Finite Group Databasebeta

← C2×#22561414 #30883305

Subgroups — C2×#22561414 #30883305

Order 17152.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524023C211yesmaximal, characteristic
2C22×#14523968C211yesmaximal, characteristic, complemented
3C22×#14523966C211yesmaximal, characteristic, complemented
4C22×(C2×D67)⋊C8C211yesmaximal, characteristic
5C2×#16863253C211yesmaximal, characteristic
6#22561414C218yesmaximal, direct product factor
7C2×#16863158C211yesmaximal, characteristic, complemented
8C22×#107×C67C211yesmaximal, characteristic, complemented
9C23×Dic134C2211yescharacteristic
10C23×C4×D67C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C2×#14524023—24no—
13C2×#14524023—22no—
14C2×#14524023—24no—
15C2×#14524023C2214yes—
16C22×C4⋊Dic67—21no—
17C22×C4⋊Dic67C2211yescharacteristic
18C22×(C2×D67)⋊C4—21no—
19C22×(C2×D67)⋊C4—21no—
20C2×#14523968—24no—
21C2×#14523968C2214yescomplemented
22C2×#14523966—24no—
23C2×#14523966C2214yescomplemented
24C2×(C2×D67)⋊C8—24no—
25C2×(C2×D67)⋊C8C2214yes—
26#16863253—22no—
27#16863253C2214yes—
28C2×C67⋊#95—21no—
29#16863158C2214yescomplemented
30C23×C8×C67C2211yescharacteristic
31C22×C4⋊C4×C67C2211yescharacteristic
32C22×C4⋊C4×C67C2211yescharacteristic
33C2×#107×C67—24no—
34C2×#107×C67C2214yescomplemented
35C24×D67D411yescharacteristic
36C22×Dic134—21no—
37C22×Dic134—22no—
38C22×Dic134—22no—
39C22×Dic134—22no—
40C22×Dic134—22no—
41C22×Dic134—22no—
42C22×Dic134D414yes—
43C22×Dic134C2312yes—
44C23×Dic67—21no—
45C23×Dic67—21no—
46C23×Dic67D411yescharacteristic
47C22×C4×D67—22no—
48C22×C4×D67—22no—
49C22×C4×D67—22no—
50C22×C4×D67—22no—
51C22×C4×D67—22no—
52C22×C4×D67—22no—
53C22×C4×D67D412yes—
54C22×C4×D67C2312yes—
55C22×C67⋊C8—22no—
56C22×C67⋊C8—22no—
57C22×C67⋊C8—22no—
58C22×C67⋊C8C2312yes—
59#14524023—216no—
60#14524023—28no—
61#14524023—28no—
62C2×C4⋊Dic67—28no—
63C2×C4⋊Dic67—24no—
64C2×C4⋊Dic67—22no—
65C2×C4⋊Dic67—21no—
66C2×C4⋊Dic67—22no—
67C2×C4⋊Dic67C2312yes—
68C2×(C2×D67)⋊C4—28no—
69C2×(C2×D67)⋊C4—24no—
70C2×(C2×D67)⋊C4—24no—
71C2×(C2×D67)⋊C4—44no—
72#14523968—28no—
73#14523966—28no—
74(C2×D67)⋊C8—28no—
75C67⋊#95—24no—
76C23×C4×C67—21no—
77C23×C4×C67—21no—
78C23×C4×C67C2311yescharacteristic
79C22×C8×C67—22no—
80C22×C8×C67—22no—
81C22×C8×C67D414yes—
82C22×C8×C67C2312yes—
83C2×C4⋊C4×C67—21no—
84C2×C4⋊C4×C67—22no—
85C2×C4⋊C4×C67—22no—
86C2×C4⋊C4×C67—22no—
87C2×C4⋊C4×C67—21no—
88C2×C4⋊C4×C67D414yes—
89C2×C4⋊C4×C67C2312yes—
90C2×C4⋊C4×C67C2312yes—
91#107×C67—28no—
92C23×D67—22no—
93C23×D67—22no—
94C23×D67—22no—
95C23×D67—22no—
96C23×D67—22no—
97C23×D67—22no—
98C23×D67Q1614yes—
99C23×D67C2×D412yes—
