Finite Group Databasebeta

← #31998676

Subgroups — #31998676

Order 18176.

2258 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986506C211yesmaximal, characteristic
2C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3C2×#17986000C211yesmaximal, characteristic, complemented
4C2×#17985962C211yesmaximal, characteristic, complemented
5#22588654C214yesmaximal, complemented
6#22588644C214yesmaximal, complemented
7#22588060C211yesmaximal, characteristic, complemented
8#22588059C211yesmaximal, characteristic, complemented
9C2×#417×C71C211yesmaximal, characteristic, complemented
10C23×C4×D71—21no—
11C23×C4×D71C2211yescharacteristic
12C22×#11765849C2211yescharacteristic
13#17986506—24no—
14#17986506C418yescomplemented
15C2×#14556109—21no—
16C2×#14556104C2212yes—
17C2×#14556099C2212yes—
18C2×#14556096—21no—
19C2×C4×C22⋊D71—22no—
20C2×#14556087C2211yescharacteristic
21C22×C4⋊Dic71C2211yescharacteristic
22C22×Dic71⋊C4—21no—
23C2×C4⋊C4×D71—24no—
24C2×C4⋊C4×D71—24no—
25C2×C4⋊C4×D71—22no—
26C2×C4⋊C4×D71C2214yes—
27#17986000—24no—
28C2×#14556052C2212yescomplemented
29C2×#14556049C2212yescomplemented
30#17985962—24no—
31C2×(C2×D71)⋊C8C2214yes—
32#17985762C2214yescomplemented
33#17985761C2214yescomplemented
34C22×#41×C71C2211yescharacteristic
35C22×M16×C71C2211yescharacteristic
36C22×C4⋊C4×C71C2211yescharacteristic
37#417×C71—24no—
38C2×#103×C71C2212yescomplemented
39C2×#102×C71C2212yescomplemented
40C24×D71D411yescharacteristic
41C22×C22⋊D71—22no—
42C23×Dic71—21no—
43C23×Dic71D411yescharacteristic
44C22×C4×D71—28no—
45C22×C4×D71—24no—
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71—22no—
50C22×C4×D71—48no—
51C22×C4×D71C2×C414yes—
52C22×C4×D71D412yes—
53C22×C4×D71D414yes—
54C22×C4×D71C2312yes—
55C2×#11765849—22no—
56C2×#11765849—22no—
57C2×#11765849—22no—
58C22×C71⋊C8C2312yes—
59#14556109—28no—
60#14556104—24no—
61#14556104C2×C418yescomplemented
62#14556099—24no—
63#14556099C2×C418yescomplemented
64#14556096—28no—
65C4×C22⋊D71—216no—
66#14556087—22no—
67#14556087C2×C414yescomplemented
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71—22no—
70C2×C4⋊Dic71D414yescomplemented
71C2×C4⋊Dic71C2312yes—
72C2×C4⋊Dic71C2312yes—
73C2×Dic71⋊C4—28no—
74C2×Dic71⋊C4—24no—
75C2×Dic71⋊C4—22no—
76C4⋊C4×D71—216no—
77C4⋊C4×D71—28no—
78C4⋊C4×D71—28no—
79C2×C22⋊Dic71—22no—
80C2×C22⋊Dic71—21no—
81C2×(C2×D71)⋊C4—22no—
82C2×C4×Dic71—21no—
83#14556052—24no—
84#14556049—24no—
85(C2×D71)⋊C8—28no—
86C2×#41×C71—22no—
87C2×#41×C71—22no—
88C2×#41×C71—22no—
89C2×#41×C71—22no—
90C2×#41×C71—22no—
91C2×#41×C71C2×C414yescomplemented
92C22×Q8×C71C2311yescharacteristic
93C22×D4×C71—21no—
94C22×D4×C71C2311yescharacteristic
95C23×C4×C71—21no—
96C23×C4×C71—21no—
97C23×C4×C71C2311yescharacteristic
98C2×M16×C71—22no—
99C2×M16×C71—22no—
100C2×M16×C71—22no—
101C22×C8×C71C2312yes—
102C2×C4⋊C4×C71—21no—
103C2×C4⋊C4×C71—22no—
