Finite Group Databasebeta

← C2×#22560437 #30856687

Subgroups — C2×#22560437 #30856687

Order 17152.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 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×#16863266C211yesmaximal, characteristic, complemented
5#22560437C218yesmaximal, 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⋊Dic67—22no—
21C22×C4⋊Dic67C2211yescharacteristic
22C22×C4⋊Dic67C2211yescharacteristic
23#16863438—22no—
24#16863438C2214yescomplemented
25C2×#14523977—21no—
26C2×#14523975—21no—
27#16863397—22no—
28#16863397C2214yes—
29C2×#14523967—21no—
30C2×#14523964—21no—
31#16863266—22no—
32#16863266C2214yescomplemented
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—28no—
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⋊Dic67—21no—
73C2×C4⋊Dic67—48no—
74C2×C4⋊Dic67D412yes—
75C2×C4⋊Dic67D411yescharacteristic
76C2×C4⋊Dic67D411yescharacteristic
77C2×C4⋊Dic67C2312yes—
78C2×C4⋊Dic67C2312yes—
79C2×C22⋊Dic67—22no—
80C2×C22⋊Dic67—21no—
81C2×C4×Dic67—22no—
82C2×C4×Dic67—21no—
83#14523977—24no—
84#14523975—24no—
85#14523967—24no—
86#14523964—24no—
87C2×#41×C67—21no—
88C2×#41×C67—21no—
89C2×#41×C67—22no—
90C2×#41×C67—22no—
91C2×#41×C67C2×C414yes—
92C2×#41×C67D412yes—
93C2×#41×C67Q814yes—
94C2×#41×C67C2312yes—
95C22×Q8×C67D411yescharacteristic
96C22×D4×C67—21no—
97C22×D4×C67D411yescharacteristic
98C23×C4×C67—21no—
99C23×C4×C67—21no—
100C23×C4×C67C2311yescharacteristic
101C2×M16×C67—21no—
102C2×M16×C67—22no—
103C2×M16×C67C2×C414yescomplemented
104C2×M16×C67C2312yes—
105C22×C8×C67—21no—
106C2×C42×C67—21no—
107C2×C42×C67—22no—
108C2×C42×C67C2×C414yescomplemented
109C2×C42×C67C2312yes—
110#104×C67C2×C4116yescomplemented
111C22×Dic67—21no—
112C22×Dic67—22no—
113C22×Dic67—21no—
114C22×Dic67—28no—
115C22×Dic67—22no—
116C22×Dic67—22no—
117C22×Dic67—44no—
118C22×Dic67—44no—
119C22×Dic67—41no—
120C22×Dic67—41no—
121C22×Dic67—84no—
122C22×Dic67C2×D414yes—
123#11761612—22no—
124#11761612—22no—
125#11761612—42no—
126C2×C67⋊C8—22no—
127C2×C67⋊C8—42no—
128C4⋊Dic67—28no—
129C4⋊Dic67—22no—
130C4⋊Dic67—21no—
131C4⋊Dic67—22no—
132C4⋊Dic67—21no—
133C4⋊Dic67—22no—
134C4⋊Dic67—22no—
135C4⋊Dic67—22no—
136C4⋊Dic67—22no—
137C4⋊Dic67—416no—
138C4⋊Dic67—41no—
139C4⋊Dic67—41no—
140C4⋊Dic67C2×D412yes—
141C4⋊Dic67C2×D412yes—
142C22⋊Dic67—28no—
143C22⋊Dic67—22no—
144C22⋊Dic67—22no—
145C4×Dic67—28no—
146C4×Dic67—22no—
147C4×Dic67—22no—
148C24×C67—21no—
149C24×C67C2×D411yescharacteristic
150#41×C67—24no—
151#41×C67—24no—
152#41×C67—24no—
153#41×C67—24no—
154#41×C67—24no—
155#41×C67—22no—
156#41×C67—22no—
157#41×C67C4⋊C418yescomplemented
158#41×C67C22×C414yes—
159#41×C67C2×Q814yes—
160C2×Q8×C67—21no—
161C2×Q8×C67—22no—
162C2×Q8×C67C4⋊C414yescomplemented
163C2×Q8×C67C2×D412yes—
164C2×D4×C67—24no—
165C2×D4×C67—22no—
166C2×D4×C67—24no—
167C2×D4×C67—22no—
168C2×D4×C67—21no—
169C2×D4×C67—22no—
170C2×D4×C67C4⋊C414yescomplemented
171C2×D4×C67C2×D412yes—
172C22×C4×C67—24no—
173C22×C4×C67—24no—
174C22×C4×C67—24no—
175C22×C4×C67—21no—
176C22×C4×C67—22no—
177C22×C4×C67—24no—
178C22×C4×C67—22no—
179C22×C4×C67—24no—
180C22×C4×C67—22no—
181C22×C4×C67C22×C412yes—
182C22×C4×C67C2×D411yescharacteristic
