Finite Group Databasebeta

← C2×#22563617 #31024535

Subgroups — C2×#22563617 #31024535

Order 17152.

3570 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524024C211yesmaximal, characteristic
2C22×#14524002C211yesmaximal, characteristic, complemented
3C23×Dic67⋊C4C211yesmaximal, characteristic, complemented
4#22563617C2116yesmaximal, direct product factor
5C2×#16863329C212yesmaximal, complemented
6C2×#16863319C212yesmaximal, complemented
7C2×#16863291C212yesmaximal, complemented
8C2×#16863290C212yesmaximal, complemented
9C2×#16863276C211yesmaximal, characteristic
10C2×#16863268C211yesmaximal, characteristic, complemented
11C2×#16863257C211yesmaximal, characteristic, complemented
12C22×#123×C67C211yesmaximal, characteristic
13C24×Dic67C2211yescharacteristic
14C24×Dic67C2211yescharacteristic
15C2×#14524024—28no—
16C2×#14524024C2218yes—
17C2×#14524002—28no—
18C2×#14524002C2218yescomplemented
19C22×C4⋊Dic67C2212yes—
20C22×Dic67⋊C4—28no—
21C22×Dic67⋊C4C2212yes—
22C22×Dic67⋊C4C2212yes—
23C22×Dic67⋊C4C2212yes—
24C22×Dic67⋊C4C2212yes—
25C22×Dic67⋊C4C2218yescomplemented
26C22×C22⋊Dic67C2211yescharacteristic
27C22×C22⋊Dic67C2212yes—
28C22×C22⋊Dic67C2212yes—
29C22×C22⋊Dic67C2211yescharacteristic
30C22×C4×Dic67C2212yes—
31#16863329C22116yescomplemented
32#16863319C22116yescomplemented
33#16863291C22116yescomplemented
34#16863290C22116yescomplemented
35#16863276C2218yes—
36#16863268C2218yescomplemented
37#16863257C2218yescomplemented
38C2×C67⋊#95C2212yes—
39C2×C67⋊#95C2212yes—
40C2×C67⋊#95C2212yes—
41C2×C67⋊#95C2212yes—
42C23×D4×C67C2211yescharacteristic
43C24×C4×C67C2211yescharacteristic
44C2×#123×C67—28no—
45C2×#123×C67C2218yes—
46C22×C4⋊C4×C67C2211yescharacteristic
47C22×C4⋊C4×C67C2211yescharacteristic
48C22×C22⋊C4×C67C2211yescharacteristic
49C22×C22⋊C4×C67C2211yescharacteristic
50C22×C22⋊C4×C67C2211yescharacteristic
51C23×Dic67—28no—
52C23×Dic67—24no—
53C23×Dic67—24no—
54C23×Dic67—24no—
55C23×Dic67—24no—
56C23×Dic67C2312yes—
57C23×Dic67C2312yes—
58C23×Dic67C2314yes—
59C23×Dic67C2314yes—
60C23×Dic67C2312yes—
61C23×Dic67C2312yes—
62#14524024—232no—
63#14524002—232no—
64C2×C4⋊Dic67—28no—
65C2×C4⋊Dic67C2318yes—
66C2×Dic67⋊C4—28no—
67C2×Dic67⋊C4—28no—
68C2×Dic67⋊C4—28no—
69C2×Dic67⋊C4—28no—
70C2×Dic67⋊C4—232no—
71C2×Dic67⋊C4C2318yes—
72C2×Dic67⋊C4C2318yes—
73C2×Dic67⋊C4C2318yes—
74C2×Dic67⋊C4C2318yes—
75C2×C22⋊Dic67—28no—
76C2×C22⋊Dic67—28no—
77C2×C22⋊Dic67—24no—
78C2×C22⋊Dic67D418yes—
79C2×C22⋊Dic67C2314yes—
80C2×C22⋊Dic67C2318yes—
81C2×C22⋊Dic67C2318yes—
82C2×C22⋊Dic67C2314yes—
83C2×C4×Dic67—28no—
84C2×C4×Dic67C2318yes—
85C67⋊#95C2318yes—
86C67⋊#95C2318yes—
87C67⋊#95C2318yes—
88C67⋊#95C2318yes—
