Finite Group Databasebeta

← C2×#22561395 #30883292

Subgroups — C2×#22561395 #30883292

Order 17152.

2006 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523974C211yesmaximal, characteristic, complemented
2C22×#14523964C211yesmaximal, characteristic, complemented
3C22×#14523964C211yesmaximal, characteristic, complemented
4C22×C22⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
5C2×#16863256C211yesmaximal, characteristic, complemented
6#22561395C218yesmaximal, direct product factor
7C2×#16863158C211yesmaximal, characteristic, complemented
8C22×#121×C67C211yesmaximal, characteristic
9C23×C67⋊C8C2211yescharacteristic
10C23×C67⋊C8C2211yescharacteristic
11C22×C4⋊Dic67C2211yescharacteristic
12C22×C4⋊Dic67C2211yescharacteristic
13C22×C22⋊Dic67—21no—
14C2×#14523974—24no—
15C2×#14523974C2214yescomplemented
16C2×#14523964—24no—
17C2×#14523964—24no—
18C2×#14523964C2214yescomplemented
19C2×#14523964C2214yescomplemented
20C2×C22⋊(C67⋊C8)—24no—
21C2×C22⋊(C67⋊C8)C2214yescomplemented
22#16863256—22no—
23#16863256C2214yescomplemented
24C2×C67⋊#95—21no—
25#16863158C2214yescomplemented
26C23×D4×C67—21no—
27C23×D4×C67C2211yescharacteristic
28C24×C4×C67C2211yescharacteristic
29C2×#121×C67—24no—
30C2×#121×C67—22no—
31C2×#121×C67—24no—
32C2×#121×C67C2214yes—
33C22×C4⋊C4×C67C2211yescharacteristic
34C22×C22⋊C4×C67—21no—
35C23×Dic67—21no—
36C23×Dic67—21no—
37C22×C67⋊C8—22no—
38C22×C67⋊C8—22no—
39C22×C67⋊C8—22no—
40C22×C67⋊C8—22no—
41C22×C67⋊C8—22no—
42C22×C67⋊C8D414yescomplemented
43C22×C67⋊C8C2312yes—
44C22×C67⋊C8C2312yes—
45C2×C4⋊Dic67—22no—
46C2×C4⋊Dic67—22no—
47C2×C4⋊Dic67—22no—
48C2×C4⋊Dic67—21no—
49C2×C4⋊Dic67—21no—
50C2×C4⋊Dic67—22no—
51C2×C4⋊Dic67C2312yes—
52C2×C4⋊Dic67C2312yes—
53C2×C22⋊Dic67—24no—
54C2×C22⋊Dic67—44no—
55#14523974—28no—
56#14523964—28no—
57#14523964—28no—
58C22⋊(C67⋊C8)—28no—
59C67⋊#95—24no—
60C25×C67—21no—
61C25×C67D411yescharacteristic
62C22×D4×C67—28no—
63C22×D4×C67—24no—
64C22×D4×C67—22no—
65C22×D4×C67—21no—
66C22×D4×C67—22no—
67C22×D4×C67—22no—
68C22×D4×C67—22no—
69C22×D4×C67—22no—
70C22×D4×C67—44no—
71C22×D4×C67—44no—
72C22×D4×C67D414yes—
73C22×D4×C67C2312yes—
74C23×C4×C67—21no—
75C23×C4×C67—22no—
76C23×C4×C67—22no—
77C23×C4×C67—22no—
78C23×C4×C67—22no—
79C23×C4×C67—22no—
80C23×C4×C67—22no—
81C23×C4×C67D411yescharacteristic
82C23×C4×C67D412yes—
83C23×C4×C67C2312yes—
84C23×C4×C67C2311yescharacteristic
85#121×C67—216no—
86#121×C67—28no—
87#121×C67—28no—
88C2×C4⋊C4×C67—21no—
89C2×C4⋊C4×C67—22no—
90C2×C4⋊C4×C67D414yes—
91C2×C4⋊C4×C67C2312yes—
92C2×C22⋊C4×C67—28no—
93C2×C22⋊C4×C67—24no—
94C22×Dic67—22no—
95C22×Dic67—22no—
96C22×Dic67—42no—
97C22×Dic67—42no—
98C22×Dic67—42no—
