Finite Group Databasebeta

← C2×#22588564 #31998304

Subgroups — C2×#22588564 #31998304

Order 18176.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986482C211yesmaximal, characteristic
2C2×#17986039C211yesmaximal, characteristic, complemented
3C22×#14556055C211yesmaximal, characteristic
4C2×#17985868C211yesmaximal, characteristic, complemented
5#22588564C218yesmaximal, direct product factor
6C2×#17985767C211yesmaximal, characteristic, complemented
7C2×#17985760C211yesmaximal, characteristic, complemented
8C2×#417×C71C211yesmaximal, characteristic
9C22×#11765849C2211yescharacteristic
10#17986482—22no—
11#17986482—22no—
12#17986482—22no—
13#17986482C2214yes—
14C2×Q8×Dic71—21no—
15C2×D4×Dic71—21no—
16C2×D4×Dic71—22no—
17C2×#14556087—21no—
18C2×#14556087C2211yescharacteristic
19C22×C4⋊Dic71—21no—
20C22×C4⋊Dic71C2211yescharacteristic
21C22×Dic71⋊C4—22no—
22C22×C4×Dic71C2211yescharacteristic
23#17986039—22no—
24#17986039C2214yescomplemented
25C2×#14556061—21no—
26C2×#14556059—21no—
27C2×#14556055C418yescomplemented
28C2×#14556055C2214yes—
29#17985868—22no—
30#17985868C2214yescomplemented
31#17985767C2214yescomplemented
32#17985760C2214yescomplemented
33C22×#41×C71C2211yescharacteristic
34C22×M16×C71C2211yescharacteristic
35C22×C4⋊C4×C71C2211yescharacteristic
36#417×C71—22no—
37#417×C71C2214yes—
38C2×#103×C71—21no—
39C2×#102×C71—21no—
40C23×Dic71—22no—
41C23×Dic71—21no—
42C23×Dic71D412yes—
43C2×#11765849—21no—
44C2×#11765849—22no—
45C2×#11765849C2×C414yescomplemented
46C2×#11765849C2312yes—
47C22×C71⋊C8—21no—
48Q8×Dic71—24no—
49Q8×Dic71—24no—
50D4×Dic71—28no—
51D4×Dic71—24no—
52D4×Dic71—28no—
53D4×Dic71—24no—
54#14556087—24no—
55#14556087—24no—
56#14556087—21no—
57#14556087—21no—
58#14556087D412yes—
59#14556087C2312yes—
60C2×C4⋊Dic71—22no—
61C2×C4⋊Dic71—24no—
62C2×C4⋊Dic71—24no—
63C2×C4⋊Dic71—21no—
64C2×C4⋊Dic71—21no—
65C2×C4⋊Dic71—21no—
66C2×C4⋊Dic71—21no—
67C2×C4⋊Dic71D412yes—
68C2×C4⋊Dic71D411yescharacteristic
69C2×C4⋊Dic71D411yescharacteristic
70C2×C4⋊Dic71C2312yes—
71C2×Dic71⋊C4—28no—
72C2×Dic71⋊C4—48no—
73C2×C22⋊Dic71—22no—
74C2×C22⋊Dic71—21no—
75C2×C4×Dic71—22no—
76C2×C4×Dic71—21no—
77C2×C4×Dic71—21no—
78C2×C4×Dic71—22no—
79C2×C4×Dic71C2×C414yescomplemented
80C2×C4×Dic71C2312yes—
81#14556061—24no—
82#14556059—24no—
83#14556055C2×C4116yescomplemented
84C2×#41×C71—21no—
85C2×#41×C71—21no—
86C2×#41×C71—22no—
87C2×#41×C71—22no—
88C2×#41×C71C2×C414yes—
89C2×#41×C71D412yes—
90C2×#41×C71Q814yes—
91C2×#41×C71C2312yes—
92C22×Q8×C71D411yescharacteristic
93C22×D4×C71—21no—
94C22×D4×C71D411yescharacteristic
95C23×C4×C71—21no—
96C23×C4×C71—21no—
97C23×C4×C71C2311yescharacteristic
98C2×M16×C71—21no—
99C2×M16×C71—22no—
100C2×M16×C71—22no—
101C2×M16×C71C2312yes—
102C22×C8×C71—21no—
