Finite Group Databasebeta

← C2×#22560525 #30857030

Subgroups — C2×#22560525 #30857030

Order 17152.

2614 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524018C211yesmaximal, characteristic
2C22×C4⋊C4×D67C211yesmaximal, characteristic, complemented
3C22×#14523965C211yesmaximal, characteristic, complemented
4C22×#14523963C211yesmaximal, characteristic, complemented
5C22×(C2×D67)⋊C8C211yesmaximal, characteristic, complemented
6#22560525C218yesmaximal, direct product factor
7C2×#16863158C211yesmaximal, characteristic, complemented
8C2×#340×C67C211yesmaximal, characteristic, complemented
9C23×D268—21no—
10C23×D268C2211yescharacteristic
11C23×C4×D67—21no—
12C23×C4×D67C2211yescharacteristic
13C23×C67⋊C8C2211yescharacteristic
14C2×#14524018—24no—
15C2×#14524018—22no—
16C2×#14524018C418yescomplemented
17C2×#14524018C2214yes—
18C22×C4⋊Dic67C2211yescharacteristic
19C22×Dic67⋊C4—21no—
20C2×C4⋊C4×D67—22no—
21C2×C4⋊C4×D67—24no—
22C2×C4⋊C4×D67—24no—
23C2×C4⋊C4×D67C2214yescomplemented
24C22×(C2×D67)⋊C4—21no—
25C2×#14523965—24no—
26C2×#14523965C2214yescomplemented
27C2×#14523963—24no—
28C2×#14523963C2214yescomplemented
29C2×(C2×D67)⋊C8—24no—
30C2×(C2×D67)⋊C8C2214yescomplemented
31#16863158C2214yescomplemented
32C23×C8×C67C2211yescharacteristic
33C22×C4⋊C4×C67C2211yescharacteristic
34C22×C4⋊C4×C67C2211yescharacteristic
35#340×C67C2214yescomplemented
36C24×D67—21no—
37C24×D67D411yescharacteristic
38C22×D268—28no—
39C22×D268—24no—
40C22×D268—22no—
41C22×D268—22no—
42C22×D268—21no—
43C22×D268—22no—
44C22×D268—22no—
45C22×D268—22no—
46C22×D268—44no—
47C22×D268—44no—
48C22×D268C2×C414yescomplemented
49C22×D268C2312yes—
50C23×Dic67—21no—
51C23×Dic67D411yescharacteristic
52C22×C4×D67—28no—
53C22×C4×D67—24no—
54C22×C4×D67—22no—
55C22×C4×D67—22no—
56C22×C4×D67—22no—
57C22×C4×D67—22no—
58C22×C4×D67—44no—
59C22×C4×D67—44no—
60C22×C4×D67C2×C414yes—
61C22×C4×D67D412yes—
62C22×C4×D67D414yes—
63C22×C4×D67C2312yes—
64C22×C67⋊C8—22no—
65C22×C67⋊C8—22no—
66C22×C67⋊C8—22no—
67C22×C67⋊C8C2312yes—
68#14524018—216no—
69#14524018—28no—
70#14524018C2×C4116yescomplemented
71C2×C4⋊Dic67—21no—
72C2×C4⋊Dic67—22no—
73C2×C4⋊Dic67D414yescomplemented
74C2×C4⋊Dic67C2312yes—
75C2×Dic67⋊C4—28no—
76C2×Dic67⋊C4—24no—
77C4⋊C4×D67—216no—
78C4⋊C4×D67—28no—
79C4⋊C4×D67—28no—
80C2×(C2×D67)⋊C4—28no—
81C2×(C2×D67)⋊C4—24no—
82#14523965—28no—
83#14523963—28no—
84(C2×D67)⋊C8—28no—
85C23×C4×C67—21no—
86C23×C4×C67—21no—
87C23×C4×C67C2311yescharacteristic
88C22×C8×C67—22no—
89C22×C8×C67—22no—
90C22×C8×C67—22no—
91C22×C8×C67C2312yes—
