Finite Group Databasebeta

← C2×#22565157 #31027607

Subgroups — C2×#22565157 #31027607

Order 17152.

4006 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524042C211yesmaximal, characteristic, complemented
2C22×#14524021C212yesmaximal, complemented
3C22×#14524020C212yesmaximal, complemented
4C22×(C22⋊D67)⋊C4C211yesmaximal, characteristic
5#22565157C2116yesmaximal, direct product factor
6C2×#16863333C211yesmaximal, characteristic, complemented
7C2×#16863332C211yesmaximal, characteristic, complemented
8C2×#16863331C211yesmaximal, characteristic, complemented
9C2×#16863296C212yesmaximal
10C2×#16863265C212yesmaximal, complemented
11C2×#16863247C211yesmaximal, characteristic, complemented
12C2×#399×C67C211yesmaximal, characteristic, complemented
13C23×Dic134C2212yes—
14C23×C22⋊D67C2211yescharacteristic
15C24×Dic67C2211yescharacteristic
16C23×C4×D67C2211yescharacteristic
17C2×#14524042—28no—
18C2×#14524042C2218yescomplemented
19C2×#14524021—216no—
20C2×#14524021C22116yescomplemented
21C2×#14524020—216no—
22C2×#14524020C22116yescomplemented
23C2×(C22⋊D67)⋊C4—28no—
24C2×(C22⋊D67)⋊C4C2218yes—
25C22×C4⋊Dic67C2212yes—
26C22×C4⋊Dic67C2211yescharacteristic
27C22×Dic67⋊C4C2211yescharacteristic
28C22×Dic67⋊C4C2212yes—
29C22×Dic67⋊C4C2212yes—
30C22×Dic67⋊C4C2212yes—
31C22×Dic67⋊C4C2212yes—
32C22×C22⋊Dic67C2212yes—
33C22×(C2×D67)⋊C4C2211yescharacteristic
34C22×(C2×D67)⋊C4C2212yes—
35C22×(C2×D67)⋊C4C2212yes—
36C22×C4×Dic67C2212yes—
37C22×C4×Dic67C2211yescharacteristic
38#16863333C2218yescomplemented
39#16863332C2218yescomplemented
40#16863331C2218yescomplemented
41#16863296C22116yes—
42#16863265C22116yescomplemented
43#16863247C2218yescomplemented
44C2×C67⋊#95C2211yescharacteristic
45C22×C4⋊C4×C67C2212yes—
46C22×C4⋊C4×C67C2211yescharacteristic
47C22×C22⋊C4×C67C2211yescharacteristic
48C22×C22⋊C4×C67C2212yes—
49#399×C67C2218yescomplemented
50C2×#95×C67C2211yescharacteristic
51C24×D67C2311yescharacteristic
52C22×Dic134—28no—
53C22×Dic134—28no—
54C22×Dic134—28no—
55C22×Dic134C2318yes—
56C22×C22⋊D67—24no—
57C22×C22⋊D67—24no—
58C22×C22⋊D67Q818yescomplemented
59C22×C22⋊D67C2314yes—
60C23×Dic67—24no—
61C23×Dic67—28no—
62C23×Dic67C2312yes—
63C23×Dic67C2311yescharacteristic
64C23×Dic67C2314yes—
65C23×Dic67C2312yes—
66C23×Dic67C2311yescharacteristic
67C23×Dic67C2311yescharacteristic
68C22×C4×D67—24no—
69C22×C4×D67—28no—
70C22×C4×D67C2314yes—
71#14524042—232no—
72#14524021—264no—
73#14524020—264no—
74(C22⋊D67)⋊C4—232no—
75C2×C4⋊Dic67—28no—
76C2×C4⋊Dic67—24no—
77C2×C4⋊Dic67C2318yes—
78C2×C4⋊Dic67C2314yes—
79C2×Dic67⋊C4—24no—
80C2×Dic67⋊C4—28no—
81C2×Dic67⋊C4—28no—
82C2×Dic67⋊C4—28no—
83C2×Dic67⋊C4D4116yescomplemented
84C2×Dic67⋊C4C2314yes—
85C2×Dic67⋊C4C2318yes—
86C2×Dic67⋊C4C2318yes—
87C2×Dic67⋊C4C2318yes—
88C2×Dic67⋊C4C2318yes—
