Finite Group Databasebeta

← C2×#74880 #10954680

Subgroups — C2×#74880 #10954680

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×#937C211yesmaximal, characteristic
2C22×C4×#67C212yesmaximal, complemented
3C22×D3×C22⋊C4C212yesmaximal, complemented
4C2×C3×#386C211yesmaximal, characteristic, complemented
5C2×#5221C211yesmaximal, characteristic, complemented
6C2×#4956C212yesmaximal, complemented
7C2×#4945C212yesmaximal, complemented
8#74880C2116yesmaximal, direct product factor
9C2×#4668C212yesmaximal, complemented
10C2×#4666C212yesmaximal, complemented
11C2×#2358—31nomaximal, Sylow
12C25×D3C2211yescharacteristic
13C23×C3×D4C2211yescharacteristic
14C24×C3×C4C2211yescharacteristic, complemented
15C23×#67C2212yes—
16C23×C4×D3C2212yes—
17C23×C4×D3C2212yes—
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic, complemented
20C22×#217C2212yes—
21C22×#217C2211yescharacteristic
22C22×#217C2212yescomplemented
23C2×#937C4116yescomplemented
24C2×#937C2218yes—
25C2×C4×#67—216no—
26C2×C4×#67C22116yescomplemented
27C22×(C2×D3)⋊C4C2212yes—
28C22×(C2×D3)⋊C4C2212yes—
29C22×(C2×D3)⋊C4C2212yes—
30C22×C4⋊Dic3C2211yescharacteristic, complemented
31C22×Dic3⋊C4C2212yescomplemented
32C22×C4×Dic3C2212yescomplemented
33C22×C4×Dic3C2211yescharacteristic, complemented
34C2×D3×C22⋊C4—216no—
35C2×D3×C22⋊C4C22116yescomplemented
36C3×#386C2218yescomplemented
37C2×C3×#95C2212yescomplemented
38#5221C2218yescomplemented
39C2×C3⋊#95C2212yescomplemented
40C2×C3⋊#95C2212yescomplemented
41#4956C22116yescomplemented
42#4945C22116yescomplemented
43#4668C22116yescomplemented
44#4666C22116yescomplemented
45C22×#120—31no—
46C22×C4×D4—32no—
47C23×C22⋊C4—32no—
48#2358—316no—
49C2×#390—31no—
50C2×#388—32no—
51C2×#386—32no—
52C2×#386D311yescharacteristic, complemented
53C2×#385—32no—
54C2×C4×C22⋊C4—32no—
55C25×C3C2311yescharacteristic
56C25×C3C2311yescharacteristic
57C24×D3—28no—
58C24×D3—28no—
59C24×D3—42no—
60C24×D3—42no—
61C24×D3C2×C418yescomplemented
62C24×D3C2314yes—
63C24×D3C2312yes—
64C22×C3×D4—42no—
65C22×C3×D4—42no—
66C22×C3×D4C2×C418yescomplemented
67C22×C3×D4C2314yes—
68C23×C3×C4—28no—
69C23×C3×C4—42no—
70C23×C3×C4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2312yes—
73C23×C3×C4C2314yescomplemented
74C22×#67—28no—
75C22×#67—28no—
76C22×#67C2×C4116yescomplemented
77C22×#67C2318yes—
78C23×Dic3C2312yes—
79C23×Dic3C2312yes—
80C23×Dic3C2312yes—
81C22×C4×D3—28no—
82C22×C4×D3—28no—
83C22×C4×D3—28no—
84C22×C4×D3—28no—
85C22×C4×D3—44no—
86C22×C4×D3—44no—
87C22×C4×D3C2318yes—
88C22×C4×D3C2318yes—
89C2×C3×C22⋊C4—28no—
90C2×C3×C22⋊C4—42no—
91C2×C3×C22⋊C4C2318yes—
92C2×C3×C22⋊C4C2314yescomplemented
93C2×#217—28no—
94C2×#217—28no—
95C2×#217C2×C418yescomplemented
96C2×#217C2318yes—
97C2×#217C2314yes—
98C2×#217C2318yescomplemented
99#937C2×C4164yescomplemented
100C4×#67—264no—