89C25×C67C2311yescharacteristic
90C25×C67C2311yescharacteristic
91C22×D4×C67—24no—
92C22×D4×C67—24no—
93C22×D4×C67—24no—
94C22×D4×C67C2314yes—
95C23×C4×C67—24no—
96C23×C4×C67—24no—
97C23×C4×C67D418yes—
98C23×C4×C67C2311yescharacteristic
99C23×C4×C67C2311yescharacteristic
100C23×C4×C67C2311yescharacteristic
101C23×C4×C67C2311yescharacteristic
102C23×C4×C67C2311yescharacteristic
103C23×C4×C67C2314yes—
104#123×C67—232no—
105C2×C4⋊C4×C67—24no—
106C2×C4⋊C4×C67—24no—
107C2×C4⋊C4×C67C2314yes—
108C2×C4⋊C4×C67C2314yes—
109C2×C22⋊C4×C67—24no—
110C2×C22⋊C4×C67—24no—
111C2×C22⋊C4×C67—24no—
112C2×C22⋊C4×C67C2314yes—
113C2×C22⋊C4×C67C2314yes—
114C2×C22⋊C4×C67C2314yes—
115C22×Dic67—232no—
116C22×Dic67—28no—
117C22×Dic67—24no—
118C22×Dic67—24no—
119C22×Dic67—24no—
120C22×Dic67—24no—
121C22×Dic67—216no—
122C22×Dic67—24no—
123C22×Dic67—24no—
124C22×Dic67—28no—
125C22×Dic67—24no—
126C22×Dic67—28no—
127C22×Dic67—24no—
128C22×Dic67—28no—
129C22×Dic67—24no—
130C22×Dic67—24no—
131C22×Dic67—416no—
132C22×Dic67C2×D418yes—
133C22×Dic67#4118yes—
134C22×Dic67C2414yes—
135C22×Dic67C2414yes—
136C22×Dic67C2414yes—
137C22×Dic67C2414yes—
138C4⋊Dic67—216no—
139Dic67⋊C4—216no—
140Dic67⋊C4—216no—
141Dic67⋊C4—216no—
142Dic67⋊C4—216no—
143C22⋊Dic67—28no—
144C22⋊Dic67—216no—
145C22⋊Dic67—216no—
146C22⋊Dic67C2×D4116yes—
147C4×Dic67—216no—
148C24×C67—24no—
149C24×C67—24no—
150C24×C67—22no—
151C24×C67—22no—
152C24×C67—22no—
153C24×C67—24no—
154C24×C67—24no—
155C24×C67C2×D414yes—
156C24×C67C2×D414yes—
157C24×C67#4114yes—
158C24×C67C2412yes—
159C24×C67C2412yes—
160C24×C67C2411yescharacteristic
161C2×D4×C67—28no—
162C2×D4×C67—28no—
163C2×D4×C67—28no—
164C2×D4×C67—416no—
165C22×C4×C67—232no—
166C22×C4×C67—22no—
167C22×C4×C67—22no—
168C22×C4×C67—28no—
169C22×C4×C67—28no—
170C22×C4×C67—22no—
171C22×C4×C67—22no—
172C22×C4×C67—22no—
173C22×C4×C67—22no—
174C22×C4×C67—22no—
175C22×C4×C67—22no—
176C22×C4×C67—22no—
177C22×C4×C67—24no—
178C22×C4×C67—24no—
179C22×C4×C67C2×D414yes—
180C22×C4×C67C2×D414yes—
181C22×C4×C67C2×D4116yes—
182C22×C4×C67C2412yes—
183C22×C4×C67C2412yes—
184C22×C4×C67C2412yes—
185C22×C4×C67C2412yes—
186C22×C4×C67C2412yes—
187C4⋊C4×C67—28no—
188C4⋊C4×C67—28no—
189C22⋊C4×C67—28no—
190C22⋊C4×C67—28no—
191C22⋊C4×C67—28no—
192C23×C67—28no—
193C23×C67—28no—
194C23×C67—216no—
195C23×C67—216no—
196C23×C67—24no—
197C23×C67—24no—
198C23×C67—22no—
199C23×C67—22no—
200C23×C67—22no—
201C23×C67—24no—
202C23×C67—24no—
203C23×C67—44no—
204C23×C67—44no—
205C23×C67—44no—
206C23×C67—48no—
207C23×C67—48no—
208C23×C67#121116yes—
209C23×C67C22×D414yes—
210C23×C67C22×D412yes—
211C23×C67C22×D414yes—
212C23×C67C22×D412yes—
213C23×C67C2×#4114yes—
214C23×C67C2×#4114yes—
