Finite Group Databasebeta

← C2×#22560444 #30856695

Subgroups — C2×#22560444 #30856695

Order 17152.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863879C211yesmaximal, characteristic
2C2×#16863438C211yesmaximal, characteristic, complemented
3C2×#16863397C211yesmaximal, characteristic
4C2×#16863262C211yesmaximal, characteristic, complemented
5#22560444C218yesmaximal, direct product factor
6C2×#16863164C211yesmaximal, characteristic, complemented
7C2×#16863156C211yesmaximal, characteristic, complemented
8C22×#104×C67C211yesmaximal, characteristic
9C22×#11761612C2211yescharacteristic
10#16863879—22no—
11#16863879—22no—
12#16863879—22no—
13#16863879C2214yes—
14C2×Q8×Dic67—21no—
15C2×D4×Dic67—22no—
16C2×D4×Dic67—21no—
17C2×#14524002—21no—
18C2×#14524002C2211yescharacteristic
19C22×C4⋊Dic67—21no—
20C22×C4⋊Dic67C2211yescharacteristic
21C22×C4⋊Dic67C2211yescharacteristic
22#16863438—22no—
23#16863438C2214yescomplemented
24C2×#14523977—21no—
25C2×#14523975—21no—
26#16863397—22no—
27#16863397C2214yes—
28C2×#14523967—21no—
29C2×#14523964—21no—
30#16863262—22no—
31#16863262C2214yescomplemented
32C2×C67⋊#95—22no—
33#16863164C2214yescomplemented
34#16863156C2214yescomplemented
35C22×#41×C67C2211yescharacteristic
36C22×M16×C67C2211yescharacteristic
37C22×C42×C67C2211yescharacteristic
38C2×#104×C67C418yescomplemented
39C2×#104×C67C2214yes—
40C23×Dic67—22no—
41C23×Dic67—21no—
42C23×Dic67D412yes—
43C2×#11761612—21no—
44C2×#11761612—22no—
45C2×#11761612—22no—
46C2×#11761612C2312yes—
47C22×C67⋊C8—21no—
48Q8×Dic67—24no—
49Q8×Dic67—24no—
50D4×Dic67—28no—
51D4×Dic67—24no—
52D4×Dic67—28no—
53D4×Dic67—24no—
54#14524002—24no—
55#14524002—24no—
56#14524002—21no—
57#14524002—21no—
58#14524002D412yes—
59#14524002C2312yes—
60C2×C4⋊Dic67—22no—
61C2×C4⋊Dic67—24no—
62C2×C4⋊Dic67—24no—
63C2×C4⋊Dic67—21no—
64C2×C4⋊Dic67—21no—
65C2×C4⋊Dic67—21no—
66C2×C4⋊Dic67—21no—
67C2×C4⋊Dic67—21no—
68C2×C4⋊Dic67—21no—
69C2×C4⋊Dic67—21no—
70C2×C4⋊Dic67—21no—
71C2×C4⋊Dic67—21no—
72C2×C4⋊Dic67D412yes—
73C2×C4⋊Dic67D411yescharacteristic
74C2×C4⋊Dic67D411yescharacteristic
75C2×C4⋊Dic67C2312yes—
76C2×C4⋊Dic67C2312yes—
77C2×C22⋊Dic67—22no—
78C2×C22⋊Dic67—21no—
79C2×C4×Dic67—22no—
80C2×C4×Dic67—21no—
81#14523977—24no—
82#14523975—24no—
83#14523967—24no—
84#14523964—24no—
85C67⋊#95—28no—
86C2×#41×C67—21no—
87C2×#41×C67—21no—
88C2×#41×C67—22no—
89C2×#41×C67—22no—
90C2×#41×C67C2×C414yes—
91C2×#41×C67D412yes—
92C2×#41×C67Q814yes—
93C2×#41×C67C2312yes—
94C22×Q8×C67D411yescharacteristic
95C22×D4×C67—21no—
96C22×D4×C67D411yescharacteristic
97C23×C4×C67—21no—
98C23×C4×C67—21no—
99C23×C4×C67C2311yescharacteristic
100C2×M16×C67—21no—
101C2×M16×C67—22no—
102C2×M16×C67C2×C414yescomplemented
103C2×M16×C67C2312yes—
104C22×C8×C67—21no—