104C2×C4⋊C4×C71D414yescomplemented
105C2×C4⋊C4×C71C2312yes—
106#103×C71—24no—
107#102×C71—24no—
108C23×D71—22no—
109C23×D71—22no—
110C23×D71—24no—
111C23×D71—24no—
112C23×D71C22⋊C414yes—
113C23×D71C2×D412yes—
114C2×C22⋊D71—28no—
115C2×C22⋊D71—44no—
116C2×C22⋊D71—44no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—24no—
120C22×Dic71—22no—
121C22×Dic71—22no—
122C22×Dic71—44no—
123C22×Dic71C22⋊C414yes—
124C22×Dic71C2×D412yes—
125C2×C4×D71—28no—
126C2×C4×D71—28no—
127C2×C4×D71—24no—
128C2×C4×D71—28no—
129C2×C4×D71—28no—
130C2×C4×D71—24no—
131C2×C4×D71—22no—
132C2×C4×D71—22no—
133C2×C4×D71—22no—
134C2×C4×D71—22no—
135C2×C4×D71—216no—
136C2×C4×D71—432no—
137C2×C4×D71—48no—
138C2×C4×D71—48no—
139C2×C4×D71C22⋊C418yes—
140C2×C4×D71C22×C414yes—
141C2×C4×D71C2×D414yes—
142#11765849—44no—
143C2×C71⋊C8—22no—
144C2×C71⋊C8—22no—
145C2×C71⋊C8—22no—
146C4⋊Dic71—24no—
147C4⋊Dic71—22no—
148C4⋊Dic71—22no—
149C4⋊Dic71C22×C414yescomplemented
150C4⋊Dic71C2×D414yes—
151Dic71⋊C4—216no—
152Dic71⋊C4—28no—
153C22⋊Dic71—24no—
154C22⋊Dic71—28no—
155(C2×D71)⋊C4—28no—
156C4×Dic71—24no—
157C24×C71—21no—
158C24×C71C2×D411yescharacteristic
159#41×C71—28no—
160#41×C71—28no—
161#41×C71—44no—
162#41×C71—44no—
163C2×Q8×C71—21no—
164C2×Q8×C71—21no—
165C2×Q8×C71—21no—
166C2×Q8×C71—21no—
167C2×Q8×C71—21no—
168C2×Q8×C71C22×C412yescomplemented
169C2×D4×C71—24no—
170C2×D4×C71—21no—
171C2×D4×C71—21no—
172C2×D4×C71—21no—
173C2×D4×C71—21no—
174C2×D4×C71—21no—
175C2×D4×C71—42no—
176C2×D4×C71—42no—
177C2×D4×C71C22×C412yescomplemented
178C22×C4×C71—24no—
179C22×C4×C71—22no—
180C22×C4×C71—21no—
181C22×C4×C71—24no—
182C22×C4×C71—44no—
183C22×C4×C71—42no—
184C22×C4×C71—42no—
185C22×C4×C71C22×C412yes—
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D412yes—
188C22×C4×C71C2×D412yes—
189C22×C4×C71C2×D412yes—
190C22×C4×C71#4112yes—
191C22×C4×C71#4112yes—
192C22×C4×C71C2411yescharacteristic
193M16×C71—44no—
194C2×C8×C71—22no—
195C2×C8×C71—22no—
196C2×C8×C71—22no—
197C4⋊C4×C71—24no—
198C4⋊C4×C71—22no—
199C4⋊C4×C71C2×D414yes—
200C23×C71—22no—
201C23×C71—22no—
202C23×C71—22no—
203C23×C71—41no—
204C23×C71—41no—
205C23×C71—42no—
206C23×C71—42no—
207C23×C71C2×C22⋊C412yes—
208C23×C71#12012yes—
209C23×C71#12312yes—
210C23×C71C22×D411yescharacteristic
211C22×D71—28no—
212C22×D71—216no—
213C22×D71—28no—
214C22×D71—22no—
215C22×D71—22no—
216C22×D71—22no—
217C22×D71—22no—
218C22×D71—28no—
219C22×D71—216no—
220C22×D71C2×C22⋊C414yes—
221C22×D71AGL(1,ℤ/8)14yes—
222C22×D71#13714yes—
223Q8×C71—22no—
224Q8×C71—22no—
225Q8×C71—41no—
226Q8×C71—41no—
227D4×C71—22no—
228D4×C71—22no—
229D4×C71—48no—
230D4×C71—41no—
231D4×C71—41no—
232C2×C4×C71—22no—
233C2×C4×C71—24no—
234C2×C4×C71—48no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—44no—
238C2×C4×C71—48no—
239C2×C4×C71C2×C22⋊C412yes—
240C2×C4×C71C4×D414yes—
241C2×C4×C71C4×D414yes—