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4C2×C4116yescomplemented
103C2×(C2×D3)⋊C4D4116yescomplemented
104C2×(C2×D3)⋊C4C2318yes—
105C2×(C2×D3)⋊C4C2318yes—
106C2×(C2×D3)⋊C4C2318yes—
107C2×C4⋊Dic3—42no—
108C2×C4⋊Dic3C2314yescomplemented
109C2×Dic3⋊C4—28no—
110C2×Dic3⋊C4C2318yescomplemented
111C2×C4×Dic3—28no—
112C2×C4×Dic3—42no—
113C2×C4×Dic3C2318yescomplemented
114C2×C4×Dic3C2314yescomplemented
115D3×C22⋊C4—264no—
116C3×#95C2318yescomplemented
117C3⋊#95C2318yescomplemented
118C3⋊#95C2318yescomplemented
119C26—31no—
120C23×D4—32no—
121C23×D4C2×D311yescharacteristic
122C24×C4—32no—
123C24×C4—32no—
124C24×C4C2×D311yescharacteristic, complemented
125C2×#120—316no—
126C2×#120—38no—
127C2×C4×D4—316no—
128C2×C4×D4—616no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—31no—
131C22×C22⋊C4—31no—
132C22×C22⋊C4—32no—
133C22×C22⋊C4—32no—
134C22×C22⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—316no—
138C22×C22⋊C4—616no—
139C22×C22⋊C4C2×D312yes—
140C22×C22⋊C4C2×D311yescharacteristic, complemented
141C22×C42—32no—
142C22×C42—31no—
143#390—38no—
144#388—316no—
145#386—316no—
146#386C2×D318yescomplemented
147#385—316no—
148C4×C22⋊C4—316no—
149C2×#95—32no—
150C2×#95—32no—
151C2×#95C2×D312yescomplemented
152C24×C3—24no—
153C24×C3—24no—
154C24×C3—24no—
155C24×C3—24no—
156C24×C3—22no—
157C24×C3—22no—
158C24×C3—22no—
159C24×C3—22no—
160C24×C3C22×C414yes—
161C24×C3C22×C414yes—
162C24×C3C2412yes—
163C24×C3C2412yes—
164C24×C3C2411yescharacteristic
165C23×D3—24no—
166C23×D3—24no—
167C23×D3—24no—
168C23×D3—24no—
169C23×D3—264no—
170C23×D3—216no—
171C23×D3—216no—
172C23×D3—416no—
173C23×D3—44no—
174C23×D3—44no—
175C23×D3—416no—
176C23×D3—432no—
177C23×D3—416no—
178C23×D3C22×C418yes—
179C23×D3C22×C4116yescomplemented
180C23×D3C2×D418yes—
181C23×D3#4118yes—
182C23×D3C2414yes—
183C2×C3×D4—416no—
184C2×C3×D4—44no—
185C2×C3×D4—44no—
186C2×C3×D4C22×C4116yescomplemented
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—216no—
195C22×C3×C4—44no—
196C22×C3×C4—416no—
197C22×C3×C4C22×C414yes—
198C22×C3×C4C2×D418yes—
199C22×C3×C4C2414yes—
200C22×C3×C4C2412yes—
201C22×C3×C4C2414yes—
202C2×#67—264no—
203C2×#67—216no—
204C2×#67—216no—
205C2×#67C22×C4132yescomplemented
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3C22×C418yes—
214C22×Dic3C2×D418yes—
215C22×Dic3C2414yes—
216C22×Dic3C2414yes—
217C22×Dic3C2414yes—
218C2×C4×D3—216no—
219C2×C4×D3—216no—
220C2×C4×D3—216no—
221C2×C4×D3—216no—
222C2×C4×D3—48no—
223C2×C4×D3—48no—
224C2×C4×D3—432no—
225C2×C4×D3—432no—
226C3×C22⋊C4—216no—
227C3×C22⋊C4—44no—
228#217—216no—
229#217—216no—
230#217C22×C4116yescomplemented
231(C2×D3)⋊C4—216no—
232(C2×D3)⋊C4C22×C4132yescomplemented
233(C2×D3)⋊C4C2×D4132yescomplemented
234C4⋊Dic3—44no—
235Dic3⋊C4—216no—
236C4×Dic3—216no—
237C4×Dic3—44no—
238C25—32no—
239C25—38no—
240C25—34no—
241C25—68no—
242C25—68no—
