Finite Group Databasebeta

← C2×#22588656 #31998651

Subgroups — C2×#22588656 #31998651

Order 18176.

2614 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556103C211yesmaximal, characteristic
2C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3C22×#14556050C211yesmaximal, characteristic, complemented
4C22×#14556048C211yesmaximal, characteristic, complemented
5C22×(C2×D71)⋊C8C211yesmaximal, characteristic, complemented
6#22588656C218yesmaximal, direct product factor
7C2×#17985761C211yesmaximal, characteristic, complemented
8C2×#340×C71C211yesmaximal, characteristic, complemented
9C23×D284—21no—
10C23×D284C2211yescharacteristic
11C23×C4×D71—21no—
12C23×C4×D71C2211yescharacteristic
13C23×C71⋊C8C2211yescharacteristic
14C2×#14556103—22no—
15C2×#14556103—24no—
16C2×#14556103C418yescomplemented
17C2×#14556103C2214yes—
18C22×C4⋊Dic71C2211yescharacteristic
19C22×Dic71⋊C4—21no—
20C2×C4⋊C4×D71—24no—
21C2×C4⋊C4×D71—24no—
22C2×C4⋊C4×D71—22no—
23C2×C4⋊C4×D71C2214yescomplemented
24C22×(C2×D71)⋊C4—21no—
25C2×#14556050—24no—
26C2×#14556050C2214yescomplemented
27C2×#14556048—24no—
28C2×#14556048C2214yescomplemented
29C2×(C2×D71)⋊C8—24no—
30C2×(C2×D71)⋊C8C2214yescomplemented
31#17985761C2214yescomplemented
32C23×C8×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C22×C4⋊C4×C71C2211yescharacteristic
35#340×C71C2214yescomplemented
36C24×D71—21no—
37C24×D71D411yescharacteristic
38C22×D284—28no—
39C22×D284—24no—
40C22×D284—22no—
41C22×D284—22no—
42C22×D284—21no—
43C22×D284—22no—
44C22×D284—22no—
45C22×D284—22no—
46C22×D284—44no—
47C22×D284—44no—
48C22×D284C2×C414yescomplemented
49C22×D284C2312yes—
50C23×Dic71—21no—
51C23×Dic71D411yescharacteristic
52C22×C4×D71—28no—
53C22×C4×D71—24no—
54C22×C4×D71—22no—
55C22×C4×D71—22no—
56C22×C4×D71—22no—
57C22×C4×D71—22no—
58C22×C4×D71—44no—
59C22×C4×D71—44no—
60C22×C4×D71C2×C414yes—
61C22×C4×D71D412yes—
62C22×C4×D71D414yes—
63C22×C4×D71C2312yes—
64C22×C71⋊C8—22no—
65C22×C71⋊C8—22no—
66C22×C71⋊C8—22no—
67C22×C71⋊C8C2312yes—
68#14556103—216no—
69#14556103—28no—
70#14556103C2×C4116yescomplemented
71C2×C4⋊Dic71—21no—
72C2×C4⋊Dic71—22no—
73C2×C4⋊Dic71D414yescomplemented
74C2×C4⋊Dic71C2312yes—
75C2×Dic71⋊C4—28no—
76C2×Dic71⋊C4—24no—
77C4⋊C4×D71—216no—
78C4⋊C4×D71—28no—
79C4⋊C4×D71—28no—
80C2×(C2×D71)⋊C4—28no—
81C2×(C2×D71)⋊C4—24no—
82#14556050—28no—
83#14556048—28no—
84(C2×D71)⋊C8—28no—
85C23×C4×C71—21no—
86C23×C4×C71—21no—
87C23×C4×C71C2311yescharacteristic
88C22×C8×C71—22no—
89C22×C8×C71—22no—
90C22×C8×C71—22no—
91C22×C8×C71C2312yes—
92C2×C4⋊C4×C71—21no—
93C2×C4⋊C4×C71—22no—
94C2×C4⋊C4×C71—21no—
95C2×C4⋊C4×C71—22no—
96C2×C4⋊C4×C71C2×C414yescomplemented
97C2×C4⋊C4×C71D414yescomplemented
98C2×C4⋊C4×C71C2312yes—
99C2×C4⋊C4×C71C2312yes—
100C23×D71—22no—
101C23×D71—22no—
102C23×D71—22no—
103C23×D71—22no—
104C23×D71—24no—
105C23×D71—22no—
106C23×D71—42no—
107C23×D71—42no—
108C23×D71—42no—
109C23×D71—42no—
110C23×D71—42no—
111C23×D71—42no—
112C23×D71C22⋊C414yes—
113C23×D71D814yes—
114C23×D71SD1614yes—
