Finite Group Databasebeta

← C2×#82579 #10954681

Subgroups — C2×#82579 #10954681

Order 768.

5942 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#120C211yesmaximal, characteristic
2C23×#217C212yesmaximal, complemented
3C22×D4×Dic3C212yesmaximal, complemented
4C2×#5221C211yesmaximal, characteristic, complemented
5C2×#5220C212yesmaximal, complemented
6C2×#5220C211yesmaximal, characteristic, complemented
7#82579C2116yesmaximal, direct product factor
8C2×#4951C212yesmaximal, complemented
9C2×#4950C212yesmaximal, complemented
10C2×Dic3×C22⋊C4C212yesmaximal, complemented
11C2×#2358—31nomaximal, Sylow
12C26×C3C2211yescharacteristic
13C23×C3×D4C2212yes—
14C23×C3×D4C2211yescharacteristic
15C24×Dic3C2212yes—
16C24×Dic3C2212yes—
17C24×Dic3C2211yescharacteristic, complemented
18C2×C3×#120C4116yescomplemented
19C2×C3×#120C2218yes—
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×#217—216no—
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C22×#217C22116yescomplemented
28C22×#217C2212yescomplemented
29C22×#217C2211yescharacteristic, complemented
30C2×D4×Dic3—216no—
31C2×D4×Dic3C22116yescomplemented
32C22×C4⋊Dic3C2212yescomplemented
33C22×C4⋊Dic3C2211yescharacteristic, complemented
34C22×C4×Dic3C2212yescomplemented
35C22×C4×Dic3C2211yescharacteristic, complemented
36#5221C2218yescomplemented
37#5220C2218yescomplemented
38#5220C22116yescomplemented
39C2×C3⋊#95C2212yescomplemented
40C2×C3⋊#95C2212yescomplemented
41C2×C3⋊#95C2212yescomplemented
42#4951C22116yescomplemented
43#4950C22116yescomplemented
44Dic3×C22⋊C4C22116yescomplemented
45C22×#120D311yescharacteristic
46C22×C4×D4—32no—
47C23×C22⋊C4—32no—
48#2358—316no—
49C2×#390—31no—
50C2×#388—32no—
51C2×#386—32no—
52C2×#386—31no—
53C2×#385—32no—
54C2×C4×C22⋊C4—32no—
55C25×C3—28no—
56C25×C3—28no—
57C25×C3—42no—
58C25×C3—42no—
59C25×C3C2×C418yescomplemented
60C25×C3C2311yescharacteristic
61C25×C3C2311yescharacteristic
62C25×C3C2314yes—
63C25×C3C2312yes—
64C22×C3×D4—28no—
65C22×C3×D4—28no—
66C22×C3×D4—42no—
67C22×C3×D4—42no—
68C22×C3×D4C2×C4116yescomplemented
69C22×C3×D4C2×C418yescomplemented
70C22×C3×D4C2318yes—
71C22×C3×D4C2314yes—
72C23×C3×C4C2312yes—
73C23×C3×C4C2311yescharacteristic
74C23×Dic3—28no—
75C23×Dic3—28no—
76C23×Dic3—28no—
77C23×Dic3—28no—
78C23×Dic3—28no—
79C23×Dic3—44no—
80C23×Dic3—44no—
81C23×Dic3—42no—
82C23×Dic3C2312yes—
83C23×Dic3C2312yes—
84C23×Dic3C2318yes—
85C23×Dic3C2318yes—
86C23×Dic3C2312yes—
87C23×Dic3C2312yes—
88C23×Dic3C2314yescomplemented
89C3×#120C2×C4164yescomplemented
90C2×C3×C22⋊C4C2×C418yescomplemented
91C2×C3×C22⋊C4C2×C4116yescomplemented
92C2×C3×C22⋊C4C2318yes—
93C2×C3×C22⋊C4C2314yes—
94C2×#217—28no—
95C2×#217—28no—
96C2×#217—28no—
97C2×#217—28no—
98C2×#217—264no—
99C2×#217—42no—
100C2×#217D4116yescomplemented
101C2×#217C2318yes—
102C2×#217C2318yes—
103C2×#217C2318yes—
104C2×#217C2318yes—
105C2×#217C2318yescomplemented
106C2×#217C2314yescomplemented
107D4×Dic3—264no—
