Finite Group Databasebeta

← C2×#22564522 #31025000

Subgroups — C2×#22564522 #31025000

Order 17152.

3374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D67C211yesmaximal, characteristic
2C22×C4⋊C4×D67C211yesmaximal, characteristic, complemented
3#22564522C2116yesmaximal, direct product factor
4C2×#16863337C212yesmaximal, complemented
5C2×#16863332C212yesmaximal, complemented
6C2×#16863295C211yesmaximal, characteristic, complemented
7C2×#16863292C211yesmaximal, characteristic, complemented
8C2×#16863272C212yesmaximal
9C2×#16863270C212yesmaximal, complemented
10C2×#16863256C211yesmaximal, characteristic, complemented
11C2×#16863253C211yesmaximal, characteristic, complemented
12C2×#380×C67C211yesmaximal, characteristic, complemented
13C23×C22⋊D67C2211yescharacteristic
14C23×C4×D67C2211yescharacteristic
15C23×C4×D67C2212yes—
16C2×C4×C22⋊D67—28no—
17C2×C4×C22⋊D67C2218yes—
18C22×C4⋊Dic67C2211yescharacteristic
19C22×C4⋊Dic67C2211yescharacteristic
20C22×C4⋊Dic67C2212yes—
21C22×Dic67⋊C4C2212yes—
22C22×Dic67⋊C4C2211yescharacteristic
23C22×Dic67⋊C4C2212yes—
24C2×C4⋊C4×D67—28no—
25C2×C4⋊C4×D67C2218yescomplemented
26C22×C22⋊Dic67C2211yescharacteristic
27C22×C22⋊Dic67C2212yes—
28C22×(C2×D67)⋊C4C2212yes—
29C22×(C2×D67)⋊C4C2211yescharacteristic
30C22×C4×Dic67C2211yescharacteristic
31#16863337C22116yescomplemented
32#16863332C22116yescomplemented
33#16863295C2218yescomplemented
34#16863292C2218yescomplemented
35#16863272C22116yes—
36#16863270C22116yescomplemented
37#16863256C2218yescomplemented
38#16863253C2218yescomplemented
39C2×C67⋊#95C2212yes—
40C2×C67⋊#95C2212yes—
41C2×C67⋊#95C2212yes—
42C2×C67⋊#95C2212yes—
43C24×C4×C67C2211yescharacteristic
44C22×C4⋊C4×C67C2211yescharacteristic
45C22×C4⋊C4×C67C2211yescharacteristic
46C22×C22⋊C4×C67C2212yes—
47#380×C67C2218yescomplemented
48C2×#95×C67C2212yes—
49C24×D67C2311yescharacteristic
50C22×C22⋊D67—24no—
51C22×C22⋊D67—24no—
52C22×C22⋊D67Q818yes—
53C22×C22⋊D67C2314yes—
54C23×Dic67C2312yes—
55C23×Dic67C2312yes—
56C23×Dic67C2311yescharacteristic
57C23×Dic67C2311yescharacteristic
58C23×Dic67C2311yescharacteristic
59C22×C4×D67—28no—
60C22×C4×D67—24no—
61C22×C4×D67—24no—
62C22×C4×D67—44no—
63C22×C4×D67—44no—
64C22×C4×D67D418yes—
65C22×C4×D67C2314yes—
66C22×C4×D67C2318yes—
67C4×C22⋊D67—232no—
68C2×C4⋊Dic67—24no—
69C2×C4⋊Dic67—24no—
70C2×C4⋊Dic67—44no—
71C2×C4⋊Dic67C2314yes—
72C2×C4⋊Dic67C2314yes—
73C2×C4⋊Dic67C2318yes—
74C2×Dic67⋊C4—28no—
75C2×Dic67⋊C4—24no—
76C2×Dic67⋊C4—44no—
77C2×Dic67⋊C4C2318yes—
78C2×Dic67⋊C4C2314yes—
79C2×Dic67⋊C4C2318yes—
80C4⋊C4×D67—232no—
81C2×C22⋊Dic67—24no—
82C2×C22⋊Dic67—44no—
83C2×C22⋊Dic67C2314yes—
84C2×C22⋊Dic67C2318yes—
85C2×(C2×D67)⋊C4—24no—
86C2×(C2×D67)⋊C4—28no—
87C2×(C2×D67)⋊C4C2318yes—
88C2×(C2×D67)⋊C4C2314yes—
89C2×C4×Dic67—24no—
90C2×C4×Dic67C2314yes—
91C67⋊#95C2318yes—
92C67⋊#95C2318yes—
93C67⋊#95C2318yes—
94C67⋊#95C2318yes—
