Finite Group Databasebeta

← C2×#22561398 #30883295

Subgroups — C2×#22561398 #30883295

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×#14524017C211yesmaximal, characteristic
2C22×#14523974C211yesmaximal, characteristic, complemented
3C22×#14523963C211yesmaximal, characteristic, complemented
4C22×#14523963C211yesmaximal, characteristic, complemented
5C22×C22⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
6#22561398C218yesmaximal, direct product factor
7C2×#16863158C211yesmaximal, characteristic, complemented
8C2×#380×C67C211yesmaximal, characteristic, complemented
9C23×D268C2211yescharacteristic
10C23×C22⋊D67—21no—
11C23×C67⋊C8C2211yescharacteristic
12C23×C67⋊C8C2211yescharacteristic
13C2×#14524017—24no—
14C2×#14524017—24no—
15C2×#14524017—22no—
16C2×#14524017C2214yes—
17C22×C4⋊Dic67C2211yescharacteristic
18C22×(C2×D67)⋊C4—21no—
19C2×#14523974—24no—
20C2×#14523974C2214yescomplemented
21C2×#14523963—24no—
22C2×#14523963—24no—
23C2×#14523963C2214yescomplemented
24C2×#14523963C2214yescomplemented
25C2×C22⋊(C67⋊C8)—24no—
26C2×C22⋊(C67⋊C8)C2214yescomplemented
27#16863158C2214yescomplemented
28C24×C4×C67C2211yescharacteristic
29C22×C4⋊C4×C67C2211yescharacteristic
30C22×C4⋊C4×C67C2211yescharacteristic
31C22×C22⋊C4×C67—21no—
32#380×C67—22no—
33#380×C67C2214yescomplemented
34C2×#95×C67—21no—
35C24×D67—21no—
36C22×D268—22no—
37C22×D268—22no—
38C22×D268—21no—
39C22×D268—22no—
40C22×D268—22no—
41C22×D268—22no—
42C22×D268D414yes—
43C22×D268C2312yes—
44C22×C22⋊D67—28no—
45C22×C22⋊D67—24no—
46C22×C22⋊D67—44no—
47C22×C22⋊D67—44no—
48C23×Dic67—21no—
49C22×C67⋊C8—22no—
50C22×C67⋊C8—22no—
51C22×C67⋊C8—22no—
52C22×C67⋊C8—22no—
53C22×C67⋊C8—22no—
54C22×C67⋊C8D414yescomplemented
55C22×C67⋊C8C2312yes—
56C22×C67⋊C8C2312yes—
57#14524017—216no—
58#14524017—28no—
59#14524017—28no—
60C2×C4⋊Dic67—22no—
61C2×C4⋊Dic67—21no—
62C2×C4⋊Dic67D414yes—
63C2×C4⋊Dic67C2312yes—
64C2×(C2×D67)⋊C4—28no—
65C2×(C2×D67)⋊C4—24no—
66#14523974—28no—
67#14523963—28no—
68#14523963—28no—
69C22⋊(C67⋊C8)—28no—
70C25×C67D411yescharacteristic
71C23×C4×C67—22no—
72C23×C4×C67—22no—
73C23×C4×C67—22no—
74C23×C4×C67—22no—
75C23×C4×C67—22no—
76C23×C4×C67—22no—
77C23×C4×C67—21no—
78C23×C4×C67—21no—
79C23×C4×C67D411yescharacteristic
80C23×C4×C67D412yes—
81C23×C4×C67C2312yes—
82C23×C4×C67C2311yescharacteristic
83C2×C4⋊C4×C67—22no—
84C2×C4⋊C4×C67—22no—
85C2×C4⋊C4×C67—22no—
86C2×C4⋊C4×C67—22no—
87C2×C4⋊C4×C67—21no—
88C2×C4⋊C4×C67—21no—
89C2×C4⋊C4×C67C2312yes—
90C2×C4⋊C4×C67C2312yes—
91C2×C22⋊C4×C67—24no—
92C2×C22⋊C4×C67—44no—
93#95×C67—24no—
94C23×D67—24no—
95C23×D67—22no—
96C23×D67—42no—
97C23×D67—42no—
98C23×D67—42no—
99C23×D67—42no—
100C23×D67—42no—
101C23×D67—42no—
102C2×D268—28no—
103C2×D268—28no—
104C2×D268—22no—
105C2×D268—22no—
106C2×D268—22no—
107C2×D268—22no—
108C2×D268—24no—
109C2×D268—22no—
110C2×D268—42no—
111C2×D268—44no—
112C2×D268—42no—
113C2×D268—44no—
114C2×D268C2×D414yes—
115C2×C22⋊D67—216no—
116C2×C22⋊D67—48no—
117C2×C22⋊D67—432no—
118C2×C22⋊D67—48no—
119C22×Dic67—24no—
120C22×Dic67—22no—
121C22×Dic67—42no—
122C22×Dic67—42no—
123C2×C67⋊C8—28no—
124C2×C67⋊C8—22no—
125C2×C67⋊C8—22no—
126C2×C67⋊C8—22no—
127C2×C67⋊C8—22no—