242C2×C4×C71#12012yes—
243C2×C4×C71#12114yes—
244C2×C4×C71#12114yes—
245C2×C4×C71#12312yes—
246C2×C4×C71C23×C411yescharacteristic
247C2×C4×C71C22×D411yescharacteristic
248C2×C4×C71C22×D411yescharacteristic
249C2×C4×C71C22×D411yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×C4×C71C2×#4111yescharacteristic
252C22⋊D71—416no—
253C2×Dic71—22no—
254C2×Dic71—22no—
255C2×Dic71—28no—
256C2×Dic71—24no—
257C2×Dic71—24no—
258C2×Dic71—48no—
259C2×Dic71—44no—
260C2×Dic71—42no—
261C2×Dic71—42no—
262C2×Dic71C2×C22⋊C414yes—
263C4×D71—24no—
264C4×D71—24no—
265C4×D71—24no—
266C4×D71—24no—
267C4×D71—216no—
268C4×D71—432no—
269C8×C71—42no—
270C71⋊C8—42no—
271C22×C71—24no—
272C22×C71—24no—
273C22×C71—22no—
274C22×C71—22no—
275C22×C71—44no—
276C22×C71—42no—
277C22×C71—82no—
278C22×C71—81no—
279C22×C71—81no—
280C22×C71#38614yes—
281C22×C71C22×C22⋊C411yescharacteristic
282C22×C71C2×#12011yescharacteristic
283C22×C71C2×#12311yescharacteristic
284C22×C71C2×AGL(1,ℤ/8)12yes—
285C22×C71C2×#13712yes—
286C2×D71—232no—
287C2×D71—24no—
288C2×D71—24no—
289C2×D71—24no—
290C2×D71—24no—
291C2×D71—24no—
292C2×D71—24no—
293C2×D71#41718yescomplemented
294C4×C71—44no—
295C4×C71—42no—
296C4×C71#38614yes—
297C4×C71C2×C4×D411yescharacteristic
298C4×C71C2×C4×D411yescharacteristic
299C4×C71C2×#12111yescharacteristic
300C4×C71C2×#12111yescharacteristic
301Dic71—24no—
302Dic71—44no—
303Dic71—44no—
304#20909—711nomaximal, Sylow
305C2×C71—24no—
306C2×C71—82no—
307C2×C71C2×#38611yescharacteristic
308C2×C71C2×#41712yes—
309C2×C71#300311yescharacteristic
310C2×C71#300411yescharacteristic
311C2×C71#309212yes—
312D71—28no—
313C2×#525—711no—
314C23×C4⋊C4—711no—
315C2×#417—711no—
316C2×#417D7111yescharacteristic, complemented
317C2×#407—711no—
318#1874—714no—
319#1872—714no—
320#1736—711no—
321#1736—711no—
322C71#2090911yescharacteristic, complemented, Sylow
323C22×#41C2×D7111yescharacteristic
324C24×C4—711no—
325C24×C4—1421no—
326C22×M16—711no—
327C22×M16C2×D7111yescharacteristic
328#525—718no—
329#525—1424no—
330C2×#123—1421no—
331C2×C22⋊Q8—712no—
332C2×#121—712no—
333C2×#120—1421no—
334C2×C4×D4—1422no—
335C2×#117—711no—
336C22×C4⋊C4—711no—
337C22×C4⋊C4—714no—
338C22×C4⋊C4—1424no—
339C22×C4⋊C4—1422no—
340C22×C4⋊C4—1421no—
341C22×C4⋊C4—1424no—
342C22×C4⋊C4C2×D7111yescharacteristic
343#417—24no—
344#417—1424no—
345C2×#103—712no—
346C2×#103C2×D7112yescomplemented
347C2×#102—712no—
348C2×#102C2×D7112yescomplemented
349#407—1424no—
350C2×C22⋊C8—714no—
351#340—714no—
352#340—714no—
353C25—711no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41C4×D7114yescomplemented
360C22×Q8C22×D7111yescharacteristic
361C22×D4—21no—
362C22×D4—1422no—
363C22×D4C22×D7111yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—2848no—
367C23×C4—711no—
368C23×C4—714no—
369C23×C4—712no—
370C23×C4—714no—
371C23×C4—712no—
372C23×C4—1424no—