215C23×C67C2×#4112yes—
216C23×C672−1+414yes—
217C23×C67C2511yescharacteristic
218D4×C67—416no—
219C2×C4×C67—232no—
220C2×C4×C67—28no—
221C2×C4×C67—28no—
222C2×C4×C67—22no—
223C2×C4×C67—22no—
224C2×C4×C67—22no—
225C2×C4×C67—22no—
226C2×C4×C67—22no—
227C2×C4×C67—22no—
228C2×C4×C67—22no—
229C2×C4×C67—22no—
230C2×C4×C67—22no—
231C2×C4×C67—24no—
232C2×C4×C67—24no—
233C2×C4×C67—44no—
234C2×C4×C67—44no—
235C2×C4×C67—44no—
236C2×C4×C67C22×D414yes—
237C2×C4×C67C22×D414yes—
238C2×Dic67—232no—
239C2×Dic67—216no—
240C2×Dic67—216no—
241C2×Dic67—24no—
242C2×Dic67—24no—
243C2×Dic67—24no—
244C2×Dic67—24no—
245C2×Dic67—24no—
246C2×Dic67—24no—
247C2×Dic67—24no—
248C2×Dic67—24no—
249C2×Dic67—24no—
250C2×Dic67—24no—
251C2×Dic67—416no—
252C2×Dic67—48no—
253C2×Dic67—48no—
254C2×Dic67C22×D418yes—
255C2×Dic67C2×#4118yes—
256C22×C67—24no—
257C22×C67—232no—
258C22×C67—24no—
259C22×C67—28no—
260C22×C67—28no—
261C22×C67—42no—
262C22×C67—42no—
263C22×C67—42no—
264C22×C67—44no—
265C22×C67—44no—
266C22×C67—88no—
267C22×C67C2×#12118yes—
268C22×C67C2×#12114yes—
269C22×C67#52514yes—
270C22×C67#53614yes—
271C22×C67#54718yes—
272C22×C67#55218yes—
273C22×C67C23×D411yescharacteristic
274C22×C67C23×D411yescharacteristic
275C22×C67C22×#4112yes—
276C22×C67C22×#4111yescharacteristic
277C22×C67C2×2−1+412yes—
278C4×C67—24no—
279C4×C67—24no—
280C4×C67—42no—
281C4×C67—42no—
282C4×C67—42no—
283Dic67—28no—
284Dic67—28no—
285Dic67—44no—
286Dic67—44no—
287C2×#2413—671nomaximal, Sylow
288C2×C67—28no—
289C2×C67—82no—
290C2×C67#241318yescomplemented
291C2×C67C22×#12111yescharacteristic
292C2×C67C2×#52511yescharacteristic
293C2×C67C2×#53611yescharacteristic
294C2×C67C2×#54712yes—
295C2×C67C2×#55212yes—
296C22×#123—671no—
297C22×#123D6711yescharacteristic
298C22×#117—671no—
299C23×C4⋊C4—671no—
300#2413—6716no—
301C2×#398—672no—
302C2×#396—672no—
303C2×#389—672no—
304C2×#388—672no—
305C2×#386—671no—
306C2×#385—671no—
307C2×#381—671no—
308C67C2×#241311yescharacteristic, complemented, Sylow
309C23×D4C2×D6711yescharacteristic
310C24×C4—671no—
311C24×C4—671no—
312C24×C4C2×D6711yescharacteristic
313C2×#123—28no—
314C2×#123—678no—
315C2×#123—1348no—
316C2×#123C2×D6718yes—
317C2×#117—678no—
318C2×#117—1348no—
319C22×C4⋊C4—672no—
320C22×C4⋊C4—672no—
321C22×C4⋊C4—672no—
322C22×C4⋊C4—672no—
323C22×C4⋊C4—672no—
324C22×C4⋊C4—678no—
325C22×C4⋊C4—1348no—
326C22×C4⋊C4C2×D6711yescharacteristic
327C22×C4⋊C4C2×D6711yescharacteristic
328C22×C22⋊C4—671no—
329C22×C22⋊C4—672no—
330C22×C22⋊C4—672no—
331C22×C22⋊C4—671no—
332C22×C22⋊C4C2×D6711yescharacteristic
333C22×C22⋊C4C2×D6711yescharacteristic
334C22×C22⋊C4C2×D6711yescharacteristic
335C22×C42—672no—
336#398—6716no—
337#396—6716no—