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71C2312yes—
109#103×C71—24no—
110#102×C71—24no—
111C22×Dic71—24no—
112C22×Dic71—24no—
113C22×Dic71—24no—
114C22×Dic71—22no—
115C22×Dic71—21no—
116C22×Dic71—22no—
117C22×Dic71—21no—
118C22×Dic71—28no—
119C22×Dic71—22no—
120C22×Dic71—44no—
121C22×Dic71—84no—
122C22×Dic71C2×D414yes—
123#11765849—24no—
124#11765849—22no—
125#11765849C22×C414yescomplemented
126C2×C71⋊C8—24no—
127C2×C71⋊C8—22no—
128C4⋊Dic71—28no—
129C4⋊Dic71—22no—
130C4⋊Dic71—21no—
131C4⋊Dic71—22no—
132C4⋊Dic71—21no—
133C4⋊Dic71—22no—
134C4⋊Dic71—22no—
135C4⋊Dic71C2×D412yes—
136C4⋊Dic71C2×D412yes—
137Dic71⋊C4—416no—
138C22⋊Dic71—28no—
139C22⋊Dic71—22no—
140C22⋊Dic71—22no—
141C4×Dic71—28no—
142C4×Dic71—24no—
143C4×Dic71—22no—
144C4×Dic71—22no—
145C4×Dic71—22no—
146C4×Dic71C22×C414yescomplemented
147C24×C71—21no—
148C24×C71C2×D411yescharacteristic
149#41×C71—24no—
150#41×C71—24no—
151#41×C71—24no—
152#41×C71—24no—
153#41×C71—24no—
154#41×C71—22no—
155#41×C71—22no—
156#41×C71C4⋊C418yescomplemented
157#41×C71C22×C414yes—
158#41×C71C2×Q814yes—
159C2×Q8×C71—21no—
160C2×Q8×C71—22no—
161C2×Q8×C71C4⋊C414yescomplemented
162C2×Q8×C71C2×D412yes—
163C2×D4×C71—24no—
164C2×D4×C71—22no—
165C2×D4×C71—24no—
166C2×D4×C71—22no—
167C2×D4×C71—21no—
168C2×D4×C71—22no—
169C2×D4×C71C4⋊C414yescomplemented
170C2×D4×C71C2×D412yes—
171C22×C4×C71—21no—
172C22×C4×C71—24no—
173C22×C4×C71—22no—
174C22×C4×C71—24no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—44no—
178C22×C4×C71—41no—
179C22×C4×C71—41no—
180C22×C4×C71C22×C412yes—
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×Q812yes—
187C22×C4×C71#4112yes—
188C22×C4×C71#4112yes—
189C22×C4×C71C2411yescharacteristic
190M16×C71—22no—
191M16×C71—22no—
192M16×C71—42no—
193C2×C8×C71—22no—
194C2×C8×C71—42no—
195C4⋊C4×C71—22no—
196C4⋊C4×C71—22no—
197C4⋊C4×C71—41no—
198C4⋊C4×C71—41no—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71—22no—
202C23×C71—24no—
203C23×C71—42no—
204C23×C71—42no—
205C23×C71C4×D412yes—
206C23×C71#12011yescharacteristic
207C23×C71#12011yescharacteristic
208C23×C71C22⋊Q812yes—
209C23×C71C22×D411yescharacteristic
210Q8×C71—22no—
211Q8×C71—24no—
212Q8×C71C2×C4⋊C414yescomplemented
213D4×C71—24no—
214D4×C71—24no—
215D4×C71—28no—
216D4×C71—22no—
217D4×C71—24no—
218D4×C71C2×C4⋊C414yescomplemented
219C2×C4×C71—22no—
220C2×C4×C71—24no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—28no—
224C2×C4×C71—44no—
225C2×C4×C71—42no—
226C2×C4×C71—82no—
227C2×C4×C71—82no—
228C2×C4×C71C2×C4⋊C412yes—
229C2×C4×C71C2×C4⋊C412yes—
230C2×C4×C71C4×D412yes—
231C2×C4×C71C4×D412yes—
232C2×C4×C71C4×D412yes—
233C2×C4×C71#12011yescharacteristic
234C2×C4×C71#12011yescharacteristic
235C2×C4×C71C22⋊Q812yes—
236C2×C4×C71C22⋊Q812yes—
237C2×C4×C71C22⋊Q812yes—
238C2×C4×C71#12312yes—
