Finite Group Databasebeta

← C2×#22563616 #31024533

Subgroups — C2×#22563616 #31024533

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×#14524026C211yesmaximal, characteristic
2C22×#14524026C211yesmaximal, characteristic
3C22×#14524002C211yesmaximal, characteristic, complemented
4#22563616C2116yesmaximal, direct product factor
5C2×#16863329C212yesmaximal, complemented
6C2×#16863318C212yesmaximal, complemented
7C2×#16863291C212yesmaximal, complemented
8C2×#16863286C212yesmaximal, complemented
9C2×#16863275C211yesmaximal, characteristic
10C2×#16863269C211yesmaximal, characteristic, complemented
11C2×#16863259C211yesmaximal, characteristic, complemented
12C23×C4⋊C4×C67C211yesmaximal, characteristic, complemented
13C23×C22⋊D67C2211yescharacteristic
14C2×#14524026—28no—
15C2×#14524026—28no—
16C2×#14524026C2218yes—
17C2×#14524026C2218yes—
18C2×#14524002—28no—
19C2×#14524002C2218yescomplemented
20C22×C4⋊Dic67C2212yes—
21C22×Dic67⋊C4C2212yes—
22C22×Dic67⋊C4C2211yescharacteristic
23C22×Dic67⋊C4C2212yes—
24C22×Dic67⋊C4C2211yescharacteristic
25C22×C22⋊Dic67C2211yescharacteristic
26C22×C22⋊Dic67C2211yescharacteristic
27C22×C22⋊Dic67C2211yescharacteristic
28C22×(C2×D67)⋊C4C2212yes—
29C22×(C2×D67)⋊C4C2211yescharacteristic
30C22×(C2×D67)⋊C4C2212yes—
31C22×(C2×D67)⋊C4C2211yescharacteristic
32C22×C4×Dic67C2212yes—
33#16863329C22116yescomplemented
34#16863318C22116yescomplemented
35#16863291C22116yescomplemented
36#16863286C22116yescomplemented
37#16863275C2218yes—
38#16863269C2218yescomplemented
39#16863259C2218yescomplemented
40C2×C67⋊#95C2212yes—
41C2×C67⋊#95C2212yes—
42C2×C67⋊#95C2212yes—
43C2×C67⋊#95C2212yes—
44C24×C4×C67C2211yescharacteristic
45C24×C4×C67C2211yescharacteristic
46C24×C4×C67C2211yescharacteristic
47C22×C4⋊C4×C67—28no—
48C22×C4⋊C4×C67C2212yes—
49C22×C4⋊C4×C67C2212yes—
50C22×C4⋊C4×C67C2218yescomplemented
51C24×D67C2311yescharacteristic
52C22×C22⋊D67—24no—
53C22×C22⋊D67—24no—
54C22×C22⋊D67—24no—
55C22×C22⋊D67C2314yes—
56C23×Dic67C2311yescharacteristic
57C23×Dic67C2312yes—
58C23×Dic67C2311yescharacteristic
59C23×Dic67C2311yescharacteristic
60C23×Dic67C2312yes—
61#14524026—232no—
62#14524026—232no—
63#14524002—232no—
64C2×C4⋊Dic67—28no—
65C2×C4⋊Dic67C2318yes—
66C2×Dic67⋊C4—28no—
67C2×Dic67⋊C4—24no—
68C2×Dic67⋊C4—24no—
69C2×Dic67⋊C4—28no—
70C2×Dic67⋊C4C2318yes—
71C2×Dic67⋊C4C2314yes—
72C2×Dic67⋊C4C2318yes—
73C2×Dic67⋊C4C2314yes—
74C2×C22⋊Dic67—24no—
75C2×C22⋊Dic67—24no—
76C2×C22⋊Dic67D418yes—
77C2×C22⋊Dic67C2314yes—
78C2×C22⋊Dic67C2314yes—
79C2×C22⋊Dic67C2314yes—
80C2×(C2×D67)⋊C4—28no—
81C2×(C2×D67)⋊C4—24no—
82C2×(C2×D67)⋊C4—24no—
83C2×(C2×D67)⋊C4—28no—
84C2×(C2×D67)⋊C4C2318yes—
