Finite Group Databasebeta

← C2×#22560513 #30857022

Subgroups — C2×#22560513 #30857022

Order 17152.

2270 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524019C211yesmaximal, characteristic
2C22×C4⋊C4×D67C211yesmaximal, characteristic
3C22×#14523967C211yesmaximal, characteristic, complemented
4C22×(C2×D67)⋊C8C211yesmaximal, characteristic
5#22560513C218yesmaximal, direct product factor
6C2×#16863159C211yesmaximal, characteristic, complemented
7C2×#16863158C211yesmaximal, characteristic, complemented
8C22×#103×C67C211yesmaximal, characteristic, complemented
9C23×C4×D67—21no—
10C23×C4×D67C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C2×#14524019—24no—
13C2×#14524019—22no—
14C2×#14524019C418yescomplemented
15C2×#14524019C2214yes—
16C22×C4⋊Dic67C2211yescharacteristic
17C22×C4⋊Dic67C2211yescharacteristic
18C22×Dic67⋊C4—21no—
19C22×Dic67⋊C4—21no—
20C2×C4⋊C4×D67—22no—
21C2×C4⋊C4×D67—24no—
22C2×C4⋊C4×D67—24no—
23C2×C4⋊C4×D67C2214yes—
24C22×(C2×D67)⋊C4—21no—
25C2×#14523967—24no—
26C2×#14523967C2214yescomplemented
27C2×(C2×D67)⋊C8—24no—
28C2×(C2×D67)⋊C8C2214yes—
29#16863159C2214yescomplemented
30#16863158C2214yescomplemented
31C23×Q8×C67C2211yescharacteristic
32C23×C8×C67C2211yescharacteristic
33C22×C4⋊C4×C67C2211yescharacteristic
34C2×#103×C67—24no—
35C2×#103×C67C2214yescomplemented
36C24×D67D411yescharacteristic
37C23×Dic67—21no—
38C23×Dic67—21no—
39C23×Dic67D411yescharacteristic
40C22×C4×D67—28no—
41C22×C4×D67—24no—
42C22×C4×D67—22no—
43C22×C4×D67—22no—
44C22×C4×D67—22no—
45C22×C4×D67—22no—
46C22×C4×D67—44no—
47C22×C4×D67—44no—
48C22×C4×D67C2×C414yes—
49C22×C4×D67D412yes—
50C22×C4×D67D414yes—
51C22×C4×D67C2312yes—
52C22×C67⋊C8—22no—
53C22×C67⋊C8—22no—
54C22×C67⋊C8—22no—
55C22×C67⋊C8C2312yes—
56#14524019—216no—
57#14524019—28no—
58#14524019C2×C4116yescomplemented
59C2×C4⋊Dic67—21no—
60C2×C4⋊Dic67—22no—
61C2×C4⋊Dic67—21no—
62C2×C4⋊Dic67—22no—
63C2×C4⋊Dic67C2×C414yescomplemented
64C2×C4⋊Dic67D414yes—
65C2×C4⋊Dic67C2312yes—
66C2×C4⋊Dic67C2312yes—
67C2×Dic67⋊C4—28no—
68C2×Dic67⋊C4—28no—
69C2×Dic67⋊C4—24no—
70C2×Dic67⋊C4—24no—
71C4⋊C4×D67—216no—
72C4⋊C4×D67—28no—
73C4⋊C4×D67—28no—
74C2×(C2×D67)⋊C4—28no—
75C2×(C2×D67)⋊C4—24no—
76#14523967—28no—
77(C2×D67)⋊C8—28no—
78C22×Q8×C67—22no—
79C22×Q8×C67—21no—
80C22×Q8×C67—22no—
81C22×Q8×C67—22no—
82C22×Q8×C67—22no—
83C22×Q8×C67—22no—
84C22×Q8×C67C2×C414yescomplemented
85C22×Q8×C67C2312yes—
86C23×C4×C67—21no—
87C23×C4×C67—21no—
88C23×C4×C67C2311yescharacteristic
89C22×C8×C67—22no—
90C22×C8×C67—22no—
91C22×C8×C67—22no—
92C22×C8×C67C2312yes—
93C2×C4⋊C4×C67—21no—
94C2×C4⋊C4×C67—22no—
95C2×C4⋊C4×C67D414yes—
96C2×C4⋊C4×C67C2312yes—
97#103×C67—28no—
98C23×D67—22no—
99C23×D67—22no—
100C23×D67—22no—
101C23×D67—22no—
102C23×D67C22⋊C414yes—
103C23×D67SD1614yes—
104C23×D67Q1614yes—
