Finite Group Databasebeta

← C2×#22560639 #30857510

Subgroups — C2×#22560639 #30857510

Order 17152.

1966 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11761611C211yesmaximal, characteristic
2C22×#14524019C211yesmaximal, characteristic
3C22×#14523967C211yesmaximal, characteristic, complemented
4C2×#16863253C211yesmaximal, characteristic
5#22560639C218yesmaximal, direct product factor
6C2×#16863159C211yesmaximal, characteristic, complemented
7C2×#16863158C211yesmaximal, characteristic, complemented
8C22×#103×C67C211yesmaximal, characteristic, complemented
9C23×C4×D67C2211yescharacteristic
10C22×#11761611—24no—
11C22×#11761611—24no—
12C22×#11761611—24no—
13C22×#11761611C2214yes—
14C23×C67⋊C8C2211yescharacteristic
15C2×#14524019—24no—
16C2×#14524019—22no—
17C2×#14524019C418yescomplemented
18C2×#14524019C2214yes—
19C22×C4⋊Dic67C2211yescharacteristic
20C22×C4⋊Dic67C2211yescharacteristic
21C22×Dic67⋊C4—21no—
22C22×(C2×D67)⋊C4—21no—
23C22×(C2×D67)⋊C4—21no—
24C2×#14523967—24no—
25C2×#14523967C2214yescomplemented
26#16863253—22no—
27#16863253C2214yes—
28C2×C67⋊#95—21no—
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—22no—
41C22×C4×D67—22no—
42C22×C4×D67—22no—
43C22×C4×D67—22no—
44C22×C4×D67C2×C414yes—
45C22×C4×D67D412yes—
46C22×C4×D67D414yes—
47C22×C4×D67C2312yes—
48C2×#11761611—232no—
49C2×#11761611—28no—
50C2×#11761611—28no—
51C2×#11761611—28no—
52C22×C67⋊C8—22no—
53C22×C67⋊C8—22no—
54C22×C67⋊C8D414yes—
55C22×C67⋊C8C2312yes—
56#14524019—216no—
57#14524019—28no—
58#14524019C2×C4116yescomplemented
59C2×C4⋊Dic67—21no—
60C2×C4⋊Dic67—22no—
61C2×C4⋊Dic67—22no—
62C2×C4⋊Dic67—21no—
63C2×C4⋊Dic67—22no—
64C2×C4⋊Dic67C2×C414yes—
65C2×C4⋊Dic67C2312yes—
66C2×C4⋊Dic67C2312yes—
67C2×Dic67⋊C4—28no—
68C2×Dic67⋊C4—24no—
69C2×(C2×D67)⋊C4—28no—
70C2×(C2×D67)⋊C4—24no—
71C2×(C2×D67)⋊C4—24no—
72C2×(C2×D67)⋊C4—44no—
73#14523967—28no—
74C67⋊#95—24no—
75C22×Q8×C67—22no—
76C22×Q8×C67—21no—
77C22×Q8×C67—22no—
78C22×Q8×C67—22no—
79C22×Q8×C67—22no—
80C22×Q8×C67—22no—
81C22×Q8×C67C2×C414yes—
82C22×Q8×C67C2312yes—
83C23×C4×C67—21no—
84C23×C4×C67—21no—
85C23×C4×C67C2311yescharacteristic
86C22×C8×C67—22no—
87C22×C8×C67—22no—
88C22×C8×C67D414yes—
89C22×C8×C67C2312yes—
90C2×C4⋊C4×C67—22no—
91C2×C4⋊C4×C67—22no—
92C2×C4⋊C4×C67—21no—
93C2×C4⋊C4×C67C2312yes—
94#103×C67—28no—
95C23×D67—22no—
96C23×D67—22no—
97C23×D67—22no—
98C23×D67—22no—
99C23×D67—22no—
100C23×D67—22no—
101C23×D67C22⋊C414yes—
102C23×D67C2×D412yes—
103C22×Dic67—22no—
104C22×Dic67—22no—
105C22×Dic67—24no—
106C22×Dic67—22no—
107C22×Dic67—22no—
108C22×Dic67—42no—
