Finite Group Databasebeta

← C2×#22570132 #31723609

Subgroups — C2×#22570132 #31723609

Order 17152.

2650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863907C211yesmaximal, characteristic
2#22570132C2116yesmaximal, direct product factor
3C2×#16863731C212yesmaximal, complemented
4C2×#16863714C212yesmaximal, complemented
5C2×#16863726C212yesmaximal, complemented
6C2×#16863709C212yesmaximal, complemented
7C2×#16863436C212yesmaximal, complemented
8C2×#16863395C212yesmaximal, complemented
9C2×#16863360C211yesmaximal, characteristic, complemented
10C23×C4⋊C4×C67C211yesmaximal, characteristic, complemented
11C22×#11761628C2211yescharacteristic
12C22×#11761612C2212yes—
13#16863907—24no—
14#16863907C2218yes—
15C2×#14524026—21no—
16C2×#14524022C2211yescharacteristic
17C2×#14524022C2211yescharacteristic
18C2×#14524017C2211yescharacteristic
19C2×#14524017C2211yescharacteristic
20C2×#14524012—21no—
21C2×C4×C22⋊D67—21no—
22C2×C4×C22⋊D67—21no—
23C2×#14524002C2211yescharacteristic
24#16863731C22116yescomplemented
25#16863726C22116yescomplemented
26#16863714C22116yescomplemented
27#16863709C22116yescomplemented
28#16863436C22116yescomplemented
29C2×#14523976C2214yescomplemented
30C2×#14523974C2214yescomplemented
31#16863395C22116yescomplemented
32C2×#14523966C2214yescomplemented
33C2×#14523963C2214yescomplemented
34#16863360C2218yescomplemented
35C2×C22⋊(C67⋊C8)C2212yes—
36C2×C22⋊(C67⋊C8)C2212yes—
37C24×C4×C67—21no—
38C24×C4×C67C2211yescharacteristic
39C22×C4⋊C4×C67—24no—
40C22×C4⋊C4×C67—21no—
41C22×C4⋊C4×C67—21no—
42C22×C4⋊C4×C67—21no—
43C22×C4⋊C4×C67C2212yes—
44C22×C4⋊C4×C67C2212yes—
45C22×C4⋊C4×C67C2212yes—
46C22×C4⋊C4×C67C2218yescomplemented
47C2×#11761628—22no—
48C2×#11761628—28no—
49C2×#11761628C2314yes—
50C22×Dic134C2311yescharacteristic
51C22×D268C2311yescharacteristic
52C22×C22⋊D67—21no—
53C22×C22⋊D67—21no—
54C22×C22⋊D67—21no—
55C22×C4×D67—21no—
56C2×#11761612—28no—
57C2×#11761612C2318yes—
58C22×C67⋊C8C2314yes—
59#14524026—28no—
60#14524022—22no—
61#14524022—22no—
62#14524022C2314yes—
63#14524022C2314yes—
64#14524017—22no—
65#14524017—22no—
66#14524017C2314yes—
67#14524017C2314yes—
68#14524012—28no—
69C4×C22⋊D67—28no—
70C4×C22⋊D67—28no—
71#14524002—22no—
72#14524002C2314yes—
73C2×C4⋊Dic67C2311yescharacteristic
74C2×C4⋊Dic67C2311yescharacteristic
75C2×Dic67⋊C4—21no—
76C2×Dic67⋊C4—21no—
77C2×C22⋊Dic67—21no—
78C2×C22⋊Dic67—21no—
79C2×C22⋊Dic67—21no—
80C2×(C2×D67)⋊C4—21no—
81C2×(C2×D67)⋊C4—21no—
82C2×C4×Dic67—21no—
83#14523976C23116yescomplemented
84#14523974C23116yescomplemented
85#14523966C23116yescomplemented
86#14523963C23116yescomplemented
87C22⋊(C67⋊C8)C2318yes—
88C22⋊(C67⋊C8)C2318yes—
89C25×C67D411yescharacteristic