100C2×Dic134—28no—
101C2×Dic134—28no—
102C2×Dic134—22no—
103C2×Dic134—22no—
104C2×Dic134—22no—
105C2×Dic134—22no—
106C2×Dic134—24no—
107C2×Dic134—22no—
108C2×Dic134—44no—
109C2×Dic134—42no—
110C2×Dic134—42no—
111C2×Dic134—44no—
112C2×Dic134C2×D414yes—
113C22×Dic67—24no—
114C22×Dic67—22no—
115C22×Dic67—22no—
116C22×Dic67—22no—
117C22×Dic67—22no—
118C22×Dic67—22no—
119C22×Dic67—42no—
120C22×Dic67—42no—
121C22×Dic67—42no—
122C22×Dic67—42no—
123C22×Dic67—42no—
124C22×Dic67C2×D412yes—
125C2×C4×D67—24no—
126C2×C4×D67—24no—
127C2×C4×D67—24no—
128C2×C4×D67—22no—
129C2×C4×D67—22no—
130C2×C4×D67—22no—
131C2×C4×D67—22no—
132C2×C4×D67—22no—
133C2×C4×D67—22no—
134C2×C4×D67—44no—
135C2×C4×D67—44no—
136C2×C4×D67—44no—
137C2×C4×D67—44no—
138C2×C67⋊C8—22no—
139C2×C67⋊C8—22no—
140C2×C67⋊C8—22no—
141C2×C67⋊C8—44no—
142C4⋊Dic67—216no—
143C4⋊Dic67—22no—
144C4⋊Dic67—22no—
145C4⋊Dic67—42no—
146(C2×D67)⋊C4—216no—
147(C2×D67)⋊C4—48no—
148C24×C67C2×D411yescharacteristic
149C22×C4×C67—24no—
150C22×C4×C67—22no—
151C22×C4×C67—22no—
152C22×C4×C67—42no—
153C22×C4×C67—42no—
154C22×C4×C67—42no—
155C22×C4×C67—42no—
156C22×C4×C67—42no—
157C22×C4×C67C2×D411yescharacteristic
158C22×C4×C67C2×D412yes—
159C22×C4×C67C2×D412yes—
160C22×C4×C67C2×D412yes—
161C22×C4×C67#4112yes—
162C22×C4×C67#4112yes—
163C22×C4×C67#4112yes—
164C22×C4×C67C2411yescharacteristic
165C2×C8×C67—28no—
166C2×C8×C67—22no—
167C2×C8×C67—22no—
168C2×C8×C67C2×D414yes—
169C4⋊C4×C67—24no—
170C4⋊C4×C67—22no—
171C4⋊C4×C67—22no—
172C4⋊C4×C67—22no—
173C4⋊C4×C67—42no—
174C4⋊C4×C67C2×D414yes—
175C23×C67#12012yes—
176C23×C67#12112yes—
177C23×C67#12312yes—
178C23×C67C2×Q1612yes—
179C23×C67#13512yes—
180C23×C67AGL(1,ℤ/8)12yes—
181C23×C67#13712yes—
182C23×C67C22×D411yescharacteristic
183C22×D67—22no—
184C22×D67—22no—
185C22×D67—22no—
186C22×D67—22no—
187C22×D67—22no—
188C22×D67—22no—
189C22×D67—28no—
190C22×D67—28no—
191C22×D67—28no—
192C22×D67—44no—
193C22×D67—44no—
194C22×D67—44no—
195C22×D67—44no—
196C22×D67C2×Q1614yes—
197C2×C4×C67—24no—
198C2×C4×C67—42no—
199C2×C4×C67—42no—
200C2×C4×C67—48no—
201C2×C4×C67—42no—
202C2×C4×C67—42no—
203C2×C4×C67—42no—
204C2×C4×C67—84no—
205C2×C4×C67#12012yes—
206C2×C4×C67#12114yes—
207C2×C4×C67#12114yes—
208C2×C4×C67#12112yes—
209C2×C4×C67#12314yes—
210C2×C4×C67#12312yes—
211C2×C4×C67#12414yes—
212C2×C4×C67C22×D411yescharacteristic
213C2×C4×C67C22×D411yescharacteristic
214C2×C4×C67C22×D411yescharacteristic
215C2×C4×C67C2×#4111yescharacteristic
216C2×C4×C67C2×#4111yescharacteristic
217C2×C4×C67C2×#4111yescharacteristic
218C2×Dic67—24no—
219C2×Dic67—22no—
220C2×Dic67—22no—
221C2×Dic67—22no—
222C2×Dic67—42no—