95C25×C67C2311yescharacteristic
96C23×C4×C67—24no—
97C23×C4×C67—24no—
98C23×C4×C67—24no—
99C23×C4×C67C2312yes—
100C23×C4×C67C2311yescharacteristic
101C23×C4×C67C2312yes—
102C23×C4×C67C2314yes—
103C23×C4×C67C2311yescharacteristic
104C2×C4⋊C4×C67—24no—
105C2×C4⋊C4×C67—24no—
106C2×C4⋊C4×C67C2314yes—
107C2×C4⋊C4×C67C2314yes—
108C2×C22⋊C4×C67—44no—
109C2×C22⋊C4×C67C2318yes—
110#95×C67C2318yes—
111C23×D67—22no—
112C23×D67—22no—
113C23×D67—22no—
114C23×D67—22no—
115C23×D67C2×D414yes—
116C23×D67C2×Q814yes—
117C23×D67#4114yes—
118C23×D67C2412yes—
119C2×C22⋊D67—232no—
120C2×C22⋊D67—28no—
121C2×C22⋊D67—28no—
122C2×C22⋊D67C2×Q8116yes—
123C22×Dic67—22no—
124C22×Dic67—22no—
125C22×Dic67—22no—
126C22×Dic67—22no—
127C22×Dic67—22no—
128C22×Dic67—24no—
129C22×Dic67—24no—
130C22×Dic67—24no—
131C22×Dic67—24no—
132C22×Dic67—24no—
133C22×Dic67—22no—
134C22×Dic67—22no—
135C22×Dic67—24no—
136C22×Dic67C2×D414yes—
137C22×Dic67C2×Q814yes—
138C22×Dic67C2414yes—
139C22×Dic67C2414yes—
140C22×Dic67C2412yes—
141C22×Dic67C2412yes—
142C22×Dic67C2412yes—
143C2×C4×D67—232no—
144C2×C4×D67—28no—
145C2×C4×D67—28no—
146C2×C4×D67—216no—
147C2×C4×D67—48no—
148C2×C4×D67—48no—
149C2×C4×D67—432no—
150C2×C4×D67C2×D4116yes—
151C4⋊Dic67—28no—
152C4⋊Dic67—28no—
153C4⋊Dic67—48no—
154Dic67⋊C4—216no—
155Dic67⋊C4—28no—
156Dic67⋊C4—48no—
157C22⋊Dic67—28no—
158C22⋊Dic67—48no—
159(C2×D67)⋊C4—28no—
160(C2×D67)⋊C4—216no—
161C4×Dic67—28no—
162C24×C67—22no—
163C24×C67—22no—
164C24×C67—22no—
165C24×C67—22no—
166C24×C67—22no—
167C24×C67—22no—
168C24×C67C2×Q814yes—
169C24×C67C2412yes—
170C24×C67C2411yescharacteristic
171C22×C4×C67—24no—
172C22×C4×C67—22no—
173C22×C4×C67—24no—
174C22×C4×C67—22no—
175C22×C4×C67—22no—
176C22×C4×C67—24no—
177C22×C4×C67—24no—
178C22×C4×C67—24no—
179C22×C4×C67—24no—
180C22×C4×C67—28no—
181C22×C4×C67—28no—
182C22×C4×C67—28no—
183C22×C4×C67—416no—
184C22×C4×C67C2×D414yes—
185C22×C4×C67#4114yes—
186C22×C4×C67#4114yes—
187C22×C4×C67C2414yes—
188C22×C4×C67C2414yes—
189C22×C4×C67C2412yes—
190C22×C4×C67C2412yes—
191C4⋊C4×C67—28no—
192C4⋊C4×C67—28no—
193C22⋊C4×C67—48no—
194C23×C67—28no—
195C23×C67—28no—
196C23×C67—28no—
197C23×C67—22no—
198C23×C67—22no—
199C23×C67—22no—
200C23×C67—22no—
201C23×C67—22no—
202C23×C67—22no—
203C23×C67—44no—
204C23×C67—44no—
205C23×C67—44no—
206C23×C67—44no—
207C23×C67C22×D412yes—
208C23×C67C22×Q814yes—
209C23×C67C22×Q812yes—
210C23×C67C2×#4112yes—
211C23×C67C2×#4112yes—
212C23×C67C2×#4112yes—
213C23×C67H(5,𝔽2)12yes—
214C23×C67H(5,𝔽2)12yes—
215C23×C67C2511yescharacteristic
216C22×D67—28no—
217C22×D67—28no—
218C22×D67—28no—
219C22×D67—28no—
220C22×D67—28no—
221C22×D67—28no—
222C22×D67—22no—
223C22×D67—22no—
224C22×D67—22no—
225C22×D67—22no—
226C22×D67C22⋊Q8116yes—
227C22×D67C22×D414yes—
228C22×D67C22×Q814yes—
229C22×D67C2×#4114yes—
230C2×C4×C67—24no—
231C2×C4×C67—22no—
232C2×C4×C67—24no—
233C2×C4×C67—22no—
234C2×C4×C67—22no—