85C2×(C2×D67)⋊C4C2314yes—
86C2×(C2×D67)⋊C4C2318yes—
87C2×(C2×D67)⋊C4C2314yes—
88C2×C4×Dic67—28no—
89C2×C4×Dic67C2318yes—
90C67⋊#95C2318yes—
91C67⋊#95C2318yes—
92C67⋊#95C2318yes—
93C67⋊#95C2318yes—
94C25×C67C2311yescharacteristic
95C23×C4×C67—24no—
96C23×C4×C67—24no—
97C23×C4×C67—24no—
98C23×C4×C67—28no—
99C23×C4×C67—24no—
100C23×C4×C67—24no—
101C23×C4×C67—24no—
102C23×C4×C67D418yes—
103C23×C4×C67C2312yes—
104C23×C4×C67C2314yes—
105C23×C4×C67C2314yes—
106C23×C4×C67C2311yescharacteristic
107C23×C4×C67C2311yescharacteristic
108C23×C4×C67C2314yes—
109C23×C4×C67C2312yes—
110C2×C4⋊C4×C67—28no—
111C2×C4⋊C4×C67—28no—
112C2×C4⋊C4×C67—232no—
113C2×C4⋊C4×C67C2318yes—
114C2×C4⋊C4×C67C2318yes—
115C23×D67—22no—
116C23×D67—22no—
117C23×D67—24no—
118C23×D67—22no—
119C23×D67—24no—
120C23×D67C2412yes—
121C2×C22⋊D67—28no—
122C2×C22⋊D67—28no—
123C2×C22⋊D67—28no—
124C2×C22⋊D67—416no—
125C22×Dic67—24no—
126C22×Dic67—24no—
127C22×Dic67—24no—
128C22×Dic67—22no—
129C22×Dic67—22no—
130C22×Dic67—22no—
131C22×Dic67—22no—
132C22×Dic67—24no—
133C22×Dic67—24no—
134C22×Dic67—22no—
135C22×Dic67—24no—
136C22×Dic67—24no—
137C22×Dic67C2×D418yes—
138C22×Dic67C2412yes—
139C22×Dic67C2414yes—
140C22×Dic67C2412yes—
141C22×Dic67C2412yes—
142C22×Dic67C2414yes—
143C4⋊Dic67—216no—
144Dic67⋊C4—28no—
145Dic67⋊C4—28no—
146Dic67⋊C4—216no—
147Dic67⋊C4—216no—
148C22⋊Dic67—28no—
149C22⋊Dic67—28no—
150C22⋊Dic67C2×D4116yes—
151(C2×D67)⋊C4—28no—
152(C2×D67)⋊C4—28no—
153(C2×D67)⋊C4—216no—
154(C2×D67)⋊C4—216no—
155C4×Dic67—216no—
156C24×C67—24no—
157C24×C67—24no—
158C24×C67C2×D414yes—
159C24×C67C2×D414yes—
160C24×C67#4114yes—
161C24×C67C2412yes—
162C24×C67C2411yescharacteristic
163C22×C4×C67—232no—
164C22×C4×C67—232no—
165C22×C4×C67—28no—
166C22×C4×C67—24no—
167C22×C4×C67—22no—
168C22×C4×C67—22no—
169C22×C4×C67—28no—
170C22×C4×C67—28no—
171C22×C4×C67—22no—
172C22×C4×C67—22no—
173C22×C4×C67—24no—
174C22×C4×C67—216no—
175C22×C4×C67—28no—
176C22×C4×C67—24no—
177C22×C4×C67—24no—
178C22×C4×C67—28no—
179C22×C4×C67—28no—
180C22×C4×C67—24no—
181C22×C4×C67—416no—
182C22×C4×C67C2×D414yes—
183C22×C4×C67C2×D414yes—
184C22×C4×C67C2×D4116yes—
185C22×C4×C67#4118yes—
186C22×C4×C67C2414yes—
187C22×C4×C67C2412yes—
188C22×C4×C67C2412yes—
189C22×C4×C67C2414yes—
190C4⋊C4×C67—216no—
191C4⋊C4×C67—216no—
192C23×C67—28no—
193C23×C67—28no—
194C23×C67—216no—
195C23×C67—216no—
196C23×C67—24no—
197C23×C67—24no—
198C23×C67#121116yes—
199C23×C67C22×D414yes—
200C23×C67C22×D412yes—
201C23×C67C22×D414yes—
202C23×C67C22×D412yes—
203C23×C67C2×#4114yes—
204C23×C67C2×#4112yes—
205C23×C67C2×#4114yes—
206C23×C672−1+414yes—
207C23×C67C2511yescharacteristic
208C22×D67—22no—