223C2×Dic67—42no—
224C2×Dic67—44no—
225C2×Dic67—48no—
226C2×Dic67—42no—
227C2×Dic67—42no—
228C2×Dic67—42no—
229C2×Dic67—84no—
230C4×D67—42no—
231C4×D67—42no—
232C4×D67—42no—
233C4×D67—42no—
234C4×D67—48no—
235Dic134—24no—
236Dic134—24no—
237Dic134—42no—
238Dic134—42no—
239Dic134—48no—
240C8×C67—24no—
241C67⋊C8—42no—
242C22×C67#39414yes—
243C22×C67#44914yes—
244C22×C67#45114yes—
245C22×C67#46714yes—
246C22×C67#46814yes—
247C22×C67#48414yes—
248C22×C67#48514yes—
249C22×C67C2×#12011yescharacteristic
250C22×C67C2×#12111yescharacteristic
251C22×C67C2×#12311yescharacteristic
252C22×C67C22×Q1611yescharacteristic
253C22×C67C2×#13511yescharacteristic
254C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C67C2×#13711yescharacteristic
256C2×D67—24no—
257C2×D67—24no—
258C2×D67—24no—
259C2×D67—42no—
260C2×D67—42no—
261C2×D67—42no—
262C2×D67—42no—
263C2×D67—416no—
264C4×C67—44no—
265C4×C67—82no—
266C4×C67#39414yes—
267C4×C67C2×#12111yescharacteristic
268C4×C67C2×#12111yescharacteristic
269C4×C67C2×#12311yescharacteristic
270C4×C67C2×#12411yescharacteristic
271Dic67—44no—
272Dic67—42no—
273Dic67—82no—
274C2×#2036—671nomaximal, Sylow
275C2×C67#203618yescomplemented
276C2×C67C2×#39411yescharacteristic
277C2×C67C2×#45111yescharacteristic
278C2×C67C2×#44911yescharacteristic
279C2×C67C2×#46711yescharacteristic
280C2×C67C2×#46811yescharacteristic
281C2×C67C2×#48511yescharacteristic
282C2×C67C2×#48411yescharacteristic
283D67—44no—
284C22×C22⋊Q8—671no—
285C22×#107D6711yescharacteristic, complemented
286C22×#103—671no—
287C22×#103—671no—
288C22×C22⋊C8—671no—
289C2×#380—671no—
290#2036—678no—
291C2×#340—671no—
292C67C2×#203611yescharacteristic, complemented, Sylow
293C23×Q8—671no—
294C24×C4—671no—
295C23×C8—671no—
296C23×C8C2×D6711yescharacteristic
297C2×C22⋊Q8—674no—
298C2×C22⋊Q8—1342no—
299C2×C22⋊Q8—1344no—
300C2×C22⋊Q8—1344no—
301C22×C4⋊C4—671no—
302C22×C4⋊C4—1341no—
303C22×C4⋊C4C2×D6711yescharacteristic
304C22×C4⋊C4C2×D6711yescharacteristic
305C22×C22⋊C4—1341no—
306C22×C22⋊C4—1341no—
307C2×#107—24no—
308C2×#107C2×D6714yescomplemented
309C2×#103—674no—
310C2×#103—674no—
311C2×#103—1344no—
312C2×#103—1344no—
313C2×C22⋊C8—674no—
314C2×C22⋊C8—1344no—
315#380—674no—
316#380—1342no—
317C2×#95—1341no—
318#340—674no—
319C25—671no—
320C22×Q8—674no—
321C22×Q8—672no—
322C22×Q8—1342no—
323C22×Q8—1342no—
324C22×Q8—1342no—
325C22×Q8—1342no—
326C22×Q8—1341no—
327C22×Q8—1342no—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—671no—
331C23×C4—672no—
332C23×C4—672no—
333C23×C4—1341no—
334C23×C4—1341no—
335C23×C4—1342no—
336C23×C4—1342no—
337C23×C4—1342no—
338C23×C4—1342no—
