Finite Group Databasebeta

← C2×#22560441 #30856692

Subgroups — C2×#22560441 #30856692

Order 17152.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863877C211yesmaximal, characteristic
2C2×#16863436C211yesmaximal, characteristic, complemented
3C22×#14523971C211yesmaximal, characteristic
4C2×#16863395C211yesmaximal, characteristic
5#22560441C218yesmaximal, direct product factor
6C2×#16863165C211yesmaximal, characteristic, complemented
7C2×#16863156C211yesmaximal, characteristic, complemented
8C2×#382×C67C211yesmaximal, characteristic, complemented
9C22×#11761628C2211yescharacteristic
10C22×#11761612C2211yescharacteristic
11C22×#11761612C2211yescharacteristic
12#16863877—22no—
13#16863877—22no—
14#16863877—22no—
15#16863877C2214yes—
16C2×#14524008—21no—
17C2×#14524005—21no—
18C2×C4×C22⋊D67—22no—
19C2×#14524002C2211yescharacteristic
20C2×#14524001—21no—
21C2×C4⋊C4×D67—21no—
22#16863436—22no—
23#16863436C2214yescomplemented
24C2×#14523976—21no—
25C2×#14523974—21no—
26C2×#14523971C418yescomplemented
27C2×#14523971C2214yes—
28#16863395—22no—
29#16863395C2214yes—
30C2×#14523966—21no—
31C2×#14523963—21no—
32#16863165C2214yescomplemented
33#16863156C2214yescomplemented
34C22×C4⋊C4×C67C2211yescharacteristic
35C22×C4⋊C4×C67C2211yescharacteristic
36C22×C42×C67C2211yescharacteristic
37#382×C67—22no—
38#382×C67C2214yescomplemented
39C2×#95×C67—22no—
40C2×#11761628—21no—
41C2×#11761628—21no—
42C2×#11761628—22no—
43C2×#11761628—22no—
44C2×#11761628C2×C414yes—
45C2×#11761628D412yes—
46C2×#11761628Q814yes—
47C2×#11761628C2312yes—
48C22×Dic134D411yescharacteristic
49C22×D268D411yescharacteristic
50C22×C22⋊D67—21no—
51C22×C4×D67—22no—
52C22×C4×D67—21no—
53C2×#11761612—21no—
54C2×#11761612—21no—
55C2×#11761612—22no—
56C2×#11761612—22no—
57C2×#11761612—22no—
58C2×#11761612C2×C414yescomplemented
59C2×#11761612C2312yes—
60C2×#11761612C2312yes—
61C22×C67⋊C8—21no—
62C22×C67⋊C8—21no—
63#14524008—24no—
64#14524008—24no—
65#14524005—24no—
66#14524005—24no—
67C4×C22⋊D67—28no—
68C4×C22⋊D67—28no—
69#14524002—21no—
70#14524002—21no—
71#14524002D412yes—
72#14524002C2312yes—
73#14524001—24no—
74#14524001—24no—
75C2×C4⋊Dic67D411yescharacteristic
76C2×C4⋊Dic67D411yescharacteristic
77C2×Dic67⋊C4—22no—
78C4⋊C4×D67—24no—
79C4⋊C4×D67—24no—
80C2×C22⋊Dic67—21no—
81C2×(C2×D67)⋊C4—22no—
82C2×C4×Dic67—22no—
83C2×C4×Dic67—21no—
84#14523976—24no—
85#14523974—24no—
86#14523971C2×C4116yescomplemented
87#14523966—24no—
88#14523963—24no—
89C23×C4×C67—21no—
90C23×C4×C67—21no—
91C23×C4×C67D412yes—
92C23×C4×C67C2311yescharacteristic
93C2×C4⋊C4×C67—21no—
94C2×C4⋊C4×C67—21no—
95C2×C4⋊C4×C67—21no—
96C2×C4⋊C4×C67—21no—
97C2×C4⋊C4×C67—21no—
98C2×C4⋊C4×C67—21no—
99C2×C4⋊C4×C67—21no—