92C2×C4⋊C4×C67—21no—
93C2×C4⋊C4×C67—22no—
94C2×C4⋊C4×C67—21no—
95C2×C4⋊C4×C67—22no—
96C2×C4⋊C4×C67C2×C414yescomplemented
97C2×C4⋊C4×C67D414yescomplemented
98C2×C4⋊C4×C67C2312yes—
99C2×C4⋊C4×C67C2312yes—
100C23×D67—22no—
101C23×D67—22no—
102C23×D67—22no—
103C23×D67—22no—
104C23×D67—24no—
105C23×D67—22no—
106C23×D67—42no—
107C23×D67—42no—
108C23×D67—42no—
109C23×D67—42no—
110C23×D67—42no—
111C23×D67—42no—
112C23×D67C22⋊C414yes—
113C23×D67D814yes—
114C23×D67SD1614yes—
115C23×D67C2×D412yes—
116C2×D268—28no—
117C2×D268—28no—
118C2×D268—22no—
119C2×D268—22no—
120C2×D268—22no—
121C2×D268—22no—
122C2×D268—216no—
123C2×D268—24no—
124C2×D268—22no—
125C2×D268—432no—
126C2×D268—48no—
127C2×D268—48no—
128C2×D268—42no—
129C2×D268—44no—
130C2×D268—44no—
131C2×D268—42no—
132C2×D268C22×C414yescomplemented
133C22×Dic67—22no—
134C22×Dic67—22no—
135C22×Dic67—24no—
136C22×Dic67—22no—
137C22×Dic67—42no—
138C22×Dic67—42no—
139C22×Dic67C22⋊C414yes—
140C22×Dic67C2×D412yes—
141C2×C4×D67—28no—
142C2×C4×D67—28no—
143C2×C4×D67—24no—
144C2×C4×D67—28no—
145C2×C4×D67—28no—
146C2×C4×D67—24no—
147C2×C4×D67—22no—
148C2×C4×D67—22no—
149C2×C4×D67—22no—
150C2×C4×D67—22no—
151C2×C4×D67—216no—
152C2×C4×D67—48no—
153C2×C4×D67—432no—
154C2×C4×D67—48no—
155C2×C4×D67C22⋊C418yes—
156C2×C4×D67C22×C414yes—
157C2×C4×D67C2×D414yes—
158C2×C67⋊C8—22no—
159C2×C67⋊C8—22no—
160C2×C67⋊C8—22no—
161C2×C67⋊C8—44no—
162C4⋊Dic67—24no—
163C4⋊Dic67—22no—
164C4⋊Dic67C2×D414yescomplemented
165Dic67⋊C4—216no—
166(C2×D67)⋊C4—216no—
167C24×C67C2×D411yescharacteristic
168C22×C4×C67—24no—
169C22×C4×C67—24no—
170C22×C4×C67—22no—
171C22×C4×C67—22no—
172C22×C4×C67—42no—
173C22×C4×C67—42no—
174C22×C4×C67—42no—
175C22×C4×C67—42no—
176C22×C4×C67C22×C412yes—
177C22×C4×C67C2×D411yescharacteristic
178C22×C4×C67C2×D412yes—
179C22×C4×C67C2×D412yes—
180C22×C4×C67C2×D412yes—
181C22×C4×C67#4112yes—
182C22×C4×C67#4112yes—
183C22×C4×C67C2411yescharacteristic
184C2×C8×C67—22no—
185C2×C8×C67—22no—
186C2×C8×C67—22no—
187C2×C8×C67—44no—
188C4⋊C4×C67—24no—
189C4⋊C4×C67—24no—
190C4⋊C4×C67—22no—
191C4⋊C4×C67—22no—
192C4⋊C4×C67C22×C414yescomplemented
193C4⋊C4×C67C2×D414yescomplemented
194C23×C67C2×C22⋊C412yes—
195C23×C67#12012yes—
196C23×C67#12312yes—
197C23×C67C2×D812yes—
198C23×C67C2×SD1612yes—
199C23×C67AGL(1,ℤ/8)12yes—
200C23×C67#13712yes—
201C23×C67C22×D411yescharacteristic
202C22×D67—22no—
203C22×D67—22no—
204C22×D67—22no—
205C22×D67—22no—
206C22×D67—28no—
207C22×D67—28no—
208C22×D67—28no—
209C22×D67—28no—
210C22×D67—28no—
211C22×D67—28no—
