Finite Group Databasebeta

← C2×#22560560 #30857158

Subgroups — C2×#22560560 #30857158

Order 17152.

2470 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11761613C211yesmaximal, characteristic
2C22×C4×D268C211yesmaximal, characteristic, complemented
3C22×#14523964C211yesmaximal, characteristic, complemented
4C22×#14523963C211yesmaximal, characteristic, complemented
5#22560560C218yesmaximal, direct product factor
6C2×#16863158C211yesmaximal, characteristic, complemented
7C2×#16863151C211yesmaximal, characteristic, complemented
8C2×#390×C67C211yesmaximal, characteristic, complemented
9C23×D268C2211yescharacteristic
10C23×C4×D67—21no—
11C22×#11761613—24no—
12C22×#11761613—24no—
13C22×#11761613C418yescomplemented
14C22×#11761613C2214yes—
15C23×C67⋊C8C2211yescharacteristic
16C23×C67⋊C8C2211yescharacteristic
17C2×C4×D268—24no—
18C2×C4×D268—24no—
19C2×C4×D268—22no—
20C2×C4×D268C2214yescomplemented
21C22×C4⋊Dic67C2211yescharacteristic
22C22×(C2×D67)⋊C4—21no—
23C2×#14523964—24no—
24C2×#14523964C2214yescomplemented
25C2×#14523963—24no—
26C2×#14523963C2214yescomplemented
27#16863158C2214yescomplemented
28#16863151C2214yescomplemented
29C23×D4×C67C2211yescharacteristic
30C22×C4⋊C4×C67C2211yescharacteristic
31C22×C22⋊C4×C67—21no—
32C22×C22⋊C4×C67—21no—
33C22×C42×C67C2211yescharacteristic
34#390×C67—22no—
35#390×C67C2214yescomplemented
36C24×D67—21no—
37C22×D268—22no—
38C22×D268—21no—
39C22×D268—22no—
40C22×D268—22no—
41C22×D268C2×C414yes—
42C22×D268D414yes—
43C22×D268D414yes—
44C22×D268C2312yes—
45C23×Dic67—21no—
46C22×C4×D67—28no—
47C22×C4×D67—24no—
48C22×C4×D67—44no—
49C22×C4×D67—44no—
50C2×#11761613—232no—
51C2×#11761613—28no—
52C2×#11761613—28no—
53C2×#11761613C2×C4116yescomplemented
54C22×C67⋊C8—22no—
55C22×C67⋊C8—22no—
56C22×C67⋊C8—22no—
57C22×C67⋊C8—22no—
58C22×C67⋊C8—22no—
59C22×C67⋊C8C2×C414yescomplemented
60C22×C67⋊C8C2312yes—
61C22×C67⋊C8C2312yes—
62C4×D268—216no—
63C4×D268—28no—
64C4×D268—28no—
65C2×C4⋊Dic67—22no—
66C2×C4⋊Dic67—21no—
67C2×C4⋊Dic67D414yes—
68C2×C4⋊Dic67C2312yes—
69C2×(C2×D67)⋊C4—28no—
70C2×(C2×D67)⋊C4—24no—
71#14523964—28no—
72#14523963—28no—
73C25×C67—21no—
74C22×D4×C67—22no—
75C22×D4×C67—22no—
76C22×D4×C67—22no—
77C22×D4×C67—21no—
78C22×D4×C67—22no—
79C22×D4×C67—22no—
80C22×D4×C67C2×C414yescomplemented
81C22×D4×C67C2312yes—
82C23×C4×C67—21no—
83C23×C4×C67D411yescharacteristic
84C23×C4×C67D411yescharacteristic
85C23×C4×C67C2311yescharacteristic
86C2×C4⋊C4×C67—22no—
87C2×C4⋊C4×C67—22no—
88C2×C4⋊C4×C67—21no—
89C2×C4⋊C4×C67C2312yes—
90C2×C22⋊C4×C67—24no—
91C2×C22⋊C4×C67—24no—
92C2×C22⋊C4×C67—44no—
93C2×C22⋊C4×C67—44no—
94C2×C42×C67—22no—
95C2×C42×C67—22no—
96C2×C42×C67D412yes—
97C2×C42×C67C2312yes—
98C23×D67—24no—
99C23×D67—24no—
100C23×D67—24no—
101C23×D67—22no—
102C23×D67—42no—
103C23×D67—42no—
104C23×D67—42no—
105C23×D67—42no—
106C2×D268—28no—
107C2×D268—24no—
108C2×D268—24no—
109C2×D268—24no—
110C2×D268—28no—
111C2×D268—28no—
112C2×D268—22no—