373C23×C4—1428no—
374C23×C4—1421no—
375C23×C4—1422no—
376C23×C4—1422no—
377C23×C4—1422no—
378C23×C4—1422no—
379C23×C4C22×D7111yescharacteristic
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—1422no—
384C2×M16—1422no—
385C2×M16—1422no—
386C22×C8—712no—
387C22×C8C22×D7112yes—
388#123—1428no—
389C22⋊Q8—718no—
390C22⋊Q8—1424no—
391#121—718no—
392#121—1424no—
393#120—1428no—
394C4×D4—14216no—
395#117—714no—
396#117—1422no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—714no—
400C2×C4⋊C4—712no—
401C2×C4⋊C4—712no—
402C2×C4⋊C4—1428no—
403C2×C4⋊C4—14216no—
404C2×C4⋊C4—1424no—
405C2×C4⋊C4—1422no—
406C2×C4⋊C4—1428no—
407C2×C4⋊C4—1421no—
408C2×C4⋊C4—1422no—
409C2×C4⋊C4—1428no—
410C2×C4⋊C4C22⋊D7114yescomplemented
411C2×C4⋊C4C22×D7112yes—
412C2×C22⋊C4—1422no—
413C2×C22⋊C4—1422no—
414C2×C22⋊C4—1421no—
415C2×C42—1421no—
416#103—24no—
417#103—1424no—
418#102—24no—
419#102—1424no—
420C22⋊C8—1428no—
421C24—21no—
422C24—712no—
423C24—714no—
424C24—1424no—
425C24—1424no—
426C24—1422no—
427C24—1422no—
428C24D4×D7111yescharacteristic
429#41—28no—
430#41—28no—
431#41—44no—
432#41—44no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D7112yescomplemented
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—2844no—
448C2×D4—2844no—
449C2×D4—1428no—
450C2×D4C2×C4×D7112yescomplemented
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—2848no—
459C22×C4—28432no—
460C22×C4—2848no—
461C22×C4—2844no—
462C22×C4—712no—
463C22×C4—714no—
464C22×C4—714no—
465C22×C4—714no—
466C22×C4—718no—
467C22×C4—14216no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1424no—
471C22×C4—1422no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1422no—
476C22×C4—1422no—
477C22×C4—1428no—
478C22×C4—1428no—
479C22×C4—1424no—
480C22×C4—1428no—
481C22×C4—1428no—
482C22×C4—1424no—
483C22×C4C2×C4×D7112yes—
484C22×C4D4×D7112yes—
485C22×C4D4×D7112yes—
486C22×C4D4×D7111yescharacteristic
487C22×C4C2×C22⋊D7112yes—
488C22×C4#1176586412yes—
489C22×C4#1176586512yes—
490C22×C4C23×D7111yescharacteristic
491M16—44no—
492M16—2844no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1422no—
497C2×C8—1422no—
498C2×C8—1422no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—714no—
502C4⋊C4—714no—
503C4⋊C4—1428no—
504C4⋊C4—14216no—
505C4⋊C4—1422no—
506C4⋊C4—1422no—
507C4⋊C4—1424no—
508C4⋊C4C2×C22⋊D7114yes—
509C22⋊C4—1428no—
510C22⋊C4—1428no—
511C22⋊C4—1424no—
512C42—1424no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—714no—
521C23—714no—
522C23—714no—
523C23—14216no—
524C23—1428no—
525C23—14216no—
526C23—1428no—
527C23—1428no—
528C23—1422no—
529C23—1422no—
530C23—1422no—
531C23—1422no—
532C23C22⋊C4×D7112yes—
533C23#1455609612yes—
534C23#1455611012yes—
535C23C2×D4×D7111yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545D4—28416no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—28432no—
554C2×C4—2848no—