338#389—6716no—
339#388—6716no—
340#386—678no—
341#385—678no—
342#381—678no—
343C2×#95—672no—
344C2×#95—672no—
345C2×#95—672no—
346C2×#95—672no—
347C25C22×D6711yescharacteristic
348C25C22×D6711yescharacteristic
349C22×D4—24no—
350C22×D4—24no—
351C22×D4—24no—
352C22×D4C22×D6714yes—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—672no—
356C23×C4—672no—
357C23×C4—674no—
358C23×C4—674no—
359C23×C4—672no—
360C23×C4—672no—
361C23×C4—1344no—
362C23×C4—1344no—
363C23×C4—1344no—
364C23×C4—1348no—
365C23×C4—1344no—
366C23×C4C22⋊D6718yes—
367C23×C4C22×D6711yescharacteristic
368C23×C4C22×D6711yescharacteristic
369C23×C4C22×D6711yescharacteristic
370C23×C4C22×D6714yes—
371C23×C4C22×D6711yescharacteristic
372C23×C4C22×D6711yescharacteristic
373#123—232no—
374#123—13432no—
375#117—13432no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—678no—
379C2×C4⋊C4—678no—
380C2×C4⋊C4—678no—
381C2×C4⋊C4—678no—
382C2×C4⋊C4—678no—
383C2×C4⋊C4—1348no—
384C2×C4⋊C4—1348no—
385C2×C4⋊C4—1348no—
386C2×C4⋊C4—1348no—
387C2×C4⋊C4—1348no—
388C2×C4⋊C4—13432no—
389C2×C4⋊C4C22×D6714yes—
390C2×C4⋊C4C22×D6714yes—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—678no—
395C2×C22⋊C4—674no—
396C2×C22⋊C4—674no—
397C2×C22⋊C4—678no—
398C2×C22⋊C4—678no—
399C2×C22⋊C4—1348no—
400C2×C22⋊C4—1348no—
401C2×C22⋊C4—1344no—
402C2×C22⋊C4C22×D6714yes—
403C2×C22⋊C4C22×D6714yes—
404C2×C22⋊C4C22×D6714yes—
405C2×C42—678no—
406C2×C42—1348no—
407#95—678no—
408#95—678no—
409#95—678no—
410#95—678no—
411C24—24no—
412C24—24no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—22no—
417C24—24no—
418C24D4×D6714yes—
419C24C2×C22⋊D6714yes—
420C24#1176162714yes—
421C24C23×D6712yes—
422C24C23×D6712yes—
423C24C23×D6711yescharacteristic
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—416no—
428C22×C4—232no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—26816no—
443C22×C4—678no—
444C22×C4—678no—
445C22×C4—674no—
446C22×C4—674no—
447C22×C4—674no—
448C22×C4—674no—
449C22×C4—1344no—
450C22×C4—1344no—
451C22×C4—1348no—
452C22×C4—1348no—
453C22×C4—1344no—
454C22×C4—1344no—
455C22×C4—1348no—
456C22×C4—1344no—
457C22×C4—1344no—
458C22×C4—13416no—
459C22×C4—13432no—
460C22×C4—1348no—
461C22×C4—1344no—
462C22×C4—1344no—
463C22×C4—1344no—
464C22×C4—1344no—
465C22×C4C2×C22⋊D6714yes—
466C22×C4C2×C22⋊D6714yes—
467C22×C4C2×C22⋊D67116yes—
468C22×C4C23×D6712yes—
469C22×C4C23×D6712yes—
470C22×C4C23×D6712yes—
471C22×C4C23×D6712yes—
472C22×C4C23×D6712yes—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—13416no—
476C4⋊C4—13416no—
477C4⋊C4—13416no—
478C4⋊C4—13416no—
479C4⋊C4—13416no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—6716no—
484C22⋊C4—13416no—
485C22⋊C4—13416no—