99C22×Dic67—42no—
100C22×Dic67—42no—
101C22×Dic67—42no—
102C2×C67⋊C8—28no—
103C2×C67⋊C8—22no—
104C2×C67⋊C8—22no—
105C2×C67⋊C8—22no—
106C2×C67⋊C8—22no—
107C2×C67⋊C8—22no—
108C2×C67⋊C8—44no—
109C2×C67⋊C8C2×D414yescomplemented
110C4⋊Dic67—22no—
111C4⋊Dic67—22no—
112C4⋊Dic67—22no—
113C4⋊Dic67—22no—
114C4⋊Dic67—42no—
115C4⋊Dic67—42no—
116C22⋊Dic67—48no—
117C24×C67—22no—
118C24×C67—22no—
119C24×C67—22no—
120C24×C67—22no—
121C24×C67—22no—
122C24×C67—22no—
123C24×C67—24no—
124C24×C67—22no—
125C24×C67—42no—
126C24×C67—42no—
127C24×C67—42no—
128C24×C67—42no—
129C24×C67—42no—
130C24×C67—42no—
131C24×C67SD1614yes—
132C24×C67C2×D412yes—
133C24×C67C2×D411yescharacteristic
134C2×D4×C67—216no—
135C2×D4×C67—24no—
136C2×D4×C67—28no—
137C2×D4×C67—28no—
138C2×D4×C67—22no—
139C2×D4×C67—22no—
140C2×D4×C67—22no—
141C2×D4×C67—22no—
142C2×D4×C67—22no—
143C2×D4×C67—48no—
144C2×D4×C67—432no—
145C2×D4×C67—48no—
146C2×D4×C67—44no—
147C2×D4×C67—42no—
148C2×D4×C67—42no—
149C2×D4×C67—44no—
150C2×D4×C67C2×D414yes—
151C22×C4×C67—22no—
152C22×C4×C67—24no—
153C22×C4×C67—22no—
154C22×C4×C67—22no—
155C22×C4×C67—22no—
156C22×C4×C67—24no—
157C22×C4×C67—24no—
158C22×C4×C67—24no—
159C22×C4×C67—22no—
160C22×C4×C67—22no—
161C22×C4×C67—22no—
162C22×C4×C67—22no—
163C22×C4×C67—22no—
164C22×C4×C67—22no—
165C22×C4×C67—42no—
166C22×C4×C67—42no—
167C22×C4×C67—44no—
168C22×C4×C67—44no—
169C22×C4×C67—44no—
170C22×C4×C67—44no—
171C22×C4×C67C2×D412yes—
172C22×C4×C67C2×D412yes—
173C22×C4×C67C2×D411yescharacteristic
174C22×C4×C67C2×D412yes—
175C22×C4×C67C2×D412yes—
176C22×C4×C67#4112yes—
177C22×C4×C67#4112yes—
178C22×C4×C67#4112yes—
179C22×C4×C67C2411yescharacteristic
180C4⋊C4×C67—24no—
181C4⋊C4×C67—22no—
182C4⋊C4×C67C2×D414yes—
183C22⋊C4×C67—216no—
184C23×C67—22no—
185C23×C67—22no—
186C23×C67—22no—
187C23×C67—22no—
188C23×C67—22no—
189C23×C67—22no—
190C23×C67—28no—
191C23×C67—28no—
192C23×C67—28no—
193C23×C67—24no—
194C23×C67—44no—
195C23×C67—44no—
196C23×C67—44no—
197C23×C67—44no—
198C23×C67—42no—
199C23×C67—42no—
200C23×C67—42no—
201C23×C67—42no—
202C23×C67—42no—
203C23×C67—42no—
204C23×C67—48no—
205C23×C67—48no—
206C23×C67—48no—
207C23×C67—84no—
208C23×C67—84no—
209C23×C67—84no—
210C23×C67—84no—
211C23×C67#12012yes—
212C23×C67#12112yes—
213C23×C67#12312yes—
214C23×C67C2×SD1614yes—
215C23×C67C2×SD1612yes—
216C23×C67#13512yes—
217C23×C67AGL(1,ℤ/8)12yes—
218C23×C67AGL(1,ℤ/8)12yes—
219C23×C67C22×D411yescharacteristic
220D4×C67—24no—
221D4×C67—24no—
222D4×C67—432no—
223D4×C67—42no—
224D4×C67—42no—
225D4×C67—48no—
226C2×C4×C67—22no—
227C2×C4×C67—24no—
228C2×C4×C67—22no—
229C2×C4×C67—22no—
230C2×C4×C67—42no—
231C2×C4×C67—42no—
232C2×C4×C67—44no—