109C22×Dic67—42no—
110C22×Dic67—42no—
111C22×Dic67—42no—
112C22×Dic67—42no—
113C22×Dic67C22⋊C414yes—
114C22×Dic67C2×D412yes—
115C2×C4×D67—28no—
116C2×C4×D67—28no—
117C2×C4×D67—28no—
118C2×C4×D67—24no—
119C2×C4×D67—24no—
120C2×C4×D67—28no—
121C2×C4×D67—22no—
122C2×C4×D67—22no—
123C2×C4×D67—22no—
124C2×C4×D67—22no—
125C2×C4×D67C22⋊C418yes—
126C2×C4×D67C22×C414yes—
127C2×C4×D67C2×D414yes—
128#11761611—232no—
129C2×C67⋊C8—28no—
130C2×C67⋊C8—22no—
131C2×C67⋊C8—22no—
132C2×C67⋊C8C2×D414yes—
133C4⋊Dic67—24no—
134C4⋊Dic67—22no—
135C4⋊Dic67—22no—
136C4⋊Dic67—22no—
137C4⋊Dic67—42no—
138C4⋊Dic67C22×C414yes—
139Dic67⋊C4—216no—
140(C2×D67)⋊C4—216no—
141(C2×D67)⋊C4—48no—
142C24×C67C2×D411yescharacteristic
143C2×Q8×C67—24no—
144C2×Q8×C67—22no—
145C2×Q8×C67—28no—
146C2×Q8×C67—28no—
147C2×Q8×C67—22no—
148C2×Q8×C67—22no—
149C2×Q8×C67—22no—
150C2×Q8×C67—22no—
151C2×Q8×C67—44no—
152C2×Q8×C67—42no—
153C2×Q8×C67—42no—
154C2×Q8×C67—44no—
155C2×Q8×C67C22×C414yes—
156C22×C4×C67—24no—
157C22×C4×C67—22no—
158C22×C4×C67—22no—
159C22×C4×C67—42no—
160C22×C4×C67—42no—
161C22×C4×C67—42no—
162C22×C4×C67—42no—
163C22×C4×C67—42no—
164C22×C4×C67C22×C412yes—
165C22×C4×C67C2×D411yescharacteristic
166C22×C4×C67C2×D412yes—
167C22×C4×C67C2×D412yes—
168C22×C4×C67C2×D412yes—
169C22×C4×C67#4112yes—
170C22×C4×C67#4112yes—
171C22×C4×C67C2411yescharacteristic
172C2×C8×C67—28no—
173C2×C8×C67—22no—
174C2×C8×C67—22no—
175C2×C8×C67C2×D414yes—
176C4⋊C4×C67—22no—
177C4⋊C4×C67—22no—
178C4⋊C4×C67—42no—
179C23×C67C2×C22⋊C412yes—
180C23×C67#12112yes—
181C23×C67#12112yes—
182C23×C67AGL(1,ℤ/8)12yes—
183C23×C67#13712yes—
184C23×C67#13712yes—
185C23×C67#13712yes—
186C23×C67C22×D411yescharacteristic
187C22×D67—22no—
188C22×D67—22no—
189C22×D67—22no—
190C22×D67—22no—
191C22×D67—22no—
192C22×D67—22no—
193C22×D67—28no—
194C22×D67—28no—
195C22×D67—28no—
196C22×D67—44no—
197C22×D67—44no—
198C22×D67—44no—
199C22×D67—44no—
200C22×D67C2×C22⋊C414yes—
201Q8×C67—24no—
202Q8×C67—24no—
203Q8×C67—42no—
204Q8×C67—42no—
205Q8×C67—48no—
206C2×C4×C67—24no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—48no—
210C2×C4×C67—42no—
211C2×C4×C67—42no—
212C2×C4×C67—42no—
213C2×C4×C67—84no—
214C2×C4×C67C2×C22⋊C412yes—
215C2×C4×C67C4×D414yes—
216C2×C4×C67C4×D414yes—
217C2×C4×C67#12014yes—
218C2×C4×C67#12112yes—
219C2×C4×C67#12112yes—
220C2×C4×C67#12314yes—
221C2×C4×C67C23×C411yescharacteristic
222C2×C4×C67C22×D411yescharacteristic
223C2×C4×C67C22×D411yescharacteristic
224C2×C4×C67C22×D411yescharacteristic
225C2×C4×C67C2×#4111yescharacteristic
226C2×C4×C67C2×#4111yescharacteristic
