Finite Group Databasebeta

← C2×#22570159 #31723639

Subgroups — C2×#22570159 #31723639

Order 17152.

2326 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524023C211yesmaximal, characteristic
2C2×#117×D67C211yesmaximal, characteristic
3#22570159C2116yesmaximal, direct product factor
4C2×#16863733C212yesmaximal, complemented
5C2×#16863740C212yesmaximal, complemented
6C2×#16863739C212yesmaximal, complemented
7C2×#16863728C212yesmaximal, complemented
8C2×#16863436C211yesmaximal, characteristic, complemented
9C2×#16863406C211yesmaximal, characteristic, complemented
10C2×#16863404C211yesmaximal, characteristic, complemented
11C2×#16863359C211yesmaximal, characteristic
12C2×#428×C67C211yesmaximal, characteristic, complemented
13C23×Dic134C2211yescharacteristic
14C23×C4×D67C2211yescharacteristic
15C22×#11761612C2211yescharacteristic
16C2×#14524023—21no—
17C2×#14524023—24no—
18C2×#14524023—21no—
19C2×#14524023C2212yes—
20C2×#14524023C2212yes—
21C2×#14524023C2218yes—
22C2×#14524002C2211yescharacteristic
23C2×#14524001C2211yescharacteristic
24C2×#14524001C2211yescharacteristic
25C2×#14524000—21no—
26C2×#14523999—21no—
27#117×D67—24no—
28#117×D67C2218yes—
29C22×C4⋊Dic67—21no—
30C2×C4⋊C4×D67C2211yescharacteristic
31C2×C4⋊C4×D67C2211yescharacteristic
32C22×(C2×D67)⋊C4—21no—
33C2×C22⋊C4×D67—21no—
34C2×C42×D67—21no—
35#16863740C22116yescomplemented
36#16863739C22116yescomplemented
37#16863733C22116yescomplemented
38#16863728C22116yescomplemented
39#16863436C2218yescomplemented
40C2×#14523976C2212yescomplemented
41C2×#14523974C2212yescomplemented
42#16863406C2218yescomplemented
43#16863404C2218yescomplemented
44C2×#14523968C2212yescomplemented
45C2×#14523968C2212yescomplemented
46C2×#14523966C2212yescomplemented
47C2×#14523966C2212yescomplemented
48#16863359C2218yes—
49C2×(C2×D67)⋊C8C2212yes—
50C2×(C2×D67)⋊C8C2212yes—
51C22×M16×C67C2211yescharacteristic
52C2×#117×C67C2211yescharacteristic
53C22×C4⋊C4×C67C2211yescharacteristic
54#428×C67C2218yescomplemented
55C2×#107×C67C2212yescomplemented
56C2×#106×C67C2212yescomplemented
57C24×D67D411yescharacteristic
58C22×Dic134—21no—
59C22×Dic134—22no—
60C22×Dic134—24no—
61C22×Dic134—24no—
62C22×Dic134C2312yes—
63C22×Dic134C2314yes—
64C23×Dic67—21no—
65C23×Dic67D411yescharacteristic
66C22×C4×D67—21no—
67C22×C4×D67—21no—
68C22×C4×D67—24no—
69C22×C4×D67—24no—
70C22×C4×D67D411yescharacteristic
71C22×C4×D67D414yes—
72C22×C4×D67D411yescharacteristic
73C22×C4×D67C2311yescharacteristic
74C22×C4×D67C2314yes—
75C22×C4×D67C2311yescharacteristic
76C2×#11761612—24no—
77C2×#11761612C2314yes—
78C22×C67⋊C8C2312yes—
79#14524023—28no—