105C23×D67C2×D412yes—
106C22×Dic67—22no—
107C22×Dic67—22no—
108C22×Dic67—24no—
109C22×Dic67—24no—
110C22×Dic67—22no—
111C22×Dic67—22no—
112C22×Dic67—42no—
113C22×Dic67—42no—
114C22×Dic67—42no—
115C22×Dic67—42no—
116C22×Dic67C22⋊C414yes—
117C22×Dic67C2×D412yes—
118C2×C4×D67—28no—
119C2×C4×D67—28no—
120C2×C4×D67—24no—
121C2×C4×D67—28no—
122C2×C4×D67—28no—
123C2×C4×D67—24no—
124C2×C4×D67—22no—
125C2×C4×D67—22no—
126C2×C4×D67—22no—
127C2×C4×D67—22no—
128C2×C4×D67—216no—
129C2×C4×D67—48no—
130C2×C4×D67—432no—
131C2×C4×D67—48no—
132C2×C4×D67C22⋊C418yes—
133C2×C4×D67C22×C414yes—
134C2×C4×D67C2×D414yes—
135C2×C67⋊C8—22no—
136C2×C67⋊C8—22no—
137C2×C67⋊C8—22no—
138C2×C67⋊C8—44no—
139C4⋊Dic67—24no—
140C4⋊Dic67—24no—
141C4⋊Dic67—22no—
142C4⋊Dic67—22no—
143C4⋊Dic67C22×C414yescomplemented
144C4⋊Dic67C2×D414yes—
145Dic67⋊C4—216no—
146Dic67⋊C4—216no—
147(C2×D67)⋊C4—216no—
148C24×C67C2×D411yescharacteristic
149C2×Q8×C67—24no—
150C2×Q8×C67—22no—
151C2×Q8×C67—28no—
152C2×Q8×C67—28no—
153C2×Q8×C67—22no—
154C2×Q8×C67—22no—
155C2×Q8×C67—22no—
156C2×Q8×C67—22no—
157C2×Q8×C67—42no—
158C2×Q8×C67—44no—
159C2×Q8×C67—44no—
160C2×Q8×C67—42no—
161C2×Q8×C67C22×C414yescomplemented
162C22×C4×C67—24no—
163C22×C4×C67—24no—
164C22×C4×C67—22no—
165C22×C4×C67—22no—
166C22×C4×C67—42no—
167C22×C4×C67—42no—
168C22×C4×C67—42no—
169C22×C4×C67—42no—
170C22×C4×C67C22×C412yes—
171C22×C4×C67C2×D411yescharacteristic
172C22×C4×C67C2×D412yes—
173C22×C4×C67C2×D412yes—
174C22×C4×C67C2×D412yes—
175C22×C4×C67#4112yes—
176C22×C4×C67#4112yes—
177C22×C4×C67C2411yescharacteristic
178C2×C8×C67—22no—
179C2×C8×C67—22no—
180C2×C8×C67—22no—
181C2×C8×C67—44no—
182C4⋊C4×C67—24no—
183C4⋊C4×C67—22no—
184C4⋊C4×C67C2×D414yes—
185C23×C67C2×C22⋊C412yes—
186C23×C67#12012yes—
187C23×C67#12312yes—
188C23×C67C2×SD1612yes—
189C23×C67C2×Q1612yes—
190C23×C67AGL(1,ℤ/8)12yes—
191C23×C67#13712yes—
192C23×C67C22×D411yescharacteristic
193C22×D67—22no—
194C22×D67—22no—
195C22×D67—22no—
196C22×D67—22no—
197C22×D67—28no—
198C22×D67—28no—
199C22×D67—28no—
200C22×D67—28no—
201C22×D67—28no—
202C22×D67—28no—
203C22×D67#103116yescomplemented
204C22×D67C2×C22⋊C414yes—
205C22×D67C2×SD1614yes—
206C22×D67C2×Q1614yes—
207Q8×C67—24no—
208Q8×C67—24no—
209Q8×C67—42no—
210Q8×C67—42no—
211Q8×C67—48no—
212C2×C4×C67—24no—
213C2×C4×C67—24no—
214C2×C4×C67—42no—
215C2×C4×C67—42no—
216C2×C4×C67—42no—
217C2×C4×C67—42no—
218C2×C4×C67—48no—
219C2×C4×C67—48no—
220C2×C4×C67C2×C22⋊C412yes—
221C2×C4×C67C4×D414yes—
222C2×C4×C67C4×D414yes—
223C2×C4×C67#12012yes—
224C2×C4×C67#12114yes—
225C2×C4×C67#12114yes—
226C2×C4×C67#12312yes—
227C2×C4×C67C23×C411yescharacteristic
228C2×C4×C67C22×D411yescharacteristic
229C2×C4×C67C22×D411yescharacteristic
230C2×C4×C67C22×D411yescharacteristic
231C2×C4×C67C2×#4111yescharacteristic