243C25—122no—
244C25—122no—
245C25C22×D311yescharacteristic
246C25C22×D311yescharacteristic
247C22×D4—316no—
248C22×D4—38no—
249C22×D4—42no—
250C22×D4—42no—
251C22×D4—68no—
252C22×D4—68no—
253C22×D4C4×D318yescomplemented
254C22×D4C22×D314yes—
255C23×C4—28no—
256C23×C4—32no—
257C23×C4—32no—
258C23×C4—32no—
259C23×C4—38no—
260C23×C4—38no—
261C23×C4—42no—
262C23×C4—68no—
263C23×C4—68no—
264C23×C4—68no—
265C23×C4—68no—
266C23×C4—124no—
267C23×C4—124no—
268C23×C4C22×D312yes—
269C23×C4C22×D311yescharacteristic
270C23×C4C22×D312yes—
271C23×C4C22×D314yescomplemented
272#120—364no—
273C4×D4—664no—
274C2×C4⋊C4—38no—
275C2×C4⋊C4—34no—
276C2×C4⋊C4—68no—
277C2×C4⋊C4—122no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—38no—
280C2×C22⋊C4—34no—
281C2×C22⋊C4—316no—
282C2×C22⋊C4—316no—
283C2×C22⋊C4—38no—
284C2×C22⋊C4—38no—
285C2×C22⋊C4—38no—
286C2×C22⋊C4—38no—
287C2×C22⋊C4—38no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—664no—
290C2×C22⋊C4—68no—
291C2×C22⋊C4—68no—
292C2×C22⋊C4—68no—
293C2×C22⋊C4C22×D318yes—
294C2×C22⋊C4C22×D314yescomplemented
295C2×C42—38no—
296C2×C42—34no—
297C2×C42—68no—
298C2×C42—122no—
299#95—38no—
300#95—38no—
301#95C22×D318yescomplemented
302C23×C3—216no—
303C23×C3—216no—
304C23×C3—28no—
305C23×C3—28no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—48no—
317C23×C3—48no—
318C23×C3—44no—
319C23×C3—44no—
320C23×C3C23×C414yes—
321C23×C3C23×C414yes—
322C23×C3C23×C412yes—
323C23×C3C22×D414yes—
324C23×C3C2×#4114yes—
325C23×C3H(5,𝔽2)12yes—
326C23×C3H(5,𝔽2)12yes—
327C23×C3C2511yescharacteristic
328C22×D3—216no—
329C22×D3—216no—
330C22×D3—216no—
331C22×D3—216no—
332C22×D3—216no—
333C22×D3—216no—
334C22×D3—24no—
335C22×D3—24no—
336C22×D3—24no—
337C22×D3—24no—
338C22×D3—264no—
339C22×D3—416no—
340C22×D3—4128no—
341C22×D3—416no—
342C22×D3—432no—
343C22×D3—416no—
344C22×D3C4×D4132yescomplemented
345C22×D3C23×C418yes—
346C22×D3C22×D418yes—
347C22×D3C2×#4118yes—
348C3×D4—416no—
349C2×C3×C4—28no—
350C2×C3×C4—216no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—416no—
359C2×C3×C4—48no—
360C2×C3×C4C23×C414yes—
361C2×C3×C4C22×D418yes—
362#67—264no—
363C2×Dic3—216no—
364C2×Dic3—216no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—48no—
373C2×Dic3C23×C418yes—
374C2×Dic3C22×D418yes—
375C4×D3—432no—
376C4×D3—432no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—24no—
385C24—38no—
386C24—38no—
387C24—38no—
388C24—34no—
389C24—316no—
390C24—616no—
391C24—616no—
392C24—664no—
393C24—64no—
394C24—64no—
395C24—64no—
396C24—64no—
397C24—124no—
398C24—124no—
399C24—1216no—
400C24—1216no—
401C24—1216no—
402C24—1232no—
403C24C2×C4×D314yes—
404C24C2×C4×D314yes—
405C24C23×D312yes—
406C24C23×D312yes—
407C24C23×D311yescharacteristic
408C2×D4—332no—
409C2×D4—416no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—616no—
413C2×D4—616no—
414C2×D4—664no—