100C2×C4⋊C4×C67—21no—
101C2×C4⋊C4×C67—21no—
102C2×C4⋊C4×C67D412yes—
103C2×C4⋊C4×C67C2312yes—
104C2×C4⋊C4×C67C2312yes—
105C2×C42×C67—21no—
106C2×C42×C67—22no—
107C2×C42×C67C2×C414yescomplemented
108C2×C42×C67C2312yes—
109#95×C67—28no—
110C23×D67—21no—
111#11761628—24no—
112#11761628—24no—
113#11761628—24no—
114#11761628—24no—
115#11761628—24no—
116#11761628—22no—
117#11761628—22no—
118#11761628C4⋊C418yes—
119#11761628C22×C414yes—
120#11761628C2×Q814yes—
121C2×Dic134—21no—
122C2×Dic134—22no—
123C2×Dic134C4⋊C414yes—
124C2×Dic134C2×D412yes—
125C2×D268—21no—
126C2×D268—22no—
127C2×D268C4⋊C414yes—
128C2×D268C2×D412yes—
129C2×C22⋊D67—24no—
130C2×C22⋊D67—22no—
131C2×C22⋊D67—24no—
132C2×C22⋊D67—22no—
133C22×Dic67—21no—
134C22×Dic67—21no—
135C22×Dic67—21no—
136C2×C4×D67—24no—
137C2×C4×D67—22no—
138C2×C4×D67—24no—
139C2×C4×D67—22no—
140C2×C4×D67—28no—
141C2×C4×D67—84no—
142#11761612—24no—
143#11761612—22no—
144#11761612—22no—
145#11761612—22no—
146#11761612—42no—
147#11761612C22×C414yescomplemented
148C2×C67⋊C8—24no—
149C2×C67⋊C8—22no—
150C2×C67⋊C8—22no—
151C2×C67⋊C8—42no—
152C4⋊Dic67—21no—
153C4⋊Dic67—21no—
154C4⋊Dic67C2×D412yes—
155C4⋊Dic67C2×D412yes—
156Dic67⋊C4—28no—
157C22⋊Dic67—22no—
158C22⋊Dic67—22no—
159(C2×D67)⋊C4—28no—
160C4×Dic67—28no—
161C4×Dic67—22no—
162C4×Dic67—22no—
163C24×C67C2×D411yescharacteristic
164C22×C4×C67—24no—
165C22×C4×C67—22no—
166C22×C4×C67—22no—
167C22×C4×C67—22no—
168C22×C4×C67—22no—
169C22×C4×C67—42no—
170C22×C4×C67—42no—
171C22×C4×C67—44no—
172C22×C4×C67—44no—
173C22×C4×C67—41no—
174C22×C4×C67—41no—
175C22×C4×C67C22×C412yes—
176C22×C4×C67C2×D411yescharacteristic
177C22×C4×C67C2×D411yescharacteristic
178C22×C4×C67C2×D411yescharacteristic
179C22×C4×C67C2×D411yescharacteristic
180C22×C4×C67C2×D414yes—
181C22×C4×C67C2×D411yescharacteristic
182C22×C4×C67C2×Q812yes—
183C22×C4×C67#4112yes—
184C22×C4×C67#4112yes—
185C22×C4×C67C2411yescharacteristic
186C4⋊C4×C67—22no—
187C4⋊C4×C67—22no—
188C4⋊C4×C67—22no—
189C4⋊C4×C67—22no—
190C4⋊C4×C67—22no—
191C4⋊C4×C67—22no—
192C4⋊C4×C67—41no—
193C4⋊C4×C67—41no—
194C42×C67—24no—
195C42×C67—22no—
196C42×C67C22×C414yescomplemented
197C23×C67—22no—
198C23×C67—22no—
199C23×C67C4×D412yes—
200C23×C67#12011yescharacteristic
201C23×C67#12011yescharacteristic
202C23×C67C22⋊Q812yes—
203C23×C67C22×D411yescharacteristic
204C22×D67—22no—
205C22×D67—24no—
206C22×D67—42no—
207C22×D67—42no—
208C2×C4×C67—24no—
209C2×C4×C67—24no—
210C2×C4×C67—48no—
211C2×C4×C67—42no—
212C2×C4×C67—42no—
213C2×C4×C67—44no—
214C2×C4×C67—44no—
215C2×C4×C67—44no—
216C2×C4×C67—42no—
217C2×C4×C67—82no—
218C2×C4×C67—82no—
219C2×C4×C67—82no—
220C2×C4×C67C2×C4⋊C412yes—
221C2×C4×C67C2×C4⋊C412yes—
222C2×C4×C67C4×D412yes—
223C2×C4×C67C4×D412yes—