115C23×D71C2×D412yes—
116C2×D284—28no—
117C2×D284—28no—
118C2×D284—22no—
119C2×D284—22no—
120C2×D284—22no—
121C2×D284—22no—
122C2×D284—216no—
123C2×D284—24no—
124C2×D284—22no—
125C2×D284—432no—
126C2×D284—48no—
127C2×D284—48no—
128C2×D284—42no—
129C2×D284—44no—
130C2×D284—44no—
131C2×D284—42no—
132C2×D284C22×C414yescomplemented
133C22×Dic71—22no—
134C22×Dic71—22no—
135C22×Dic71—24no—
136C22×Dic71—22no—
137C22×Dic71—42no—
138C22×Dic71—42no—
139C22×Dic71C22⋊C414yes—
140C22×Dic71C2×D412yes—
141C2×C4×D71—28no—
142C2×C4×D71—28no—
143C2×C4×D71—24no—
144C2×C4×D71—28no—
145C2×C4×D71—28no—
146C2×C4×D71—24no—
147C2×C4×D71—22no—
148C2×C4×D71—22no—
149C2×C4×D71—22no—
150C2×C4×D71—22no—
151C2×C4×D71—216no—
152C2×C4×D71—48no—
153C2×C4×D71—432no—
154C2×C4×D71—48no—
155C2×C4×D71C22⋊C418yes—
156C2×C4×D71C22×C414yes—
157C2×C4×D71C2×D414yes—
158C2×C71⋊C8—22no—
159C2×C71⋊C8—22no—
160C2×C71⋊C8—22no—
161C2×C71⋊C8—44no—
162C4⋊Dic71—24no—
163C4⋊Dic71—22no—
164C4⋊Dic71C2×D414yescomplemented
165Dic71⋊C4—216no—
166(C2×D71)⋊C4—216no—
167C24×C71C2×D411yescharacteristic
168C22×C4×C71—24no—
169C22×C4×C71—24no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—42no—
173C22×C4×C71—42no—
174C22×C4×C71—42no—
175C22×C4×C71—42no—
176C22×C4×C71C22×C412yes—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D412yes—
179C22×C4×C71C2×D412yes—
180C22×C4×C71C2×D412yes—
181C22×C4×C71#4112yes—
182C22×C4×C71#4112yes—
183C22×C4×C71C2411yescharacteristic
184C2×C8×C71—22no—
185C2×C8×C71—22no—
186C2×C8×C71—22no—
187C2×C8×C71—44no—
188C4⋊C4×C71—24no—
189C4⋊C4×C71—24no—
190C4⋊C4×C71—22no—
191C4⋊C4×C71—22no—
192C4⋊C4×C71C22×C414yescomplemented
193C4⋊C4×C71C2×D414yescomplemented
194C23×C71C2×C22⋊C412yes—
195C23×C71#12012yes—
196C23×C71#12312yes—
197C23×C71C2×D812yes—
198C23×C71C2×SD1612yes—
199C23×C71AGL(1,ℤ/8)12yes—
200C23×C71#13712yes—
201C23×C71C22×D411yescharacteristic
202C22×D71—22no—
203C22×D71—22no—
204C22×D71—22no—
205C22×D71—22no—
206C22×D71—28no—
207C22×D71—28no—
208C22×D71—28no—
209C22×D71—28no—
210C22×D71—28no—
211C22×D71—28no—
212C22×D71—24no—
213C22×D71—42no—
214C22×D71—42no—
215C22×D71—42no—
216C22×D71—42no—
217C22×D71—42no—
218C22×D71—42no—
219C22×D71—48no—
220C22×D71—48no—
221C22×D71—48no—
222C22×D71—84no—
223C22×D71—84no—
224C22×D71—84no—
225C22×D71—84no—
226C22×D71#102116yescomplemented
227C22×D71C2×C22⋊C414yes—
228C22×D71C2×D814yes—
229C22×D71C2×SD1614yes—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—42no—
233C2×C4×C71—42no—
234C2×C4×C71—42no—
235C2×C4×C71—42no—
236C2×C4×C71—48no—
237C2×C4×C71—48no—
238C2×C4×C71C2×C22⋊C412yes—
239C2×C4×C71C4×D414yes—
240C2×C4×C71C4×D414yes—
241C2×C4×C71#12012yes—
242C2×C4×C71#12114yes—
243C2×C4×C71#12114yes—
244C2×C4×C71#12312yes—
245C2×C4×C71C23×C411yescharacteristic
246C2×C4×C71C22×D411yescharacteristic
247C2×C4×C71C22×D411yescharacteristic
248C2×C4×C71C22×D411yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—28no—
254C2×Dic71—24no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—48no—
258C2×Dic71C2×C22⋊C414yes—