232C2×C4×C67C2×#4111yescharacteristic
233C2×Dic67—22no—
234C2×Dic67—22no—
235C2×Dic67—28no—
236C2×Dic67—24no—
237C2×Dic67—24no—
238C2×Dic67—42no—
239C2×Dic67—42no—
240C2×Dic67—42no—
241C2×Dic67—42no—
242C2×Dic67—48no—
243C2×Dic67—48no—
244C2×Dic67C2×C22⋊C414yes—
245C4×D67—24no—
246C4×D67—24no—
247C4×D67—24no—
248C4×D67—24no—
249C4×D67—216no—
250C4×D67—432no—
251C8×C67—42no—
252C67⋊C8—42no—
253C22×C67#38614yes—
254C22×C67C2×#10314yes—
255C22×C67#41714yes—
256C22×C67#45014yes—
257C22×C67#45114yes—
258C22×C67#48314yes—
259C22×C67#48414yes—
260C22×C67C22×C22⋊C411yescharacteristic
261C22×C67C2×#12011yescharacteristic
262C22×C67C2×#12311yescharacteristic
263C22×C67C22×SD1611yescharacteristic
264C22×C67C22×Q1611yescharacteristic
265C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C67C2×#13711yescharacteristic
267C2×D67—24no—
268C2×D67—24no—
269C2×D67—24no—
270C2×D67—232no—
271C2×D67—24no—
272C2×D67—24no—
273C2×D67—24no—
274C2×D67C2×#10318yescomplemented
275C4×C67—44no—
276C4×C67—44no—
277C4×C67#38614yes—
278C4×C67C2×C4×D411yescharacteristic
279C4×C67C2×C4×D411yescharacteristic
280C4×C67C2×#12111yescharacteristic
281C4×C67C2×#12111yescharacteristic
282Dic67—24no—
283Dic67—44no—
284Dic67—44no—
285C2×#1872—671nomaximal, Sylow
286C2×C67#187218yescomplemented
287C2×C67C2×#38611yescharacteristic
288C2×C67C22×#10311yescharacteristic
289C2×C67C2×#41711yescharacteristic
290C2×C67C2×#45011yescharacteristic
291C2×C67C2×#45111yescharacteristic
292C2×C67C2×#48311yescharacteristic
293C2×C67C2×#48411yescharacteristic
294D67—28no—
295C22×C22⋊Q8—671no—
296C23×C4⋊C4—671no—
297C22×#103—671no—
298C22×#103D6711yescharacteristic, complemented
299C22×C22⋊C8—671no—
300#1872—678no—
301C2×#340—671no—
302C2×#340—671no—
303C67C2×#187211yescharacteristic, complemented, Sylow
304C23×Q8C2×D6711yescharacteristic
305C24×C4—671no—
306C24×C4—1341no—
307C23×C8—671no—
308C23×C8C2×D6711yescharacteristic
309C2×C22⋊Q8—678no—
310C2×C22⋊Q8—674no—
311C2×C22⋊Q8—1342no—
312C2×C22⋊Q8—1344no—
313C22×C4⋊C4—671no—
314C22×C4⋊C4—671no—
315C22×C4⋊C4—674no—
316C22×C4⋊C4—1344no—
317C22×C4⋊C4—1342no—
318C22×C4⋊C4—1341no—
319C22×C4⋊C4—1341no—
320C22×C4⋊C4—1344no—
321C22×C4⋊C4C2×D6711yescharacteristic
322C22×C22⋊C4—1341no—
323C2×#103—24no—
324C2×#103—674no—
325C2×#103—1344no—
326C2×#103C2×D6714yescomplemented
327C2×C22⋊C8—674no—
328C2×C22⋊C8—1344no—
329#340—674no—
330#340—674no—
331C25—671no—
332C22×Q8—22no—
333C22×Q8—21no—
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8C4×D6714yescomplemented
339C22×Q8C22×D6712yes—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—2684no—
343C23×C4—2684no—
344C23×C4—671no—
345C23×C4—674no—
346C23×C4—672no—
347C23×C4—674no—
348C23×C4—672no—
349C23×C4—1344no—
350C23×C4—1348no—
351C23×C4—1341no—
352C23×C4—1341no—
353C23×C4—1342no—