80#14524023—24no—
81#14524023—28no—
82#14524023—24no—
83#14524023C2318yes—
84#14524023C2318yes—
85#14524002—22no—
86#14524002C2314yes—
87#14524001—22no—
88#14524001—22no—
89#14524001C2314yes—
90#14524001C2314yes—
91#14524000—28no—
92#14523999—28no—
93C2×C4⋊Dic67—22no—
94C2×C4⋊Dic67—22no—
95C2×C4⋊Dic67—28no—
96C2×C4⋊Dic67C2311yescharacteristic
97C2×C4⋊Dic67C2311yescharacteristic
98C2×Dic67⋊C4—21no—
99C2×Dic67⋊C4—21no—
100C4⋊C4×D67—22no—
101C4⋊C4×D67—22no—
102C4⋊C4×D67C2314yes—
103C4⋊C4×D67C2314yes—
104C2×C22⋊Dic67—21no—
105C2×(C2×D67)⋊C4—21no—
106C2×(C2×D67)⋊C4—21no—
107C2×(C2×D67)⋊C4—22no—
108C2×(C2×D67)⋊C4—28no—
109C2×(C2×D67)⋊C4—22no—
110C22⋊C4×D67—28no—
111C2×C4×Dic67—21no—
112C2×C4×Dic67—21no—
113C2×C4×Dic67—21no—
114C42×D67—28no—
115#14523976C2318yescomplemented
116#14523974C2318yescomplemented
117#14523968C2318yescomplemented
118#14523968C2318yescomplemented
119#14523966C2318yescomplemented
120#14523966C2318yescomplemented
121(C2×D67)⋊C8C2318yes—
122(C2×D67)⋊C8C2318yes—
123C23×C4×C67—21no—
124C23×C4×C67C2311yescharacteristic
125C2×M16×C67—24no—
126C2×M16×C67C2314yes—
127C22×C8×C67C2312yes—
128#117×C67—22no—
129#117×C67C2314yes—
130C2×C4⋊C4×C67—21no—
131C2×C4⋊C4×C67—22no—
132C2×C4⋊C4×C67C2312yes—
133C2×C4⋊C4×C67C2314yes—
134C2×C4⋊C4×C67C2311yescharacteristic
135C2×C4⋊C4×C67C2311yescharacteristic
136C2×C22⋊C4×C67—21no—
137C2×C42×C67—21no—
138#107×C67C2318yescomplemented
139#106×C67C2318yescomplemented
140C23×D67—24no—
141C23×D67—24no—
142C23×D67—42no—
143C23×D67C2×D411yescharacteristic
144C23×D67C2×D414yes—
145C23×D67C2×D411yescharacteristic
146C2×Dic134—24no—
147C2×Dic134—28no—
148C2×Dic134—24no—
149C2×Dic134—28no—
150C2×Dic134—24no—
151C2×Dic134—22no—
152C2×Dic134—48no—
153C2×Dic134—44no—
154C2×Dic134C2414yes—
155C22×Dic67—24no—
156C22×Dic67—21no—
157C22×Dic67—21no—
158C22×Dic67—24no—
159C22×Dic67—22no—
160C22×Dic67—44no—
161C22×Dic67C2×D411yescharacteristic
162C22×Dic67C2×D411yescharacteristic
163C22×Dic67C2×D414yes—
164C2×C4×D67—24no—
165C2×C4×D67—216no—
166C2×C4×D67—24no—
167C2×C4×D67—22no—
168C2×C4×D67—22no—
169C2×C4×D67—28no—
170C2×C4×D67—22no—
171C2×C4×D67—22no—
172C2×C4×D67—28no—
173C2×C4×D67—24no—
174C2×C4×D67—24no—
175C2×C4×D67—44no—
176C2×C4×D67—44no—
177C2×C4×D67C2×D414yes—
178C2×C4×D67C2×D412yes—
179C2×C4×D67C2×D414yes—
180C2×C4×D67C2×D412yes—
181C2×C4×D67C2412yes—
182C2×C4×D67C2412yes—
183#11761612—28no—
184C2×C67⋊C8—24no—
185C2×C67⋊C8C2414yes—
186C4⋊Dic67—21no—
187C4⋊Dic67—21no—
188C4⋊Dic67—28no—
189C4⋊Dic67—28no—
190C4⋊Dic67C2412yes—
191C4⋊Dic67C2412yes—
192Dic67⋊C4—24no—
193Dic67⋊C4—24no—
194C22⋊Dic67—24no—