239C2×C4×C71#12312yes—
240C2×C4×C71C23×C411yescharacteristic
241C2×C4×C71C22×D411yescharacteristic
242C2×C4×C71C22×D411yescharacteristic
243C2×C4×C71C22×Q811yescharacteristic
244C2×C4×C71C2×#4111yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×Dic71—24no—
247C2×Dic71—216no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—22no—
251C2×Dic71—24no—
252C2×Dic71—22no—
253C2×Dic71—44no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71—44no—
257C2×Dic71—82no—
258C2×Dic71—88no—
259C8×C71—42no—
260C71⋊C8—24no—
261C22×C71—24no—
262C22×C71—24no—
263C22×C71—21no—
264C22×C71—21no—
265C22×C71—21no—
266C22×C71—21no—
267C22×C71—24no—
268C22×C71—48no—
269C22×C71—42no—
270C22×C71—42no—
271C22×C71#38512yes—
272C22×C71#38512yes—
273C22×C71#45312yes—
274C22×C71#45612yes—
275C22×C71C2×C4×D411yescharacteristic
276C22×C71C2×#12011yescharacteristic
277C22×C71C2×C22⋊Q811yescharacteristic
278C4×C71—24no—
279C4×C71—82no—
280C4×C71#38512yes—
281C4×C71#38512yes—
282C4×C71C22×C4⋊C411yescharacteristic
283C4×C71C2×C4×D411yescharacteristic
284C4×C71C2×C22⋊Q811yescharacteristic
285C4×C71C2×#12311yescharacteristic
286Dic71—28no—
287Dic71—44no—
288Dic71—42no—
289Dic71—42no—
290C2×#1859—711nomaximal, Sylow
291C2×C71—21no—
292C2×C71—21no—
293C2×C71—22no—
294C2×C71—44no—
295C2×C71#185914yescomplemented
296C2×C71C2×#38511yescharacteristic
297C2×C71C2×#45311yescharacteristic
298C2×C71C2×#45611yescharacteristic
299C2×#520—711no—
300C2×#428—711no—
301C22×#104—711no—
302C2×#417D7111yescharacteristic
303C2×#384—711no—
304#1859—718no—
305C2×#343—711no—
306C2×#339—711no—
307C71C2×#185911yescharacteristic, complemented, Sylow
308C22×#41C2×D7111yescharacteristic
309C22×M16—711no—
310C22×M16C2×D7111yescharacteristic
311#520—714no—
312#520—1422no—
313#520—1422no—
314#520—1422no—
315C2×C4×Q8—1421no—
316C2×C4×D4—1422no—
317C2×C4×D4—1421no—
318C2×#117—711no—
319C2×#117—1421no—
320C22×C4⋊C4—711no—
321C22×C4⋊C4—1422no—
322C22×C4⋊C4—1421no—
323C22×C4⋊C4C2×D7111yescharacteristic
324C22×C42—711no—
325#428—714no—
326#428—1422no—
327C2×#107—1421no—
328C2×#106—1421no—
329C2×#104—718no—
330C2×#104—714no—
331#417—22no—
332#417C2×D7114yes—
333C2×#103—21no—
334C2×#102—21no—
335#384—714no—
336#384—1422no—
337#343—714no—
338#339—714no—
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41Dic14214yes—
344C2×#41C4×D7114yes—
345C2×#41C22⋊D7112yes—
346C2×#41C22×D7112yes—
347C22×Q8C22⋊D7111yescharacteristic
348C22×D4—21no—
349C22×D4C22⋊D7111yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4—712no—
353C23×C4—1422no—
354C23×C4—1421no—
355C23×C4C22×D7111yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—22no—
359C2×M16—714no—
360C2×M16—712no—
361C2×M16—1421no—
362C2×M16—1422no—
363C2×M16C22×D7112yes—
364C22×C8—21no—
365C22×C8—1421no—