212C22×D67—24no—
213C22×D67—42no—
214C22×D67—42no—
215C22×D67—42no—
216C22×D67—42no—
217C22×D67—42no—
218C22×D67—42no—
219C22×D67—48no—
220C22×D67—48no—
221C22×D67—48no—
222C22×D67—84no—
223C22×D67—84no—
224C22×D67—84no—
225C22×D67—84no—
226C22×D67#102116yescomplemented
227C22×D67C2×C22⋊C414yes—
228C22×D67C2×D814yes—
229C22×D67C2×SD1614yes—
230C2×C4×C67—24no—
231C2×C4×C67—24no—
232C2×C4×C67—42no—
233C2×C4×C67—42no—
234C2×C4×C67—42no—
235C2×C4×C67—42no—
236C2×C4×C67—48no—
237C2×C4×C67—48no—
238C2×C4×C67C2×C22⋊C412yes—
239C2×C4×C67C4×D414yes—
240C2×C4×C67C4×D414yes—
241C2×C4×C67#12012yes—
242C2×C4×C67#12114yes—
243C2×C4×C67#12114yes—
244C2×C4×C67#12312yes—
245C2×C4×C67C23×C411yescharacteristic
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C22×D411yescharacteristic
248C2×C4×C67C22×D411yescharacteristic
249C2×C4×C67C2×#4111yescharacteristic
250C2×C4×C67C2×#4111yescharacteristic
251C2×Dic67—22no—
252C2×Dic67—22no—
253C2×Dic67—28no—
254C2×Dic67—24no—
255C2×Dic67—42no—
256C2×Dic67—42no—
257C2×Dic67—48no—
258C2×Dic67C2×C22⋊C414yes—
259D268—24no—
260D268—24no—
261D268—432no—
262D268—42no—
263D268—42no—
264D268—48no—
265C4×D67—24no—
266C4×D67—24no—
267C4×D67—24no—
268C4×D67—24no—
269C4×D67—216no—
270C4×D67—432no—
271C8×C67—42no—
272C67⋊C8—42no—
273C22×C67#38614yes—
274C22×C67C2×#10214yes—
275C22×C67#41714yes—
276C22×C67#44714yes—
277C22×C67#44814yes—
278C22×C67#48014yes—
279C22×C67#48114yes—
280C22×C67C22×C22⋊C411yescharacteristic
281C22×C67C2×#12011yescharacteristic
282C22×C67C2×#12311yescharacteristic
283C22×C67C22×D811yescharacteristic
284C22×C67C22×SD1611yescharacteristic
285C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C67C2×#13711yescharacteristic
287C2×D67—24no—
288C2×D67—24no—
289C2×D67—24no—
290C2×D67—232no—
291C2×D67—24no—
292C2×D67—24no—
293C2×D67—24no—
294C2×D67—44no—
295C2×D67—44no—
296C2×D67—44no—
297C2×D67—82no—
298C2×D67—82no—
299C2×D67—816no—
300C2×D67—82no—
301C2×D67—82no—
302C2×D67C2×#10218yescomplemented
303C4×C67—44no—
304C4×C67—44no—
305C4×C67#38614yes—
306C4×C67C2×C4×D411yescharacteristic
307C4×C67C2×C4×D411yescharacteristic
308C4×C67C2×#12111yescharacteristic
309C4×C67C2×#12111yescharacteristic
310Dic67—24no—
311Dic67—44no—
312C2×#1874—671nomaximal, Sylow
313C2×C67#187418yescomplemented
314C2×C67C2×#38611yescharacteristic
315C2×C67C22×#10211yescharacteristic
316C2×C67C2×#41711yescharacteristic
317C2×C67C2×#44711yescharacteristic
318C2×C67C2×#44811yescharacteristic
319C2×C67C2×#48011yescharacteristic
320C2×C67C2×#48111yescharacteristic
321D67—28no—