233C2×C4×C67—48no—
234C2×C4×C67—42no—
235C2×C4×C67—42no—
236C2×C4×C67—42no—
237C2×C4×C67—42no—
238C2×C4×C67—48no—
239C2×C4×C67#12012yes—
240C2×C4×C67#12114yes—
241C2×C4×C67#12114yes—
242C2×C4×C67#12112yes—
243C2×C4×C67#12314yes—
244C2×C4×C67#12312yes—
245C2×C4×C67#12414yes—
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C22×D411yescharacteristic
248C2×C4×C67C22×D411yescharacteristic
249C2×C4×C67C2×#4111yescharacteristic
250C2×C4×C67C2×#4111yescharacteristic
251C2×C4×C67C2×#4111yescharacteristic
252C2×Dic67—42no—
253C2×Dic67—42no—
254C2×Dic67—42no—
255C2×Dic67—42no—
256C2×Dic67—42no—
257C2×Dic67—42no—
258C2×Dic67—84no—
259C2×Dic67—84no—
260C67⋊C8—24no—
261C67⋊C8—42no—
262C22×C67—24no—
263C22×C67—24no—
264C22×C67—24no—
265C22×C67—42no—
266C22×C67—42no—
267C22×C67—42no—
268C22×C67—42no—
269C22×C67—416no—
270C22×C67—44no—
271C22×C67—44no—
272C22×C67—44no—
273C22×C67—82no—
274C22×C67—82no—
275C22×C67—82no—
276C22×C67—82no—
277C22×C67—816no—
278C22×C67#39414yes—
279C22×C67#44914yes—
280C22×C67#45014yes—
281C22×C67#46514yes—
282C22×C67#46914yes—
283C22×C67#48114yes—
284C22×C67#48214yes—
285C22×C67C2×#12011yescharacteristic
286C22×C67C2×#12111yescharacteristic
287C22×C67C2×#12311yescharacteristic
288C22×C67C22×SD1611yescharacteristic
289C22×C67C2×#13511yescharacteristic
290C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C67—42no—
293C4×C67—44no—
294C4×C67#39414yes—
295C4×C67C2×#12111yescharacteristic
296C4×C67C2×#12111yescharacteristic
297C4×C67C2×#12311yescharacteristic
298C4×C67C2×#12411yescharacteristic
299Dic67—82no—
300Dic67—82no—
301C2×#2034—671nomaximal, Sylow
302C2×C67—44no—
303C2×C67—84no—
304C2×C67#203418yescomplemented
305C2×C67C2×#39411yescharacteristic
306C2×C67C2×#45011yescharacteristic
307C2×C67C2×#44911yescharacteristic
308C2×C67C2×#46511yescharacteristic
309C2×C67C2×#46911yescharacteristic
310C2×C67C2×#48211yescharacteristic
311C2×C67C2×#48111yescharacteristic
312C22×#121D6711yescharacteristic
313C22×#106—671no—
314C22×#102—671no—
315C22×#102—671no—
316C22×C22⋊C8—671no—
317C2×#380—671no—
318#2034—678no—
319C2×#340—671no—
320C67C2×#203411yescharacteristic, complemented, Sylow
321C23×D4—21no—
322C23×D4C2×D6711yescharacteristic
323C24×C4C2×D6711yescharacteristic
324C23×C8—671no—
325C23×C8—671no—
326C2×#121—24no—
327C2×#121—22no—
328C2×#121—24no—
329C2×#121C2×D6714yes—
330C22×C4⋊C4—671no—
331C22×C4⋊C4—671no—
332C22×C4⋊C4C2×D6711yescharacteristic
333C22×C22⋊C4—21no—
334C22×C22⋊C4—1341no—
335C2×#106—674no—
336C2×#106—1344no—
337C2×#102—674no—
338C2×#102—674no—
339C2×#102—1344no—
340C2×#102—1344no—
341C2×C22⋊C8—674no—
342C2×C22⋊C8—1344no—
343#380—674no—
344#380—1342no—