486C22⋊C4—1348no—
487C42—13416no—
488C23—216no—
489C23—216no—
490C23—28no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—48no—
500C23—48no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23#14524015116yes—
505C23C2×D4×D6714yes—
506C23C2×D4×D6712yes—
507C23C22×C22⋊D6714yes—
508C23C22×C22⋊D6712yes—
509C23#41×D6714yes—
510C23C2×#1176162714yes—
511C23C2×#1176162712yes—
512C23#1452409014yes—
513C23C24×D6711yescharacteristic
514D4—416no—
515C2×C4—232no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—2688no—
533C2×C4—26816no—
534C2×C4—2688no—
535C2×C4—678no—
536C2×C4—678no—
537C2×C4—1344no—
538C2×C4—1344no—
539C2×C4—1344no—
540C2×C4—1344no—
541C2×C4—1344no—
542C2×C4—1344no—
543C2×C4—13432no—
544C2×C4—13416no—
545C2×C4—13416no—
546C2×C4—1344no—
547C2×C4—1344no—
548C2×C4—1344no—
549C2×C4—1344no—
550C2×C4C22×C22⋊D6714yes—
551C2×C4C22×C22⋊D6714yes—
552C22—232no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—24no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—88no—
563C22C2×#1452401518yes—
564C22C2×#1452401514yes—
565C22#1686390314yes—
566C22#1686396714yes—
567C22#1686404418yes—
568C22#1686408318yes—
569C22C22×D4×D6711yescharacteristic
570C22C23×C22⋊D6711yescharacteristic
571C22C2×#41×D6712yes—
572C22C22×#1176162711yescharacteristic
573C22C2×#1452409012yes—
574C4—24no—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—2684no—
580C4—2684no—
581C4—1348no—
582C4—1348no—
583C2—28no—
584C2—82no—
585C2#2256361718yesdirect product factor
586C2C22×#1452401511yescharacteristic
587C2C2×#1686390311yescharacteristic
588C2C2×#1686396711yescharacteristic
589C2C2×#1686404412yes—
590C2C2×#1686408312yes—
subgroupG/H
centerC24C23×D67
commutator, cosocleC23×C67C25
FrattiniC23C24×D67
FittingC22×#123×C67C2
socleC24×C67C24
prime radicalC23×D4×C67C22
hypercenterC22×#123D67
hypocenterC67C2×#2413
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. C22×#14524002C2semidirect
3. C23×Dic67⋊C4C2semidirect
4. #22563617C2direct (N × H)
5. C2×#16863329C2semidirect
6. C2×#16863319C2semidirect
7. C2×#16863291C2semidirect
8. C2×#16863290C2semidirect
10. C2×#16863268C2semidirect
11. C2×#16863257C2semidirect
18. C2×#14524002C22semidirect
25. C22×Dic67⋊C4C22semidirect
31. #16863329C22semidirect
32. #16863319C22semidirect
33. #16863291C22semidirect
34. #16863290C22semidirect
36. #16863268C22semidirect
37. #16863257C22semidirect
290. C2×C67#2413semidirect
308. C67C2×#2413semidirect
585. C2#22563617direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240242
C22×#145240022
C23×Dic67⋊C42
#225636172
C2×#168633292
C2×#168633192
C2×#168632912
C2×#168632902
C2×#168632762
C2×#168632682
C2×#168632572
C22×#123×C672
C2×#241367