322D67—84no—
323C22×#121—671no—
324C23×C4⋊C4—671no—
325C22×#102—671no—
326C22×#102—671no—
327C22×C22⋊C8—671no—
328#1874—678no—
329C2×#340—671no—
330C2×#340D6711yescharacteristic, complemented
331C67C2×#187411yescharacteristic, complemented, Sylow
332C23×D4—671no—
333C23×D4—1341no—
334C24×C4—671no—
335C24×C4—1341no—
336C23×C8—671no—
337C23×C8C2×D6711yescharacteristic
338C2×#121—678no—
339C2×#121—674no—
340C2×#121—1342no—
341C2×#121—1344no—
342C22×C4⋊C4—671no—
343C22×C4⋊C4—674no—
344C22×C4⋊C4—1344no—
345C22×C4⋊C4—1342no—
346C22×C4⋊C4—1341no—
347C22×C4⋊C4—1344no—
348C22×C4⋊C4C2×D6711yescharacteristic
349C22×C4⋊C4C2×D6711yescharacteristic
350C22×C22⋊C4—1341no—
351C2×#102—674no—
352C2×#102—674no—
353C2×#102—1344no—
354C2×#102—1344no—
355C2×C22⋊C8—674no—
356C2×C22⋊C8—1344no—
357#340—674no—
358#340C2×D6714yescomplemented
359C25—671no—
360C25—1341no—
361C22×D4—2684no—
362C22×D4—2684no—
363C22×D4—674no—
364C22×D4—672no—
365C22×D4—1348no—
366C22×D4—1344no—
367C22×D4—1341no—
368C22×D4—1342no—
369C22×D4—1342no—
370C22×D4—1342no—
371C22×D4—1342no—
372C22×D4—1342no—
373C23×C4—21no—
374C23×C4—21no—
375C23×C4—2684no—
376C23×C4—2684no—
377C23×C4—671no—
378C23×C4—674no—
379C23×C4—672no—
380C23×C4—674no—
381C23×C4—672no—
382C23×C4—1344no—
383C23×C4—1348no—
384C23×C4—1341no—
385C23×C4—1342no—
386C23×C4—1342no—
387C23×C4—1342no—
388C23×C4—1342no—
389C23×C4C22×D6711yescharacteristic
390C22×C8—22no—
391C22×C8—22no—
392C22×C8—22no—
393C22×C8—672no—
394C22×C8—1342no—
395C22×C8—1342no—
396C22×C8—1342no—
397C22×C8C22×D6712yes—
398#121—6716no—
399#121—13416no—
400#121—1348no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—674no—
406C2×C4⋊C4—672no—
407C2×C4⋊C4—1348no—
408C2×C4⋊C4—13416no—
409C2×C4⋊C4—1344no—
410C2×C4⋊C4—1348no—
411C2×C4⋊C4—1341no—
412C2×C4⋊C4—1342no—
413C2×C4⋊C4—1348no—
414C2×C4⋊C4C4×D6714yescomplemented
415C2×C4⋊C4D26814yescomplemented
416C2×C4⋊C4C22×D6712yes—
417C2×C4⋊C4C22×D6712yes—
418C2×C22⋊C4—1348no—
419C2×C22⋊C4—1344no—
420#102—1348no—
421#102—1348no—
422C22⋊C8—1348no—
423C24—2682no—
424C24—2682no—
425C24—2682no—
426C24—2682no—
427C24—2682no—
428C24—2682no—
429C24—674no—
430C24—674no—
431C24—674no—
432C24—672no—
433C24—1344no—
434C24—1342no—
435C24—1342no—
436C24—1342no—
437C24—1342no—
438C24—1342no—
439C24D4×D6711yescharacteristic
440C2×D4—26832no—
441C2×D4—2688no—
442C2×D4—2688no—
443C2×D4—2682no—
444C2×D4—2684no—
445C2×D4—2684no—
446C2×D4—2682no—
447C2×D4—674no—
448C2×D4—13416no—
449C2×D4—1344no—
450C2×D4—1348no—
451C2×D4—1348no—
452C2×D4—1342no—
453C2×D4—1342no—