345C2×#95—1341no—
346#340—674no—
347C25—21no—
348C25C22⋊D6711yescharacteristic
349C22×D4—28no—
350C22×D4—24no—
351C22×D4—22no—
352C22×D4—21no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—22no—
356C22×D4—22no—
357C22×D4—44no—
358C22×D4—44no—
359C22×D4C22⋊D6714yes—
360C22×D4C22×D6712yes—
361C23×C4—21no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—1341no—
369C23×C4—1341no—
370C23×C4C22⋊D6712yes—
371C23×C4C22⋊D6711yescharacteristic
372C23×C4C22×D6712yes—
373C23×C4C22×D6711yescharacteristic
374C22×C8—674no—
375C22×C8—672no—
376C22×C8—672no—
377C22×C8—1342no—
378C22×C8—1342no—
379C22×C8—1342no—
380C22×C8—1342no—
381C22×C8—1342no—
382#121—216no—
383#121—28no—
384#121—28no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—672no—
388C2×C4⋊C4—672no—
389C2×C4⋊C4—1341no—
390C2×C4⋊C4—1341no—
391C2×C4⋊C4—1342no—
392C2×C4⋊C4—1342no—
393C2×C4⋊C4—1342no—
394C2×C4⋊C4—1342no—
395C2×C4⋊C4C22⋊D6714yes—
396C2×C4⋊C4C22×D6712yes—
397C2×C22⋊C4—28no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—2684no—
400C2×C22⋊C4—1344no—
401#106—1348no—
402#102—1348no—
403#102—1348no—
404C22⋊C8—1348no—
405#95—1344no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—24no—
413C24—22no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24#1176161614yes—
421C24C2×C22⋊D6712yes—
422C24C2×C22⋊D6711yescharacteristic
423C2×D4—216no—
424C2×D4—24no—
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—432no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4C2×C22⋊D6714yes—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—2682no—
461C22×C4—2682no—
462C22×C4—2682no—
463C22×C4—2682no—
464C22×C4—2682no—
465C22×C4—2682no—
466C22×C4—1342no—
467C22×C4—1342no—
468C22×C4D4×D6712yes—
469C22×C4C2×C22⋊D6712yes—
470C22×C4C2×C22⋊D6711yescharacteristic
471C22×C4C2×C22⋊D6712yes—
472C22×C4C2×C22⋊D6712yes—
473C22×C4#1176162712yes—
474C22×C4#1176162712yes—
475C22×C4#1176162712yes—
476C22×C4C23×D6711yescharacteristic
477C2×C8—2684no—
478C2×C8—674no—
479C2×C8—1342no—
480C2×C8—1342no—
481C2×C8—1342no—
482C2×C8—1348no—
483C2×C8—1342no—
484C2×C8—1342no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—2682no—
488C4⋊C4—2682no—
489C4⋊C4—1342no—
490C4⋊C4—1342no—
491C4⋊C4—1342no—
492C4⋊C4—1342no—
493C4⋊C4C2×C22⋊D6714yes—
494C22⋊C4—216no—
495C22⋊C4—2688no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—24no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23#1452401212yes—
524C23#1452401512yes—
525C23#1452402412yes—
526C23C2×#1176161614yes—
527C23C2×#1176161612yes—
528C23#1452406312yes—
529C23#1452406812yes—
530C23#1452406912yes—
531C23C22×C22⋊D6711yescharacteristic
532D4—24no—
533D4—24no—
534D4—432no—
535D4—48no—