259D284—24no—
260D284—24no—
261D284—432no—
262D284—42no—
263D284—42no—
264D284—48no—
265C4×D71—24no—
266C4×D71—24no—
267C4×D71—24no—
268C4×D71—24no—
269C4×D71—216no—
270C4×D71—432no—
271C8×C71—42no—
272C71⋊C8—42no—
273C22×C71#38614yes—
274C22×C71C2×#10214yes—
275C22×C71#41714yes—
276C22×C71#44714yes—
277C22×C71#44814yes—
278C22×C71#48014yes—
279C22×C71#48114yes—
280C22×C71C22×C22⋊C411yescharacteristic
281C22×C71C2×#12011yescharacteristic
282C22×C71C2×#12311yescharacteristic
283C22×C71C22×D811yescharacteristic
284C22×C71C22×SD1611yescharacteristic
285C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C71C2×#13711yescharacteristic
287C2×D71—24no—
288C2×D71—24no—
289C2×D71—24no—
290C2×D71—232no—
291C2×D71—24no—
292C2×D71—24no—
293C2×D71—24no—
294C2×D71—44no—
295C2×D71—44no—
296C2×D71—44no—
297C2×D71—82no—
298C2×D71—82no—
299C2×D71—816no—
300C2×D71—82no—
301C2×D71—82no—
302C2×D71C2×#10218yescomplemented
303C4×C71—44no—
304C4×C71—44no—
305C4×C71#38614yes—
306C4×C71C2×C4×D411yescharacteristic
307C4×C71C2×C4×D411yescharacteristic
308C4×C71C2×#12111yescharacteristic
309C4×C71C2×#12111yescharacteristic
310Dic71—24no—
311Dic71—44no—
312C2×#1874—711nomaximal, Sylow
313C2×C71#187418yescomplemented
314C2×C71C2×#38611yescharacteristic
315C2×C71C22×#10211yescharacteristic
316C2×C71C2×#41711yescharacteristic
317C2×C71C2×#44711yescharacteristic
318C2×C71C2×#44811yescharacteristic
319C2×C71C2×#48011yescharacteristic
320C2×C71C2×#48111yescharacteristic
321D71—28no—
322D71—84no—
323C22×#121—711no—
324C23×C4⋊C4—711no—
325C22×#102—711no—
326C22×#102—711no—
327C22×C22⋊C8—711no—
328#1874—718no—
329C2×#340—711no—
330C2×#340D7111yescharacteristic, complemented
331C71C2×#187411yescharacteristic, complemented, Sylow
332C23×D4—711no—
333C23×D4—1421no—
334C24×C4—711no—
335C24×C4—1421no—
336C23×C8—711no—
337C23×C8C2×D7111yescharacteristic
338C2×#121—718no—
339C2×#121—714no—
340C2×#121—1422no—
341C2×#121—1424no—
342C22×C4⋊C4—711no—
343C22×C4⋊C4—714no—
344C22×C4⋊C4—1424no—
345C22×C4⋊C4—1422no—
346C22×C4⋊C4—1421no—
347C22×C4⋊C4—1424no—
348C22×C4⋊C4C2×D7111yescharacteristic
349C22×C4⋊C4C2×D7111yescharacteristic
350C22×C22⋊C4—1421no—
351C2×#102—714no—
352C2×#102—714no—
353C2×#102—1424no—
354C2×#102—1424no—
355C2×C22⋊C8—714no—
356C2×C22⋊C8—1424no—
357#340—714no—
358#340C2×D7114yescomplemented
359C25—711no—
360C25—1421no—
361C22×D4—2844no—
362C22×D4—2844no—
363C22×D4—714no—
364C22×D4—712no—
365C22×D4—1424no—
366C22×D4—1428no—
367C22×D4—1422no—
368C22×D4—1421no—
369C22×D4—1422no—
370C22×D4—1422no—
371C22×D4—1422no—
372C22×D4—1422no—
373C23×C4—21no—
374C23×C4—21no—
375C23×C4—2844no—
376C23×C4—2844no—
377C23×C4—711no—
378C23×C4—714no—
379C23×C4—712no—
380C23×C4—714no—
381C23×C4—712no—
382C23×C4—1424no—
383C23×C4—1428no—
384C23×C4—1421no—
385C23×C4—1422no—
386C23×C4—1422no—
387C23×C4—1422no—
388C23×C4—1422no—
389C23×C4C22×D7111yescharacteristic
390C22×C8—22no—
391C22×C8—22no—
392C22×C8—22no—
393C22×C8—712no—
394C22×C8—1422no—
395C22×C8—1422no—
396C22×C8—1422no—