195(C2×D67)⋊C4—24no—
196(C2×D67)⋊C4—24no—
197(C2×D67)⋊C4—28no—
198(C2×D67)⋊C4—28no—
199C4×Dic67—24no—
200C4×Dic67—24no—
201C4×Dic67—24no—
202C24×C67C2×D411yescharacteristic
203C22×C4×C67—21no—
204C22×C4×C67—21no—
205C22×C4×C67—24no—
206C22×C4×C67—22no—
207C22×C4×C67—44no—
208C22×C4×C67C2×D411yescharacteristic
209C22×C4×C67C2×D412yes—
210C22×C4×C67#4114yes—
211C22×C4×C67#4114yes—
212C22×C4×C67C2411yescharacteristic
213C22×C4×C67C2412yes—
214M16×C67—28no—
215C2×C8×C67—24no—
216C2×C8×C67C2414yes—
217C4⋊C4×C67—24no—
218C4⋊C4×C67—22no—
219C4⋊C4×C67—21no—
220C4⋊C4×C67—21no—
221C4⋊C4×C67C2412yes—
222C4⋊C4×C67C2414yes—
223C4⋊C4×C67C2412yes—
224C22⋊C4×C67—24no—
225C42×C67—24no—
226C23×C67—22no—
227C23×C67—22no—
228C23×C67#12314yes—
229C23×C67C22×D412yes—
230C23×C67C22×D411yescharacteristic
231C22×D67—22no—
232C22×D67—22no—
233C22×D67—28no—
234C22×D67—22no—
235C22×D67—22no—
236C22×D67—22no—
237C22×D67—28no—
238C22×D67—44no—
239C22×D67—416no—
240C22×D67#13714yes—
241C22×D67C22×D412yes—
242C22×D67C22×D412yes—
243C2×C4×C67—22no—
244C2×C4×C67—22no—
245C2×C4×C67—24no—
246C2×C4×C67—48no—
247C2×C4×C67—42no—
248C2×C4×C67—44no—
249C2×C4×C67—42no—
250C2×C4×C67#12318yes—
251C2×C4×C67#12314yes—
252C2×C4×C67C22×D412yes—
253C2×C4×C67C22×D411yescharacteristic
254C2×C4×C67C2×#4112yes—
255C2×C4×C67C2×#4114yes—
256C2×C4×C67C2×#4112yes—
257C2×C4×C67C2×#4114yes—
258C2×C4×C67C2511yescharacteristic
259C2×Dic67—28no—
260C2×Dic67—22no—
261C2×Dic67—22no—
262C2×Dic67—22no—
263C2×Dic67—22no—
264C2×Dic67—24no—
265C2×Dic67—48no—
266C2×Dic67—44no—
267C2×Dic67—42no—
268C2×Dic67—42no—
269C2×Dic67C22×D412yes—
270C2×Dic67C22×D412yes—
271C4×D67—28no—
272C4×D67—24no—
273C4×D67—28no—
274C4×D67—24no—
275C4×D67—22no—
276C4×D67—22no—
277C4×D67—22no—
278C4×D67—22no—
279C4×D67—48no—
280C4×D67—48no—
281Dic134—24no—
282Dic134—24no—
283Dic134—48no—
284Dic134—44no—
285C8×C67—24no—
286C67⋊C8—24no—
287C22×C67—28no—
288C22×C67—22no—
289C22×C67—22no—
290C22×C67C2×#12314yes—
291C22×C67C2×#12312yes—
292C22×C67C2×#13712yes—
293C22×C67#57512yes—
294C22×C67C23×D411yescharacteristic
295C2×D67—28no—
296C2×D67—22no—
297C2×D67—22no—
298C2×D67—22no—
299C2×D67—22no—
300C2×D67—22no—
301C2×D67—416no—
302C2×D67—44no—
303C2×D67C2×#13714yes—
304C4×C67—42no—
305C4×C67—42no—
306C4×C67—44no—
307C4×C67C2×#12314yes—
308C4×C67C2×#12312yes—
309C4×C67C22×#4111yescharacteristic
310C4×C67C22×#4111yescharacteristic
311Dic67—22no—
312Dic67—22no—
313Dic67—44no—
314Dic67—42no—
315Dic67—42no—
316C2×#3095—671nomaximal, Sylow
317C2×C67—24no—
318C2×C67#309514yescomplemented
319C2×C67C22×#12311yescharacteristic
320C2×C67C22×#13711yescharacteristic