113C2×D268—22no—
114C2×D268—22no—
115C2×D268C22⋊C4116yescomplemented
116C2×D268C22×C414yes—
117C2×D268C2×D414yes—
118C2×D268C2×D414yes—
119C22×Dic67—24no—
120C22×Dic67—22no—
121C22×Dic67—42no—
122C22×Dic67—42no—
123C2×C4×D67—216no—
124C2×C4×D67—48no—
125C2×C4×D67—432no—
126C2×C4×D67—48no—
127#11761613—232no—
128C2×C67⋊C8—28no—
129C2×C67⋊C8—22no—
130C2×C67⋊C8—22no—
131C2×C67⋊C8—22no—
132C2×C67⋊C8—22no—
133C2×C67⋊C8—22no—
134C2×C67⋊C8—44no—
135C2×C67⋊C8C22×C414yescomplemented
136C4⋊Dic67—24no—
137C4⋊Dic67—22no—
138C4⋊Dic67C2×D414yes—
139(C2×D67)⋊C4—216no—
140C24×C67—24no—
141C24×C67—22no—
142C24×C67—42no—
143C24×C67—42no—
144C24×C67—42no—
145C24×C67—42no—
146C24×C67—42no—
147C24×C67—42no—
148C24×C67C2×D411yescharacteristic
149C2×D4×C67—28no—
150C2×D4×C67—28no—
151C2×D4×C67—24no—
152C2×D4×C67—22no—
153C2×D4×C67—22no—
154C2×D4×C67—22no—
155C2×D4×C67—22no—
156C2×D4×C67—22no—
157C2×D4×C67—42no—
158C2×D4×C67—42no—
159C2×D4×C67—44no—
160C2×D4×C67—44no—
161C2×D4×C67C22×C414yescomplemented
162C22×C4×C67—22no—
163C22×C4×C67—22no—
164C22×C4×C67—22no—
165C22×C4×C67—22no—
166C22×C4×C67—22no—
167C22×C4×C67—22no—
168C22×C4×C67—42no—
169C22×C4×C67—42no—
170C22×C4×C67—42no—
171C22×C4×C67D814yescomplemented
172C22×C4×C67C22×C412yes—
173C22×C4×C67C2×D412yes—
174C22×C4×C67C2×D412yes—
175C22×C4×C67C2×D412yes—
176C22×C4×C67C2×D412yes—
177C22×C4×C67C2×D412yes—
178C22×C4×C67C2×D411yescharacteristic
179C22×C4×C67#4112yes—
180C22×C4×C67#4112yes—
181C22×C4×C67C2411yescharacteristic
182C4⋊C4×C67—22no—
183C4⋊C4×C67—22no—
184C4⋊C4×C67—42no—
185C22⋊C4×C67—48no—
186C22⋊C4×C67—48no—
187C42×C67—24no—
188C42×C67—22no—
189C42×C67—22no—
190C23×C67—24no—
191C23×C67—42no—
192C23×C67—42no—
193C23×C67—42no—
194C23×C67—42no—
195C23×C67—42no—
196C23×C67—42no—
197C23×C67—48no—
198C23×C67—48no—
199C23×C67—48no—
200C23×C67—84no—
201C23×C67—84no—
202C23×C67—84no—
203C23×C67—84no—
204C23×C67C4×D412yes—
205C23×C67#12012yes—
206C23×C67#12112yes—
207C23×C67C2×D812yes—
208C23×C67#13512yes—
209C23×C67AGL(1,ℤ/8)12yes—
210C23×C67AGL(1,ℤ/8)12yes—
211C23×C67C22×D411yescharacteristic
212C22×D67—24no—
213C22×D67—24no—
214C22×D67—24no—
215C22×D67—216no—
216C22×D67—42no—
217C22×D67—42no—
218C22×D67—42no—
219C22×D67—42no—
220C22×D67—48no—
221C22×D67—48no—
222C22×D67—48no—
223C22×D67—48no—
224C22×D67—48no—
225C22×D67—48no—
226D4×C67—24no—
227D4×C67—24no—
228D4×C67—42no—
229D4×C67—42no—
230D4×C67—48no—
231C2×C4×C67—22no—
232C2×C4×C67—22no—
233C2×C4×C67—28no—
234C2×C4×C67—22no—
235C2×C4×C67—22no—
236C2×C4×C67—22no—
237C2×C4×C67—44no—
238C2×C4×C67—42no—
239C2×C4×C67—42no—
240C2×C4×C67—42no—
241C2×C4×C67—84no—
242C2×C4×C67C2×C22⋊C414yes—
243C2×C4×C67C4×D414yes—
244C2×C4×C67C4×D412yes—
245C2×C4×C67#12012yes—
246C2×C4×C67#12112yes—
247C2×C4×C67#12114yes—
248C2×C4×C67#12314yes—
249C2×C4×C67C2×D814yescomplemented
250C2×C4×C67C23×C411yescharacteristic
251C2×C4×C67C22×D411yescharacteristic
252C2×C4×C67C22×D411yescharacteristic