339C23×C4—1342no—
340C23×C4—1342no—
341C23×C4C22×D6711yescharacteristic
342C22×C8—22no—
343C22×C8—22no—
344C22×C8—672no—
345C22×C8—1342no—
346C22×C8—1342no—
347C22×C8—1342no—
348C22×C8D26814yes—
349C22×C8C22×D6712yes—
350C22⋊Q8—1348no—
351C22⋊Q8—13416no—
352C22⋊Q8—1348no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—672no—
359C2×C4⋊C4—1341no—
360C2×C4⋊C4—1342no—
361C2×C4⋊C4—1342no—
362C2×C4⋊C4—1344no—
363C2×C4⋊C4—1348no—
364C2×C4⋊C4C22⋊D6714yes—
365C2×C4⋊C4C22×D6712yes—
366C2×C4⋊C4C22×D6712yes—
367C2×C22⋊C4—2684no—
368C2×C22⋊C4—1344no—
369C2×C22⋊C4—1344no—
370C2×C22⋊C4—1348no—
371#107—28no—
372#103—1348no—
373#103—1348no—
374C22⋊C8—1348no—
375#95—1344no—
376C24—674no—
377C24—672no—
378C24—1342no—
379C24—1342no—
380C24—1342no—
381C24—1342no—
382C24—1342no—
383C24—1342no—
384C24D4×D6711yescharacteristic
385C2×Q8—2684no—
386C2×Q8—2682no—
387C2×Q8—2682no—
388C2×Q8—2684no—
389C2×Q8—674no—
390C2×Q8—1348no—
391C2×Q8—1348no—
392C2×Q8—1342no—
393C2×Q8—1342no—
394C2×Q8—1342no—
395C2×Q8—1342no—
396C2×Q8—1344no—
397C2×Q8—1342no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—2682no—
407C22×C4—2682no—
408C22×C4—2684no—
409C22×C4—2684no—
410C22×C4—2684no—
411C22×C4—2684no—
412C22×C4—2682no—
413C22×C4—2682no—
414C22×C4—2682no—
415C22×C4—672no—
416C22×C4—1342no—
417C22×C4—1344no—
418C22×C4—1342no—
419C22×C4—1342no—
420C22×C4—1342no—
421C22×C4—1342no—
422C22×C4—1344no—
423C22×C4—1344no—
424C22×C4—1344no—
425C22×C4—1342no—
426C22×C4—1342no—
427C22×C4—1342no—
428C22×C4—1342no—
429C22×C4—1342no—
430C22×C4—1342no—
431C22×C4D4×D6711yescharacteristic
432C22×C4D4×D6712yes—
433C22×C4C2×C22⋊D6712yes—
434C22×C4C2×D26812yes—
435C22×C4#1176162812yes—
436C22×C4#1176162912yes—
437C22×C4#1176162912yes—
438C22×C4C23×D6711yescharacteristic
439C2×C8—28no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—2684no—
443C2×C8—1342no—
444C2×C8—1342no—
445C2×C8—1342no—
446C2×C8C2×D26814yes—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—2682no—
453C4⋊C4—13416no—
454C4⋊C4—1342no—
455C4⋊C4—1342no—
456C4⋊C4C2×C22⋊D6714yes—
457C22⋊C4—2688no—
458C22⋊C4—13416no—
459C23—2684no—
460C23—2684no—
461C23—2684no—
462C23—2684no—
463C23—674no—
464C23—1348no—
465C23—1342no—
466C23—1342no—
467C23—1348no—
468C23—1342no—
469C23—1342no—
470C23—1342no—
471C23—1342no—
472C23—1348no—
473C23#1452401112yes—
474C23#1452401812yes—
475C23#1452402812yes—
476C23Q16×D6712yes—
477C23#1452406212yes—
478C23#1452406712yes—
479C23#1452407512yes—
480C23C2×D4×D6711yescharacteristic
481Q8—2688no—
482Q8—2682no—
483Q8—2682no—
484Q8—1344no—
485Q8—1344no—
486C2×C4—24no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—2684no—