90C23×C4×C67—28no—
91C23×C4×C67—22no—
92C23×C4×C67—22no—
93C23×C4×C67—22no—
94C23×C4×C67—28no—
95C23×C4×C67—48no—
96C23×C4×C67D411yescharacteristic
97C23×C4×C67D411yescharacteristic
98C23×C4×C67D411yescharacteristic
99C23×C4×C67D414yes—
100C23×C4×C67C2311yescharacteristic
101C23×C4×C67C2311yescharacteristic
102C23×C4×C67C2314yes—
103C23×C4×C67C2311yescharacteristic
104C2×C4⋊C4×C67—22no—
105C2×C4⋊C4×C67—22no—
106C2×C4⋊C4×C67—24no—
107C2×C4⋊C4×C67—24no—
108C2×C4⋊C4×C67—22no—
109C2×C4⋊C4×C67—24no—
110C2×C4⋊C4×C67—28no—
111C2×C4⋊C4×C67—28no—
112C2×C4⋊C4×C67—28no—
113C2×C4⋊C4×C67C2314yes—
114C2×C4⋊C4×C67C2318yes—
115C2×C4⋊C4×C67C2318yes—
116C2×C4⋊C4×C67C2318yes—
117C23×D67—21no—
118#11761628—216no—
119#11761628—48no—
120C2×Dic134—24no—
121C2×Dic134—21no—
122C2×Dic134C2412yes—
123C2×D268—24no—
124C2×D268—21no—
125C2×D268C2412yes—
126C2×C22⋊D67—24no—
127C2×C22⋊D67—24no—
128C2×C22⋊D67—24no—
129C2×C22⋊D67—44no—
130C2×C22⋊D67—44no—
131C2×C22⋊D67—44no—
132C22×Dic67—21no—
133C22×Dic67—21no—
134C22×Dic67—21no—
135C2×C4×D67—24no—
136C2×C4×D67—44no—
137#11761612—216no—
138C2×C67⋊C8—28no—
139C2×C67⋊C8C2418yes—
140C4⋊Dic67—21no—
141C4⋊Dic67—21no—
142C4⋊Dic67C2412yes—
143C4⋊Dic67C2412yes—
144Dic67⋊C4—24no—
145Dic67⋊C4—24no—
146C22⋊Dic67—24no—
147C22⋊Dic67—24no—
148C22⋊Dic67—24no—
149(C2×D67)⋊C4—24no—
150(C2×D67)⋊C4—24no—
151C4×Dic67—24no—
152C24×C67—24no—
153C24×C67—24no—
154C24×C67—24no—
155C24×C67C2×D414yes—
156C24×C67C2×D411yescharacteristic
157C24×C67C2×D411yescharacteristic
158C24×C67C2×D411yescharacteristic
159C22×C4×C67—24no—
160C22×C4×C67—24no—
161C22×C4×C67—24no—
162C22×C4×C67—216no—
163C22×C4×C67—24no—
164C22×C4×C67—24no—
165C22×C4×C67—216no—
166C22×C4×C67—28no—
167C22×C4×C67—28no—
168C22×C4×C67—28no—
169C22×C4×C67—24no—
170C22×C4×C67—48no—
171C22×C4×C67—48no—
172C22×C4×C67—432no—
173C22×C4×C67—48no—
174C22×C4×C67C2×D411yescharacteristic
175C22×C4×C67C2×D411yescharacteristic
176C22×C4×C67C2×D414yes—
177C22×C4×C67C2×D414yes—
178C22×C4×C67C2×D412yes—
179C22×C4×C67C2×D412yes—
180C22×C4×C67C2×D414yes—
181C22×C4×C67C2×D411yescharacteristic
182C22×C4×C67C2×D412yes—
183C22×C4×C67#4118yes—
184C22×C4×C67C2412yes—
185C22×C4×C67C2412yes—
186C22×C4×C67C2411yescharacteristic
187C22×C4×C67C2412yes—
188C4⋊C4×C67—28no—
189C4⋊C4×C67—28no—
190C4⋊C4×C67—28no—
191C4⋊C4×C67—24no—
192C4⋊C4×C67C2418yes—
193C23×C67—232no—
194C23×C67—28no—
195C23×C67—22no—
196C23×C67—22no—
197C23×C67—28no—
198C23×C67—22no—
199C23×C67—22no—
200C23×C67—28no—
201C23×C67—22no—
202C23×C67—22no—
203C23×C67#12314yes—
204C23×C67AGL(1,ℤ/8)14yes—
205C23×C67#13714yes—
206C23×C67C22×D412yes—
207C23×C67C22×D412yes—
208C23×C67C22×D412yes—
209C23×C67C22×D411yescharacteristic