183C22×C4×C67C2×D411yescharacteristic
184C22×C4×C67C2×D411yescharacteristic
185C22×C4×C67C2×D411yescharacteristic
186C22×C4×C67C2×D411yescharacteristic
187C22×C4×C67C2×Q812yes—
188C22×C4×C67#4112yes—
189C22×C4×C67#4112yes—
190C22×C4×C67C2411yescharacteristic
191M16×C67—24no—
192M16×C67—22no—
193M16×C67C22×C414yescomplemented
194C2×C8×C67—24no—
195C2×C8×C67—22no—
196C42×C67—24no—
197C42×C67—22no—
198C42×C67C22×C414yescomplemented
199C23×C67—22no—
200C23×C67—22no—
201C23×C67—22no—
202C23×C67—24no—
203C23×C67—42no—
204C23×C67—42no—
205C23×C67C4×D412yes—
206C23×C67#12011yescharacteristic
207C23×C67#12011yescharacteristic
208C23×C67C22⋊Q812yes—
209C23×C67C22×D411yescharacteristic
210Q8×C67—22no—
211Q8×C67—24no—
212Q8×C67C2×C4⋊C414yescomplemented
213D4×C67—24no—
214D4×C67—24no—
215D4×C67—28no—
216D4×C67—22no—
217D4×C67—24no—
218D4×C67C2×C4⋊C414yescomplemented
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—28no—
228C2×C4×C67C2×C4⋊C412yes—
229C2×C4×C67C2×C4⋊C412yes—
230C2×C4×C67C4×D412yes—
231C2×C4×C67C4×D412yes—
232C2×C4×C67C4×D412yes—
233C2×C4×C67#12011yescharacteristic
234C2×C4×C67#12011yescharacteristic
235C2×C4×C67C22⋊Q812yes—
236C2×C4×C67C22⋊Q812yes—
237C2×C4×C67C22⋊Q812yes—
238C2×C4×C67#12312yes—
239C2×C4×C67#12312yes—
240C2×C4×C67C23×C411yescharacteristic
241C2×C4×C67C22×D411yescharacteristic
242C2×C4×C67C22×D411yescharacteristic
243C2×C4×C67C22×Q811yescharacteristic
244C2×C4×C67C2×#4111yescharacteristic
245C2×C4×C67C2×#4111yescharacteristic
246C2×Dic67—22no—
247C2×Dic67—24no—
248C2×Dic67—22no—
249C2×Dic67—44no—
250C2×Dic67—42no—
251C2×Dic67—42no—
252C2×Dic67—44no—
253C2×Dic67—44no—
254C2×Dic67—42no—
255C2×Dic67—82no—
256C2×Dic67—88no—
257C2×Dic67—82no—
258C2×Dic67—82no—
259C8×C67—24no—
260C67⋊C8—42no—
261C22×C67—24no—
262C22×C67—24no—
263C22×C67—21no—
264C22×C67—21no—
265C22×C67—21no—
266C22×C67—21no—
267C22×C67—24no—
268C22×C67—48no—
269C22×C67—42no—
270C22×C67—42no—
271C22×C67#38512yes—
272C22×C67#38512yes—
273C22×C67#45312yes—
274C22×C67#45612yes—
275C22×C67C2×C4×D411yescharacteristic
276C22×C67C2×#12011yescharacteristic
277C22×C67C2×C22⋊Q811yescharacteristic
278C4×C67—28no—
279C4×C67—24no—
280C4×C67#38512yes—
281C4×C67#38512yes—
282C4×C67C22×C4⋊C411yescharacteristic
283C4×C67C2×C4×D411yescharacteristic
284C4×C67C2×C22⋊Q811yescharacteristic
285C4×C67C2×#12311yescharacteristic
286Dic67—44no—
287Dic67—42no—
288Dic67—42no—
289Dic67—82no—
290C2×#1859—671nomaximal, Sylow
291C2×C67—21no—
292C2×C67—21no—
293C2×C67—22no—
294C2×C67—44no—
295C2×C67#185914yescomplemented
296C2×C67C2×#38511yescharacteristic
297C2×C67C2×#45311yescharacteristic
298C2×C67C2×#45611yescharacteristic
299C2×#520—671no—
300C2×#428—671no—
301C22×#104D6711yescharacteristic
302C2×#417—671no—
303C2×#384—671no—
304#1859—678no—
305C2×#343—671no—
306C2×#339—671no—
307C67C2×#185911yescharacteristic, complemented, Sylow
308C22×#41C2×D6711yescharacteristic
309C22×M16—671no—
310C22×M16C2×D6711yescharacteristic
311#520—674no—
312#520—1342no—
313#520—1342no—
314#520—1342no—
315C2×C4×Q8—1341no—
316C2×C4×D4—1342no—
317C2×C4×D4—1341no—
318C2×#117—671no—
319C2×#117—1341no—
320C22×C4⋊C4—671no—
321C22×C4⋊C4—671no—
322C22×C4⋊C4—1342no—