105C2×C42×C67—21no—
106C2×C42×C67—22no—
107C2×C42×C67C2×C414yescomplemented
108C2×C42×C67C2312yes—
109#104×C67C2×C4116yescomplemented
110C22×Dic67—21no—
111C22×Dic67—22no—
112C22×Dic67—21no—
113C22×Dic67—28no—
114C22×Dic67—22no—
115C22×Dic67—22no—
116C22×Dic67—44no—
117C22×Dic67—44no—
118C22×Dic67—41no—
119C22×Dic67—41no—
120C22×Dic67—84no—
121C22×Dic67C2×D414yes—
122#11761612—22no—
123#11761612—22no—
124#11761612—42no—
125C2×C67⋊C8—22no—
126C2×C67⋊C8—42no—
127C4⋊Dic67—28no—
128C4⋊Dic67—22no—
129C4⋊Dic67—21no—
130C4⋊Dic67—22no—
131C4⋊Dic67—21no—
132C4⋊Dic67—22no—
133C4⋊Dic67—22no—
134C4⋊Dic67—22no—
135C4⋊Dic67—22no—
136C4⋊Dic67—41no—
137C4⋊Dic67—41no—
138C4⋊Dic67C2×D412yes—
139C4⋊Dic67C2×D412yes—
140C22⋊Dic67—28no—
141C22⋊Dic67—22no—
142C22⋊Dic67—22no—
143C4×Dic67—28no—
144C4×Dic67—22no—
145C4×Dic67—22no—
146C24×C67—21no—
147C24×C67C2×D411yescharacteristic
148#41×C67—24no—
149#41×C67—24no—
150#41×C67—24no—
151#41×C67—24no—
152#41×C67—24no—
153#41×C67—22no—
154#41×C67—22no—
155#41×C67C4⋊C418yes—
156#41×C67C22×C414yes—
157#41×C67C2×Q814yes—
158C2×Q8×C67—21no—
159C2×Q8×C67—22no—
160C2×Q8×C67C4⋊C414yes—
161C2×Q8×C67C2×D412yes—
162C2×D4×C67—24no—
163C2×D4×C67—22no—
164C2×D4×C67—24no—
165C2×D4×C67—22no—
166C2×D4×C67—21no—
167C2×D4×C67—22no—
168C2×D4×C67C4⋊C414yes—
169C2×D4×C67C2×D412yes—
170C22×C4×C67—21no—
171C22×C4×C67—24no—
172C22×C4×C67—22no—
173C22×C4×C67—24no—
174C22×C4×C67—22no—
175C22×C4×C67—24no—
176C22×C4×C67—22no—
177C22×C4×C67—42no—
178C22×C4×C67—42no—
179C22×C4×C67C22×C412yes—
180C22×C4×C67C2×D411yescharacteristic
181C22×C4×C67C2×D411yescharacteristic
182C22×C4×C67C2×D411yescharacteristic
183C22×C4×C67C2×D411yescharacteristic
184C22×C4×C67C2×D411yescharacteristic
185C22×C4×C67C2×Q812yes—
186C22×C4×C67#4112yes—
187C22×C4×C67#4112yes—
188C22×C4×C67C2411yescharacteristic
189M16×C67—24no—
190M16×C67—22no—
191M16×C67C22×C414yescomplemented
192C2×C8×C67—24no—
193C2×C8×C67—22no—
194C42×C67—24no—
195C42×C67—22no—
196C42×C67C22×C414yescomplemented
197C23×C67—22no—
198C23×C67—22no—
199C23×C67—22no—
200C23×C67—24no—
201C23×C67—42no—
202C23×C67—42no—
203C23×C67C4×D412yes—
204C23×C67#12011yescharacteristic
205C23×C67#12011yescharacteristic
206C23×C67C22⋊Q812yes—
207C23×C67C22×D411yescharacteristic
208Q8×C67—22no—
209Q8×C67—24no—
210Q8×C67C2×C4⋊C414yes—
211D4×C67—24no—
212D4×C67—24no—
213D4×C67—28no—
214D4×C67—22no—
215D4×C67—24no—
216D4×C67C2×C4⋊C414yes—
217C2×C4×C67—22no—
218C2×C4×C67—24no—
219C2×C4×C67—24no—
220C2×C4×C67—24no—
221C2×C4×C67—28no—
222C2×C4×C67—24no—
223C2×C4×C67—48no—
224C2×C4×C67—42no—
225C2×C4×C67—42no—
226C2×C4×C67C2×C4⋊C412yes—
227C2×C4×C67C2×C4⋊C412yes—
228C2×C4×C67C4×D412yes—
229C2×C4×C67C4×D412yes—
230C2×C4×C67C4×D412yes—
231C2×C4×C67#12011yescharacteristic