495C2×C4—2688no—
496C2×C4—2682no—
497C2×C4—2682no—
498C2×C4—2688no—
499C2×C4—2682no—
500C2×C4—2682no—
501C2×C4—2682no—
502C2×C4—2682no—
503C2×C4—2682no—
504C2×C4—2682no—
505C2×C4—2682no—
506C2×C4—5364no—
507C2×C4—1344no—
508C2×C4—1342no—
509C2×C4—1342no—
510C2×C4—1342no—
511C2×C4#1452401112yes—
512C2×C4#1452401714yes—
513C2×C4#1452401812yes—
514C2×C4#1452401814yes—
515C2×C4#1452402812yes—
516C2×C4#1452402814yes—
517C2×C4#1452403214yes—
518C2×C4C2×D4×D6711yescharacteristic
519C2×C4C22×C22⋊D6711yescharacteristic
520C2×C4C22×D26811yescharacteristic
521C2×C4C2×#1176162811yescharacteristic
522C2×C4C2×#1176162911yescharacteristic
523C2×C4C2×#1176162911yescharacteristic
524C8—24no—
525C8—2682no—
526C22—2682no—
527C22—2682no—
528C22—2682no—
529C22—26816no—
530C22—2682no—
531C22—1344no—
532C22—1344no—
533C22—1344no—
534C22#1686331314yes—
535C22#1686353814yes—
536C22#1686355014yes—
537C22#1686363814yes—
538C22#1686364314yes—
539C22#1686373314yes—
540C22#1686373914yes—
541C22C2×#1452401111yescharacteristic
542C22C2×#1452401811yescharacteristic
543C22C2×#1452402811yescharacteristic
544C22C2×Q16×D6711yescharacteristic
545C22C2×#1452406211yescharacteristic
546C22C2×#1452406711yescharacteristic
547C22C2×#1452407511yescharacteristic
548C4—44no—
549C4—82no—
550C4—2682no—
551C4—2684no—
552C4—5362no—
553C4#1686331314yes—
554C4C2×#1452401711yescharacteristic
555C4C2×#1452401811yescharacteristic
556C4C2×#1452402811yescharacteristic
557C4C2×#1452403211yescharacteristic
558C2—2684no—
559C2#2256141418yesdirect product factor
560C2C2×#1686331311yescharacteristic
561C2C2×#1686355011yescharacteristic
562C2C2×#1686353811yescharacteristic
563C2C2×#1686363811yescharacteristic
564C2C2×#1686364311yescharacteristic
565C2C2×#1686373911yescharacteristic
566C2C2×#1686373311yescharacteristic
subgroupG/H
centerC24D4×D67
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
FittingC22×#107×C67C2
socleC24×C67C2×D4
prime radicalC24×D67D4
hypercenterC22×#107D67
hypocenterC67C2×#2036
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
2. C22×#14523968C2semidirect
3. C22×#14523966C2semidirect
6. #22561414C2direct (N × H)
7. C2×#16863158C2semidirect
8. C22×#107×C67C2semidirect
21. C2×#14523968C22semidirect
23. C2×#14523966C22semidirect
29. #16863158C22semidirect
34. C2×#107×C67C22semidirect
275. C2×C67#2036semidirect
285. C22×#107D67semidirect
292. C67C2×#2036semidirect
308. C2×#107C2×D67semidirect
559. C2#22561414direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240232
C22×#145239682
C22×#145239662
C22×(C2×D67)⋊C82
C2×#168632532
#225614142
C2×#168631582
C22×#107×C672
C2×#203667