89C2×C22⋊Dic67—28no—
90C2×C22⋊Dic67C2318yes—
91C2×(C2×D67)⋊C4—28no—
92C2×(C2×D67)⋊C4—24no—
93C2×(C2×D67)⋊C4—28no—
94C2×(C2×D67)⋊C4C2318yes—
95C2×(C2×D67)⋊C4C2318yes—
96C2×(C2×D67)⋊C4C2314yes—
97C2×C4×Dic67—28no—
98C2×C4×Dic67D418yes—
99C2×C4×Dic67C2318yes—
100C2×C4×Dic67C2314yes—
101C67⋊#95C2314yes—
102C25×C67C2311yescharacteristic
103C23×C4×C67C2312yes—
104C23×C4×C67C2311yescharacteristic
105C23×C4×C67C2311yescharacteristic
106C23×C4×C67C2312yes—
107C2×C4⋊C4×C67—28no—
108C2×C4⋊C4×C67—24no—
109C2×C4⋊C4×C67C2318yes—
110C2×C4⋊C4×C67C2314yes—
111C2×C22⋊C4×C67—24no—
112C2×C22⋊C4×C67—28no—
113C2×C22⋊C4×C67C2318yes—
114C2×C22⋊C4×C67C2314yes—
115#95×C67C2314yes—
116C23×D67—24no—
117C23×D67—22no—
118C23×D67—22no—
119C23×D67—24no—
120C23×D67C2×Q814yes—
121C23×D67C2412yes—
122C2×Dic134—216no—
123C2×Dic134—216no—
124C2×Dic134—216no—
125C2×Dic134—432no—
126C2×C22⋊D67—232no—
127C2×C22⋊D67—28no—
128C2×C22⋊D67—28no—
129C2×C22⋊D67C2×Q8116yescomplemented
130C22×Dic67—24no—
131C22×Dic67—24no—
132C22×Dic67—22no—
133C22×Dic67—24no—
134C22×Dic67—24no—
135C22×Dic67—24no—
136C22×Dic67—24no—
137C22×Dic67—28no—
138C22×Dic67—216no—
139C22×Dic67—416no—
140C22×Dic67C2×D418yes—
141C22×Dic67C2×D414yes—
142C22×Dic67C2×D414yes—
143C22×Dic67C2×D418yes—
144C22×Dic67C2×Q814yes—
145C22×Dic67#4114yes—
146C22×Dic67C2414yes—
147C22×Dic67C2412yes—
148C22×Dic67C2414yes—
149C22×Dic67C2412yes—
150C22×Dic67C2412yes—
151C2×C4×D67—28no—
152C2×C4×D67—216no—
153C2×C4×D67—416no—
154C4⋊Dic67—216no—
155C4⋊Dic67—28no—
156Dic67⋊C4—28no—
157Dic67⋊C4—216no—
158Dic67⋊C4—216no—
159Dic67⋊C4—216no—
160Dic67⋊C4C2×D4132yescomplemented
161C22⋊Dic67—216no—
162(C2×D67)⋊C4—28no—
163(C2×D67)⋊C4—216no—
164(C2×D67)⋊C4—216no—
165C4×Dic67—216no—
166C4×Dic67C2×D4116yes—
167C24×C67—24no—
168C24×C67—22no—
169C24×C67—22no—
170C24×C67—24no—
171C24×C67C2×Q814yes—
172C24×C67C2412yes—
173C24×C67C2411yescharacteristic
174C22×C4×C67—24no—
175C22×C4×C67—24no—
176C22×C4×C67—24no—
177C22×C4×C67—22no—
178C22×C4×C67—24no—
179C22×C4×C67—24no—
180C22×C4×C67—24no—
181C22×C4×C67—24no—
182C22×C4×C67C2×D414yes—
183C22×C4×C67C2×D418yes—
184C22×C4×C67C2414yes—
185C22×C4×C67C2412yes—
186C22×C4×C67C2412yes—
187C22×C4×C67C2414yes—
188C4⋊C4×C67—216no—
189C4⋊C4×C67—28no—
190C22⋊C4×C67—28no—
191C22⋊C4×C67—216no—
192C23×C67—28no—
193C23×C67—216no—
194C23×C67—24no—
195C23×C67—22no—
196C23×C67—22no—
197C23×C67—24no—
198C23×C67—44no—
199C23×C67—44no—
200C23×C67—48no—
201C23×C67C22×D412yes—
202C23×C67C22×D414yes—
203C23×C67C22×Q814yes—
204C23×C67C22×Q812yes—
205C23×C67C2×#4112yes—
206C23×C672−1+414yes—
207C23×C67C2511yescharacteristic
208C22×D67—28no—
209C22×D67—216no—
210C22×D67—24no—
211C22×D67—22no—
212C22×D67—22no—
213C22×D67—24no—
214C22×D67—44no—
215C22×D67—44no—
216C22×D67—48no—
217C22×D67C22×Q814yes—