366C4×Q8—1424no—
367C4×Q8—1424no—
368C4×D4—1428no—
369C4×D4—1428no—
370C4×D4—1424no—
371C4×D4—1424no—
372#117—712no—
373#117—712no—
374#117—1424no—
375#117—1424no—
376#117—1421no—
377#117—1421no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—2848no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—711no—
386C2×C4⋊C4—712no—
387C2×C4⋊C4—712no—
388C2×C4⋊C4—1428no—
389C2×C4⋊C4—1424no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4—1424no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4—1421no—
395C2×C4⋊C4—1421no—
396C2×C4⋊C4C22×D7112yes—
397C2×C22⋊C4—1422no—
398C2×C22⋊C4—1421no—
399C2×C42—714no—
400C2×C42—712no—
401C2×C42—1422no—
402C2×C42—1421no—
403C2×C42—1421no—
404C2×C42—1422no—
405#107—1424no—
406#106—1424no—
407#104—7116no—
408#103—24no—
409#102—24no—
410C24—21no—
411C24D4×D7111yescharacteristic
412#41—22no—
413#41—22no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—24no—
418#41—24no—
419#41Dic71⋊C418yescomplemented
420#41C2×C4×D7114yes—
421#41C2×Dic14214yes—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8Dic71⋊C414yescomplemented
425C2×Q8C2×C22⋊D7112yes—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4Dic71⋊C414yescomplemented
433C2×D4C2×C22⋊D7112yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—44no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—2844no—
444C22×C4—5684no—
445C22×C4—714no—
446C22×C4—1421no—
447C22×C4—1421no—
448C22×C4—1422no—
449C22×C4—1428no—
450C22×C4—1422no—
451C22×C4—1424no—
452C22×C4—1424no—
453C22×C4—1422no—
454C22×C4—1424no—
455C22×C4C2×C4×D7112yes—
456C22×C4D4×D7111yescharacteristic
457C22×C4D4×D7111yescharacteristic
458C22×C4D4×D7111yescharacteristic
459C22×C4C2×C22⋊D7111yescharacteristic
460C22×C4C2×C22⋊D7111yescharacteristic
461C22×C4C2×Dic14212yes—
462C22×C4#1176586412yes—
463C22×C4#1176586412yes—
464C22×C4C23×D7111yescharacteristic
465M16—22no—
466M16—22no—
467M16—42no—
468M16—714no—
469M16—1422no—
470M16—1424no—
471C2×C8—22no—
472C2×C8—42no—
473C2×C8—1422no—
474C2×C8—1424no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—28416no—
480C4⋊C4—712no—
481C4⋊C4—712no—
482C4⋊C4—1428no—
483C4⋊C4—1422no—
484C4⋊C4—1422no—
485C4⋊C4—1421no—
486C4⋊C4—1422no—
487C4⋊C4—1422no—
488C4⋊C4—1421no—
489C22⋊C4—1428no—
490C22⋊C4—1422no—
491C22⋊C4—1422no—
492C42—714no—
493C42—1428no—
494C42—1422no—
495C42—1422no—
496C42—1424no—
497C42—1422no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23(C22⋊D71)⋊C412yes—
505C23#1455609611yescharacteristic
506C23#1455609611yescharacteristic
507C23#1455610512yes—
508C23C2×D4×D7111yescharacteristic
509Q8—24no—
510Q8—22no—
511Q8C2×Dic71⋊C414yescomplemented
512D4—28no—
513D4—24no—
514D4—24no—
515D4—24no—
516D4—22no—
517D4C2×Dic71⋊C414yescomplemented
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—2844no—
