C22×Q8—64no—
239C22×Q8—122no—
240C22×Q8—122no—
241C22×D4—28no—
242C22×D4—28no—
243C22×D4—42no—
244C22×D4—42no—
245C22×D4—44no—
246C22×D4#6718yes—
247C22×D4C22×D318yes—
248C22×D4C22×D314yes—
249C23×C4—32no—
250C23×C4—32no—
251C23×C4—32no—
252C23×C4—32no—
253C23×C4—34no—
254C23×C4—122no—
255C23×C4—124no—
256C23×C4C22×D312yes—
257C23×C4C22×D311yescharacteristic
258#124—632no—
259#120—232no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—124no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—38no—
265C2×C22⋊C4—38no—
266C2×C22⋊C4—38no—
267C2×C22⋊C4—38no—
268C2×C22⋊C4—38no—
269C2×C22⋊C4—38no—
270C2×C22⋊C4—38no—
271C2×C22⋊C4—34no—
272C2×C22⋊C4—68no—
273C2×C22⋊C4—68no—
274C2×C22⋊C4—68no—
275C2×C22⋊C4—122no—
276C2×C22⋊C4—124no—
277C2×C22⋊C4—124no—
278C2×C22⋊C4—124no—
279C2×C22⋊C4—124no—
280C2×C22⋊C4C22×D318yes—
281C2×C22⋊C4C22×D314yes—
282C2×C42—34no—
283C2×C42—64no—
284#95—38no—
285#95—38no—
286#95—38no—
287#95—38no—
288C23×C3—28no—
289C23×C3—28no—
290C23×C3—216no—
291C23×C3—216no—
292C23×C3—216no—
293C23×C3—216no—
294C23×C3—216no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—22no—
305C23×C3—24no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—24no—
309C23×C3—432no—
310C23×C3—416no—
311C23×C3—416no—
312C23×C3—432no—
313C23×C3—4128no—
314C23×C3—416no—
315C23×C3—48no—
316C23×C3—48no—
317C23×C3—48no—
318C23×C3—48no—
319C23×C3—48no—
320C23×C3—44no—
321C23×C3—44no—
322C23×C3—44no—
323C23×C3—44no—
324C23×C3—48no—
325C23×C3C22×D414yes—
326C23×C3C22×D414yes—
327C23×C3C22×D412yes—
328C23×C3C2×#4118yes—
329C23×C3C2×#4114yes—
330C23×C3H(5,𝔽2)12yes—
331C23×C3H(5,𝔽2)12yes—
332C23×C3H(5,𝔽2)14yes—
333C23×C3C2511yescharacteristic
334C3×D4—416no—
335C3×D4—432no—
336C2×C3×C4—28no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—48no—
343C2×C3×C4C22×D414yes—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—416no—
357C2×Dic3—48no—
358C2×Dic3—48no—
359C2×Dic3—48no—
360C2×Dic3—48no—
361Dic6—416no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—216no—
377C24—28no—
378C24—48no—
379C24—44no—
380C24—44no—
381C24—432no—
382C24—416no—
383C24—416no—
384C24—432no—
385C24—416no—
386C24#23718yes—
387C24C2×#6714yes—
388C24C2×#6714yes—
389C24C23×D314yes—
390C24C23×D312yes—
391C24C23×D312yes—
392C24C23×D311yescharacteristic
393C2×Q8—68no—
394C2×Q8—124no—
395C2×Q8—124no—
396C2×Q8—1216no—
397C2×D4—216no—
398C2×D4—216no—
399C2×D4—416no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—432no—
404C2×D4C2×#67116yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—34no—
411C22×C4—34no—
412C22×C4—34no—
413C22×C4—34no—
414C22×C4—64no—
415C22×C4—64no—
416C22×C4—64no—
417C22×C4—64no—
418C22×C4—64no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—64no—
422C22×C4—64no—
423C22×C4—64no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4—128no—
427C22×C4—124no—
428C22×C4—1216no—
429C22×C4C2×#6714yes—
430C22×C4C23×D314yes—
431C22×C4C23×D312yes—
432C4⋊C4—128no—
433C22⋊C4—28no—
434C22⋊C4—216no—
435C22⋊C4—616no—
436C22⋊C4—616no—
437C22⋊C4—616no—
438C22⋊C4—124no—
439C22⋊C4—128no—
440C22⋊C4—128no—
441C22⋊C4—128no—
442C22⋊C4—128no—
443C42—68no—
444C22×C3—24no—
445C22×C3—24no—
446C22×C3—28no—
447C22×C3—28no—
448C22×C3—28no—
449C22×C3—28no—
450C22×C3—28no—
451C22×C3—464no—
452C22×C3—44no—
453C22×C3—44no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—416no—
457C22×C3—416no—
458C22×C3—432no—
459C22×C3—44no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—44no—
465C22×C3#53414yes—
466C22×C3#53818yes—
467C22×C3#54618yes—
468C22×C3#55114yes—
469C22×C3H(3,𝔽4)14yes—
470C22×C3C23×D411yescharacteristic
471C22×C3C22×#4112yes—
472C22×C3C2×H(5,𝔽2)11yescharacteristic
473C22×C3C2×H(5,𝔽2)11yescharacteristic
474C22×C3C2×H(5,𝔽2)12yes—
475C3×C4—24no—
476C3×C4—44no—
477Dic3—44no—
478Dic3—44no—
479Dic3—44no—
480Dic3—44no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—216no—
498C23—216no—
499C23—216no—
500C23—216no—
501C23—216no—
502C23—416no—
503C23—432no—
504C23—432no—
505C23—416no—
506C23—416no—
507C23—4128no—
508C23—48no—
509C23—48no—
510C23—44no—
511C23—44no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23C2×#23718yes—
519C23C2×#23714yes—
520C23#100412yes—
521C23#100412yes—
522C23#100414yes—
523C23C22×#6714yes—
524C23C22×#6714yes—
525C23C22×#6712yes—
526C23C24×D311yescharacteristic
527Q8—1216no—
528D4—416no—
529D4—432no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—48no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—1216no—
550C2×C4—128no—
551C2×C4—128no—
552C2×C4—128no—
553C2×C4—128no—
554C2×C4C22×#6714yes—
555C2×C3—44no—
556C2×C3—44no—
557C2×C3—48no—
558C2×C3#268418yescomplemented
559C2×C3C2×#53411yescharacteristic
560C2×C3C2×#53812yes—
561C2×C3C2×#54612yes—
562C2×C3C2×#55111yescharacteristic
563C2×C3C2×H(3,𝔽4)11yescharacteristic
564C22—28no—
565C22—28no—
566C22—28no—
567C22—28no—
568C22—28no—
569C22—24no—
570C22—24no—
571C22—464no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22—416no—
581C22—432no—
582C22—416no—
583C22—44no—
584C22—44no—
585C22#558714yes—
586C22#559318yes—
587C22#559414yes—
588C22#580414yes—
589C22#580618yes—
590C22C22×#23712yes—
591C22C2×#100411yescharacteristic
592C22C2×#100412yes—
593C22C2×#100411yescharacteristic
594C22C23×#6711yescharacteristic
595C4—24no—
596C4—44no—
597C4—124no—
598C4—124no—
599C4—124no—
600C4—124no—
601C3C2×#268411yescharacteristic, complemented, Sylow
602C2—48no—
603C2—44no—
604C2—44no—
605C2#8259818yesdirect product factor
606C2C2×#559312yes—
607C2C2×#558711yescharacteristic
608C2C2×#559411yescharacteristic
609C2C2×#580612yes—
610C2C2×#580411yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
Fitting, prime radicalC22×C3×#120C2
socleC24×C3C24
hypercenterC22×#120D3
hypocenterC3C2×#2684
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#120