253C2×C4×C67C22×D411yescharacteristic
254C2×C4×C67C2×#4111yescharacteristic
255C2×C4×C67C2×#4111yescharacteristic
256C2×Dic67—24no—
257C2×Dic67—42no—
258C2×Dic67—42no—
259C2×Dic67—48no—
260D268—24no—
261D268—24no—
262D268—24no—
263D268—216no—
264D268C2×C22⋊C418yescomplemented
265C4×D67—432no—
266C67⋊C8—24no—
267C67⋊C8—42no—
268C22×C67—44no—
269C22×C67—44no—
270C22×C67—44no—
271C22×C67—816no—
272C22×C67—82no—
273C22×C67—82no—
274C22×C67—82no—
275C22×C67—82no—
276C22×C67#38614yes—
277C22×C67C4×D814yes—
278C22×C67#44214yes—
279C22×C67#44714yes—
280C22×C67#44914yes—
281C22×C67#45914yes—
282C22×C67#46314yes—
283C22×C67C2×C4×D411yescharacteristic
284C22×C67C2×#12011yescharacteristic
285C22×C67C2×#12111yescharacteristic
286C22×C67C22×D811yescharacteristic
287C22×C67C2×#13511yescharacteristic
288C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
290C2×D67—28no—
291C2×D67—432no—
292C2×D67—44no—
293C2×D67—44no—
294C2×D67—44no—
295C2×D67—44no—
296C2×D67—44no—
297C2×D67—44no—
298C4×C67—24no—
299C4×C67—42no—
300C4×C67—82no—
301C4×C67#38614yes—
302C4×C67C22×C22⋊C411yescharacteristic
303C4×C67C2×C4×D411yescharacteristic
304C4×C67C2×#12111yescharacteristic
305C4×C67C2×#12311yescharacteristic
306Dic67—44no—
307C2×#1879—671nomaximal, Sylow
308C2×C67—84no—
309C2×C67#187918yescomplemented
310C2×C67C2×#38611yescharacteristic
311C2×C67C2×C4×D811yescharacteristic
312C2×C67C2×#44211yescharacteristic
313C2×C67C2×#44711yescharacteristic
314C2×C67C2×#44911yescharacteristic
315C2×C67C2×#45911yescharacteristic
316C2×C67C2×#46311yescharacteristic
317D67—48no—
318C23×D8—671no—
319C22×C4×D4—671no—
320C22×#102—671no—
321C22×#102—671no—
322C2×#390D6711yescharacteristic, complemented
323#1879—678no—
324C2×#340—671no—
325C2×#336—671no—
326C67C2×#187911yescharacteristic, complemented, Sylow
327C23×D4—671no—
328C23×D4C2×D6711yescharacteristic
329C24×C4—1341no—
330C22×D8—678no—
331C22×D8—674no—
332C22×D8—1344no—
333C22×D8—1344no—
334C23×C8—671no—
335C23×C8—671no—
336C2×C4×D4—674no—
337C2×C4×D4—1344no—
338C2×C4×D4—1344no—
339C2×C4×D4—1342no—
340C22×C4⋊C4—671no—
341C22×C4⋊C4C2×D6711yescharacteristic
342C22×C22⋊C4—21no—
343C22×C22⋊C4—21no—
344C22×C22⋊C4—1341no—
345C22×C42C2×D6711yescharacteristic
346C2×#102—674no—
347C2×#102—674no—
348C2×#102—1344no—
349C2×#102—1344no—
350#390—22no—
351#390C2×D6714yescomplemented
352#340—674no—
353#336—674no—
354C25—21no—
355C25—1341no—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4—21no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—674no—
363C22×D4—674no—
364C22×D4—674no—
365C22×D4—672no—
366C22×D4—1342no—
367C22×D4—1341no—
368C22×D4—1342no—
369C22×D4—1342no—
370C22×D4C4×D6714yescomplemented
371C22×D4C22×D6712yes—
372C23×C4—21no—
373C23×C4—2684no—
374C23×C4—2684no—
375C23×C4—1348no—
376C23×C4—1341no—
377C23×C4—1344no—
378C23×C4C22⋊D6711yescharacteristic
379C23×C4C22⋊D6711yescharacteristic
380C23×C4C22×D6711yescharacteristic
381C2×D8—6716no—
382C2×D8—13432no—