227C2×Dic67—22no—
228C2×Dic67—22no—
229C2×Dic67—28no—
230C2×Dic67—24no—
231C2×Dic67—42no—
232C2×Dic67—42no—
233C2×Dic67—48no—
234C2×Dic67—42no—
235C2×Dic67—42no—
236C2×Dic67—42no—
237C2×Dic67—84no—
238C2×Dic67C2×C22⋊C414yes—
239C4×D67—24no—
240C4×D67—24no—
241C4×D67—24no—
242C4×D67—24no—
243C4×D67—216no—
244C8×C67—24no—
245C67⋊C8—24no—
246C22×C67#38614yes—
247C22×C67#41714yes—
248C22×C67#41914yes—
249C22×C67#46814yes—
250C22×C67#46814yes—
251C22×C67#47014yes—
252C22×C67#47014yes—
253C22×C67C22×C22⋊C411yescharacteristic
254C22×C67C2×#12111yescharacteristic
255C22×C67C2×#12111yescharacteristic
256C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C67C2×#13711yescharacteristic
258C22×C67C2×#13711yescharacteristic
259C22×C67C2×#13711yescharacteristic
260C2×D67—24no—
261C2×D67—24no—
262C2×D67—24no—
263C2×D67—416no—
264C2×D67—42no—
265C2×D67—42no—
266C2×D67—42no—
267C2×D67—42no—
268C4×C67—44no—
269C4×C67—82no—
270C4×C67#38614yes—
271C4×C67C2×C4×D411yescharacteristic
272C4×C67C2×C4×D411yescharacteristic
273C4×C67C2×#12011yescharacteristic
274C4×C67C2×#12311yescharacteristic
275Dic67—24no—
276Dic67—44no—
277Dic67—82no—
278C2×#1894—671nomaximal, Sylow
279C2×C67#189418yescomplemented
280C2×C67C2×#38611yescharacteristic
281C2×C67C2×#41911yescharacteristic
282C2×C67C2×#41711yescharacteristic
283C2×C67C2×#46811yescharacteristic
284C2×C67C2×#46811yescharacteristic
285C2×C67C2×#47011yescharacteristic
286C2×C67C2×#47011yescharacteristic
287D67—44no—
288C23×M16—671no—
289C22×C22⋊Q8—671no—
290C22×#103—671no—
291C22×#103D6711yescharacteristic, complemented
292C2×#380—671no—
293#1894—678no—
294C2×#340—671no—
295C2×#340—671no—
296C67C2×#189411yescharacteristic, complemented, Sylow
297C23×Q8C2×D6711yescharacteristic
298C24×C4—671no—
299C22×M16—674no—
300C22×M16—1344no—
301C22×M16—1344no—
302C22×M16—1344no—
303C23×C8—671no—
304C23×C8C2×D6711yescharacteristic
305C2×C22⋊Q8—678no—
306C2×C22⋊Q8—674no—
307C2×C22⋊Q8—1344no—
308C2×C22⋊Q8—1342no—
309C22×C4⋊C4—671no—
310C22×C4⋊C4—671no—
311C22×C4⋊C4—1341no—
312C22×C4⋊C4C2×D6711yescharacteristic
313C22×C22⋊C4—1341no—
314C22×C22⋊C4—1341no—
315C2×#103—24no—
316C2×#103—674no—
317C2×#103—1344no—
318C2×#103C2×D6714yescomplemented
319#380—674no—
320#380—1342no—
321C2×#95—1341no—
322#340—674no—
323#340—674no—
324C25—671no—
325C22×Q8—22no—
326C22×Q8—21no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8C4×D6714yes—
332C22×Q8C22×D6712yes—
333C23×C4—21no—
334C23×C4—21no—
335C23×C4—671no—
336C23×C4—674no—
337C23×C4—674no—
338C23×C4—672no—
339C23×C4—672no—
340C23×C4—1341no—
341C23×C4—1341no—