108C2×C4⋊Dic3—28no—
109C2×C4⋊Dic3—42no—
110C2×C4⋊Dic3C2318yescomplemented
111C2×C4⋊Dic3C2314yescomplemented
112C2×C4×Dic3—28no—
113C2×C4×Dic3—42no—
114C2×C4×Dic3C2318yescomplemented
115C2×C4×Dic3C2314yescomplemented
116C3⋊#95C2318yescomplemented
117C3⋊#95C2318yescomplemented
118C3⋊#95C2318yescomplemented
119C26C2×D311yescharacteristic
120C23×D4C2×D312yes—
121C23×D4C2×D311yescharacteristic
122C24×C4—32no—
123C24×C4—32no—
124C24×C4—31no—
125C2×#120Dic3116yescomplemented
126C2×#120C2×D318yes—
127C2×C4×D4—316no—
128C2×C4×D4—616no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—31no—
131C22×C22⋊C4—32no—
132C22×C22⋊C4—316no—
133C22×C22⋊C4—32no—
134C22×C22⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—616no—
139C22×C22⋊C4C2×D312yes—
140C22×C22⋊C4C2×D311yescharacteristic
141C22×C42—32no—
142C22×C42—31no—
143#390—38no—
144#388—316no—
145#386—38no—
146#386—316no—
147#385—316no—
148C4×C22⋊C4—316no—
149C2×#95—32no—
150C2×#95—32no—
151C2×#95—32no—
152C24×C3—24no—
153C24×C3—24no—
154C24×C3—24no—
155C24×C3—24no—
156C24×C3—264no—
157C24×C3—216no—
158C24×C3—24no—
159C24×C3—24no—
160C24×C3—216no—
161C24×C3—24no—
162C24×C3—24no—
163C24×C3—22no—
164C24×C3—22no—
165C24×C3—22no—
166C24×C3—22no—
167C24×C3—416no—
168C24×C3—44no—
169C24×C3—44no—
170C24×C3—416no—
171C24×C3—432no—
172C24×C3—416no—
173C24×C3C22×C414yes—
174C24×C3C22×C414yes—
175C24×C3C22×C418yes—
176C24×C3C22×C4116yescomplemented
177C24×C3C2×D418yes—
178C24×C3#4118yes—
179C24×C3C2412yes—
180C24×C3C2412yes—
181C24×C3C2414yes—
182C24×C3C2411yescharacteristic
183C2×C3×D4—264no—
184C2×C3×D4—216no—
185C2×C3×D4—216no—
186C2×C3×D4—416no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4C22×C4132yescomplemented
190C2×C3×D4C22×C4116yescomplemented
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4C22×C418yes—
196C22×C3×C4C22×C414yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2412yes—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—216no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—216no—
208C22×Dic3—24no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—216no—
212C22×Dic3—216no—
213C22×Dic3—216no—
214C22×Dic3—48no—
215C22×Dic3—48no—
216C22×Dic3—44no—
217C22×Dic3—416no—
218C22×Dic3—432no—
219C22×Dic3—432no—
220C22×Dic3C2×D418yes—
221C22×Dic3C2×D418yes—
222C22×Dic3C2414yes—
223C22×Dic3C2414yes—
224C22×Dic3C2414yes—
225C22×Dic3C2414yes—
226C3×C22⋊C4C22×C4116yescomplemented
227C3×C22⋊C4C22×C4132yescomplemented
228#217—216no—
229#217—216no—
230#217—216no—
231#217—216no—
232#217—44no—
233#217C2×D4132yescomplemented
234C4⋊Dic3—216no—
235C4⋊Dic3—44no—
236C4×Dic3—216no—
237C4×Dic3—44no—
238C25—28no—
239C25—28no—
240C25—42no—
241C25—42no—
242C25C2×Dic318yescomplemented
243C25C22×D311yescharacteristic
244C25C22×D311yescharacteristic
245C25C22×D314yes—
246C25C22×D312yes—
247C22×D4—28no—
248C22×D4—28no—
249C22×D4—42no—
250C22×D4—42no—
251C22×D4C2×Dic3116yescomplemented
252C22×D4C2×Dic318yescomplemented