397C22×C8C22×D7112yes—
398#121—7116no—
399#121—1428no—
400#121—14216no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—712no—
407C2×C4⋊C4—1428no—
408C2×C4⋊C4—14216no—
409C2×C4⋊C4—1424no—
410C2×C4⋊C4—1428no—
411C2×C4⋊C4—1421no—
412C2×C4⋊C4—1422no—
413C2×C4⋊C4—1428no—
414C2×C4⋊C4C4×D7114yescomplemented
415C2×C4⋊C4D28414yescomplemented
416C2×C4⋊C4C22×D7112yes—
417C2×C4⋊C4C22×D7112yes—
418C2×C22⋊C4—1424no—
419C2×C22⋊C4—1428no—
420#102—1428no—
421#102—1428no—
422C22⋊C8—1428no—
423C24—2842no—
424C24—2842no—
425C24—2842no—
426C24—2842no—
427C24—2842no—
428C24—2842no—
429C24—712no—
430C24—714no—
431C24—714no—
432C24—714no—
433C24—1422no—
434C24—1424no—
435C24—1422no—
436C24—1422no—
437C24—1422no—
438C24—1422no—
439C24D4×D7111yescharacteristic
440C2×D4—28432no—
441C2×D4—2848no—
442C2×D4—2848no—
443C2×D4—2842no—
444C2×D4—2844no—
445C2×D4—2844no—
446C2×D4—2842no—
447C2×D4—714no—
448C2×D4—14216no—
449C2×D4—1422no—
450C2×D4—1422no—
451C2×D4—1422no—
452C2×D4—1422no—
453C2×D4—1422no—
454C2×D4—1424no—
455C2×D4—1428no—
456C2×D4—1428no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—2848no—
466C22×C4—28432no—
467C22×C4—2848no—
468C22×C4—2842no—
469C22×C4—2842no—
470C22×C4—712no—
471C22×C4—714no—
472C22×C4—714no—
473C22×C4—714no—
474C22×C4—718no—
475C22×C4—14216no—
476C22×C4—1422no—
477C22×C4—1424no—
478C22×C4—1422no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1428no—
485C22×C4—1428no—
486C22×C4—1424no—
487C22×C4—1428no—
488C22×C4—1428no—
489C22×C4—1424no—
490C22×C4C2×C4×D7112yes—
491C22×C4D4×D7112yes—
492C22×C4D4×D7112yes—
493C22×C4D4×D7111yescharacteristic
494C22×C4C2×D28412yes—
495C22×C4#1176586512yes—
496C22×C4#1176586612yes—
497C22×C4C23×D7111yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8—2844no—
503C2×C8—1422no—
504C2×C8—1422no—
505C2×C8—1422no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—714no—
511C4⋊C4—14216no—
512C4⋊C4—1422no—
513C4⋊C4—1424no—
514C4⋊C4C2×C4×D7114yescomplemented
515C4⋊C4C2×D28414yescomplemented
516C22⋊C4—14216no—
517C23—2848no—
518C23—2842no—
519C23—2848no—
520C23—2842no—
521C23—2848no—
522C23—2842no—
523C23—2842no—
524C23—2842no—
525C23—2842no—
526C23—5684no—
527C23—5684no—
528C23—5684no—
529C23—5684no—
530C23—714no—
531C23—7116no—
532C23—714no—
533C23—714no—
534C23—1424no—
535C23—1428no—
536C23—1428no—
537C23—1428no—
538C23—1428no—
539C23—1428no—
540C23—1422no—
541C23—1422no—
542C23—1428no—
543C23—1422no—
544C23—1422no—
545C23C22⋊C4×D7112yes—
546C23#1455609812yes—
547C23#1455611312yes—
548C23D8×D7112yes—
549C23SD16×D7112yes—
550C23#1455615512yes—
551C23#1455616012yes—
552C23C2×D4×D7111yescharacteristic
553D4—28432no—
554D4—2848no—
555D4—2842no—
556D4—2842no—
557D4—1424no—
558D4—1424no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—28432no—
568C2×C4—2848no—
569C2×C4—2842no—
570C2×C4—2842no—
571C2×C4—714no—
572C2×C4—1424no—
573C2×C4—1428no—
574C2×C4—1422no—
575C2×C4—1422no—
576C2×C4—14216no—
577C2×C4—1424no—
578C2×C4—1424no—
579C2×C4—1424no—
580C2×C4—1424no—
581C2×C4C22⋊C4×D7112yes—