528C2×C4—2842no—
529C2×C4—2842no—
530C2×C4—2844no—
531C2×C4—5682no—
532C2×C4—5688no—
533C2×C4—1422no—
534C2×C4—1422no—
535C2×C4—1424no—
536C2×C4—1424no—
537C2×C4—1424no—
538C2×C4—14216no—
539C2×C4—1424no—
540C2×C4C2×Dic71⋊C412yes—
541C2×C4C2×Dic71⋊C412yes—
542C2×C4(C22⋊D71)⋊C412yes—
543C2×C4(C22⋊D71)⋊C412yes—
544C2×C4(C22⋊D71)⋊C412yes—
545C2×C4#1455609611yescharacteristic
546C2×C4#1455609611yescharacteristic
547C2×C4#1455610512yes—
548C2×C4#1455610512yes—
549C2×C4#1455610512yes—
550C2×C4#1455610912yes—
551C2×C4#1455610912yes—
552C2×C4C22×C4×D7111yescharacteristic
553C2×C4C2×D4×D7111yescharacteristic
554C2×C4C22×C22⋊D7111yescharacteristic
555C2×C4C22×Dic14211yescharacteristic
556C2×C4C2×#1176586411yescharacteristic
557C2×C4C2×#1176586411yescharacteristic
558C8—42no—
559C8—1424no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22#1798587112yes—
571C22#1798587112yes—
572C22#1798616512yes—
573C22#1798618012yes—
574C22C2×(C22⋊D71)⋊C411yescharacteristic
575C22C2×#1455609611yescharacteristic
576C22C2×#1455610511yescharacteristic
577C4—24no—
578C4—82no—
579C4—2844no—
580C4—2842no—
581C4—2842no—
582C4—1428no—
583C4#1798587112yes—
584C4#1798587112yes—
585C4C22×Dic71⋊C411yescharacteristic
586C4C2×(C22⋊D71)⋊C411yescharacteristic
587C4C2×#1455610511yescharacteristic
588C4C2×#1455610911yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—44no—
593C2#2258856414yesdirect product factor
594C2C2×#1798587111yescharacteristic
595C2C2×#1798616511yescharacteristic
596C2C2×#1798618011yescharacteristic
subgroupG/H
centerC23#14556096
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#417×C71C2
socleC23×C71#120
prime radicalC22×#41×C71C22
cosocleC22×C4×C71C24
hypercenterC2×#417D71
hypocenterC71C2×#1859
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#17986039C2semidirect
4. C2×#17985868C2semidirect
5. #22588564C2direct (N × H)
6. C2×#17985767C2semidirect
7. C2×#17985760C2semidirect
24. #17986039C22semidirect
27. C2×#14556055C4semidirect
30. #17985868C22semidirect
31. #17985767C22semidirect
32. #17985760C22semidirect
45. C2×#11765849C2×C4semidirect
79. C2×C4×Dic71C2×C4semidirect
83. #14556055C2×C4semidirect
125. #11765849C22×C4semidirect
146. C4×Dic71C22×C4semidirect
156. #41×C71C4⋊C4semidirect
161. C2×Q8×C71C4⋊C4semidirect
169. C2×D4×C71C4⋊C4semidirect
212. Q8×C71C2×C4⋊C4semidirect
218. D4×C71C2×C4⋊C4semidirect
295. C2×C71#1859semidirect
307. C71C2×#1859semidirect
419. #41Dic71⋊C4semidirect
424. C2×Q8Dic71⋊C4semidirect
432. C2×D4Dic71⋊C4semidirect
511. Q8C2×Dic71⋊C4semidirect
517. D4C2×Dic71⋊C4semidirect
593. C2#22588564direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179864822
C2×#179860392
C22×#145560552
C2×#179858682
#225885642
C2×#179857672
C2×#179857602
C2×#417×C712
C2×#185971