253C22×D4C22×D318yes—
254C22×D4C22×D314yes—
255C23×C4—32no—
256C23×C4—32no—
257C23×C4—32no—
258C23×C4—32no—
259C23×C4—38no—
260C23×C4—38no—
261C23×C4—34no—
262C23×C4—68no—
263C23×C4—68no—
264C23×C4—68no—
265C23×C4—68no—
266C23×C4—68no—
267C23×C4—124no—
268C23×C4—122no—
269C23×C4—124no—
270C23×C4C22×D312yes—
271C23×C4C22×D311yescharacteristic
272#120C2×Dic3164yescomplemented
273C4×D4—664no—
274C2×C4⋊C4—38no—
275C2×C4⋊C4—34no—
276C2×C4⋊C4—68no—
277C2×C4⋊C4—122no—
278C2×C22⋊C4—316no—
279C2×C22⋊C4—38no—
280C2×C22⋊C4—38no—
281C2×C22⋊C4—38no—
282C2×C22⋊C4—38no—
283C2×C22⋊C4—38no—
284C2×C22⋊C4—34no—
285C2×C22⋊C4—664no—
286C2×C22⋊C4—68no—
287C2×C22⋊C4—68no—
288C2×C22⋊C4—68no—
289C2×C22⋊C4—68no—
290C2×C22⋊C4—122no—
291C2×C22⋊C4C2×Dic318yescomplemented
292C2×C22⋊C4C2×Dic3116yescomplemented
293C2×C22⋊C4C22×D318yes—
294C2×C22⋊C4C22×D314yes—
295C2×C42—38no—
296C2×C42—34no—
297C2×C42—68no—
298C2×C42—122no—
299#95—38no—
300#95—38no—
301#95—38no—
302C23×C3—216no—
303C23×C3—216no—
304C23×C3—216no—
305C23×C3—216no—
306C23×C3—216no—
307C23×C3—216no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—216no—
313C23×C3—216no—
314C23×C3—264no—
315C23×C3—28no—
316C23×C3—28no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—24no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—416no—
326C23×C3—4128no—
327C23×C3—416no—
328C23×C3—432no—
329C23×C3—416no—
330C23×C3—44no—
331C23×C3—44no—
332C23×C3—48no—
333C23×C3—48no—
334C23×C3—44no—
335C23×C3—44no—
336C23×C3C4×D4132yescomplemented
337C23×C3C23×C414yes—
338C23×C3C23×C414yes—
339C23×C3C23×C418yes—
340C23×C3C23×C412yes—
341C23×C3C22×D418yes—
342C23×C3C22×D414yes—
343C23×C3C2×#4118yes—
344C23×C3C2×#4114yes—
345C23×C3H(5,𝔽2)12yes—
346C23×C3H(5,𝔽2)12yes—
347C23×C3C2511yescharacteristic
348C3×D4—264no—
349C3×D4—416no—
350C2×C3×C4—216no—
351C2×C3×C4—28no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4C23×C418yes—
357C2×C3×C4C23×C414yes—
358C2×Dic3—216no—
359C2×Dic3—216no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—416no—
371C2×Dic3—432no—
372C2×Dic3—432no—
373C2×Dic3—48no—
374C2×Dic3—48no—
375C2×Dic3C22×D418yes—
376C2×Dic3C22×D418yes—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—264no—
382C24—216no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—216no—
387C24—24no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—416no—
393C24—44no—
394C24—44no—
395C24—416no—
396C24—416no—
397C24—432no—
398C24D3×D418yes—
399C24#23718yes—
400C24C22×Dic314yes—
401C24C22×Dic314yes—
402C24C22×Dic318yes—
403C24C22×Dic3116yescomplemented
404C24C23×D312yes—
405C24C23×D312yes—
406C24C23×D314yes—
407C24C23×D311yescharacteristic
408C2×D4—264no—
409C2×D4—216no—
410C2×D4—216no—
411C2×D4—416no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4C22×Dic3132yescomplemented
415C2×D4C22×Dic3116yescomplemented
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—38no—
421C22×C4—38no—
422C22×C4—34no—
423C22×C4—34no—
424C22×C4—34no—