218C2×C4×C67—28no—
219C2×C4×C67—216no—
220C2×C4×C67—24no—
221C2×C4×C67—24no—
222C2×C4×C67—24no—
223C2×C4×C67—22no—
224C2×C4×C67—24no—
225C2×C4×C67—24no—
226C2×C4×C67—24no—
227C2×C4×C67—24no—
228C2×C4×C67—44no—
229C2×C4×C67—48no—
230C2×C4×C67C22×D414yes—
231C2×C4×C67C22×D418yes—
232C22⋊D67—232no—
233C2×Dic67—216no—
234C2×Dic67—24no—
235C2×Dic67—24no—
236C2×Dic67—22no—
237C2×Dic67—24no—
238C2×Dic67—24no—
239C2×Dic67—24no—
240C2×Dic67—24no—
241C2×Dic67—44no—
242C2×Dic67—48no—
243C2×Dic67—416no—
244C2×Dic67C22⋊Q8116yescomplemented
245C2×Dic67#127116yes—
246C2×Dic67C22×D418yes—
247C2×Dic67C22×D414yes—
248C2×Dic67C22×D414yes—
249C2×Dic67C22×D418yes—
250C2×Dic67C22×Q814yes—
251C2×Dic67C2×#4114yes—
252C4×D67—416no—
253Dic134—432no—
254C22×C67—24no—
255C22×C67—28no—
256C22×C67—42no—
257C22×C67—42no—
258C22×C67—44no—
259C22×C67—416no—
260C22×C67C2×C22⋊Q814yes—
261C22×C67C2×#12714yes—
262C22×C67#53614yes—
263C22×C67#54718yes—
264C22×C67D4×Q818yes—
265C22×C67C23×D411yescharacteristic
266C22×C67C23×D412yes—
267C22×C67C23×Q811yescharacteristic
268C22×C67C22×#4111yescharacteristic
269C22×C67C2×2−1+412yes—
270C2×D67—24no—
271C2×D67—28no—
272C2×D67—42no—
273C2×D67—42no—
274C2×D67—44no—
275C2×D67—416no—
276C4×C67—24no—
277C4×C67—28no—
278C4×C67—44no—
279C4×C67—42no—
280Dic67—28no—
281Dic67—44no—
282Dic67—42no—
283Dic67C2×C22⋊Q818yescomplemented
284Dic67C2×#12718yes—
285C2×#2542—671nomaximal, Sylow
286C2×C67—44no—
287C2×C67#254218yescomplemented
288C2×C67C22×C22⋊Q811yescharacteristic
289C2×C67C22×#12711yescharacteristic
290C2×C67C2×#53611yescharacteristic
291C2×C67C2×#54712yes—
292C2×C67C2×D4×Q812yes—
293D67—44no—
294C22×C4⋊Q8—671no—
295C22×C22⋊Q8—672no—
296C22×C22⋊Q8—672no—
297C22×C4×D4—671no—
298#2542—6716no—
299C2×#399—671no—
300C2×#399—671no—
301C2×#399—671no—
302C2×#399D6711yescharacteristic, complemented
303C2×#390—672no—
304C2×#384—672no—
305C2×C4×C4⋊C4—671no—
306C67C2×#254211yescharacteristic, complemented, Sylow
307C23×Q8—672no—
308C23×D4—671no—
309C24×C4—671no—
310C24×C4—671no—
311C2×C4⋊Q8—678no—
312C2×C4⋊Q8—1348no—
313C2×C22⋊Q8—6716no—
314C2×C22⋊Q8—6716no—
315C2×C22⋊Q8—13416no—
316C2×C22⋊Q8—13416no—
317C2×C4×D4—678no—
318C2×C4×D4—1348no—
319C22×C4⋊C4—672no—
320C22×C4⋊C4—671no—
321C22×C4⋊C4—672no—
322C22×C4⋊C4—672no—
323C22×C4⋊C4—672no—
324C22×C4⋊C4—671no—
325C22×C4⋊C4—672no—
326C22×C4⋊C4C2×D6711yescharacteristic
327C22×C4⋊C4C2×D6712yes—
328C22×C22⋊C4—672no—
329C22×C22⋊C4—671no—
330C22×C22⋊C4—672no—
331C22×C22⋊C4—672no—
332C22×C22⋊C4C2×D6711yescharacteristic
333C22×C22⋊C4C2×D6712yes—
334C22×C42—671no—
335C22×C42—672no—
336#399—678no—
337#399—678no—
338#399—678no—
339#399C2×D6718yescomplemented
340#390—6716no—
341#384—6716no—
342C4×C4⋊C4—678no—
343C2×#95—671no—
344C2×#95C2×D6711yescharacteristic
345C25—671no—
346C25C22×D6711yescharacteristic