354C23×C4—1342no—
355C23×C4—1342no—
356C23×C4—1342no—
357C23×C4C22×D6711yescharacteristic
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—672no—
362C22×C8—1342no—
363C22×C8—1342no—
364C22×C8—1342no—
365C22×C8C22×D6712yes—
366C22⋊Q8—6716no—
367C22⋊Q8—13416no—
368C22⋊Q8—1348no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—674no—
372C2×C4⋊C4—674no—
373C2×C4⋊C4—672no—
374C2×C4⋊C4—672no—
375C2×C4⋊C4—1348no—
376C2×C4⋊C4—13416no—
377C2×C4⋊C4—1344no—
378C2×C4⋊C4—1348no—
379C2×C4⋊C4—1344no—
380C2×C4⋊C4—1348no—
381C2×C4⋊C4—1341no—
382C2×C4⋊C4—1341no—
383C2×C4⋊C4—1342no—
384C2×C4⋊C4—1342no—
385C2×C4⋊C4—1348no—
386C2×C4⋊C4C22⋊D6714yes—
387C2×C4⋊C4C22×D6712yes—
388C2×C22⋊C4—1348no—
389C2×C22⋊C4—1344no—
390#103—28no—
391#103—1348no—
392C22⋊C8—1348no—
393C24—674no—
394C24—674no—
395C24—674no—
396C24—672no—
397C24—1342no—
398C24—1342no—
399C24—1342no—
400C24—1342no—
401C24D4×D6711yescharacteristic
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8C2×C4×D6714yescomplemented
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—2688no—
424C22×C4—26832no—
425C22×C4—2688no—
426C22×C4—2682no—
427C22×C4—2682no—
428C22×C4—2682no—
429C22×C4—2682no—
430C22×C4—674no—
431C22×C4—672no—
432C22×C4—678no—
433C22×C4—674no—
434C22×C4—674no—
435C22×C4—13416no—
436C22×C4—1344no—
437C22×C4—1344no—
438C22×C4—1342no—
439C22×C4—1342no—
440C22×C4—1342no—
441C22×C4—1342no—
442C22×C4—1348no—
443C22×C4—1348no—
444C22×C4—1344no—
445C22×C4—1348no—
446C22×C4—1348no—
447C22×C4—1344no—
448C22×C4—1342no—
449C22×C4—1342no—
450C22×C4—1342no—
451C22×C4—1342no—
452C22×C4C2×C4×D6712yes—
453C22×C4D4×D6711yescharacteristic
454C22×C4D4×D6712yes—
455C22×C4D4×D6712yes—
456C22×C4C2×C22⋊D6712yes—
457C22×C4#1176162712yes—
458C22×C4#1176162812yes—
459C22×C4C23×D6711yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C2×C8—2684no—
465C2×C8—1342no—
466C2×C8—1342no—
467C2×C8—1342no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—674no—
471C4⋊C4—674no—
472C4⋊C4—13416no—
473C4⋊C4—13416no—
474C4⋊C4—1344no—
475C4⋊C4—1344no—
476C4⋊C4—1342no—
477C4⋊C4—1342no—
478C4⋊C4C2×C22⋊D6714yes—
479C22⋊C4—13416no—
480C23—674no—
481C23—6716no—
482C23—674no—
483C23—674no—
484C23—1348no—
485C23—1348no—
486C23—1348no—
487C23—1342no—
488C23—1342no—
489C23—1348no—
490C23—1342no—
491C23—1342no—
492C23—1348no—
493C23—1348no—
494C23C22⋊C4×D6712yes—
495C23#1452401112yes—
496C23#1452402512yes—
497C23SD16×D6712yes—
498C23Q16×D6712yes—
499C23#1452407012yes—
500C23#1452407512yes—
501C23C2×D4×D6711yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—48no—
505Q8—42no—
506Q8—42no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—26832no—
516C2×C4—2688no—
517C2×C4—2688no—
518C2×C4—2682no—
519C2×C4—2682no—
520C2×C4—2682no—
521C2×C4—2682no—
522C2×C4—674no—
523C2×C4—1344no—