536D4—42no—
537D4—42no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—2682no—
552C2×C4—2682no—
553C2×C4—2682no—
554C2×C4—2682no—
555C2×C4—2682no—
556C2×C4—2682no—
557C2×C4—5364no—
558C2×C4—5364no—
559C2×C4#1452401212yes—
560C2×C4#1452401414yes—
561C2×C4#1452401512yes—
562C2×C4#1452401514yes—
563C2×C4#1452402412yes—
564C2×C4#1452402414yes—
565C2×C4#1452403014yes—
566C2×C4C2×D4×D6711yescharacteristic
567C2×C4C22×C22⋊D6711yescharacteristic
568C2×C4C22×C22⋊D6711yescharacteristic
569C2×C4C2×#1176162711yescharacteristic
570C2×C4C2×#1176162711yescharacteristic
571C2×C4C2×#1176162711yescharacteristic
572C8—2682no—
573C8—1344no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—416no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—816no—
589C22—82no—
590C22#1686331014yes—
591C22#1686354114yes—
592C22#1686354714yes—
593C22#1686362314yes—
594C22#1686364714yes—
595C22#1686371214yes—
596C22#1686371714yes—
597C22C2×#1452401211yescharacteristic
598C22C2×#1452401511yescharacteristic
599C22C2×#1452402411yescharacteristic
600C22C22×#1176161611yescharacteristic
601C22C2×#1452406311yescharacteristic
602C22C2×#1452406811yescharacteristic
603C22C2×#1452406911yescharacteristic
604C4—42no—
605C4—44no—
606C4—5362no—
607C4—5362no—
608C4#1686331014yes—
609C4C2×#1452401411yescharacteristic
610C4C2×#1452401511yescharacteristic
611C4C2×#1452402411yescharacteristic
612C4C2×#1452403011yescharacteristic
613C2—44no—
614C2—84no—
615C2#2256139518yesdirect product factor
616C2C2×#1686331011yescharacteristic
617C2C2×#1686354711yescharacteristic
618C2C2×#1686354111yescharacteristic
619C2C2×#1686362311yescharacteristic
620C2C2×#1686364711yescharacteristic
621C2C2×#1686371711yescharacteristic
622C2C2×#1686371211yescharacteristic
subgroupG/H
centerC24C2×C22⋊D67
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
Fitting, prime radicalC22×#121×C67C2
socleC24×C67C2×D4
hypercenterC22×#121D67
hypocenterC67C2×#2034
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14523974C2semidirect
2. C22×#14523964C2semidirect
3. C22×#14523964C2semidirect
4. C22×C22⋊(C67⋊C8)C2semidirect
5. C2×#16863256C2semidirect
6. #22561395C2direct (N × H)
7. C2×#16863158C2semidirect
15. C2×#14523974C22semidirect
18. C2×#14523964C22semidirect
19. C2×#14523964C22semidirect
21. C2×C22⋊(C67⋊C8)C22semidirect
23. #16863256C22semidirect
25. #16863158C22semidirect
42. C22×C67⋊C8D4semidirect
109. C2×C67⋊C8C2×D4semidirect
304. C2×C67#2034semidirect
320. C67C2×#2034semidirect
615. C2#22561395direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145239742
C22×#145239642
C22×C22⋊(C67⋊C8)2
C2×#168632562
#225613952
C2×#168631582
C22×#121×C672
C2×#203467