321C2×C67C2×#57511yescharacteristic
322D67—24no—
323D67—42no—
324C22×C22⋊Q8—671no—
325C22×#117—671no—
326#3095—6716no—
327C2×#484—672no—
328C2×#485—672no—
329C2×#485—672no—
330C2×#483—672no—
331C2×#428—671no—
332C2×#428D6711yescharacteristic, complemented
333C2×#419—671no—
334C2×#419—671no—
335C2×#407—671no—
336C67C2×#309511yescharacteristic, complemented, Sylow
337C23×Q8—671no—
338C24×C4—671no—
339C22×M16—671no—
340C22×M16C2×D6711yescharacteristic
341C2×C22⋊Q8—672no—
342C2×C22⋊Q8—672no—
343C2×C22⋊Q8—678no—
344C2×C22⋊Q8—1344no—
345C2×C22⋊Q8—1341no—
346C2×C22⋊Q8—1341no—
347C2×#117—671no—
348C2×#117—678no—
349C2×#117—671no—
350C2×#117—671no—
351C2×#117—1341no—
352C2×#117—1341no—
353C2×#117—1344no—
354C2×#117C2×D6711yescharacteristic
355C22×C4⋊C4—671no—
356C22×C4⋊C4—671no—
357C22×C4⋊C4—1341no—
358C22×C4⋊C4C2×D6711yescharacteristic
359C22×C22⋊C4—1341no—
360C22×C22⋊C4—1341no—
361C22×C42—1341no—
362#485—6716no—
363#485—6716no—
364#484—6716no—
365#483—6716no—
366#428—678no—
367#428C2×D6718yescomplemented
368C2×#107—672no—
369C2×#107C2×D6712yescomplemented
370C2×#106—672no—
371C2×#106C2×D6712yescomplemented
372#419—678no—
373#419—678no—
374C2×#103—672no—
375C2×#103—672no—
376C2×#103—672no—
377C2×#103—672no—
378#407—678no—
379C2×C22⋊C8—672no—
380C2×C22⋊C8—672no—
381C25—671no—
382C22×Q8—672no—
383C22×Q8—674no—
384C22×Q8—1344no—
385C22×Q8—1344no—
386C22×Q8—1342no—
387C22×Q8—1341no—
388C23×C4—21no—
389C23×C4—671no—
390C23×C4—674no—
391C23×C4—671no—
392C23×C4—671no—
393C23×C4—671no—
394C23×C4—671no—
395C23×C4—674no—
396C23×C4—1341no—
397C23×C4—1341no—
398C23×C4—1341no—
399C23×C4—1344no—
400C23×C4—1344no—
401C23×C4C22×D6711yescharacteristic
402C2×M16—24no—
403C2×M16—674no—
404C2×M16—1344no—
405C2×M16C22×D6714yes—
406C22×C8—672no—
407C22×C8C22×D6712yes—
408C22⋊Q8—678no—
409C22⋊Q8—678no—
410C22⋊Q8—1348no—
411C22⋊Q8—1344no—
412C22⋊Q8—1344no—
413C22⋊Q8—1348no—
414#117—22no—
415#117—674no—
416#117—674no—
417#117—674no—
418#117—1348no—
419#117—1348no—
420#117—1342no—
421#117—1342no—
422#117—1342no—
423#117C22×D6714yes—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—671no—
427C2×C4⋊C4—671no—
428C2×C4⋊C4—674no—
429C2×C4⋊C4—674no—
430C2×C4⋊C4—1342no—
431C2×C4⋊C4—1342no—
432C2×C4⋊C4—1341no—
433C2×C4⋊C4—1341no—
434C2×C4⋊C4—1348no—
435C2×C4⋊C4—1342no—
436C2×C4⋊C4—1342no—
437C2×C4⋊C4C22×D6711yescharacteristic
438C2×C4⋊C4C22×D6711yescharacteristic
439C2×C4⋊C4C22×D6712yes—
440C2×C4⋊C4C22×D6714yes—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1348no—
443C2×C22⋊C4—1341no—
444C2×C22⋊C4—1341no—
445C2×C22⋊C4—1348no—
446C2×C22⋊C4—1342no—