323C22×C4⋊C4—1341no—
324C22×C42C2×D6711yescharacteristic
325#428—674no—
326#428—1342no—
327C2×#107—1341no—
328C2×#106—1341no—
329C2×#104Dic6718yescomplemented
330C2×#104C2×D6714yes—
331#417—674no—
332#417—1342no—
333C2×#103—1341no—
334C2×#102—1341no—
335#384—674no—
336#384—1342no—
337#343—674no—
338#339—674no—
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41Dic13414yes—
344C2×#41D26812yes—
345C2×#41C2×Dic6714yes—
346C2×#41C22×D6712yes—
347C22×Q8D26811yescharacteristic
348C22×D4—21no—
349C22×D4D26811yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4—672no—
353C23×C4—1341no—
354C23×C4—1342no—
355C23×C4C22×D6711yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—672no—
359C2×M16—1342no—
360C2×M16—1342no—
361C2×M16—1341no—
362C2×M16C2×Dic6714yescomplemented
363C2×M16C22×D6712yes—
364C22×C8—21no—
365C22×C8—1341no—
366C4×Q8—1344no—
367C4×Q8—1344no—
368C4×D4—1348no—
369C4×D4—1348no—
370C4×D4—1344no—
371C4×D4—1344no—
372#117—672no—
373#117—672no—
374#117—1344no—
375#117—1344no—
376#117—1341no—
377#117—1341no—
378C2×C4⋊C4—2688no—
379C2×C4⋊C4—671no—
380C2×C4⋊C4—671no—
381C2×C4⋊C4—672no—
382C2×C4⋊C4—672no—
383C2×C4⋊C4—672no—
384C2×C4⋊C4—1348no—
385C2×C4⋊C4—1341no—
386C2×C4⋊C4—1341no—
387C2×C4⋊C4—1341no—
388C2×C4⋊C4—1341no—
389C2×C4⋊C4—1341no—
390C2×C4⋊C4—1344no—
391C2×C4⋊C4—1342no—
392C2×C4⋊C4—1344no—
393C2×C4⋊C4—1341no—
394C2×C4⋊C4—1341no—
395C2×C4⋊C4—1341no—
396C2×C4⋊C4—1341no—
397C2×C22⋊C4—1342no—
398C2×C22⋊C4—1341no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—1342no—
402C2×C42—1341no—
403C2×C42C2×Dic6714yescomplemented
404C2×C42C22×D6712yes—
405#107—1344no—
406#106—1344no—
407#104C2×Dic67116yescomplemented
408#103—1344no—
409#102—1344no—
410C24—21no—
411C24D4×D6711yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—22no—
418#41—22no—
419#41C4⋊Dic6718yescomplemented
420#41C22×Dic6714yes—
421#41C2×Dic13414yes—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8C4⋊Dic6714yescomplemented
425C2×Q8C2×D26812yes—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4C4⋊Dic6714yescomplemented
433C2×D4C2×D26812yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—2684no—
444C22×C4—2684no—
445C22×C4—2681no—
446C22×C4—2681no—
447C22×C4—5364no—
448C22×C4—674no—
449C22×C4—1342no—
450C22×C4—1341no—
451C22×C4—1341no—
452C22×C4—1342no—
453C22×C4—1348no—
454C22×C4—1342no—
455C22×C4C22×Dic6712yes—
456C22×C4D4×D6711yescharacteristic
457C22×C4D4×D6711yescharacteristic
458C22×C4D4×D6711yescharacteristic
459C22×C4C2×D26811yescharacteristic
460C22×C4C2×D26811yescharacteristic
461C22×C4C2×Dic13412yes—
462C22×C4#1176162712yes—
463C22×C4#1176162712yes—
464C22×C4C23×D6711yescharacteristic
465M16—24no—
466M16—22no—
467M16—2682no—
468M16—1342no—
469M16—1342no—
470M16C22×Dic6714yescomplemented
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—2682no—
474C2×C8—1342no—
475C4⋊C4—26816no—
476C4⋊C4—2681no—
477C4⋊C4—2681no—
478C4⋊C4—672no—
479C4⋊C4—672no—
480C4⋊C4—1348no—
481C4⋊C4—1342no—
482C4⋊C4—1342no—
483C4⋊C4—1341no—
484C4⋊C4—1342no—
485C4⋊C4—1342no—
486C4⋊C4—1341no—
487C4⋊C4—1342no—
488C4⋊C4—1342no—
489C22⋊C4—1348no—
490C22⋊C4—1342no—
491C22⋊C4—1342no—
492C42—24no—
493C42—22no—
494C42—1348no—