235C2×C4×C67—24no—
236C2×C4×C67—24no—
237C2×C4×C67—24no—
238C2×C4×C67—24no—
239C2×C4×C67—44no—
240C2×C4×C67—48no—
241C2×C4×C67—48no—
242C2×C4×C67—416no—
243C2×C4×C67#123116yes—
244C2×C4×C67C22×D414yes—
245C2×C4×C67C2×#4114yes—
246C2×C4×C67C2×#4114yes—
247C22⋊D67—232no—
248C2×Dic67—28no—
249C2×Dic67—28no—
250C2×Dic67—22no—
251C2×Dic67—22no—
252C2×Dic67—22no—
253C2×Dic67—22no—
254C2×Dic67—22no—
255C2×Dic67—24no—
256C2×Dic67—24no—
257C2×Dic67—24no—
258C2×Dic67—24no—
259C2×Dic67—24no—
260C2×Dic67—22no—
261C2×Dic67—22no—
262C2×Dic67—24no—
263C2×Dic67—48no—
264C2×Dic67—44no—
265C2×Dic67—48no—
266C2×Dic67C22×D414yes—
267C2×Dic67C22×Q814yes—
268C4×D67—232no—
269C4×D67—432no—
270C22×C67—24no—
271C22×C67—24no—
272C22×C67—24no—
273C22×C67—42no—
274C22×C67—42no—
275C22×C67—42no—
276C22×C67—42no—
277C22×C67—416no—
278C22×C67C2×C22⋊Q814yes—
279C22×C67C2×#12314yes—
280C22×C67#53514yes—
281C22×C67#55314yes—
282C22×C67#55314yes—
283C22×C67#55414yes—
284C22×C67#55414yes—
285C22×C67C23×D411yescharacteristic
286C22×C67C23×Q811yescharacteristic
287C22×C67C22×#4111yescharacteristic
288C22×C67C22×#4111yescharacteristic
289C22×C67C22×#4111yescharacteristic
290C22×C67C2×H(5,𝔽2)11yescharacteristic
291C22×C67C2×H(5,𝔽2)11yescharacteristic
292C2×D67—24no—
293C2×D67—24no—
294C2×D67—24no—
295C2×D67—24no—
296C2×D67—24no—
297C2×D67—24no—
298C2×D67—232no—
299C2×D67C2×C22⋊Q818yes—
300C4×C67—42no—
301C4×C67—44no—
302C4×C67—44no—
303C4×C67C2×#12318yes—
304Dic67—24no—
305Dic67—24no—
306Dic67—44no—
307Dic67—42no—
308Dic67—44no—
309C2×#2487—671nomaximal, Sylow
310C2×C67—44no—
311C2×C67#248718yescomplemented
312C2×C67C22×C22⋊Q811yescharacteristic
313C2×C67C22×#12311yescharacteristic
314C2×C67C2×#53511yescharacteristic
315C2×C67C2×#55311yescharacteristic
316C2×C67C2×#55311yescharacteristic
317C2×C67C2×#55411yescharacteristic
318C2×C67C2×#55411yescharacteristic
319D67—28no—
320C22×C4×D4—671no—
321C23×C4⋊C4—671no—
322#2487—6716no—
323C2×#400—672no—
324C2×#399—672no—
325C2×#389—671no—
326C2×#389—671no—
327C2×#386—672no—
328C2×#385—672no—
329C2×#380—671no—
330C2×#380—671no—
331C2×#380D6711yescharacteristic, complemented
332C67C2×#248711yescharacteristic, complemented, Sylow
333C23×D4—671no—
334C24×C4—672no—
335C24×C4—671no—
336C24×C4C2×D6711yescharacteristic
337C2×C4×D4—678no—
338C2×C4×D4—1348no—
339C22×C4⋊C4—672no—
340C22×C4⋊C4—672no—
341C22×C4⋊C4—672no—
342C22×C4⋊C4—671no—
343C22×C4⋊C4—671no—
344C22×C4⋊C4—671no—
345C22×C4⋊C4—678no—
346C22×C4⋊C4—1348no—
347C22×C4⋊C4C2×D6711yescharacteristic
348C22×C4⋊C4C2×D6711yescharacteristic
349C22×C22⋊C4—672no—
350C22×C22⋊C4—671no—
351C22×C22⋊C4—671no—
352C22×C22⋊C4—672no—
353C22×C22⋊C4C2×D6712yes—
354C22×C42—671no—
355#400—6716no—
356#399—6716no—
357#389—678no—
358#389—678no—
359#386—6716no—
360#385—6716no—
361#380—678no—
362#380—678no—
363#380C2×D6718yescomplemented
364C2×#95—672no—
365C2×#95—672no—
366C2×#95—672no—