524C2×C4—1344no—
525C2×C4—1348no—
526C2×C4—1342no—
527C2×C4—1342no—
528C2×C4—13416no—
529C2×C4—1344no—
530C2×C4—1344no—
531C2×C4—1344no—
532C2×C4—1344no—
533C2×C4C22⋊C4×D6712yes—
534C2×C4(C22⋊D67)⋊C414yes—
535C2×C4C4×C22⋊D6714yes—
536C2×C4#1452401112yes—
537C2×C4#1452401414yes—
538C2×C4#1452401614yes—
539C2×C4#1452402512yes—
540C2×C4C22×C4×D6711yescharacteristic
541C2×C4C2×D4×D6711yescharacteristic
542C2×C4C2×D4×D6711yescharacteristic
543C2×C4C22×C22⋊D6711yescharacteristic
544C2×C4C2×#1176162711yescharacteristic
545C2×C4C2×#1176162811yescharacteristic
546C8—42no—
547C8—2682no—
548C22—678no—
549C22—13432no—
550C22—1344no—
551C22—1344no—
552C22—1344no—
553C22—1344no—
554C22—1344no—
555C22—1344no—
556C22#1686327214yes—
557C22#103×D6714yes—
558C22#1686339614yes—
559C22#1686354514yes—
560C22#1686355014yes—
561C22#1686372514yes—
562C22#1686373014yes—
563C22C2×C22⋊C4×D6711yescharacteristic
564C22C2×#1452401111yescharacteristic
565C22C2×#1452402511yescharacteristic
566C22C2×SD16×D6711yescharacteristic
567C22C2×Q16×D6711yescharacteristic
568C22C2×#1452407011yescharacteristic
569C22C2×#1452407511yescharacteristic
570C4—44no—
571C4—44no—
572C4—2684no—
573C4—2684no—
574C4—1344no—
575C4#1686327214yes—
576C4C2×(C22⋊D67)⋊C411yescharacteristic
577C4C2×C4×C22⋊D6711yescharacteristic
578C4C2×#1452401411yescharacteristic
579C4C2×#1452401611yescharacteristic
580C2—1348no—
581C2#2256051318yesdirect product factor
582C2C2×#1686327211yescharacteristic
583C2C2×#103×D6711yescharacteristic
584C2C2×#1686339611yescharacteristic
585C2C2×#1686354511yescharacteristic
586C2C2×#1686355011yescharacteristic
587C2C2×#1686372511yescharacteristic
588C2C2×#1686373011yescharacteristic
subgroupG/H
centerC24D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC22×#103×C67C2
socleC24×C67C2×D4
prime radicalC24×D67D4
cosocleC22×C4×C67C24
hypercenterC22×#103D67
hypocenterC67C2×#1872
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#14523967C2semidirect
5. #22560513C2direct (N × H)
6. C2×#16863159C2semidirect
7. C2×#16863158C2semidirect
8. C22×#103×C67C2semidirect
14. C2×#14524019C4semidirect
26. C2×#14523967C22semidirect
29. #16863159C22semidirect
30. #16863158C22semidirect
35. C2×#103×C67C22semidirect
58. #14524019C2×C4semidirect
63. C2×C4⋊Dic67C2×C4semidirect
84. C22×Q8×C67C2×C4semidirect
143. C4⋊Dic67C22×C4semidirect
161. C2×Q8×C67C22×C4semidirect
203. C22×D67#103semidirect
274. C2×D67C2×#103semidirect
286. C2×C67#1872semidirect
298. C22×#103D67semidirect
303. C67C2×#1872semidirect
326. C2×#103C2×D67semidirect
338. C22×Q8C4×D67semidirect
414. C2×Q8C2×C4×D67semidirect
581. C2#22560513direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240192
C22×C4⋊C4×D672
C22×#145239672
C22×(C2×D67)⋊C82
#225605132
C2×#168631592
C2×#168631582
C22×#103×C672
C2×#187267