232C2×C4×C67#12011yescharacteristic
233C2×C4×C67C22⋊Q812yes—
234C2×C4×C67C22⋊Q812yes—
235C2×C4×C67C22⋊Q812yes—
236C2×C4×C67#12312yes—
237C2×C4×C67#12312yes—
238C2×C4×C67C23×C411yescharacteristic
239C2×C4×C67C22×D411yescharacteristic
240C2×C4×C67C22×D411yescharacteristic
241C2×C4×C67C22×Q811yescharacteristic
242C2×C4×C67C2×#4111yescharacteristic
243C2×C4×C67C2×#4111yescharacteristic
244C2×Dic67—22no—
245C2×Dic67—24no—
246C2×Dic67—22no—
247C2×Dic67—42no—
248C2×Dic67—44no—
249C2×Dic67—42no—
250C2×Dic67—44no—
251C2×Dic67—44no—
252C2×Dic67—42no—
253C2×Dic67—82no—
254C2×Dic67—88no—
255C2×Dic67—82no—
256C2×Dic67—82no—
257C8×C67—24no—
258C67⋊C8—42no—
259C22×C67—24no—
260C22×C67—24no—
261C22×C67—21no—
262C22×C67—21no—
263C22×C67—21no—
264C22×C67—21no—
265C22×C67—24no—
266C22×C67—48no—
267C22×C67—42no—
268C22×C67—42no—
269C22×C67#38512yes—
270C22×C67#38512yes—
271C22×C67#45412yes—
272C22×C67#45512yes—
273C22×C67C2×C4×D411yescharacteristic
274C22×C67C2×#12011yescharacteristic
275C22×C67C2×C22⋊Q811yescharacteristic
276C4×C67—24no—
277C4×C67—44no—
278C4×C67#38512yes—
279C4×C67#38512yes—
280C4×C67C22×C4⋊C411yescharacteristic
281C4×C67C2×C4×D411yescharacteristic
282C4×C67C2×C22⋊Q811yescharacteristic
283C4×C67C2×#12311yescharacteristic
284Dic67—44no—
285Dic67—42no—
286Dic67—42no—
287Dic67—82no—
288C2×#1860—671nomaximal, Sylow
289C2×C67—21no—
290C2×C67—21no—
291C2×C67—22no—
292C2×C67—44no—
293C2×C67#186014yescomplemented
294C2×C67C2×#38511yescharacteristic
295C2×C67C2×#45511yescharacteristic
296C2×C67C2×#45411yescharacteristic
297C2×#520—671no—
298C2×#428—671no—
299C22×#104D6711yescharacteristic
300C2×#417—671no—
301C2×#382—671no—
302#1860—678no—
303C2×#343—671no—
304C2×#339—671no—
305C67C2×#186011yescharacteristic, complemented, Sylow
306C22×#41C2×D6711yescharacteristic
307C22×M16—671no—
308C22×M16C2×D6711yescharacteristic
309#520—674no—
310#520—1342no—
311#520—1342no—
312#520—1342no—
313C2×C4×Q8—1341no—
314C2×C4×D4—1342no—
315C2×C4×D4—1341no—
316C2×#117—671no—
317C2×#117—1341no—
318C22×C4⋊C4—671no—
319C22×C4⋊C4—671no—
320C22×C4⋊C4—1341no—
321C22×C42C2×D6711yescharacteristic
322#428—674no—
323#428—1342no—
324C2×#107—1341no—
325C2×#106—1341no—
326C2×#104Dic6718yescomplemented
327C2×#104C2×D6714yes—
328#417—674no—
329#417—1342no—
330C2×#103—1341no—
331C2×#102—1341no—
332#382—674no—
333#382—1342no—
334C2×#95—1342no—
335#343—674no—
336#339—674no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41Dic13414yes—
342C2×#41D26812yes—
343C2×#41C2×Dic6714yes—
344C2×#41C22×D6712yes—
345C22×Q8D26811yescharacteristic
346C22×D4—21no—
347C22×D4D26811yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—672no—
351C23×C4—1341no—
352C23×C4—1342no—
353C23×C4C22×D6711yescharacteristic
354C2×M16—21no—
355C2×M16—22no—