209C22×D67—22no—
210C22×D67—24no—
211C22×D67—22no—
212C22×D67—24no—
213C22×D67—44no—
214C22×D67—44no—
215C22×D67—44no—
216C22×D67—48no—
217C22×D67—48no—
218C2×C4×C67—232no—
219C2×C4×C67—232no—
220C2×C4×C67—28no—
221C2×C4×C67—28no—
222C2×C4×C67—216no—
223C2×C4×C67—24no—
224C2×C4×C67—22no—
225C2×C4×C67—22no—
226C2×C4×C67—22no—
227C2×C4×C67—22no—
228C2×C4×C67—24no—
229C2×C4×C67—24no—
230C2×C4×C67—24no—
231C2×C4×C67—24no—
232C2×C4×C67—416no—
233C2×C4×C67—48no—
234C2×C4×C67C22×D414yes—
235C2×C4×C67C22×D414yes—
236C2×C4×C67C2×#4118yes—
237C22⋊D67—416no—
238C2×Dic67—216no—
239C2×Dic67—24no—
240C2×Dic67—24no—
241C2×Dic67—24no—
242C2×Dic67—22no—
243C2×Dic67—22no—
244C2×Dic67—22no—
245C2×Dic67—22no—
246C2×Dic67—24no—
247C2×Dic67—24no—
248C2×Dic67—22no—
249C2×Dic67—24no—
250C2×Dic67—24no—
251C2×Dic67—44no—
252C2×Dic67—44no—
253C2×Dic67—44no—
254C2×Dic67—48no—
255C2×Dic67C22×D418yes—
256C22×C67—24no—
257C22×C67—232no—
258C22×C67—24no—
259C22×C67—28no—
260C22×C67—28no—
261C22×C67C2×#12118yes—
262C22×C67C2×#12114yes—
263C22×C67#52514yes—
264C22×C67#53614yes—
265C22×C67#54718yes—
266C22×C67#55218yes—
267C22×C67C23×D411yescharacteristic
268C22×C67C23×D411yescharacteristic
269C22×C67C22×#4111yescharacteristic
270C22×C67C22×#4112yes—
271C22×C67C2×2−1+412yes—
272C2×D67—42no—
273C2×D67—42no—
274C2×D67—42no—
275C2×D67—44no—
276C2×D67—44no—
277C2×D67—88no—
278C4×C67—24no—
279C4×C67—24no—
280C4×C67—28no—
281C4×C67—44no—
282Dic67—28no—
283Dic67—42no—
284Dic67—42no—
285Dic67—42no—
286Dic67—44no—
287C2×#2413—671nomaximal, Sylow
288C2×C67—28no—
289C2×C67#241318yescomplemented
290C2×C67C22×#12111yescharacteristic
291C2×C67C2×#52511yescharacteristic
292C2×C67C2×#53611yescharacteristic
293C2×C67C2×#54712yes—
294C2×C67C2×#55212yes—
295D67—82no—
296C22×#123—671no—
297C22×#123—671no—
298C22×#117—671no—
299C23×C4⋊C4D6711yescharacteristic, complemented
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×D4—671no—
310C24×C4C2×D6711yescharacteristic
311C24×C4C2×D6711yescharacteristic
312C24×C4C2×D6711yescharacteristic
313C2×#123—678no—
314C2×#123—678no—
315C2×#123—1348no—
316C2×#123—1348no—
317C2×#117—678no—
318C2×#117—1348no—
319C22×C4⋊C4—28no—
320C22×C4⋊C4—672no—
321C22×C4⋊C4—672no—
322C22×C4⋊C4—671no—
323C22×C4⋊C4—672no—
324C22×C4⋊C4—671no—
325C22×C4⋊C4C2×D6712yes—
326C22×C4⋊C4C2×D6712yes—
327C22×C4⋊C4C2×D6718yescomplemented
328C22×C22⋊C4—671no—
329C22×C22⋊C4—671no—
330C22×C22⋊C4—671no—
331C22×C22⋊C4—671no—
332C22×C22⋊C4—672no—
333C22×C22⋊C4—672no—
334C22×C22⋊C4—671no—
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—
347C25—671no—
348C25C22×D6711yescharacteristic
349C22×D4—674no—
350C22×D4—1344no—