415C2×D4C2×C4×D3116yescomplemented
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—216no—
424C22×C4—38no—
425C22×C4—38no—
426C22×C4—34no—
427C22×C4—34no—
428C22×C4—34no—
429C22×C4—44no—
430C22×C4—416no—
431C22×C4—616no—
432C22×C4—616no—
433C22×C4—616no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—616no—
442C22×C4—128no—
443C22×C4—128no—
444C22×C4—1232no—
445C22×C4—1232no—
446C22×C4C2×C4×D314yes—
447C22×C4D3×D418yes—
448C22×C4C23×D314yes—
449C22×C4C23×D312yes—
450C22×C4C23×D314yes—
451C4⋊C4—616no—
452C4⋊C4—124no—
453C22⋊C4—216no—
454C22⋊C4—316no—
455C22⋊C4—332no—
456C22⋊C4—332no—
457C22⋊C4—44no—
458C22⋊C4—616no—
459C22⋊C4—616no—
460C22⋊C4—616no—
461C42—616no—
462C42—124no—
463C22×C3—28no—
464C22×C3—28no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—416no—
468C22×C3—416no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—44no—
472C22×C3—44no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3C2×C4×D418yes—
476C22×C3#51814yes—
477C22×C3D4214yes—
478C22×C3#54618yes—
479C22×C3#55114yes—
480C22×C3C24×C411yescharacteristic
481C22×C3C23×D412yes—
482C22×C3C22×#4112yes—
483C22×C3C2×H(5,𝔽2)11yescharacteristic
484C22×C3C2×H(5,𝔽2)11yescharacteristic
485C2×D3—264no—
486C2×D3—28no—
487C2×D3—28no—
488C2×D3—28no—
489C2×D3—28no—
490C2×D3—28no—
491C2×D3—28no—
492C2×D3—464no—
493C2×D3C2×C4×D4116yescomplemented
494C3×C4—24no—
495C3×C4—28no—
496C3×C4—44no—
497Dic3—28no—
498Dic3—28no—
499Dic3—44no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—216no—
511C23—216no—
512C23—38no—
513C23—38no—
514C23—38no—
515C23—332no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23—664no—
523C23—616no—
524C23—64no—
525C23—64no—
526C23—64no—
527C23—64no—
528C23—616no—
529C23—616no—
530C23—616no—
531C23—616no—
532C23—616no—
533C23—1216no—
534C23—12128no—
535C23—1216no—
536C23—1232no—
537C23—1216no—
538C23C22×C4×D314yes—
539C23C22×C4×D314yes—
540C23C22×C4×D312yes—
541C23C2×D3×D414yes—
542C23#100412yes—
543C23#100412yes—
544C23D3×#4114yes—
545C23C24×D311yescharacteristic
546D4—416no—
547D4—664no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—216no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—38no—
558C2×C4—38no—
559C2×C4—416no—
560C2×C4—48no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—616no—
566C2×C4—616no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—1232no—
571C2×C4—1232no—
572C2×C4—128no—
573C2×C4C22×C4×D314yes—
574C2×C4C2×D3×D418yes—
575C2×C3—44no—
576C2×C3—44no—
577C2×C3#235818yescomplemented
578C2×C3C22×C4×D412yes—
579C2×C3C2×#51811yescharacteristic
580C2×C3C2×D4211yescharacteristic
581C2×C3C2×#54612yes—
582C2×C3C2×#55111yescharacteristic
583D3—216no—
584C22—28no—
585C22—28no—
586C22—24no—
587C22—24no—
588C22—316no—
589C22—416no—
590C22—416no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—68no—
598C22—664no—
599C22—68no—
600C22—68no—