495C42—1342no—
496C42—1342no—
497C42C22×Dic6714yescomplemented
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23D4×Dic6712yes—
505C23#1452401311yescharacteristic
506C23#1452401311yescharacteristic
507C23#1452402012yes—
508C23C2×D4×D6711yescharacteristic
509Q8—24no—
510Q8—22no—
511Q8C2×C4⋊Dic6714yescomplemented
512D4—28no—
513D4—24no—
514D4—24no—
515D4—24no—
516D4—22no—
517D4C2×C4⋊Dic6714yescomplemented
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—2684no—
528C2×C4—2682no—
529C2×C4—2682no—
530C2×C4—2684no—
531C2×C4—2684no—
532C2×C4—2682no—
533C2×C4—5362no—
534C2×C4—5368no—
535C2×C4—5362no—
536C2×C4—5362no—
537C2×C4—1342no—
538C2×C4—1342no—
539C2×C4—1344no—
540C2×C4C2×C4⋊Dic6712yes—
541C2×C4C2×C4⋊Dic6712yes—
542C2×C4D4×Dic6712yes—
543C2×C4D4×Dic6712yes—
544C2×C4D4×Dic6712yes—
545C2×C4#1452401311yescharacteristic
546C2×C4#1452401311yescharacteristic
547C2×C4#1452402012yes—
548C2×C4#1452402012yes—
549C2×C4#1452402012yes—
550C2×C4#1452402712yes—
551C2×C4#1452402712yes—
552C2×C4C23×Dic6711yescharacteristic
553C2×C4C2×D4×D6711yescharacteristic
554C2×C4C22×D26811yescharacteristic
555C2×C4C22×Dic13411yescharacteristic
556C2×C4C2×#1176162711yescharacteristic
557C2×C4C2×#1176162711yescharacteristic
558C8—24no—
559C8—2682no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22#1686327112yes—
571C22#1686327112yes—
572C22#1686356512yes—
573C22#1686358012yes—
574C22C2×D4×Dic6711yescharacteristic
575C22C2×#1452401311yescharacteristic
576C22C2×#1452402011yescharacteristic
577C4—28no—
578C4—24no—
579C4—2684no—
580C4—2682no—
581C4—2682no—
582C4—5362no—
583C4#1686327112yes—
584C4#1686327112yes—
585C4C22×C4⋊Dic6711yescharacteristic
586C4C2×D4×Dic6711yescharacteristic
587C4C2×#1452402011yescharacteristic
588C4C2×#1452402711yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—44no—
593C2#2256043714yesdirect product factor
594C2C2×#1686327111yescharacteristic
595C2C2×#1686356511yescharacteristic
596C2C2×#1686358011yescharacteristic
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×#1859
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×#16863266C2semidirect
5. #22560437C2direct (N × H)
6. C2×#16863164C2semidirect
7. C2×#16863156C2semidirect
24. #16863438C22semidirect
32. #16863266C22semidirect
33. #16863164C22semidirect
34. #16863156C22semidirect
38. C2×#104×C67C4semidirect
103. C2×M16×C67C2×C4semidirect
108. C2×C42×C67C2×C4semidirect
110. #104×C67C2×C4semidirect
157. #41×C67C4⋊C4semidirect
162. C2×Q8×C67C4⋊C4semidirect
170. C2×D4×C67C4⋊C4semidirect
193. M16×C67C22×C4semidirect
198. C42×C67C22×C4semidirect
212. Q8×C67C2×C4⋊C4semidirect
218. D4×C67C2×C4⋊C4semidirect
295. C2×C67#1859semidirect
307. C67C2×#1859semidirect
329. C2×#104Dic67semidirect
362. C2×M16C2×Dic67semidirect
403. C2×C42C2×Dic67semidirect
407. #104C2×Dic67semidirect
419. #41C4⋊Dic67semidirect
424. C2×Q8C4⋊Dic67semidirect
432. C2×D4C4⋊Dic67semidirect
470. M16C22×Dic67semidirect
497. C42C22×Dic67semidirect
511. Q8C2×C4⋊Dic67semidirect
517. D4C2×C4⋊Dic67semidirect
593. C2#22560437direct (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×#168632662
#225604372
C2×#168631642
C2×#168631562
C22×#104×C672
C2×#185967