351C22×D4—1344no—
352C22×D4—1344no—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—24no—
356C23×C4—28no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—24no—
360C23×C4—672no—
361C23×C4—671no—
362C23×C4—671no—
363C23×C4—672no—
364C23×C4—671no—
365C23×C4C22⋊D6718yes—
366C23×C4C22×D6714yes—
367C23×C4C22×D6714yes—
368C23×C4C22×D6712yes—
369C23×C4C22×D6714yes—
370C23×C4C22×D6711yescharacteristic
371C23×C4C22×D6711yescharacteristic
372C23×C4C22×D6712yes—
373#123—13432no—
374#123—13432no—
375#117—13432no—
376C2×C4⋊C4—28no—
377C2×C4⋊C4—28no—
378C2×C4⋊C4—232no—
379C2×C4⋊C4—678no—
380C2×C4⋊C4—674no—
381C2×C4⋊C4—678no—
382C2×C4⋊C4—674no—
383C2×C4⋊C4—678no—
384C2×C4⋊C4—1348no—
385C2×C4⋊C4—1348no—
386C2×C4⋊C4—1344no—
387C2×C4⋊C4—1344no—
388C2×C4⋊C4—1348no—
389C2×C4⋊C4C22×D6718yes—
390C2×C4⋊C4C22×D6718yes—
391C2×C22⋊C4—678no—
392C2×C22⋊C4—674no—
393C2×C22⋊C4—674no—
394C2×C22⋊C4—674no—
395C2×C22⋊C4—674no—
396C2×C22⋊C4—678no—
397C2×C22⋊C4—674no—
398C2×C22⋊C4—678no—
399C2×C22⋊C4—1348no—
400C2×C22⋊C4—1344no—
401C2×C22⋊C4—1344no—
402C2×C22⋊C4—1344no—
403C2×C22⋊C4—1344no—
404C2×C22⋊C4—1348no—
405C2×C42—678no—
406C2×C42—1348no—
407#95—678no—
408#95—678no—
409#95—678no—
410#95—678no—
411C24—24no—
412C24—24no—
413C24—672no—
414C24—1342no—
415C24—1342no—
416C24—1342no—
417C24—1344no—
418C24—1344no—
419C24D4×D6714yes—
420C24C2×C22⋊D6714yes—
421C24#1176162714yes—
422C24C23×D6712yes—
423C24C23×D6711yescharacteristic
424C2×D4—26816no—
425C2×D4—1348no—
426C2×D4—1348no—
427C2×D4—1348no—
428C22×C4—232no—
429C22×C4—232no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—216no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—416no—
447C22×C4—678no—
448C22×C4—674no—
449C22×C4—672no—
450C22×C4—674no—
451C22×C4—672no—
452C22×C4—672no—
453C22×C4—1344no—
454C22×C4—1342no—
455C22×C4—1342no—
456C22×C4—1342no—
457C22×C4—1342no—
458C22×C4—1342no—
459C22×C4—1344no—
460C22×C4—1344no—
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×C4#1176162818yes—
469C22×C4C23×D6714yes—
470C22×C4C23×D6712yes—
471C22×C4C23×D6712yes—
472C22×C4C23×D6714yes—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—13416no—
476C4⋊C4—13416no—
477C4⋊C4—1348no—
478C4⋊C4—1348no—
479C4⋊C4—13416no—
480C22⋊C4—6716no—
481C22⋊C4—13416no—
482C22⋊C4—1348no—
483C22⋊C4—1348no—
484C22⋊C4—1348no—
485C22⋊C4—1348no—
486C22⋊C4—13416no—
487C42—13416no—
488C23—216no—
489C23—216no—
490C23—28no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—2688no—
495C23—2688no—
496C23—2684no—
497C23—2684no—
498C23—2684no—
499C23—1342no—
500C23—1342no—
501C23—1342no—
502C23—1344no—
503C23—1344no—
504C23#14524015116yes—
505C23C2×D4×D6714yes—