383C2×D8—1348no—
384C2×D8—1348no—
385C22×C8—674no—
386C22×C8—672no—
387C22×C8—672no—
388C22×C8—1342no—
389C22×C8—1342no—
390C22×C8—1342no—
391C22×C8—1342no—
392C22×C8—1342no—
393C4×D4—1348no—
394C4×D4—1348no—
395C4×D4—13416no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—674no—
400C2×C4⋊C4—672no—
401C2×C4⋊C4—1342no—
402C2×C4⋊C4—1341no—
403C2×C4⋊C4C22×D6712yes—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—44no—
407C2×C22⋊C4—44no—
408C2×C22⋊C4—1348no—
409C2×C22⋊C4—1344no—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42C22⋊D6712yes—
413C2×C42C22×D6712yes—
414#102—1348no—
415#102—1348no—
416C24—24no—
417C24—22no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—2682no—
425C24—2682no—
426C24—2682no—
427C24—2682no—
428C24—1344no—
429C24—1344no—
430C24—1344no—
431C24—1342no—
432C24C2×C22⋊D6711yescharacteristic
433C2×D4—28no—
434C2×D4—28no—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—6716no—
446C2×D4—674no—
447C2×D4—674no—
448C2×D4—674no—
449C2×D4—1348no—
450C2×D4—1344no—
451C2×D4—1344no—
452C2×D4—1344no—
453C2×D4—1348no—
454C2×D4—1348no—
455C2×D4—1342no—
456C2×D4—1342no—
457C2×D4—1342no—
458C2×D4C2×C4×D6714yescomplemented
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—26832no—
469C22×C4—2688no—
470C22×C4—2682no—
471C22×C4—2682no—
472C22×C4—2688no—
473C22×C4—13416no—
474C22×C4—1344no—
475C22×C4—1342no—
476C22×C4#1176161314yescomplemented
477C22×C4C2×C4×D6712yes—
478C22×C4D4×D6712yes—
479C22×C4D4×D6712yes—
480C22×C4D4×D6712yes—
481C22×C4C2×C22⋊D6712yes—
482C22×C4C2×C22⋊D6712yes—
483C22×C4C2×C22⋊D6711yescharacteristic
484C22×C4#1176162712yes—
485C22×C4#1176162812yes—
486C22×C4C23×D6711yescharacteristic
487D8—13432no—
488C2×C8—2684no—
489C2×C8—674no—
490C2×C8—1342no—
491C2×C8—1342no—
492C2×C8—1342no—
493C2×C8—1348no—
494C2×C8—1342no—
495C2×C8—1342no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—674no—
500C4⋊C4—1344no—
501C4⋊C4—1342no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—13416no—
505C42—24no—
506C42—22no—
507C42—22no—
508C23—24no—
509C23—48no—
510C23—48no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—48no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—2688no—
523C23—2688no—
524C23—2688no—
525C23—2688no—
526C23—2688no—
527C23—2682no—
528C23—2682no—
529C23—2682no—
530C23—2682no—
531C23—2688no—
532C23—13416no—
533C23—1344no—
534C23—1344no—
535C23—1344no—
536C23C4×C22⋊D6712yes—
537C23#1452401112yes—
538C23#1452401412yes—
539C23C2×#1176161312yes—
540C23#1452406312yes—
541C23#1452406812yes—
542C23#1452406912yes—
543C23C22×C22⋊D6711yescharacteristic
544D4—24no—
545D4—24no—
546D4—48no—
547D4—42no—
548D4—42no—
549D4—678no—
550D4—13416no—
551D4—1344no—
552D4—1344no—
553D4—1344no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—84no—
565C2×C4—26832no—
566C2×C4—2688no—
567C2×C4—2682no—
568C2×C4—2682no—
569C2×C4—1344no—
570C2×C4C22⋊C4×D6714yes—
571C2×C4(C22⋊D67)⋊C414yes—
572C2×C4C4×C22⋊D6712yes—