367C2×#95—672no—
368C2×#95C2×D6712yes—
369C25—671no—
370C25C22×D6711yescharacteristic
371C22×D4—678no—
372C22×D4—674no—
373C22×D4—1344no—
374C22×D4—1344no—
375C23×C4—24no—
376C23×C4—24no—
377C23×C4—24no—
378C23×C4—2684no—
379C23×C4—2684no—
380C23×C4—678no—
381C23×C4—671no—
382C23×C4—671no—
383C23×C4—678no—
384C23×C4—674no—
385C23×C4—671no—
386C23×C4—672no—
387C23×C4—672no—
388C23×C4—1348no—
389C23×C4—1344no—
390C23×C4—1344no—
391C23×C4C22×D6714yes—
392C23×C4C22×D6712yes—
393C23×C4C22×D6711yescharacteristic
394C23×C4C22×D6712yes—
395C23×C4C22×D6711yescharacteristic
396C4×D4—13432no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—2684no—
400C2×C4⋊C4—2684no—
401C2×C4⋊C4—678no—
402C2×C4⋊C4—678no—
403C2×C4⋊C4—678no—
404C2×C4⋊C4—674no—
405C2×C4⋊C4—674no—
406C2×C4⋊C4—674no—
407C2×C4⋊C4—13432no—
408C2×C4⋊C4—1348no—
409C2×C4⋊C4—1344no—
410C2×C4⋊C4—1344no—
411C2×C4⋊C4—1344no—
412C2×C4⋊C4C22×D6714yes—
413C2×C4⋊C4C22×D6714yes—
414C2×C22⋊C4—44no—
415C2×C22⋊C4—2684no—
416C2×C22⋊C4—678no—
417C2×C22⋊C4—674no—
418C2×C22⋊C4—674no—
419C2×C22⋊C4—678no—
420C2×C22⋊C4—1344no—
421C2×C22⋊C4—1348no—
422C2×C22⋊C4—1344no—
423C2×C22⋊C4C22×D6718yes—
424C2×C42—674no—
425C2×C42—1344no—
426#95—678no—
427#95—678no—
428#95—678no—
429#95—678no—
430#95C22×D6718yes—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—674no—
438C24—674no—
439C24—674no—
440C24—672no—
441C24—1342no—
442C24—1342no—
443C24—1342no—
444C24—1342no—
445C24Q8×D6714yes—
446C24C23×D6712yes—
447C24C23×D6711yescharacteristic
448C2×D4—6716no—
449C2×D4—1348no—
450C2×D4—13432no—
451C2×D4—1348no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—416no—
465C22×C4—2688no—
466C22×C4—2688no—
467C22×C4—26832no—
468C22×C4—674no—
469C22×C4—674no—
470C22×C4—672no—
471C22×C4—672no—
472C22×C4—6716no—
473C22×C4—674no—
474C22×C4—674no—
475C22×C4—672no—
476C22×C4—13416no—
477C22×C4—1342no—
478C22×C4—1344no—
479C22×C4—1344no—
480C22×C4—1344no—
481C22×C4—1342no—
482C22×C4—1344no—
483C22×C4—1344no—
484C22×C4—1344no—
485C22×C4—1342no—
486C22×C4—1342no—
487C22×C4—1342no—
488C22×C4—1342no—
489C22×C4—1342no—
490C22×C4—13432no—
491C22×C4—1348no—
492C22×C4—1348no—
493C22×C4D4×D6714yes—
494C22×C4#1176162714yes—
495C22×C4#1176162814yes—
496C22×C4C23×D6712yes—
497C22×C4C23×D6714yes—
498C22×C4C23×D6714yes—
499C22×C4C23×D6712yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—2688no—
503C4⋊C4—2688no—
504C4⋊C4—13416no—
505C4⋊C4—1348no—
506C4⋊C4—1348no—
507C4⋊C4—1348no—
508C22⋊C4—48no—
509C22⋊C4—2688no—
510C22⋊C4—1348no—
511C22⋊C4—13416no—
512C22⋊C4—1348no—
513C42—1348no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—6716no—
528C23—674no—
529C23—674no—
530C23—674no—
531C23—1348no—
532C23—1348no—
533C23—1348no—
534C23—1348no—
535C23—1348no—
536C23—1348no—
537C23—1342no—
538C23—1342no—
539C23—1342no—
540C23—1342no—
541C23C2×D4×D6712yes—
542C23C2×Q8×D6714yes—
543C23C2×Q8×D6712yes—
544C23#41×D6712yes—