210C22×D67—22no—
211C22×D67—42no—
212C22×D67—42no—
213C22×D67—42no—
214C2×C4×C67—28no—
215C2×C4×C67—22no—
216C2×C4×C67—22no—
217C2×C4×C67—24no—
218C2×C4×C67—24no—
219C2×C4×C67—28no—
220C2×C4×C67—28no—
221C2×C4×C67—24no—
222C2×C4×C67—24no—
223C2×C4×C67—28no—
224C2×C4×C67—416no—
225C2×C4×C67—416no—
226C2×C4×C67—416no—
227C2×C4×C67—48no—
228C2×C4×C67#12314yes—
229C2×C4×C67#12318yes—
230C2×C4×C67C22×D412yes—
231C2×C4×C67C22×D412yes—
232C2×C4×C67C22×D412yes—
233C2×C4×C67C22×D411yescharacteristic
234C2×C4×C67C2×#4114yes—
235C2×C4×C67C2×#4118yes—
236C2×C4×C67C2511yescharacteristic
237C22⋊D67—48no—
238C22⋊D67—48no—
239C22⋊D67—48no—
240C2×Dic67—22no—
241C2×Dic67—22no—
242C2×Dic67—22no—
243C2×Dic67—42no—
244C2×Dic67—42no—
245C2×Dic67—42no—
246D268—24no—
247D268—42no—
248C4×D67—48no—
249Dic134—24no—
250Dic134—42no—
251C67⋊C8—28no—
252C22×C67—232no—
253C22×C67—28no—
254C22×C67—28no—
255C22×C67—28no—
256C22×C67—22no—
257C22×C67—22no—
258C22×C67—22no—
259C22×C67—22no—
260C22×C67—22no—
261C22×C67—22no—
262C22×C67C2×#12314yes—
263C22×C67C2×#12312yes—
264C22×C67C2×AGL(1,ℤ/8)14yes—
265C22×C67C2×AGL(1,ℤ/8)12yes—
266C22×C67C2×#13714yes—
267C22×C67C2×#13712yes—
268C22×C67C23×D411yescharacteristic
269C2×D67—42no—
270C2×D67—42no—
271C2×D67—42no—
272C2×D67—84no—
273C4×C67—22no—
274C4×C67—22no—
275C4×C67—48no—
276C4×C67C2×#12312yes—
277C4×C67C2×#12314yes—
278C4×C67C22×#4112yes—
279Dic67—42no—
280Dic67—42no—
281Dic67—42no—
282C2×#3092—671nomaximal, Sylow
283C2×C67—24no—
284C2×C67—24no—
285C2×C67—24no—
286C2×C67#309214yescomplemented
287C2×C67C22×#12311yescharacteristic
288C2×C67C22×AGL(1,ℤ/8)11yescharacteristic
289C2×C67C22×#13711yescharacteristic
290D67—82no—
291C2×#525—671no—
292C23×C4⋊C4D6711yescharacteristic, complemented
293#3092—6716no—
294C2×#484—672no—
295C2×#481—672no—
296C2×#483—672no—
297C2×#480—672no—
298C2×#428—672no—
299C2×#417—672no—
300C2×#407—671no—
301C67C2×#309211yescharacteristic, complemented, Sylow
302C22×#41—671no—
303C24×C4—21no—
304C24×C4C2×D6711yescharacteristic
305C22×M16—672no—
306#525—678no—
307#525—1344no—
308C2×#123—1341no—
309C2×C22⋊Q8—671no—
310C2×C22⋊Q8—671no—
311C2×#121—671no—
312C2×#121—671no—
313C2×#120—1341no—
314C2×C4×D4—1341no—
315C2×C4×D4—1341no—
316C2×#117—671no—
317C22×C4⋊C4—24no—
318C22×C4⋊C4—21no—
319C22×C4⋊C4—21no—
320C22×C4⋊C4—21no—
321C22×C4⋊C4C2×D6712yes—
322C22×C4⋊C4C2×D6712yes—
323C22×C4⋊C4C2×D6712yes—
324C22×C4⋊C4C2×D6718yescomplemented
325#484—6716no—
326#483—6716no—
327#481—6716no—
328#480—6716no—
329#428—6716no—
330C2×#107—674no—
331C2×#106—674no—
332#417—6716no—
333C2×#103—674no—
334C2×#102—674no—
335#407—678no—
336C2×C22⋊C8—672no—
337C2×C22⋊C8—672no—
338C25C22⋊D6711yescharacteristic