342C23×C4—1342no—
343C23×C4—1342no—
344C23×C4—1342no—
345C23×C4—1342no—
346C23×C4C22×D6711yescharacteristic
347C2×M16—1348no—
348C2×M16—1348no—
349C2×M16—13432no—
350C2×M16—1348no—
351C22×C8—22no—
352C22×C8—22no—
353C22×C8—674no—
354C22×C8—672no—
355C22×C8—1342no—
356C22×C8—1342no—
357C22×C8C22⋊D6714yes—
358C22×C8C22×D6712yes—
359C22⋊Q8—6716no—
360C22⋊Q8—13416no—
361C22⋊Q8—1348no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—674no—
366C2×C4⋊C4—672no—
367C2×C4⋊C4—672no—
368C2×C4⋊C4—1341no—
369C2×C4⋊C4—1342no—
370C2×C4⋊C4—1342no—
371C2×C4⋊C4—1348no—
372C2×C4⋊C4—1344no—
373C2×C4⋊C4—1341no—
374C2×C4⋊C4—1342no—
375C2×C4⋊C4C22×D6712yes—
376C2×C22⋊C4—2684no—
377C2×C22⋊C4—1344no—
378C2×C22⋊C4—1348no—
379C2×C22⋊C4—1344no—
380#103—28no—
381#103—1348no—
382#95—1344no—
383C24—674no—
384C24—672no—
385C24—1342no—
386C24—1342no—
387C24—1342no—
388C24—1342no—
389C24—1342no—
390C24—1342no—
391C24D4×D6711yescharacteristic
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8C2×C4×D6714yes—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—2682no—
414C22×C4—2682no—
415C22×C4—2682no—
416C22×C4—2682no—
417C22×C4—2682no—
418C22×C4—674no—
419C22×C4—672no—
420C22×C4—678no—
421C22×C4—674no—
422C22×C4—674no—
423C22×C4—1342no—
424C22×C4—1344no—
425C22×C4—1342no—
426C22×C4—1342no—
427C22×C4—1342no—
428C22×C4—1348no—
429C22×C4—1348no—
430C22×C4—1348no—
431C22×C4—1344no—
432C22×C4—1344no—
433C22×C4—1348no—
434C22×C4—1342no—
435C22×C4—1342no—
436C22×C4—1342no—
437C22×C4—1342no—
438C22×C4C2×C4×D6712yes—
439C22×C4D4×D6711yescharacteristic
440C22×C4D4×D6712yes—
441C22×C4D4×D6712yes—
442C22×C4C2×C22⋊D6712yes—
443C22×C4#1176162712yes—
444C22×C4#1176162812yes—
445C22×C4C23×D6711yescharacteristic
446M16—13432no—
447C2×C8—28no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—674no—
451C2×C8—1348no—
452C2×C8—1342no—
453C2×C8—1342no—
454C2×C8C2×C22⋊D6714yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—2682no—
459C4⋊C4—674no—
460C4⋊C4—13416no—
461C4⋊C4—1342no—
462C4⋊C4—1342no—
463C4⋊C4—1344no—
464C4⋊C4—1342no—
465C22⋊C4—2688no—
466C22⋊C4—13416no—
467C23—2684no—
468C23—2684no—
469C23—2684no—
470C23—2684no—
471C23—674no—
472C23—1348no—
473C23—1342no—
474C23—1342no—
475C23—1348no—
476C23—1348no—
477C23—1342no—
478C23—1342no—
479C23—1342no—
480C23—1342no—
481C23C22⋊C4×D6712yes—
482C23#1452401412yes—
483C23#1452401612yes—
484C23#1452407012yes—
485C23#1452407212yes—
486C23#1452407512yes—
487C23#1452407512yes—
488C23C2×D4×D6711yescharacteristic
489Q8—24no—
490Q8—24no—
491Q8—48no—
492Q8—42no—
493Q8—42no—