224C2×C4×C67C4×D412yes—
225C2×C4×C67#12011yescharacteristic
226C2×C4×C67#12011yescharacteristic
227C2×C4×C67C22⋊Q812yes—
228C2×C4×C67C22⋊Q812yes—
229C2×C4×C67C22⋊Q812yes—
230C2×C4×C67#12312yes—
231C2×C4×C67#12312yes—
232C2×C4×C67C23×C411yescharacteristic
233C2×C4×C67C22×D411yescharacteristic
234C2×C4×C67C22×D411yescharacteristic
235C2×C4×C67C22×Q811yescharacteristic
236C2×C4×C67C2×#4111yescharacteristic
237C2×C4×C67C2×#4111yescharacteristic
238C22⋊D67—24no—
239C22⋊D67—24no—
240C22⋊D67—28no—
241C2×Dic67—22no—
242C2×Dic67—24no—
243C2×Dic67—22no—
244C2×Dic67—22no—
245C2×Dic67—42no—
246C2×Dic67—42no—
247D268—22no—
248D268—24no—
249D268C2×C4⋊C414yes—
250C4×D67—24no—
251C4×D67—24no—
252C4×D67—28no—
253C4×D67—88no—
254Dic134—22no—
255Dic134—24no—
256Dic134C2×C4⋊C414yes—
257C67⋊C8—24no—
258C67⋊C8—42no—
259C22×C67—24no—
260C22×C67—24no—
261C22×C67—21no—
262C22×C67—21no—
263C22×C67—21no—
264C22×C67—21no—
265C22×C67#38512yes—
266C22×C67#38512yes—
267C22×C67#45412yes—
268C22×C67#45512yes—
269C22×C67C2×C4×D411yescharacteristic
270C22×C67C2×#12011yescharacteristic
271C22×C67C2×C22⋊Q811yescharacteristic
272C2×D67—24no—
273C2×D67—48no—
274C2×D67—42no—
275C2×D67—42no—
276C4×C67—44no—
277C4×C67—44no—
278C4×C67—82no—
279C4×C67#38512yes—
280C4×C67#38512yes—
281C4×C67C22×C4⋊C411yescharacteristic
282C4×C67C2×C4×D411yescharacteristic
283C4×C67C2×C22⋊Q811yescharacteristic
284C4×C67C2×#12311yescharacteristic
285Dic67—24no—
286Dic67—42no—
287Dic67—42no—
288C2×#1860—671nomaximal, Sylow
289C2×C67—21no—
290C2×C67—21no—
291C2×C67—22no—
292C2×C67#186014yescomplemented
293C2×C67C2×#38511yescharacteristic
294C2×C67C2×#45511yescharacteristic
295C2×C67C2×#45411yescharacteristic
296D67—44no—
297C2×#520—671no—
298C2×#428—671no—
299C22×#104—671no—
300C2×#417—671no—
301C2×#382D6711yescharacteristic, complemented
302#1860—678no—
303C2×#343—671no—
304C2×#339—671no—
305C67C2×#186011yescharacteristic, complemented, Sylow
306C22×#41—671no—
307C22×M16—671no—
308C22×M16—671no—
309#520—674no—
310#520—1342no—
311#520—1342no—
312#520—1342no—
313C2×C4×Q8—1341no—
314C2×C4×D4—1341no—
315C2×C4×D4—1342no—
316C2×#117—671no—
317C2×#117—1341no—
318C22×C4⋊C4—1341no—
319C22×C4⋊C4C2×D6711yescharacteristic
320C22×C4⋊C4C2×D6711yescharacteristic
321C22×C42C2×D6711yescharacteristic
322#428—674no—
323#428—1342no—
324C2×#107—1341no—
325C2×#106—1341no—
326C2×#104—678no—
327C2×#104—674no—
328#417—674no—
329#417—1342no—
330C2×#103—1341no—
331C2×#102—1341no—
332#382—22no—
333#382C2×D6714yescomplemented
334C2×#95—22no—
335#343—674no—
336#339—674no—
337C2×#41—674no—
338C2×#41—672no—
339C2×#41—674no—
340C2×#41—672no—
341C2×#41—1341no—
342C2×#41—1341no—
343C2×#41—1342no—
344C2×#41—1342no—