128C2×C67⋊C8—22no—
129C2×C67⋊C8—44no—
130C2×C67⋊C8C2×D414yescomplemented
131C4⋊Dic67—24no—
132C4⋊Dic67—22no—
133C4⋊Dic67C2×D414yes—
134(C2×D67)⋊C4—216no—
135C24×C67—22no—
136C24×C67—22no—
137C24×C67—22no—
138C24×C67—22no—
139C24×C67—22no—
140C24×C67—22no—
141C24×C67SD1614yes—
142C24×C67C2×D412yes—
143C24×C67C2×D411yescharacteristic
144C22×C4×C67—22no—
145C22×C4×C67—22no—
146C22×C4×C67—22no—
147C22×C4×C67—24no—
148C22×C4×C67—24no—
149C22×C4×C67—24no—
150C22×C4×C67—22no—
151C22×C4×C67—22no—
152C22×C4×C67—22no—
153C22×C4×C67—22no—
154C22×C4×C67—22no—
155C22×C4×C67—22no—
156C22×C4×C67—22no—
157C22×C4×C67—22no—
158C22×C4×C67—44no—
159C22×C4×C67—44no—
160C22×C4×C67—44no—
161C22×C4×C67—44no—
162C22×C4×C67—42no—
163C22×C4×C67—42no—
164C22×C4×C67—42no—
165C22×C4×C67—42no—
166C22×C4×C67—42no—
167C22×C4×C67—42no—
168C22×C4×C67C2×D412yes—
169C22×C4×C67C2×D412yes—
170C22×C4×C67C2×D412yes—
171C22×C4×C67C2×D412yes—
172C22×C4×C67C2×D411yescharacteristic
173C22×C4×C67#4112yes—
174C22×C4×C67#4112yes—
175C22×C4×C67#4112yes—
176C22×C4×C67C2411yescharacteristic
177C4⋊C4×C67—22no—
178C4⋊C4×C67—22no—
179C4⋊C4×C67—22no—
180C4⋊C4×C67—22no—
181C4⋊C4×C67—42no—
182C4⋊C4×C67—42no—
183C22⋊C4×C67—48no—
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—44no—
194C23×C67—44no—
195C23×C67—44no—
196C23×C67—44no—
197C23×C67#12012yes—
198C23×C67#12112yes—
199C23×C67#12312yes—
200C23×C67C2×SD1614yes—
201C23×C67C2×SD1612yes—
202C23×C67#13512yes—
203C23×C67AGL(1,ℤ/8)12yes—
204C23×C67AGL(1,ℤ/8)12yes—
205C23×C67C22×D411yescharacteristic
206C22×D67—24no—
207C22×D67—42no—
208C22×D67—42no—
209C22×D67—42no—
210C22×D67—42no—
211C22×D67—42no—
212C22×D67—42no—
213C22×D67—48no—
214C22×D67—48no—
215C22×D67—48no—
216C22×D67—84no—
217C22×D67—84no—
218C22×D67—84no—
219C22×D67—84no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—22no—
223C2×C4×C67—44no—
224C2×C4×C67—42no—
225C2×C4×C67—42no—
226C2×C4×C67—42no—
227C2×C4×C67—42no—
228C2×C4×C67—48no—
229C2×C4×C67—42no—
230C2×C4×C67—42no—
231C2×C4×C67—42no—
232C2×C4×C67—42no—
233C2×C4×C67—42no—
234C2×C4×C67—42no—
235C2×C4×C67—84no—
236C2×C4×C67—84no—
237C2×C4×C67#12012yes—
238C2×C4×C67#12112yes—
239C2×C4×C67#12114yes—
240C2×C4×C67#12114yes—
241C2×C4×C67#12314yes—
242C2×C4×C67#12312yes—
243C2×C4×C67#12414yes—
244C2×C4×C67C22×D411yescharacteristic
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C2×#4111yescharacteristic
248C2×C4×C67C2×#4111yescharacteristic
249C2×C4×C67C2×#4111yescharacteristic
250C22⋊D67—432no—
251C2×Dic67—24no—
252C2×Dic67—42no—
253C2×Dic67—42no—
254C2×Dic67—48no—
255D268—24no—
256D268—24no—
257D268—42no—
258D268—42no—
259D268—48no—
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#39414yes—
271C22×C67#44914yes—
272C22×C67#45014yes—
273C22×C67#46514yes—
274C22×C67#46914yes—
275C22×C67#48114yes—
276C22×C67#48214yes—
277C22×C67C2×#12011yescharacteristic
278C22×C67C2×#12111yescharacteristic
279C22×C67C2×#12311yescharacteristic
280C22×C67C22×SD1611yescharacteristic
281C22×C67C2×#13511yescharacteristic
282C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
284C2×D67—44no—
285C2×D67—44no—
286C2×D67—44no—
287C2×D67—82no—
288C2×D67—82no—
289C2×D67—82no—