506C23C2×D4×D6712yes—
507C23C22×C22⋊D6714yes—
508C23C22×C22⋊D6712yes—
509C23C2×#1176162714yes—
510C23C2×#1176162712yes—
511C23C2×#1176162814yes—
512C23#1452409114yes—
513C23C24×D6711yescharacteristic
514D4—26816no—
515C2×C4—232no—
516C2×C4—232no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—216no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—416no—
530C2×C4—48no—
531C2×C4—2688no—
532C2×C4—2684no—
533C2×C4—2684no—
534C2×C4—2684no—
535C2×C4—678no—
536C2×C4—1344no—
537C2×C4—1342no—
538C2×C4—1342no—
539C2×C4—1342no—
540C2×C4—1342no—
541C2×C4—1342no—
542C2×C4—1344no—
543C2×C4—1344no—
544C2×C4—1344no—
545C2×C4—1344no—
546C2×C4—13416no—
547C2×C4—1344no—
548C2×C4—1344no—
549C2×C4C22×C22⋊D6714yes—
550C2×C4C22×C22⋊D6714yes—
551C2×C4C2×#1176162818yes—
552C22—232no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—24no—
557C22—2684no—
558C22—2682no—
559C22—2682no—
560C22—2684no—
561C22—2682no—
562C22—5368no—
563C22C2×#1452401518yes—
564C22C2×#1452401514yes—
565C22#1686390714yes—
566C22#1686397014yes—
567C22#1686404718yes—
568C22#1686408118yes—
569C22C22×D4×D6711yescharacteristic
570C22C23×C22⋊D6711yescharacteristic
571C22C22×#1176162711yescharacteristic
572C22C22×#1176162812yes—
573C22C2×#1452409112yes—
574C4—24no—
575C4—24no—
576C4—28no—
577C4—44no—
578C4—2684no—
579C4—2682no—
580C4—2682no—
581C4—2682no—
582C4—1348no—
583C2—28no—
584C2—5362no—
585C2#2256361618yesdirect product factor
586C2C22×#1452401511yescharacteristic
587C2C2×#1686390711yescharacteristic
588C2C2×#1686397011yescharacteristic
589C2C2×#1686404712yes—
590C2C2×#1686408112yes—
subgroupG/H
centerC24C23×D67
commutator, cosocleC23×C67C25
FrattiniC23C24×D67
FittingC23×C4⋊C4×C67C2
socleC24×C67C24
prime radicalC23×C22⋊D67C22
hypercenterC23×C4⋊C4D67
hypocenterC67C2×#2413
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×#14524002C2semidirect
4. #22563616C2direct (N × H)
5. C2×#16863329C2semidirect
6. C2×#16863318C2semidirect
7. C2×#16863291C2semidirect
8. C2×#16863286C2semidirect
10. C2×#16863269C2semidirect
11. C2×#16863259C2semidirect
12. C23×C4⋊C4×C67C2semidirect
19. C2×#14524002C22semidirect
33. #16863329C22semidirect
34. #16863318C22semidirect
35. #16863291C22semidirect
36. #16863286C22semidirect
38. #16863269C22semidirect
39. #16863259C22semidirect
50. C22×C4⋊C4×C67C22semidirect
289. C2×C67#2413semidirect
299. C23×C4⋊C4D67semidirect
308. C67C2×#2413semidirect
327. C22×C4⋊C4C2×D67semidirect
585. C2#22563616direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240262
C22×#145240022
#225636162
C2×#168633292
C2×#168633182
C2×#168632912
C2×#168632862
C2×#168632752
C2×#168632692
C2×#168632592
C23×C4⋊C4×C672
C2×#241367