425C22×C4—34no—
426C22×C4—616no—
427C22×C4—616no—
428C22×C4—616no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—64no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4—616no—
437C22×C4—616no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—128no—
442C22×C4—124no—
443C22×C4—128no—
444C22×C4—1216no—
445C22×C4—1232no—
446C22×C4—1232no—
447C22×C4C22×Dic318yes—
448C22×C4C22×Dic314yes—
449C22×C4C23×D314yes—
450C22×C4C23×D312yes—
451C4⋊C4—616no—
452C4⋊C4—124no—
453C22⋊C4—332no—
454C22⋊C4—616no—
455C22⋊C4—616no—
456C22⋊C4—616no—
457C22⋊C4—616no—
458C22⋊C4—124no—
459C22⋊C4C22×Dic3116yescomplemented
460C22⋊C4C22×Dic3132yescomplemented
461C42—616no—
462C42—124no—
463C22×C3—264no—
464C22×C3—28no—
465C22×C3—28no—
466C22×C3—28no—
467C22×C3—28no—
468C22×C3—28no—
469C22×C3—28no—
470C22×C3—28no—
471C22×C3—28no—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—464no—
475C22×C3—416no—
476C22×C3—416no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3C2×C4×D418yes—
484C22×C3C2×C4×D4116yescomplemented
485C22×C3#51814yes—
486C22×C3D4214yes—
487C22×C3#54618yes—
488C22×C3#55114yes—
489C22×C3C24×C411yescharacteristic
490C22×C3C23×D412yes—
491C22×C3C22×#4112yes—
492C22×C3C2×H(5,𝔽2)11yescharacteristic
493C22×C3C2×H(5,𝔽2)11yescharacteristic
494C3×C4—28no—
495C3×C4—24no—
496Dic3—28no—
497Dic3—28no—
498Dic3—44no—
499Dic3—44no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—216no—
505C23—216no—
506C23—216no—
507C23—216no—
508C23—216no—
509C23—216no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—216no—
519C23—216no—
520C23—264no—
521C23—28no—
522C23—28no—
523C23—416no—
524C23—4128no—
525C23—416no—
526C23—416no—
527C23—432no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—48no—
533C23—48no—
534C23D4×Dic3132yescomplemented
535C23C2×D3×D418yes—
536C23C2×D3×D414yes—
537C23C2×#23718yes—
538C23C2×#23714yes—
539C23#100412yes—
540C23#100412yes—
541C23C23×Dic314yes—
542C23C23×Dic314yes—
543C23C23×Dic318yes—
544C23C23×Dic312yes—
545C23C24×D311yescharacteristic
546D4—264no—
547D4—416no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—216no—
553C2×C4—28no—
554C2×C4—38no—
555C2×C4—38no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—616no—
561C2×C4—616no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—1216no—
569C2×C4—1232no—
570C2×C4—1232no—
571C2×C4—128no—
572C2×C4—128no—
573C2×C4C23×Dic318yes—
574C2×C4C23×Dic314yes—
575C2×C3—216no—
576C2×C3—44no—
577C2×C3—44no—
578C2×C3#235818yescomplemented
579C2×C3C22×C4×D412yes—
580C2×C3C2×#51811yescharacteristic
581C2×C3C2×D4211yescharacteristic
582C2×C3C2×#54612yes—
583C2×C3C2×#55111yescharacteristic
584C22—264no—
585C22—28no—
586C22—28no—
587C22—28no—
588C22—28no—
589C22—28no—
590C22—28no—
591C22—28no—
592C22—28no—
593C22—24no—
594C22—24no—
595C22—464no—
596C22—416no—
597C22—416no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22#558714yes—
605C22#559014yes—
606C22#559118yes—
607C22C2×D4×Dic318yes—