290C2×D67—82no—
291C2×D67—816no—
292C4×C67—42no—
293C4×C67—82no—
294C4×C67—82no—
295C4×C67#39414yes—
296C4×C67C2×#12111yescharacteristic
297C4×C67C2×#12111yescharacteristic
298C4×C67C2×#12311yescharacteristic
299C4×C67C2×#12411yescharacteristic
300Dic67—44no—
301C2×#2034—671nomaximal, Sylow
302C2×C67—44no—
303C2×C67#203418yescomplemented
304C2×C67C2×#39411yescharacteristic
305C2×C67C2×#45011yescharacteristic
306C2×C67C2×#44911yescharacteristic
307C2×C67C2×#46511yescharacteristic
308C2×C67C2×#46911yescharacteristic
309C2×C67C2×#48211yescharacteristic
310C2×C67C2×#48111yescharacteristic
311D67—84no—
312C22×#121—671no—
313C22×#106—671no—
314C22×#102—671no—
315C22×#102—671no—
316C22×C22⋊C8—671no—
317C2×#380D6711yescharacteristic, complemented
318#2034—678no—
319C2×#340—671no—
320C67C2×#203411yescharacteristic, complemented, Sylow
321C23×D4—671no—
322C23×D4—1341no—
323C24×C4C2×D6711yescharacteristic
324C23×C8—671no—
325C23×C8—671no—
326C2×#121—674no—
327C2×#121—1344no—
328C2×#121—1344no—
329C2×#121—1342no—
330C22×C4⋊C4—671no—
331C22×C4⋊C4C2×D6711yescharacteristic
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—22no—
344#380C2×D6714yescomplemented
345C2×#95—21no—
346#340—674no—
347C25—1341no—
348C25C22⋊D6711yescharacteristic
349C22×D4—2684no—
350C22×D4—2684no—
351C22×D4—674no—
352C22×D4—672no—
353C22×D4—1348no—
354C22×D4—1344no—
355C22×D4—1342no—
356C22×D4—1342no—
357C22×D4—1341no—
358C22×D4—1342no—
359C22×D4—1342no—
360C22×D4—1342no—
361C23×C4—22no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—21no—
368C23×C4—21no—
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—1348no—
383#121—1348no—
384#121—13416no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—674no—
392C2×C4⋊C4—672no—
393C2×C4⋊C4—1342no—
394C2×C4⋊C4—1341no—
395C2×C4⋊C4C22×D6712yes—
396C2×C4⋊C4C22×D6712yes—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—44no—
399C2×C22⋊C4—1348no—
400C2×C22⋊C4—1344no—
401#106—1348no—
402#102—1348no—
403#102—1348no—
404C22⋊C8—1348no—
405#95—24no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—2682no—
413C24—2682no—
414C24—2682no—
415C24—2682no—
416C24—2682no—
417C24—2682no—
418C24—1344no—
419C24—1342no—
420C24#1176161514yes—
421C24C2×C22⋊D6712yes—
422C24C2×C22⋊D6711yescharacteristic
423C2×D4—26832no—
424C2×D4—2688no—
425C2×D4—2688no—
426C2×D4—2682no—
427C2×D4—2684no—
428C2×D4—2682no—
429C2×D4—2684no—
430C2×D4—674no—
431C2×D4—13416no—
432C2×D4—1344no—
433C2×D4—1348no—
434C2×D4—1348no—
435C2×D4—1342no—
436C2×D4—1342no—
437C2×D4—1342no—
438C2×D4—1342no—
439C2×D4—1342no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—2682no—
465C22×C4—2682no—
466C22×C4—1344no—
467C22×C4—1342no—
468C22×C4D4×D6712yes—
469C22×C4D4×D6712yes—
470C22×C4D4×D6712yes—
471C22×C4C2×C22⋊D6712yes—
472C22×C4C2×C22⋊D6711yescharacteristic
473C22×C4#1176162712yes—
474C22×C4#1176162812yes—
475C22×C4#1176162812yes—
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—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—674no—
492C4⋊C4—1344no—
493C4⋊C4—1342no—
494C22⋊C4—48no—
495C22⋊C4—13416no—
496C23—28no—
497C23—22no—
498C23—28no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—2688no—
510C23—2688no—