345C22×Q8—671no—
346C22×D4—671no—
347C22×D4—1341no—
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—1342no—
351C23×C4—1341no—
352C23×C4C22⋊D6712yes—
353C23×C4C22×D6711yescharacteristic
354C2×M16—674no—
355C2×M16—672no—
356C2×M16—672no—
357C2×M16—1342no—
358C2×M16—1342no—
359C2×M16—1341no—
360C2×M16—1341no—
361C2×M16—1342no—
362C22×C8—1341no—
363C22×C8—1341no—
364C4×Q8—1344no—
365C4×Q8—1344no—
366C4×D4—1344no—
367C4×D4—1344no—
368C4×D4—1348no—
369C4×D4—1348no—
370#117—672no—
371#117—672no—
372#117—1344no—
373#117—1344no—
374#117—1341no—
375#117—1341no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—671no—
386C2×C4⋊C4—671no—
387C2×C4⋊C4—1344no—
388C2×C4⋊C4—1344no—
389C2×C4⋊C4—1342no—
390C2×C4⋊C4C22⋊D6712yes—
391C2×C4⋊C4C22×D6712yes—
392C2×C4⋊C4C22×D6712yes—
393C2×C22⋊C4—1342no—
394C2×C22⋊C4—1341no—
395C2×C42—21no—
396C2×C42—22no—
397C2×C42—1342no—
398C2×C42—1341no—
399C2×C42C4×D6714yescomplemented
400C2×C42C22×D6712yes—
401#107—1344no—
402#106—1344no—
403#104—6716no—
404#103—1344no—
405#102—1344no—
406#95—28no—
407C24—1341no—
408C24C2×C22⋊D6711yescharacteristic
409#41—678no—
410#41—674no—
411#41—674no—
412#41—1344no—
413#41—1344no—
414#41—1344no—
415#41—1344no—
416#41—1344no—
417#41—1342no—
418#41—1342no—
419C2×Q8—674no—
420C2×Q8—672no—
421C2×Q8—1341no—
422C2×Q8—1342no—
423C2×D4—674no—
424C2×D4—672no—
425C2×D4—1344no—
426C2×D4—1342no—
427C2×D4—1344no—
428C2×D4—1342no—
429C2×D4—1341no—
430C2×D4—1342no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—5364no—
443C22×C4—1348no—
444C22×C4—1341no—
445C22×C4—1341no—
446C22×C4—1341no—
447C22×C4—1344no—
448C22×C4—1342no—
449C22×C4—1344no—
450C22×C4—1342no—
451C22×C4C2×C4×D6712yes—
452C22×C4D4×D6711yescharacteristic
453C22×C4D4×D6711yescharacteristic
454C22×C4D4×D6711yescharacteristic
455C22×C4D4×D6711yescharacteristic
456C22×C4C2×C22⋊D6714yes—
457C22×C4C2×C22⋊D6711yescharacteristic
458C22×C4Q8×D6712yes—
459C22×C4#1176162812yes—
460C22×C4#1176162912yes—
461C22×C4C23×D6711yescharacteristic
462M16—2682no—
463M16—674no—
464M16—1342no—
465M16—1342no—
466M16—1344no—
467M16—1342no—
468C2×C8—2682no—
469C2×C8—1342no—
470C2×C8—1344no—
471C2×C8—1342no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—672no—
481C4⋊C4—672no—
482C4⋊C4—1348no—
483C4⋊C4—1341no—
484C4⋊C4—1341no—
485C22⋊C4—1348no—
486C22⋊C4—1342no—
487C22⋊C4—1342no—
488C42—24no—
489C42—22no—
490C42—1348no—
491C42—1342no—
492C42—1342no—
493C42C2×C4×D6714yescomplemented
494C23—22no—
495C23—22no—
496C23—2682no—
497C23—2682no—
498C23—1344no—
499C23—1342no—
500C23C4×C22⋊D6712yes—
501C23#1452401111yescharacteristic
502C23#1452401111yescharacteristic
503C23#1452401912yes—
504C23C22×C22⋊D6711yescharacteristic
505Q8—674no—
506Q8—1344no—