454C2×D4—1342no—
455C2×D4—1342no—
456C2×D4—1342no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—2688no—
466C22×C4—26832no—
467C22×C4—2688no—
468C22×C4—2682no—
469C22×C4—2682no—
470C22×C4—674no—
471C22×C4—672no—
472C22×C4—678no—
473C22×C4—674no—
474C22×C4—674no—
475C22×C4—13416no—
476C22×C4—1344no—
477C22×C4—1342no—
478C22×C4—1342no—
479C22×C4—1342no—
480C22×C4—1348no—
481C22×C4—1348no—
482C22×C4—1344no—
483C22×C4—1348no—
484C22×C4—1348no—
485C22×C4—1344no—
486C22×C4—1342no—
487C22×C4—1342no—
488C22×C4—1342no—
489C22×C4—1342no—
490C22×C4C2×C4×D6712yes—
491C22×C4D4×D6711yescharacteristic
492C22×C4D4×D6712yes—
493C22×C4D4×D6712yes—
494C22×C4C2×D26812yes—
495C22×C4#1176162812yes—
496C22×C4#1176162912yes—
497C22×C4C23×D6711yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8—2684no—
503C2×C8—1342no—
504C2×C8—1342no—
505C2×C8—1342no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—674no—
511C4⋊C4—13416no—
512C4⋊C4—1344no—
513C4⋊C4—1342no—
514C4⋊C4C2×C4×D6714yescomplemented
515C4⋊C4C2×D26814yescomplemented
516C22⋊C4—13416no—
517C23—2688no—
518C23—2682no—
519C23—2688no—
520C23—2682no—
521C23—2688no—
522C23—2682no—
523C23—2682no—
524C23—2682no—
525C23—2682no—
526C23—5364no—
527C23—5364no—
528C23—5364no—
529C23—5364no—
530C23—674no—
531C23—6716no—
532C23—674no—
533C23—674no—
534C23—1344no—
535C23—1348no—
536C23—1348no—
537C23—1348no—
538C23—1342no—
539C23—1342no—
540C23—1348no—
541C23—1342no—
542C23—1342no—
543C23—1348no—
544C23—1348no—
545C23C22⋊C4×D6712yes—
546C23#1452401312yes—
547C23#1452402812yes—
548C23D8×D6712yes—
549C23SD16×D6712yes—
550C23#1452407012yes—
551C23#1452407512yes—
552C23C2×D4×D6711yescharacteristic
553D4—26832no—
554D4—2688no—
555D4—2682no—
556D4—2682no—
557D4—1344no—
558D4—1344no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—26832no—
568C2×C4—2688no—
569C2×C4—2682no—
570C2×C4—2682no—
571C2×C4—674no—
572C2×C4—1344no—
573C2×C4—1348no—
574C2×C4—1342no—
575C2×C4—1342no—
576C2×C4—13416no—
577C2×C4—1344no—
578C2×C4—1344no—
579C2×C4—1344no—
580C2×C4—1344no—
581C2×C4C22⋊C4×D6712yes—
582C2×C4#1452400514yes—
583C2×C4C4×D26814yes—
584C2×C4#1452401312yes—
585C2×C4#1452401614yes—
586C2×C4#1452401814yes—
587C2×C4#1452402812yes—
588C2×C4C22×C4×D6711yescharacteristic
589C2×C4C2×D4×D6711yescharacteristic
590C2×C4C2×D4×D6711yescharacteristic
591C2×C4C22×D26811yescharacteristic
592C2×C4C2×#1176162811yescharacteristic
593C2×C4C2×#1176162911yescharacteristic
594C8—42no—
595C8—2682no—
596C22—2684no—
597C22—2684no—
598C22—2684no—
599C22—53616no—