356C2×M16—672no—
357C2×M16—1342no—
358C2×M16—1342no—
359C2×M16—1341no—
360C2×M16C2×Dic6714yescomplemented
361C2×M16C22×D6712yes—
362C22×C8—21no—
363C22×C8—1341no—
364C4×Q8—1344no—
365C4×Q8—1344no—
366C4×D4—1348no—
367C4×D4—1348no—
368C4×D4—1344no—
369C4×D4—1344no—
370#117—672no—
371#117—672no—
372#117—1344no—
373#117—1344no—
374#117—1341no—
375#117—1341no—
376C2×C4⋊C4—671no—
377C2×C4⋊C4—671no—
378C2×C4⋊C4—672no—
379C2×C4⋊C4—672no—
380C2×C4⋊C4—672no—
381C2×C4⋊C4—1341no—
382C2×C4⋊C4—1341no—
383C2×C4⋊C4—1341no—
384C2×C4⋊C4—1341no—
385C2×C4⋊C4—1341no—
386C2×C4⋊C4—1344no—
387C2×C4⋊C4—1342no—
388C2×C4⋊C4—1344no—
389C2×C4⋊C4—1341no—
390C2×C4⋊C4—1341no—
391C2×C4⋊C4—1341no—
392C2×C4⋊C4—1341no—
393C2×C22⋊C4—1342no—
394C2×C22⋊C4—1341no—
395C2×C42—21no—
396C2×C42—22no—
397C2×C42—1342no—
398C2×C42—1341no—
399C2×C42C2×Dic6714yescomplemented
400C2×C42C22×D6712yes—
401#107—1344no—
402#106—1344no—
403#104C2×Dic67116yescomplemented
404#103—1344no—
405#102—1344no—
406#95—1348no—
407C24—21no—
408C24D4×D6711yescharacteristic
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—22no—
415#41—22no—
416#41C4⋊Dic6718yes—
417#41C22×Dic6714yes—
418#41C2×Dic13414yes—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8C4⋊Dic6714yes—
422C2×Q8C2×D26812yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4C4⋊Dic6714yes—
430C2×D4C2×D26812yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—2684no—
441C22×C4—2684no—
442C22×C4—2681no—
443C22×C4—2681no—
444C22×C4—5364no—
445C22×C4—674no—
446C22×C4—1342no—
447C22×C4—1341no—
448C22×C4—1341no—
449C22×C4—1342no—
450C22×C4—1348no—
451C22×C4—1342no—
452C22×C4C22×Dic6712yes—
453C22×C4D4×D6711yescharacteristic
454C22×C4D4×D6711yescharacteristic
455C22×C4D4×D6711yescharacteristic
456C22×C4C2×D26811yescharacteristic
457C22×C4C2×D26811yescharacteristic
458C22×C4C2×Dic13412yes—
459C22×C4#1176162712yes—
460C22×C4#1176162712yes—
461C22×C4C23×D6711yescharacteristic
462M16—24no—
463M16—22no—
464M16—2682no—
465M16—1342no—
466M16—1342no—
467M16C22×Dic6714yescomplemented
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—2682no—
471C2×C8—1342no—
472C4⋊C4—2681no—
473C4⋊C4—2681no—
474C4⋊C4—672no—
475C4⋊C4—672no—
476C4⋊C4—1348no—
477C4⋊C4—1342no—
478C4⋊C4—1342no—
479C4⋊C4—1341no—
480C4⋊C4—1342no—
481C4⋊C4—1342no—
482C4⋊C4—1341no—
483C4⋊C4—1342no—
484C4⋊C4—1342no—
485C22⋊C4—1348no—
486C22⋊C4—1342no—
487C22⋊C4—1342no—
488C42—24no—
489C42—22no—
490C42—1348no—
491C42—1342no—
492C42—1342no—
493C42C22×Dic6714yescomplemented
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23D4×Dic6712yes—
501C23#1452401311yescharacteristic
502C23#1452401311yescharacteristic
503C23#1452402012yes—
504C23C2×D4×D6711yescharacteristic
505Q8—24no—
506Q8—22no—
507Q8C2×C4⋊Dic6714yes—
508D4—28no—
509D4—24no—