555C2×C4—2844no—
556C2×C4—2842no—
557C2×C4—2842no—
558C2×C4—714no—
559C2×C4—1424no—
560C2×C4—1424no—
561C2×C4—1428no—
562C2×C4—1422no—
563C2×C4—1422no—
564C2×C4—14216no—
565C2×C4—1424no—
566C2×C4—1424no—
567C2×C4—1424no—
568C2×C4—1424no—
569C2×C4C22⋊C4×D7112yes—
570C2×C4(C22⋊D71)⋊C414yes—
571C2×C4C4×C22⋊D7114yes—
572C2×C4#1455609612yes—
573C2×C4#1455609914yes—
574C2×C4#1455610114yes—
575C2×C4#1455611012yes—
576C2×C4C22×C4×D7111yescharacteristic
577C2×C4C2×D4×D7111yescharacteristic
578C2×C4C2×D4×D7111yescharacteristic
579C2×C4C22×C22⋊D7111yescharacteristic
580C2×C4C2×#1176586411yescharacteristic
581C2×C4C2×#1176586511yescharacteristic
582C8—42no—
583C8—2842no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—24no—
588C22—42no—
589C22—44no—
590C22—82no—
591C22—81no—
592C22—81no—
593C22—718no—
594C22—14232no—
595C22—1424no—
596C22—1424no—
597C22—1424no—
598C22—1424no—
599C22—1424no—
600C22—1424no—
601C22#1798587514yes—
602C22C2×C22⋊C4×D7111yescharacteristic
603C22C2×#1455609611yescharacteristic
604C22C2×#1455611011yescharacteristic
605C22AGL(1,ℤ/8)×D7112yes—
606C22#137×D7112yes—
607C4—42no—
608C4—44no—
609C4—2844no—
610C4—2844no—
611C4—1424no—
612C4#1798587514yes—
613C4C2×(C22⋊D71)⋊C411yescharacteristic
614C4C2×C4×C22⋊D7111yescharacteristic
615C4C2×#1455609911yescharacteristic
616C4C2×#1455610111yescharacteristic
617C2—24no—
618C2—82no—
619C2—1428no—
620C2C2×#1798587511yescharacteristic
621C2#417×D7112yes—
622C2#2259756511yescharacteristic
623C2#2259757211yescharacteristic
624C2#2259826112yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#417×C71C2
socleC23×C71C22×D4
prime radicalC2×#17986506C2
cosocleC22×C4×C71C24
hypercenterC2×#417D71
hypocenterC71#20909
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. C22×C4⋊C4×D71C2semidirect
3. C2×#17986000C2semidirect
4. C2×#17985962C2semidirect
5. #22588654C2semidirect
6. #22588644C2semidirect
7. #22588060C2semidirect
8. #22588059C2semidirect
9. C2×#417×C71C2semidirect
14. #17986506C4semidirect
28. C2×#14556052C22semidirect
29. C2×#14556049C22semidirect
32. #17985762C22semidirect
33. #17985761C22semidirect
38. C2×#103×C71C22semidirect
39. C2×#102×C71C22semidirect
61. #14556104C2×C4semidirect
63. #14556099C2×C4semidirect
67. #14556087C2×C4semidirect
70. C2×C4⋊Dic71D4semidirect
91. C2×#41×C71C2×C4semidirect
104. C2×C4⋊C4×C71D4semidirect
149. C4⋊Dic71C22×C4semidirect
168. C2×Q8×C71C22×C4semidirect
177. C2×D4×C71C22×C4semidirect
293. C2×D71#417semidirect
316. C2×#417D71semidirect
322. C71#20909semidirect
346. C2×#103C2×D71semidirect
348. C2×#102C2×D71semidirect
359. C2×#41C4×D71semidirect
410. C2×C4⋊C4C22⋊D71semidirect
438. C2×Q8C2×C4×D71semidirect
450. C2×D4C2×C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179865062
C22×C4⋊C4×D712
C2×#179860002
C2×#179859622
#225886542
#225886442
#225880602
#225880592
C2×#417×C712
#2090971