511C23—2688no—
512C23—2682no—
513C23—2682no—
514C23—2682no—
515C23—2682no—
516C23—2682no—
517C23—2682no—
518C23—5364no—
519C23—5364no—
520C23—5364no—
521C23—5364no—
522C23—1344no—
523C23#1452401112yes—
524C23#1452401512yes—
525C23#1452402612yes—
526C23C2×#1176161514yes—
527C23C2×#1176161512yes—
528C23#1452406312yes—
529C23#1452406812yes—
530C23#1452406912yes—
531C23C22×C22⋊D6711yescharacteristic
532D4—26832no—
533D4—2688no—
534D4—2682no—
535D4—2682no—
536D4—1344no—
537D4—1344no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—2688no—
556C2×C4—2682no—
557C2×C4—2682no—
558C2×C4—1344no—
559C2×C4#1452401112yes—
560C2×C4#1452401512yes—
561C2×C4#1452401614yes—
562C2×C4#1452401614yes—
563C2×C4#1452402514yes—
564C2×C4#1452402612yes—
565C2×C4#1452402914yes—
566C2×C4C2×D4×D6711yescharacteristic
567C2×C4C2×D4×D6711yescharacteristic
568C2×C4C2×D4×D6711yescharacteristic
569C2×C4C2×#1176162711yescharacteristic
570C2×C4C2×#1176162811yescharacteristic
571C2×C4C2×#1176162811yescharacteristic
572C8—2682no—
573C8—1344no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—416no—
581C22—42no—
582C22—2684no—
583C22—2684no—
584C22—2684no—
585C22—5362no—
586C22—5362no—
587C22—5362no—
588C22—53616no—
589C22—5362no—
590C22#1686331114yes—
591C22#1686353914yes—
592C22#1686354614yes—
593C22#1686362714yes—
594C22#1686364714yes—
595C22#1686371414yes—
596C22#1686372014yes—
597C22C2×#1452401111yescharacteristic
598C22C2×#1452401511yescharacteristic
599C22C2×#1452402611yescharacteristic
600C22C22×#1176161511yescharacteristic
601C22C2×#1452406311yescharacteristic
602C22C2×#1452406811yescharacteristic
603C22C2×#1452406911yescharacteristic
604C4—42no—
605C4—82no—
606C4—82no—
607C4—2684no—
608C4#1686331114yes—
609C4C2×#1452401611yescharacteristic
610C4C2×#1452401611yescharacteristic
611C4C2×#1452402511yescharacteristic
612C4C2×#1452402911yescharacteristic
613C2—44no—
614C2—5364no—
615C2#2256139818yesdirect product factor
616C2C2×#1686331111yescharacteristic
617C2C2×#1686354611yescharacteristic
618C2C2×#1686353911yescharacteristic
619C2C2×#1686362711yescharacteristic
620C2C2×#1686364711yescharacteristic
621C2C2×#1686372011yescharacteristic
622C2C2×#1686371411yescharacteristic
subgroupG/H
centerC24C2×C22⋊D67
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
FittingC2×#380×C67C2
socleC24×C67C2×D4
prime radicalC22×#14524017C2
hypercenterC2×#380D67
hypocenterC67C2×#2034
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#14523974C2semidirect
3. C22×#14523963C2semidirect
4. C22×#14523963C2semidirect
5. C22×C22⋊(C67⋊C8)C2semidirect
6. #22561398C2direct (N × H)
7. C2×#16863158C2semidirect
8. C2×#380×C67C2semidirect
20. C2×#14523974C22semidirect
23. C2×#14523963C22semidirect
24. C2×#14523963C22semidirect
26. C2×C22⋊(C67⋊C8)C22semidirect
27. #16863158C22semidirect
33. #380×C67C22semidirect
54. C22×C67⋊C8D4semidirect
130. C2×C67⋊C8C2×D4semidirect
303. C2×C67#2034semidirect
317. C2×#380D67semidirect
320. C67C2×#2034semidirect
344. #380C2×D67semidirect
615. C2#22561398direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240172
C22×#145239742
C22×#145239632
C22×C22⋊(C67⋊C8)2
#225613982
C2×#168631582
C2×#380×C672
C2×#203467