339C2×#41—674no—
340C2×#41—1348no—
341C2×#41—1342no—
342C22×Q8—671no—
343C22×D4—671no—
344C22×D4—1341no—
345C22×D4—1341no—
346C22×D4—1341no—
347C23×C4—28no—
348C23×C4—22no—
349C23×C4—22no—
350C23×C4—22no—
351C23×C4—28no—
352C23×C4—48no—
353C23×C4—1341no—
354C23×C4C22⋊D6711yescharacteristic
355C23×C4C22⋊D6711yescharacteristic
356C23×C4C22⋊D6711yescharacteristic
357C23×C4C22⋊D6714yes—
358C23×C4C22×D6711yescharacteristic
359C23×C4C22×D6711yescharacteristic
360C23×C4C22×D6714yes—
361C23×C4C22×D6711yescharacteristic
362C2×M16—678no—
363C2×M16—1348no—
364C22×C8—674no—
365#123—1348no—
366C22⋊Q8—674no—
367C22⋊Q8—674no—
368C22⋊Q8—1342no—
369C22⋊Q8—1342no—
370#121—674no—
371#121—674no—
372#121—1342no—
373#121—1342no—
374#120—1348no—
375C4×D4—1348no—
376C4×D4—1348no—
377#117—674no—
378#117—1342no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—28no—
388C2×C4⋊C4—671no—
389C2×C4⋊C4—671no—
390C2×C4⋊C4—1341no—
391C2×C4⋊C4—1341no—
392C2×C4⋊C4C22×D6714yes—
393C2×C4⋊C4C22×D6718yes—
394C2×C4⋊C4C22×D6718yes—
395C2×C4⋊C4C22×D6718yes—
396C2×C22⋊C4—1341no—
397C2×C22⋊C4—1341no—
398C2×C22⋊C4—1341no—
399C2×C22⋊C4—1341no—
400C2×C22⋊C4—1341no—
401C2×C42—1341no—
402#107—6716no—
403#106—6716no—
404#103—6716no—
405#102—6716no—
406C22⋊C8—678no—
407C22⋊C8—678no—
408C24—24no—
409C24—24no—
410C24—24no—
411C24—1341no—
412C24C2×C22⋊D6714yes—
413C24C2×C22⋊D6711yescharacteristic
414C24C2×C22⋊D6711yescharacteristic
415C24C2×C22⋊D6711yescharacteristic
416#41—2688no—
417#41—13416no—
418C2×Q8—672no—
419C2×Q8—1344no—
420C2×Q8—1341no—
421C2×D4—2684no—
422C2×D4—2684no—
423C2×D4—2684no—
424C2×D4—672no—
425C2×D4—1344no—
426C2×D4—1344no—
427C2×D4—1344no—
428C2×D4—1341no—
429C2×D4—1344no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—432no—
444C22×C4—48no—
445C22×C4—2684no—
446C22×C4—1341no—
447C22×C4—1341no—
448C22×C4—1341no—
449C22×C4—1344no—
450C22×C4C2×C22⋊D6714yes—
451C22×C4C2×C22⋊D6714yes—
452C22×C4C2×C22⋊D6712yes—
453C22×C4C2×C22⋊D6712yes—
454C22×C4C2×C22⋊D6714yes—
455C22×C4C2×C22⋊D6711yescharacteristic
456C22×C4C2×C22⋊D6711yescharacteristic
457C22×C4C2×C22⋊D6712yes—
458C22×C4C2×C22⋊D6711yescharacteristic
459C22×C4#1176162818yes—
460C22×C4C23×D6712yes—
461C22×C4C23×D6712yes—
462C22×C4C23×D6712yes—
463C22×C4C23×D6711yescharacteristic
464M16—13416no—
465C2×C8—678no—
466C2×C8—1348no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—672no—
472C4⋊C4—672no—
473C4⋊C4—1341no—
474C4⋊C4—1341no—
475C4⋊C4—1344no—
476C4⋊C4—1344no—
477C4⋊C4C23×D6718yes—
478C22⋊C4—1344no—
479C22⋊C4—1344no—
480C22⋊C4—1344no—
481C22⋊C4—1344no—
482C22⋊C4—1344no—
483C42—1344no—
484C23—232no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—2682no—