573C2×C4#1452401112yes—
574C2×C4#1452401412yes—
575C2×C4#1452401614yes—
576C2×C4#1452402514yes—
577C2×C4C2×#1176161314yescomplemented
578C2×C4C22×C4×D6711yescharacteristic
579C2×C4C2×D4×D6711yescharacteristic
580C2×C4C2×D4×D6711yescharacteristic
581C2×C4C2×D4×D6711yescharacteristic
582C2×C4C2×#1176162711yescharacteristic
583C2×C4C2×#1176162811yescharacteristic
584C8—2682no—
585C8—1344no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—816no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—2684no—
595C22—26832no—
596C22—2684no—
597C22—2684no—
598C22—2684no—
599C22—2684no—
600C22—2684no—
601C22—1348no—
602C22#1686327214yes—
603C22C4×#1176161314yes—
604C22#1686350614yes—
605C22#1686352914yes—
606C22#1686353914yes—
607C22#1686358814yes—
608C22#1686360814yes—
609C22C2×C4×C22⋊D6711yescharacteristic
610C22C2×#1452401111yescharacteristic
611C22C2×#1452401411yescharacteristic
612C22C22×#1176161311yescharacteristic
613C22C2×#1452406311yescharacteristic
614C22C2×#1452406811yescharacteristic
615C22C2×#1452406911yescharacteristic
616C4—24no—
617C4—42no—
618C4—82no—
619C4—2684no—
620C4#1686327214yes—
621C4C2×C22⋊C4×D6711yescharacteristic
622C4C2×(C22⋊D67)⋊C411yescharacteristic
623C4C2×#1452401611yescharacteristic
624C4C2×#1452402511yescharacteristic
625C2—84no—
626C2—2688no—
627C2#2256056018yesdirect product factor
628C2C2×#1686327211yescharacteristic
629C2C2×C4×#1176161311yescharacteristic
630C2C2×#1686350611yescharacteristic
631C2C2×#1686352911yescharacteristic
632C2C2×#1686353911yescharacteristic
633C2C2×#1686358811yescharacteristic
634C2C2×#1686360811yescharacteristic
subgroupG/H
centerC24C2×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#390×C67C2
socleC24×C67C2×D4
prime radicalC23×#11761613C2
cosocleC22×C4×C67C24
hypercenterC2×#390D67
hypocenterC67C2×#1879
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×C4×D268C2semidirect
3. C22×#14523964C2semidirect
4. C22×#14523963C2semidirect
5. #22560560C2direct (N × H)
6. C2×#16863158C2semidirect
7. C2×#16863151C2semidirect
8. C2×#390×C67C2semidirect
13. C22×#11761613C4semidirect
20. C2×C4×D268C22semidirect
24. C2×#14523964C22semidirect
26. C2×#14523963C22semidirect
27. #16863158C22semidirect
28. #16863151C22semidirect
35. #390×C67C22semidirect
53. C2×#11761613C2×C4semidirect
59. C22×C67⋊C8C2×C4semidirect
80. C22×D4×C67C2×C4semidirect
115. C2×D268C22⋊C4semidirect
135. C2×C67⋊C8C22×C4semidirect
161. C2×D4×C67C22×C4semidirect
171. C22×C4×C67D8semidirect
249. C2×C4×C67C2×D8semidirect
264. D268C2×C22⋊C4semidirect
309. C2×C67#1879semidirect
322. C2×#390D67semidirect
326. C67C2×#1879semidirect
351. #390C2×D67semidirect
370. C22×D4C4×D67semidirect
458. C2×D4C2×C4×D67semidirect
476. C22×C4#11761613semidirect
577. C2×C4C2×#11761613semidirect
627. C2#22560560direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616132
C22×C4×D2682
C22×#145239642
C22×#145239632
#225605602
C2×#168631582
C2×#168631512
C2×#390×C672
C2×#187967