447C2×C22⋊C4—1342no—
448C2×C22⋊C4—1341no—
449C2×C42—21no—
450C2×C42—1348no—
451C2×C42—1341no—
452C2×C42—1341no—
453C2×C42—1341no—
454#107—678no—
455#107C22×D6718yescomplemented
456#106—678no—
457#106C22×D6718yescomplemented
458#103—678no—
459#103—678no—
460#103—678no—
461#103—678no—
462C22⋊C8—678no—
463C22⋊C8—678no—
464C24—2682no—
465C24—674no—
466C24—671no—
467C24—671no—
468C24—1344no—
469C24—1344no—
470C24D4×D6711yescharacteristic
471C2×Q8—2688no—
472C2×Q8—2684no—
473C2×Q8—674no—
474C2×Q8—1344no—
475C2×Q8—1348no—
476C2×Q8—1344no—
477C2×Q8—1348no—
478C2×Q8—1344no—
479C2×Q8—1342no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—2684no—
486C22×C4—2684no—
487C22×C4—2684no—
488C22×C4—674no—
489C22×C4—671no—
490C22×C4—671no—
491C22×C4—674no—
492C22×C4—672no—
493C22×C4—672no—
494C22×C4—674no—
495C22×C4—672no—
496C22×C4—672no—
497C22×C4—1344no—
498C22×C4—1344no—
499C22×C4—1344no—
500C22×C4—1341no—
501C22×C4—1341no—
502C22×C4—1342no—
503C22×C4—1344no—
504C22×C4—13416no—
505C22×C4—1344no—
506C22×C4—1344no—
507C22×C4—1342no—
508C22×C4—1342no—
509C22×C4—1348no—
510C22×C4—1342no—
511C22×C4—1342no—
512C22×C4—1348no—
513C22×C4D4×D6712yes—
514C22×C4D4×D6711yescharacteristic
515C22×C4#1176162814yes—
516C22×C4#1176162914yes—
517C22×C4C23×D6712yes—
518C22×C4C23×D6711yescharacteristic
519M16—28no—
520M16—1348no—
521C2×C8—24no—
522C2×C8—674no—
523C2×C8—1344no—
524C2×C8C23×D6714yes—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—672no—
530C4⋊C4—672no—
531C4⋊C4—1344no—
532C4⋊C4—1344no—
533C4⋊C4—1348no—
534C4⋊C4—1348no—
535C4⋊C4—1341no—
536C4⋊C4—1341no—
537C4⋊C4C23×D6712yes—
538C4⋊C4C23×D6712yes—
539C4⋊C4C23×D6714yes—
540C22⋊C4—24no—
541C22⋊C4—1344no—
542C22⋊C4—1344no—
543C22⋊C4—1348no—
544C22⋊C4—1348no—
545C22⋊C4—1344no—
546C42—24no—
547C42—1344no—
548C42—1344no—
549C42—1344no—
550C23—22no—
551C23—22no—
552C23—26816no—
553C23—2684no—
554C23—674no—
555C23—672no—
556C23—672no—
557C23—1348no—
558C23—1348no—
559C23—1342no—
560C23—1342no—
561C23—1342no—
562C23—1342no—
563C23—1342no—
564C23#1452402814yes—
565C23C2×D4×D6712yes—
566C23C2×D4×D6711yescharacteristic
567Q8—2688no—
568Q8—2684no—
569Q8—1344no—
570Q8—1344no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—2688no—
579C2×C4—2688no—
580C2×C4—2688no—
581C2×C4—2684no—
582C2×C4—2682no—
583C2×C4—2682no—
584C2×C4—672no—
585C2×C4—672no—
586C2×C4—1342no—
587C2×C4—1342no—
588C2×C4—1344no—
589C2×C4—1348no—
590C2×C4—1342no—
591C2×C4—1342no—
592C2×C4—1344no—
593C2×C4—1348no—
594C2×C4—1348no—
595C2×C4—1344no—
596C2×C4—1342no—
597C2×C4—1342no—
598C2×C4—1342no—
599C2×C4—1342no—
600C2×C4#1452402818yes—
601C2×C4#1452402814yes—