Complements

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

NH ≅ G/Ntype
2. C22×#936C2semidirect
3. C2×#5247C2semidirect
4. C2×#5222C2semidirect
5. C2×#5220C2semidirect
6. C2×#5220C2semidirect
7. #82598C2direct (N × H)
8. C2×#4952C2semidirect
9. C2×#4947C2semidirect
10. C2×#4944C2semidirect
15. C24×Dic3C22semidirect
16. C23×Dic6C22semidirect
27. C22×#217C22semidirect
28. C22×#217C22semidirect
30. C2×#936C22semidirect
31. C22×Dic3⋊C4C22semidirect
32. C22×C4×Dic3C22semidirect
33. #5247C22semidirect
34. #5222C22semidirect
35. #5220C22semidirect
36. #5220C22semidirect
37. C2×C3⋊#95C22semidirect
38. C2×C3⋊#95C22semidirect
39. C2×C3⋊#95C22semidirect
40. C2×C3⋊#95C22semidirect
41. #4952C22semidirect
42. #4947C22semidirect
43. #4944C22semidirect
79. C23×Dic3C23semidirect
83. C22×Dic6C23semidirect
103. C2×#217C23semidirect
104. C2×#217C23semidirect
107. C2×Dic3⋊C4C23semidirect
109. C2×C4×Dic3C23semidirect
110. C3⋊#95C23semidirect
111. C3⋊#95C23semidirect
112. C3⋊#95C23semidirect
113. C3⋊#95C23semidirect
558. C2×C3#2684semidirect
601. C3C2×#2684semidirect
605. C2#82598direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1202
C22×#9362
C2×#52472
C2×#52222
C2×#52202
#825982
C2×#49522
C2×#49472
C2×#49442
C2×#26843