494C2×C4—24no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—2688no—
503C2×C4—2682no—
504C2×C4—2682no—
505C2×C4—2682no—
506C2×C4—2682no—
507C2×C4—2682no—
508C2×C4—5364no—
509C2×C4—674no—
510C2×C4—1344no—
511C2×C4—1348no—
512C2×C4—1342no—
513C2×C4—1342no—
514C2×C4—13416no—
515C2×C4—1344no—
516C2×C4—1344no—
517C2×C4—1344no—
518C2×C4—1344no—
519C2×C4C22⋊C4×D6712yes—
520C2×C4(C22⋊D67)⋊C414yes—
521C2×C4C4×C22⋊D6714yes—
522C2×C4#1452401114yes—
523C2×C4#1452401412yes—
524C2×C4#1452401612yes—
525C2×C4#1452402514yes—
526C2×C4C22×C4×D6711yescharacteristic
527C2×C4C2×D4×D6711yescharacteristic
528C2×C4C2×D4×D6711yescharacteristic
529C2×C4C22×C22⋊D6711yescharacteristic
530C2×C4C2×#1176162711yescharacteristic
531C2×C4C2×#1176162811yescharacteristic
532C8—24no—
533C8—1344no—
534C22—2682no—
535C22—2682no—
536C22—2682no—
537C22—2682no—
538C22—26816no—
539C22—1344no—
540C22—1344no—
541C22—1344no—
542C22#1686327214yes—
543C22#1686339614yes—
544C22#1686340314yes—
545C22#1686363914yes—
546C22#1686364514yes—
547C22#1686365114yes—
548C22#1686365314yes—
549C22C2×C22⋊C4×D6711yescharacteristic
550C22C2×#1452401411yescharacteristic
551C22C2×#1452401611yescharacteristic
552C22C2×#1452407011yescharacteristic
553C22C2×#1452407211yescharacteristic
554C22C2×#1452407511yescharacteristic
555C22C2×#1452407511yescharacteristic
556C4—44no—
557C4—82no—
558C4—2684no—
559C4—5362no—
560C4—1344no—
561C4#1686327214yes—
562C4C2×(C22⋊D67)⋊C411yescharacteristic
563C4C2×C4×C22⋊D6711yescharacteristic
564C4C2×#1452401111yescharacteristic
565C4C2×#1452402511yescharacteristic
566C2—2684no—
567C2#2256063918yesdirect product factor
568C2C2×#1686327211yescharacteristic
569C2C2×#1686340311yescharacteristic
570C2C2×#1686339611yescharacteristic
571C2C2×#1686363911yescharacteristic
572C2C2×#1686364511yescharacteristic
573C2C2×#1686365111yescharacteristic
574C2C2×#1686365311yescharacteristic
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×#1894
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. #22560639C2direct (N × H)
6. C2×#16863159C2semidirect
7. C2×#16863158C2semidirect
8. C22×#103×C67C2semidirect
17. C2×#14524019C4semidirect
25. C2×#14523967C22semidirect
29. #16863159C22semidirect
30. #16863158C22semidirect
35. C2×#103×C67C22semidirect
58. #14524019C2×C4semidirect
279. C2×C67#1894semidirect
291. C22×#103D67semidirect
296. C67C2×#1894semidirect
318. C2×#103C2×D67semidirect
567. C2#22560639direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616112
C22×#145240192
C22×#145239672
C2×#168632532
#225606392
C2×#168631592
C2×#168631582
C22×#103×C672
C2×#189467