608C22C2×D4×Dic3116yescomplemented
609C22#579814yes—
610C22C22×D3×D412yes—
611C22C22×#23712yes—
612C22C2×#100411yescharacteristic
613C22C2×#100411yescharacteristic
614C22C24×Dic311yescharacteristic
615C4—28no—
616C4—24no—
617C4—68no—
618C4—68no—
619C4—124no—
620C4—124no—
621C3C2×#235811yescharacteristic, complemented, Sylow
622C2—216no—
623C2—44no—
624C2—44no—
625C2#8257918yesdirect product factor
626C2C2×#558711yescharacteristic
627C2C2×#559011yescharacteristic
628C2C2×#559112yes—
629C2C22×D4×Dic312yes—
630C2C2×#579811yescharacteristic
subgroupG/H
centerC24C23×D3
commutatorC22×C3C24×C4
FrattiniC23C24×D3
Fitting, prime radicalC22×C3×#120C2
socleC24×C3C24
cosocleC23×C3C25
hypercenterC22×#120D3
hypocenterC3C2×#2358
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#120

Complements

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

NH ≅ G/Ntype
2. C23×#217C2semidirect
3. C22×D4×Dic3C2semidirect
4. C2×#5221C2semidirect
5. C2×#5220C2semidirect
6. C2×#5220C2semidirect
7. #82579C2direct (N × H)
8. C2×#4951C2semidirect
9. C2×#4950C2semidirect
10. C2×Dic3×C22⋊C4C2semidirect
17. C24×Dic3C22semidirect
18. C2×C3×#120C4semidirect
27. C22×#217C22semidirect
28. C22×#217C22semidirect
29. C22×#217C22semidirect
31. C2×D4×Dic3C22semidirect
32. C22×C4⋊Dic3C22semidirect
33. C22×C4⋊Dic3C22semidirect
34. C22×C4×Dic3C22semidirect
35. C22×C4×Dic3C22semidirect
36. #5221C22semidirect
37. #5220C22semidirect
38. #5220C22semidirect
39. C2×C3⋊#95C22semidirect
40. C2×C3⋊#95C22semidirect
41. C2×C3⋊#95C22semidirect
42. #4951C22semidirect
43. #4950C22semidirect
44. Dic3×C22⋊C4C22semidirect
59. C25×C3C2×C4semidirect
68. C22×C3×D4C2×C4semidirect
69. C22×C3×D4C2×C4semidirect
88. C23×Dic3C23semidirect
89. C3×#120C2×C4semidirect
90. C2×C3×C22⋊C4C2×C4semidirect
91. C2×C3×C22⋊C4C2×C4semidirect
100. C2×#217D4semidirect
105. C2×#217C23semidirect
106. C2×#217C23semidirect
110. C2×C4⋊Dic3C23semidirect
111. C2×C4⋊Dic3C23semidirect
114. C2×C4×Dic3C23semidirect
115. C2×C4×Dic3C23semidirect
116. C3⋊#95C23semidirect
117. C3⋊#95C23semidirect
118. C3⋊#95C23semidirect
125. C2×#120Dic3semidirect
176. C24×C3C22×C4semidirect
189. C2×C3×D4C22×C4semidirect
190. C2×C3×D4C22×C4semidirect
226. C3×C22⋊C4C22×C4semidirect
227. C3×C22⋊C4C22×C4semidirect
233. #217C2×D4semidirect
242. C25C2×Dic3semidirect
251. C22×D4C2×Dic3semidirect
252. C22×D4C2×Dic3semidirect
272. #120C2×Dic3semidirect
291. C2×C22⋊C4C2×Dic3semidirect
292. C2×C22⋊C4C2×Dic3semidirect
336. C23×C3C4×D4semidirect
403. C24C22×Dic3semidirect
414. C2×D4C22×Dic3semidirect
415. C2×D4C22×Dic3semidirect
459. C22⋊C4C22×Dic3semidirect
460. C22⋊C4C22×Dic3semidirect
484. C22×C3C2×C4×D4semidirect
534. C23D4×Dic3semidirect
578. C2×C3#2358semidirect
608. C22C2×D4×Dic3semidirect
621. C3C2×#2358semidirect
625. C2#82579direct (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
C23×#2172
C22×D4×Dic32
C2×#52212
C2×#52202
#825792
C2×#49512
C2×#49502
C2×Dic3×C22⋊C42
C2×#23583