545C23C2×#1176162712yes—
546C23C2×#1176162812yes—
547C23#1452408812yes—
548C23#1452408912yes—
549C23C24×D6711yescharacteristic
550D4—13432no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—416no—
564C2×C4—26832no—
565C2×C4—2688no—
566C2×C4—2688no—
567C2×C4—2684no—
568C2×C4—674no—
569C2×C4—674no—
570C2×C4—1342no—
571C2×C4—1344no—
572C2×C4—1344no—
573C2×C4—1344no—
574C2×C4—1342no—
575C2×C4—1344no—
576C2×C4—1344no—
577C2×C4—1344no—
578C2×C4—1342no—
579C2×C4—1348no—
580C2×C4—1348no—
581C2×C4—1342no—
582C2×C4—1342no—
583C2×C4—1342no—
584C2×C4—1342no—
585C2×C4—13432no—
586C2×C4#14524025116yes—
587C2×C4C2×D4×D6714yes—
588C2×C4C2×#1176162714yes—
589C2×C4C2×#1176162814yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—416no—
597C22—42no—
598C22—678no—
599C22—1344no—
600C22—1344no—
601C22—1344no—
602C22—13432no—
603C22—1344no—
604C22—1344no—
605C22—1344no—
606C22C22⋊Q8×D6714yes—
607C22C2×#1452402514yes—
608C22#1686396214yes—
609C22#1686409014yes—
610C22#1686409314yes—
611C22#1686410114yes—
612C22#1686410414yes—
613C22C22×D4×D6711yescharacteristic
614C22C22×Q8×D6711yescharacteristic
615C22C2×#41×D6711yescharacteristic
616C22C22×#1176162711yescharacteristic
617C22C22×#1176162811yescharacteristic
618C22C2×#1452408811yescharacteristic
619C22C2×#1452408911yescharacteristic
620C4—42no—
621C4—44no—
622C4—44no—
623C4—2684no—
624C4—2684no—
625C4—2682no—
626C4—1344no—
627C4—1344no—
628C4C2×#1452402518yes—
629C2—44no—
630C2—1348no—
631C2#2256452218yesdirect product factor
632C2C2×C22⋊Q8×D6711yescharacteristic
633C2C22×#1452402511yescharacteristic
634C2C2×#1686396211yescharacteristic
635C2C2×#1686409011yescharacteristic
636C2C2×#1686409311yescharacteristic
637C2C2×#1686410111yescharacteristic
638C2C2×#1686410411yescharacteristic
subgroupG/H
centerC24C23×D67
commutator, cosocleC23×C67C25
FrattiniC23C24×D67
FittingC2×#380×C67C2
socleC24×C67C24
prime radicalC23×C22⋊D67C22
hypercenterC2×#380D67
hypocenterC67C2×#2487
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D67C2semidirect
3. #22564522C2direct (N × H)
4. C2×#16863337C2semidirect
5. C2×#16863332C2semidirect
6. C2×#16863295C2semidirect
7. C2×#16863292C2semidirect
9. C2×#16863270C2semidirect
10. C2×#16863256C2semidirect
11. C2×#16863253C2semidirect
12. C2×#380×C67C2semidirect
25. C2×C4⋊C4×D67C22semidirect
31. #16863337C22semidirect
32. #16863332C22semidirect
33. #16863295C22semidirect
34. #16863292C22semidirect
36. #16863270C22semidirect
37. #16863256C22semidirect
38. #16863253C22semidirect
47. #380×C67C22semidirect
311. C2×C67#2487semidirect
331. C2×#380D67semidirect
332. C67C2×#2487semidirect
363. #380C2×D67semidirect
631. C2#22564522direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D672
C22×C4⋊C4×D672
#225645222
C2×#168633372
C2×#168633322
C2×#168632952
C2×#168632922
C2×#168632722
C2×#168632702
C2×#168632562
C2×#168632532
C2×#380×C672
C2×#248767