582C2×C4#1455609014yes—
583C2×C4C4×D28414yes—
584C2×C4#1455609812yes—
585C2×C4#1455610114yes—
586C2×C4#1455610314yes—
587C2×C4#1455611312yes—
588C2×C4C22×C4×D7111yescharacteristic
589C2×C4C2×D4×D7111yescharacteristic
590C2×C4C2×D4×D7111yescharacteristic
591C2×C4C22×D28411yescharacteristic
592C2×C4C2×#1176586511yescharacteristic
593C2×C4C2×#1176586611yescharacteristic
594C8—42no—
595C8—2842no—
596C22—2844no—
597C22—2844no—
598C22—2844no—
599C22—5682no—
600C22—5682no—
601C22—5682no—
602C22—5682no—
603C22—56816no—
604C22—718no—
605C22—1424no—
606C22—1424no—
607C22—1424no—
608C22—1424no—
609C22—1424no—
610C22—14232no—
611C22—1424no—
612C22#1798587714yes—
613C22#102×D7114yes—
614C22#1798599914yes—
615C22#1798613414yes—
616C22#1798613914yes—
617C22#1798631414yes—
618C22#1798631914yes—
619C22C2×C22⋊C4×D7111yescharacteristic
620C22C2×#1455609811yescharacteristic
621C22C2×#1455611311yescharacteristic
622C22C2×D8×D7111yescharacteristic
623C22C2×SD16×D7111yescharacteristic
624C22C2×#1455615511yescharacteristic
625C22C2×#1455616011yescharacteristic
626C4—44no—
627C4—44no—
628C4—2844no—
629C4—1424no—
630C4#1798587714yes—
631C4C2×#1455609011yescharacteristic
632C4C2×C4×D28411yescharacteristic
633C4C2×#1455610111yescharacteristic
634C4C2×#1455610311yescharacteristic
635C2—5684no—
636C2—1428no—
637C2#2258865618yesdirect product factor
638C2C2×#1798587711yescharacteristic
639C2C2×#102×D7111yescharacteristic
640C2C2×#1798599911yescharacteristic
641C2C2×#1798613411yescharacteristic
642C2C2×#1798613911yescharacteristic
643C2C2×#1798631411yescharacteristic
644C2C2×#1798631911yescharacteristic
subgroupG/H
centerC24D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#340×C71C2
socleC24×C71C2×D4
prime radicalC22×#14556103C2
cosocleC22×C4×C71C24
hypercenterC2×#340D71
hypocenterC71C2×#1874
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D71C2semidirect
3. C22×#14556050C2semidirect
4. C22×#14556048C2semidirect
5. C22×(C2×D71)⋊C8C2semidirect
6. #22588656C2direct (N × H)
7. C2×#17985761C2semidirect
8. C2×#340×C71C2semidirect
16. C2×#14556103C4semidirect
23. C2×C4⋊C4×D71C22semidirect
26. C2×#14556050C22semidirect
28. C2×#14556048C22semidirect
30. C2×(C2×D71)⋊C8C22semidirect
31. #17985761C22semidirect
35. #340×C71C22semidirect
48. C22×D284C2×C4semidirect
70. #14556103C2×C4semidirect
73. C2×C4⋊Dic71D4semidirect
96. C2×C4⋊C4×C71C2×C4semidirect
97. C2×C4⋊C4×C71D4semidirect
132. C2×D284C22×C4semidirect
164. C4⋊Dic71C2×D4semidirect
192. C4⋊C4×C71C22×C4semidirect
193. C4⋊C4×C71C2×D4semidirect
226. C22×D71#102semidirect
302. C2×D71C2×#102semidirect
313. C2×C71#1874semidirect
330. C2×#340D71semidirect
331. C71C2×#1874semidirect
358. #340C2×D71semidirect
414. C2×C4⋊C4C4×D71semidirect
415. C2×C4⋊C4D284semidirect
514. C4⋊C4C2×C4×D71semidirect
515. C4⋊C4C2×D284semidirect
637. C2#22588656direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561032
C22×C4⋊C4×D712
C22×#145560502
C22×#145560482
C22×(C2×D71)⋊C82
#225886562
C2×#179857612
C2×#340×C712
C2×#187471