601C22—68no—
602C22—68no—
603C22—68no—
604C22—1264no—
605C22#548614yes—
606C22C4×D3×D418yes—
607C22#559014yes—
608C22#561218yes—
609C22#565914yes—
610C22C23×C4×D311yescharacteristic
611C22C22×D3×D412yes—
612C22C2×#100411yescharacteristic
613C22C2×#100411yescharacteristic
614C22C2×D3×#4112yes—
615C4—28no—
616C4—24no—
617C4—44no—
618C4—68no—
619C4—68no—
620C4—124no—
621C3C2×#235811yescharacteristic, complemented, Sylow
622C2—44no—
623C2—44no—
624C2—616no—
625C2#7488018yesdirect product factor
626C2C2×#548611yescharacteristic
627C2C2×C4×D3×D412yes—
628C2C2×#559011yescharacteristic
629C2C2×#561212yes—
630C2C2×#565911yescharacteristic
subgroupG/H
centerC24C23×D3
commutatorC22×C3C24×C4
FrattiniC23C24×D3
FittingC2×C3×#386C2
socleC24×C3C24
prime radicalC22×#937C2
cosocleC23×C3C25
hypercenterC2×#386D3
hypocenterC3C2×#2358
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C22×C4×#67C2semidirect
3. C22×D3×C22⋊C4C2semidirect
4. C2×C3×#386C2semidirect
5. C2×#5221C2semidirect
6. C2×#4956C2semidirect
7. C2×#4945C2semidirect
8. #74880C2direct (N × H)
9. C2×#4668C2semidirect
10. C2×#4666C2semidirect
14. C24×C3×C4C22semidirect
19. C22×C3×C22⋊C4C22semidirect
22. C22×#217C22semidirect
23. C2×#937C4semidirect
26. C2×C4×#67C22semidirect
30. C22×C4⋊Dic3C22semidirect
31. C22×Dic3⋊C4C22semidirect
32. C22×C4×Dic3C22semidirect
33. C22×C4×Dic3C22semidirect
35. C2×D3×C22⋊C4C22semidirect
36. C3×#386C22semidirect
37. C2×C3×#95C22semidirect
38. #5221C22semidirect
39. C2×C3⋊#95C22semidirect
40. C2×C3⋊#95C22semidirect
41. #4956C22semidirect
42. #4945C22semidirect
43. #4668C22semidirect
44. #4666C22semidirect
52. C2×#386D3semidirect
61. C24×D3C2×C4semidirect
66. C22×C3×D4C2×C4semidirect
73. C23×C3×C4C23semidirect
76. C22×#67C2×C4semidirect
92. C2×C3×C22⋊C4C23semidirect
95. C2×#217C2×C4semidirect
98. C2×#217C23semidirect
99. #937C2×C4semidirect
102. C2×(C2×D3)⋊C4C2×C4semidirect
103. C2×(C2×D3)⋊C4D4semidirect
108. C2×C4⋊Dic3C23semidirect
110. C2×Dic3⋊C4C23semidirect
113. C2×C4×Dic3C23semidirect
114. C2×C4×Dic3C23semidirect
116. C3×#95C23semidirect
117. C3⋊#95C23semidirect
118. C3⋊#95C23semidirect
124. C24×C4C2×D3semidirect
140. C22×C22⋊C4C2×D3semidirect
146. #386C2×D3semidirect
151. C2×#95C2×D3semidirect
179. C23×D3C22×C4semidirect
186. C2×C3×D4C22×C4semidirect
205. C2×#67C22×C4semidirect
230. #217C22×C4semidirect
232. (C2×D3)⋊C4C22×C4semidirect
233. (C2×D3)⋊C4C2×D4semidirect
253. C22×D4C4×D3semidirect
271. C23×C4C22×D3semidirect
294. C2×C22⋊C4C22×D3semidirect
301. #95C22×D3semidirect
344. C22×D3C4×D4semidirect
415. C2×D4C2×C4×D3semidirect
493. C2×D3C2×C4×D4semidirect
577. C2×C3#2358semidirect
621. C3C2×#2358semidirect
625. C2#74880direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9372
C22×C4×#672
C22×D3×C22⋊C42
C2×C3×#3862
C2×#52212
C2×#49562
C2×#49452
#748802
C2×#46682
C2×#46662
C2×#23583