600C22—5362no—
601C22—5362no—
602C22—5362no—
603C22—5362no—
604C22—678no—
605C22—13432no—
606C22—1344no—
607C22—1344no—
608C22—1344no—
609C22—1344no—
610C22—1344no—
611C22—1344no—
612C22#1686327414yes—
613C22#102×D6714yes—
614C22#1686339614yes—
615C22#1686353114yes—
616C22#1686353614yes—
617C22#1686371114yes—
618C22#1686371614yes—
619C22C2×C22⋊C4×D6711yescharacteristic
620C22C2×#1452401311yescharacteristic
621C22C2×#1452402811yescharacteristic
622C22C2×D8×D6711yescharacteristic
623C22C2×SD16×D6711yescharacteristic
624C22C2×#1452407011yescharacteristic
625C22C2×#1452407511yescharacteristic
626C4—44no—
627C4—44no—
628C4—2684no—
629C4—1344no—
630C4#1686327414yes—
631C4C2×#1452400511yescharacteristic
632C4C2×C4×D26811yescharacteristic
633C4C2×#1452401611yescharacteristic
634C4C2×#1452401811yescharacteristic
635C2—5364no—
636C2—1348no—
637C2#2256052518yesdirect product factor
638C2C2×#1686327411yescharacteristic
639C2C2×#102×D6711yescharacteristic
640C2C2×#1686339611yescharacteristic
641C2C2×#1686353111yescharacteristic
642C2C2×#1686353611yescharacteristic
643C2C2×#1686371111yescharacteristic
644C2C2×#1686371611yescharacteristic
subgroupG/H
centerC24D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#340×C67C2
socleC24×C67C2×D4
prime radicalC22×#14524018C2
cosocleC22×C4×C67C24
hypercenterC2×#340D67
hypocenterC67C2×#1874
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D67C2semidirect
3. C22×#14523965C2semidirect
4. C22×#14523963C2semidirect
5. C22×(C2×D67)⋊C8C2semidirect
6. #22560525C2direct (N × H)
7. C2×#16863158C2semidirect
8. C2×#340×C67C2semidirect
16. C2×#14524018C4semidirect
23. C2×C4⋊C4×D67C22semidirect
26. C2×#14523965C22semidirect
28. C2×#14523963C22semidirect
30. C2×(C2×D67)⋊C8C22semidirect
31. #16863158C22semidirect
35. #340×C67C22semidirect
48. C22×D268C2×C4semidirect
70. #14524018C2×C4semidirect
73. C2×C4⋊Dic67D4semidirect
96. C2×C4⋊C4×C67C2×C4semidirect
97. C2×C4⋊C4×C67D4semidirect
132. C2×D268C22×C4semidirect
164. C4⋊Dic67C2×D4semidirect
192. C4⋊C4×C67C22×C4semidirect
193. C4⋊C4×C67C2×D4semidirect
226. C22×D67#102semidirect
302. C2×D67C2×#102semidirect
313. C2×C67#1874semidirect
330. C2×#340D67semidirect
331. C67C2×#1874semidirect
358. #340C2×D67semidirect
414. C2×C4⋊C4C4×D67semidirect
415. C2×C4⋊C4D268semidirect
514. C4⋊C4C2×C4×D67semidirect
515. C4⋊C4C2×D268semidirect
637. C2#22560525direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240182
C22×C4⋊C4×D672
C22×#145239652
C22×#145239632
C22×(C2×D67)⋊C82
#225605252
C2×#168631582
C2×#340×C672
C2×#187467