602C2×C4C2×D4×D6712yes—
603C2×C4C2×D4×D6711yescharacteristic
604C2×C4C2×#1176162812yes—
605C2×C4C2×#1176162814yes—
606C2×C4C2×#1176162912yes—
607C2×C4C2×#1176162914yes—
608C2×C4C24×D6711yescharacteristic
609C8—24no—
610C8—1344no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—26816no—
615C22—2684no—
616C22—674no—
617C22—1342no—
618C22—1342no—
619C22—1342no—
620C22—1342no—
621C22—1342no—
622C22—1348no—
623C22C2×#1452402814yes—
624C22C2×#1452402812yes—
625C22#137×D6712yes—
626C22#1686419212yes—
627C22C22×D4×D6711yescharacteristic
628C4—42no—
629C4—44no—
630C4—42no—
631C4—2682no—
632C4—2682no—
633C4—2684no—
634C4—1342no—
635C4—1342no—
636C4C2×#1452402814yes—
637C4C2×#1452402812yes—
638C4C22×#1176162811yescharacteristic
639C4C22×#1176162911yescharacteristic
640C2—24no—
641C2—2682no—
642C2—1344no—
643C2#2257015914yesdirect product factor
644C2C22×#1452402811yescharacteristic
645C2C2×#137×D6711yescharacteristic
646C2C2×#1686419211yescharacteristic
subgroupG/H
centerC23C2×D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×#428×C67C2
socleC23×C67C22×D4
prime radicalC24×D67D4
hypercenterC2×#428D67
hypocenterC67C2×#3095
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #22570159C2direct (N × H)
4. C2×#16863733C2semidirect
5. C2×#16863740C2semidirect
6. C2×#16863739C2semidirect
7. C2×#16863728C2semidirect
8. C2×#16863436C2semidirect
9. C2×#16863406C2semidirect
10. C2×#16863404C2semidirect
12. C2×#428×C67C2semidirect
35. #16863740C22semidirect
36. #16863739C22semidirect
37. #16863733C22semidirect
38. #16863728C22semidirect
39. #16863436C22semidirect
40. C2×#14523976C22semidirect
41. C2×#14523974C22semidirect
42. #16863406C22semidirect
43. #16863404C22semidirect
44. C2×#14523968C22semidirect
45. C2×#14523968C22semidirect
46. C2×#14523966C22semidirect
47. C2×#14523966C22semidirect
54. #428×C67C22semidirect
55. C2×#107×C67C22semidirect
56. C2×#106×C67C22semidirect
115. #14523976C23semidirect
116. #14523974C23semidirect
117. #14523968C23semidirect
118. #14523968C23semidirect
119. #14523966C23semidirect
120. #14523966C23semidirect
138. #107×C67C23semidirect
139. #106×C67C23semidirect
318. C2×C67#3095semidirect
332. C2×#428D67semidirect
336. C67C2×#3095semidirect
367. #428C2×D67semidirect
369. C2×#107C2×D67semidirect
371. C2×#106C2×D67semidirect
455. #107C22×D67semidirect
457. #106C22×D67semidirect
643. C2#22570159direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240232
C2×#117×D672
#225701592
C2×#168637332
C2×#168637402
C2×#168637392
C2×#168637282
C2×#168634362
C2×#168634062
C2×#168634042
C2×#168633592
C2×#428×C672
C2×#309567