495C23—2682no—
496C23—2682no—
497C23—1342no—
498C23#1452402614yes—
499C23#1452406914yes—
500C23#1452407414yes—
501C23C22×C22⋊D6712yes—
502C23C22×C22⋊D6712yes—
503C23C22×C22⋊D6712yes—
504C23C22×C22⋊D6711yescharacteristic
505Q8—2682no—
506Q8—1344no—
507D4—2688no—
508D4—2688no—
509D4—2688no—
510D4—2682no—
511D4—1344no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—48no—
526C2×C4—2682no—
527C2×C4—2682no—
528C2×C4—2682no—
529C2×C4—2688no—
530C2×C4—1342no—
531C2×C4—1342no—
532C2×C4—1342no—
533C2×C4#1452402614yes—
534C2×C4#1452402618yes—
535C2×C4C22×C22⋊D6712yes—
536C2×C4C22×C22⋊D6712yes—
537C2×C4C22×C22⋊D6712yes—
538C2×C4C22×C22⋊D6711yescharacteristic
539C2×C4C2×#1176162814yes—
540C2×C4C2×#1176162818yes—
541C2×C4C24×D6711yescharacteristic
542C8—1348no—
543C22—232no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—28no—
553C22—2682no—
554C22—2682no—
555C22—2682no—
556C22—5364no—
557C22C2×#1452402614yes—
558C22C2×#1452402612yes—
559C22C2×#1452406914yes—
560C22C2×#1452406912yes—
561C22C2×#1452407414yes—
562C22C2×#1452407412yes—
563C22C23×C22⋊D6711yescharacteristic
564C4—22no—
565C4—22no—
566C4—48no—
567C4—2682no—
568C4—2682no—
569C4—2682no—
570C4C2×#1452402614yes—
571C4C2×#1452402612yes—
572C4C22×#1176162812yes—
573C2—24no—
574C2—24no—
575C2—24no—
576C2—5362no—
577C2#2257013214yesdirect product factor
578C2C22×#1452402611yescharacteristic
579C2C22×#1452406911yescharacteristic
580C2C22×#1452407411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC23×C4⋊C4×C67C2
socleC23×C67C22×D4
prime radicalC2×#16863907C2
hypercenterC23×C4⋊C4D67
hypocenterC67C2×#3092
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22570132C2direct (N × H)
3. C2×#16863731C2semidirect
4. C2×#16863714C2semidirect
5. C2×#16863726C2semidirect
6. C2×#16863709C2semidirect
7. C2×#16863436C2semidirect
8. C2×#16863395C2semidirect
9. C2×#16863360C2semidirect
10. C23×C4⋊C4×C67C2semidirect
24. #16863731C22semidirect
25. #16863726C22semidirect
26. #16863714C22semidirect
27. #16863709C22semidirect
28. #16863436C22semidirect
29. C2×#14523976C22semidirect
30. C2×#14523974C22semidirect
31. #16863395C22semidirect
32. C2×#14523966C22semidirect
33. C2×#14523963C22semidirect
34. #16863360C22semidirect
46. C22×C4⋊C4×C67C22semidirect
83. #14523976C23semidirect
84. #14523974C23semidirect
85. #14523966C23semidirect
86. #14523963C23semidirect
286. C2×C67#3092semidirect
292. C23×C4⋊C4D67semidirect
301. C67C2×#3092semidirect
324. C22×C4⋊C4C2×D67semidirect
577. C2#22570132direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168639072
#225701322
C2×#168637312
C2×#168637142
C2×#168637262
C2×#168637092
C2×#168634362
C2×#168633952
C2×#168633602
C23×C4⋊C4×C672
C2×#309267