507Q8—1342no—
508D4—674no—
509D4—1348no—
510D4—1344no—
511D4—1344no—
512D4—1344no—
513D4—1342no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—2682no—
527C2×C4—2682no—
528C2×C4—5368no—
529C2×C4—1342no—
530C2×C4—1342no—
531C2×C4—1344no—
532C2×C4—1342no—
533C2×C4—1348no—
534C2×C4—1344no—
535C2×C4—1344no—
536C2×C4C4⋊C4×D6712yes—
537C2×C4C4⋊C4×D6712yes—
538C2×C4C4×C22⋊D6712yes—
539C2×C4#1452400512yes—
540C2×C4#1452400512yes—
541C2×C4#1452401111yescharacteristic
542C2×C4#1452401111yescharacteristic
543C2×C4#1452401912yes—
544C2×C4#1452402112yes—
545C2×C4#1452402112yes—
546C2×C4#1452402812yes—
547C2×C4#1452402812yes—
548C2×C4C22×C4×D6711yescharacteristic
549C2×C4C2×D4×D6711yescharacteristic
550C2×C4C2×D4×D6711yescharacteristic
551C2×C4C2×Q8×D6711yescharacteristic
552C2×C4C2×#1176162811yescharacteristic
553C2×C4C2×#1176162911yescharacteristic
554C8—2682no—
555C8—1344no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—2682no—
563C22—2682no—
564C22—2688no—
565C22—1344no—
566C22#1686326712yes—
567C22#1686326712yes—
568C22#1686356812yes—
569C22#1686357312yes—
570C22C2×C4×C22⋊D6711yescharacteristic
571C22C2×#1452401111yescharacteristic
572C22C2×#1452401911yescharacteristic
573C4—44no—
574C4—44no—
575C4—82no—
576C4—2682no—
577C4—2682no—
578C4—1344no—
579C4#1686326712yes—
580C4#1686326712yes—
581C4C2×C4⋊C4×D6711yescharacteristic
582C4C2×#1452400511yescharacteristic
583C4C2×#1452402111yescharacteristic
584C4C2×#1452402811yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—2684no—
589C2#2256044114yesdirect product factor
590C2C2×#1686326711yescharacteristic
591C2C2×#1686357311yescharacteristic
592C2C2×#1686356811yescharacteristic
subgroupG/H
centerC23#14524011
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#382×C67C2
socleC23×C67#120
prime radicalC22×#11761628C22
cosocleC22×C4×C67C24
hypercenterC2×#382D67
hypocenterC67C2×#1860
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#16863436C2semidirect
5. #22560441C2direct (N × H)
6. C2×#16863165C2semidirect
7. C2×#16863156C2semidirect
8. C2×#382×C67C2semidirect
23. #16863436C22semidirect
26. C2×#14523971C4semidirect
32. #16863165C22semidirect
33. #16863156C22semidirect
38. #382×C67C22semidirect
58. C2×#11761612C2×C4semidirect
86. #14523971C2×C4semidirect
107. C2×C42×C67C2×C4semidirect
147. #11761612C22×C4semidirect
196. C42×C67C22×C4semidirect
292. C2×C67#1860semidirect
301. C2×#382D67semidirect
305. C67C2×#1860semidirect
333. #382C2×D67semidirect
399. C2×C42C4×D67semidirect
493. C42C2×C4×D67semidirect
589. C2#22560441direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168638772
C2×#168634362
C22×#145239712
C2×#168633952
#225604412
C2×#168631652
C2×#168631562
C2×#382×C672
C2×#186067