347C22×Q8—678no—
348C22×Q8—1348no—
349C22×Q8—1348no—
350C22×Q8—1348no—
351C22×D4—678no—
352C22×D4—674no—
353C22×D4—1344no—
354C22×D4—1344no—
355C23×C4—671no—
356C23×C4—671no—
357C23×C4—672no—
358C23×C4—674no—
359C23×C4—672no—
360C23×C4—671no—
361C23×C4—674no—
362C23×C4—1344no—
363C23×C4—1348no—
364C23×C4—1344no—
365C23×C4—1348no—
366C23×C4C22×D6712yes—
367C23×C4C22×D6711yescharacteristic
368C23×C4C22×D6712yes—
369C23×C4C22×D6711yescharacteristic
370C4⋊Q8—13432no—
371C22⋊Q8—13464no—
372C22⋊Q8—13464no—
373C4×D4—13432no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—6716no—
377C2×C4⋊C4—678no—
378C2×C4⋊C4—674no—
379C2×C4⋊C4—678no—
380C2×C4⋊C4—678no—
381C2×C4⋊C4—678no—
382C2×C4⋊C4—678no—
383C2×C4⋊C4—674no—
384C2×C4⋊C4—1344no—
385C2×C4⋊C4—1348no—
386C2×C4⋊C4—1348no—
387C2×C4⋊C4—1348no—
388C2×C4⋊C4—1348no—
389C2×C4⋊C4—1344no—
390C2×C4⋊C4C22×D6718yes—
391C2×C4⋊C4C22×D6714yes—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—28no—
394C2×C22⋊C4—678no—
395C2×C22⋊C4—674no—
396C2×C22⋊C4—678no—
397C2×C22⋊C4—678no—
398C2×C22⋊C4—1348no—
399C2×C22⋊C4—1348no—
400C2×C22⋊C4—1344no—
401C2×C22⋊C4—1348no—
402C2×C22⋊C4C22×D6714yes—
403C2×C22⋊C4C22×D6718yes—
404C2×C42—678no—
405C2×C42—674no—
406C2×C42—678no—
407C2×C42—1348no—
408#95—674no—
409#95C22×D6714yes—
410C24—24no—
411C24—22no—
412C24—22no—
413C24—24no—
414C24—674no—
415C24—672no—
416C24—1344no—
417C24—1342no—
418C24—1342no—
419C24—1344no—
420C24Q8×D6714yes—
421C24C23×D6712yes—
422C24C23×D6711yescharacteristic
423C2×Q8—26832no—
424C2×Q8—13416no—
425C2×Q8—13416no—
426C2×Q8—13416no—
427C2×D4—6716no—
428C2×D4—1348no—
429C2×D4—13432no—
430C2×D4—1348no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—26816no—
440C22×C4—26816no—
441C22×C4—674no—
442C22×C4—674no—
443C22×C4—678no—
444C22×C4—674no—
445C22×C4—674no—
446C22×C4—672no—
447C22×C4—672no—
448C22×C4—678no—
449C22×C4—674no—
450C22×C4—674no—
451C22×C4—672no—
452C22×C4—1348no—
453C22×C4—13416no—
454C22×C4—1348no—
455C22×C4—13416no—
456C22×C4—1342no—
457C22×C4—1344no—
458C22×C4—1344no—
459C22×C4—1344no—
460C22×C4—1344no—
461C22×C4—1344no—
462C22×C4—1344no—
463C22×C4D4×D6714yes—
464C22×C4D4×D6718yes—
465C22×C4C23×D6714yes—
466C22×C4C23×D6712yes—
467C22×C4C23×D6712yes—
468C22×C4C23×D6714yes—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—6732no—
472C4⋊C4—1348no—
473C4⋊C4—13416no—
474C4⋊C4—13416no—
475C4⋊C4—13416no—
476C4⋊C4—13416no—
477C4⋊C4—1348no—
478C22⋊C4—28no—
479C22⋊C4—216no—
480C22⋊C4—13416no—
481C22⋊C4—13416no—
482C22⋊C4—1348no—
483C22⋊C4—13416no—
484C42—6716no—
485C42—13416no—
486C23—216no—
487C23—28no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—48no—
493C23—44no—
494C23—44no—
495C23—2688no—
496C23—2684no—
497C23—2684no—
498C23—674no—
499C23—13416no—
500C23—1348no—
501C23—1344no—