510D4—24no—
511D4—24no—
512D4—22no—
513D4C2×C4⋊Dic6714yes—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—48no—
523C2×C4—2682no—
524C2×C4—2684no—
525C2×C4—2682no—
526C2×C4—2684no—
527C2×C4—2684no—
528C2×C4—2682no—
529C2×C4—5362no—
530C2×C4—5368no—
531C2×C4—5362no—
532C2×C4—5362no—
533C2×C4—1342no—
534C2×C4—1342no—
535C2×C4—1344no—
536C2×C4C2×C4⋊Dic6712yes—
537C2×C4C2×C4⋊Dic6712yes—
538C2×C4D4×Dic6712yes—
539C2×C4D4×Dic6712yes—
540C2×C4D4×Dic6712yes—
541C2×C4#1452401311yescharacteristic
542C2×C4#1452401311yescharacteristic
543C2×C4#1452402012yes—
544C2×C4#1452402012yes—
545C2×C4#1452402012yes—
546C2×C4#1452402712yes—
547C2×C4#1452402712yes—
548C2×C4C23×Dic6711yescharacteristic
549C2×C4C2×D4×D6711yescharacteristic
550C2×C4C22×D26811yescharacteristic
551C2×C4C22×Dic13411yescharacteristic
552C2×C4C2×#1176162711yescharacteristic
553C2×C4C2×#1176162711yescharacteristic
554C8—24no—
555C8—2682no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—48no—
566C22#1686327112yes—
567C22#1686327112yes—
568C22#1686357012yes—
569C22#1686357512yes—
570C22C2×D4×Dic6711yescharacteristic
571C22C2×#1452401311yescharacteristic
572C22C2×#1452402011yescharacteristic
573C4—24no—
574C4—44no—
575C4—2684no—
576C4—2682no—
577C4—2682no—
578C4—5362no—
579C4#1686327112yes—
580C4#1686327112yes—
581C4C22×C4⋊Dic6711yescharacteristic
582C4C2×D4×Dic6711yescharacteristic
583C4C2×#1452402011yescharacteristic
584C4C2×#1452402711yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—44no—
589C2#2256044414yesdirect product factor
590C2C2×#1686327111yescharacteristic
591C2C2×#1686357511yescharacteristic
592C2C2×#1686357011yescharacteristic
subgroupG/H
centerC23#14524013
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC22×#104×C67C2
socleC23×C67#120
prime radicalC22×#41×C67C22
cosocleC22×C4×C67C24
hypercenterC22×#104D67
hypocenterC67C2×#1860
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16863438C2semidirect
4. C2×#16863262C2semidirect
5. #22560444C2direct (N × H)
6. C2×#16863164C2semidirect
7. C2×#16863156C2semidirect
23. #16863438C22semidirect
31. #16863262C22semidirect
33. #16863164C22semidirect
34. #16863156C22semidirect
38. C2×#104×C67C4semidirect
102. C2×M16×C67C2×C4semidirect
107. C2×C42×C67C2×C4semidirect
109. #104×C67C2×C4semidirect
191. M16×C67C22×C4semidirect
196. C42×C67C22×C4semidirect
293. C2×C67#1860semidirect
305. C67C2×#1860semidirect
326. C2×#104Dic67semidirect
360. C2×M16C2×Dic67semidirect
399. C2×C42C2×Dic67semidirect
403. #104C2×Dic67semidirect
467. M16C22×Dic67semidirect
493. C42C22×Dic67semidirect
589. C2#22560444direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168638792
C2×#168634382
C2×#168633972
C2×#168632622
#225604442
C2×#168631642
C2×#168631562
C22×#104×C672
C2×#186067