502C23—1342no—
503C23—1342no—
504C23—1344no—
505C23C2×D4×D6714yes—
506C23C2×D4×D6712yes—
507C23C2×Q8×D6714yes—
508C23C2×Q8×D6712yes—
509C23#41×D6712yes—
510C23#1452409014yes—
511C23C24×D6711yescharacteristic
512Q8—26832no—
513D4—13432no—
514C2×C4—216no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—26816no—
527C2×C4—2684no—
528C2×C4—2688no—
529C2×C4—26816no—
530C2×C4—6716no—
531C2×C4—6716no—
532C2×C4—674no—
533C2×C4—674no—
534C2×C4—678no—
535C2×C4—674no—
536C2×C4—674no—
537C2×C4—678no—
538C2×C4—1342no—
539C2×C4—1344no—
540C2×C4—1344no—
541C2×C4—1344no—
542C2×C4—1344no—
543C2×C4—13416no—
544C2×C4—1344no—
545C2×C4—1344no—
546C2×C4C2×D4×D6714yes—
547C2×C4C2×D4×D6718yes—
548C22—28no—
549C22—24no—
550C22—416no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—26816no—
555C22—2684no—
556C22—2682no—
557C22—2682no—
558C22—1348no—
559C22—1344no—
560C22C22⋊Q8×D6714yes—
561C22#127×D6714yes—
562C22#1686396814yes—
563C22#1686404418yes—
564C22#1686405918yes—
565C22C22×D4×D6712yes—
566C22C22×D4×D6711yescharacteristic
567C22C22×Q8×D6711yescharacteristic
568C22C2×#41×D6711yescharacteristic
569C22C2×#1452409012yes—
570C4—28no—
571C4—24no—
572C4—42no—
573C4—44no—
574C4—2682no—
575C4—2684no—
576C4—678no—
577C4—678no—
578C4—1348no—
579C2—44no—
580C2—2684no—
581C2#2256515718yesdirect product factor
582C2C2×C22⋊Q8×D6711yescharacteristic
583C2C2×#127×D6711yescharacteristic
584C2C2×#1686396811yescharacteristic
585C2C2×#1686404412yes—
586C2C2×#1686405912yes—
subgroupG/H
centerC24C23×D67
commutator, cosocleC23×C67C25
FrattiniC23C24×D67
FittingC2×#399×C67C2
socleC24×C67C24
prime radicalC23×C22⋊D67C22
hypercenterC2×#399D67
hypocenterC67C2×#2542
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#14524042C2semidirect
2. C22×#14524021C2semidirect
3. C22×#14524020C2semidirect
5. #22565157C2direct (N × H)
6. C2×#16863333C2semidirect
7. C2×#16863332C2semidirect
8. C2×#16863331C2semidirect
10. C2×#16863265C2semidirect
11. C2×#16863247C2semidirect
12. C2×#399×C67C2semidirect
18. C2×#14524042C22semidirect
20. C2×#14524021C22semidirect
22. C2×#14524020C22semidirect
38. #16863333C22semidirect
39. #16863332C22semidirect
40. #16863331C22semidirect
42. #16863265C22semidirect
43. #16863247C22semidirect
49. #399×C67C22semidirect
58. C22×C22⋊D67Q8semidirect
83. C2×Dic67⋊C4D4semidirect
129. C2×C22⋊D67C2×Q8semidirect
160. Dic67⋊C4C2×D4semidirect
244. C2×Dic67C22⋊Q8semidirect
283. Dic67C2×C22⋊Q8semidirect
287. C2×C67#2542semidirect
302. C2×#399D67semidirect
306. C67C2×#2542semidirect
339. #399C2×D67semidirect
581. C2#22565157direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240422
C22×#145240212
C22×#145240202
C22×(C22⋊D67)⋊C42
#225651572
C2×#168633332
C2×#168633322
C2×#168633312
C2×#168632962
C2×#168632652
C2×#168632472
C2×#399×C672
C2×#254267