Finite Group Databasebeta

← C2×#22589349 #32024249

Subgroups — C2×#22589349 #32024249

Order 18176.

2182 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765855C211yesmaximal, characteristic, complemented
2C22×#14556108C211yesmaximal, characteristic
3C22×#14556104C211yesmaximal, characteristic
4C22×#14556053C211yesmaximal, characteristic, complemented
5C22×(C2×D71)⋊C8C211yesmaximal, characteristic
6#22589349C218yesmaximal, direct product factor
7C2×#17985761C211yesmaximal, characteristic, complemented
8C22×#103×C71C211yesmaximal, characteristic, complemented
9C23×Dic142C2211yescharacteristic
10C23×C4×D71C2211yescharacteristic
11C22×#11765855—24no—
12C22×#11765855—24no—
13C22×#11765855—24no—
14C22×#11765855C2214yescomplemented
15C23×C71⋊C8C2211yescharacteristic
16C2×#14556108—24no—
17C2×#14556108—24no—
18C2×#14556108—22no—
19C2×#14556108C2214yes—
20C2×#14556104—24no—
21C2×#14556104—24no—
22C2×#14556104—22no—
23C2×#14556104C2214yes—
24C22×C4⋊Dic71—21no—
25C22×C4⋊Dic71C2211yescharacteristic
26C22×Dic71⋊C4—21no—
27C22×(C2×D71)⋊C4—21no—
28C22×(C2×D71)⋊C4—21no—
29C2×#14556053—24no—
30C2×#14556053C2214yescomplemented
31C2×(C2×D71)⋊C8—24no—
32C2×(C2×D71)⋊C8C2214yes—
33#17985761C2214yescomplemented
34C23×Q8×C71C2211yescharacteristic
35C23×C8×C71C2211yescharacteristic
36C22×C4⋊C4×C71C2211yescharacteristic
37C2×#103×C71—24no—
38C2×#103×C71C2214yescomplemented
39C24×D71D411yescharacteristic
40C22×Dic142—21no—
41C22×Dic142—22no—
42C22×Dic142—22no—
43C22×Dic142—22no—
44C22×Dic142—22no—
45C22×Dic142—22no—
46C22×Dic142D414yes—
47C22×Dic142C2312yes—
48C23×Dic71—21no—
49C23×Dic71—21no—
50C23×Dic71D411yescharacteristic
51C22×C4×D71—22no—
52C22×C4×D71—22no—
53C22×C4×D71—22no—
54C22×C4×D71—22no—
55C22×C4×D71—22no—
56C22×C4×D71—22no—
57C22×C4×D71D412yes—
58C22×C4×D71C2312yes—
59C2×#11765855—232no—
60C2×#11765855—28no—
61C2×#11765855—28no—
62C2×#11765855—28no—
63C22×C71⋊C8—22no—
64C22×C71⋊C8—22no—
65C22×C71⋊C8D414yes—
66C22×C71⋊C8C2312yes—
67#14556108—216no—
68#14556108—28no—
69#14556108—28no—
70#14556104—216no—
71#14556104—28no—
72#14556104—28no—
73C2×C4⋊Dic71—28no—
74C2×C4⋊Dic71—24no—
75C2×C4⋊Dic71—21no—
76C2×C4⋊Dic71—22no—
77C2×C4⋊Dic71D414yes—
78C2×C4⋊Dic71C2312yes—
79C2×Dic71⋊C4—28no—
80C2×Dic71⋊C4—24no—
81C2×(C2×D71)⋊C4—28no—
82C2×(C2×D71)⋊C4—28no—
83C2×(C2×D71)⋊C4—24no—
84C2×(C2×D71)⋊C4—24no—
85#14556053—28no—
86(C2×D71)⋊C8—28no—
87C22×Q8×C71—22no—
88C22×Q8×C71—21no—
89C22×Q8×C71—22no—
90C22×Q8×C71—22no—
91C22×Q8×C71—22no—
92C22×Q8×C71—22no—
93C22×Q8×C71D414yes—
94C22×Q8×C71C2312yes—
95C23×C4×C71—21no—
96C23×C4×C71—21no—
97C23×C4×C71C2311yescharacteristic
98C22×C8×C71—22no—
99C22×C8×C71—22no—
100C22×C8×C71—22no—
101C22×C8×C71C2312yes—
102C2×C4⋊C4×C71—21no—
103C2×C4⋊C4×C71—22no—
104C2×C4⋊C4×C71D414yes—
105C2×C4⋊C4×C71C2312yes—
106#103×C71—28no—
107C23×D71—22no—
108C23×D71—22no—
109C23×D71—22no—
110C23×D71—22no—
111C23×D71—22no—
112C23×D71—22no—
113C23×D71Q1614yescomplemented
114C23×D71C2×D412yes—
115C2×Dic142—28no—
116C2×Dic142—28no—
117C2×Dic142—22no—
118C2×Dic142—22no—
119C2×Dic142—22no—
120C2×Dic142—22no—
121C2×Dic142—24no—
122C2×Dic142—22no—
123C2×Dic142—42no—
124C2×Dic142—42no—
125C2×Dic142—44no—
126C2×Dic142—44no—
127C2×Dic142C2×D414yes—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71—24no—
131C22×Dic71—24no—
132C22×Dic71—22no—
133C22×Dic71—22no—
134C22×Dic71—22no—
135C22×Dic71—42no—
136C22×Dic71—42no—
137C22×Dic71—42no—
138C22×Dic71—42no—
139C22×Dic71C2×D412yes—
140C2×C4×D71—24no—
141C2×C4×D71—24no—
142C2×C4×D71—24no—
143C2×C4×D71—22no—
144C2×C4×D71—22no—
145C2×C4×D71—22no—
146C2×C4×D71—22no—
147C2×C4×D71—22no—
148C2×C4×D71—22no—
149C2×C4×D71—44no—
150C2×C4×D71—44no—
151C2×C4×D71—44no—
152C2×C4×D71—44no—
153#11765855—232no—
154C2×C71⋊C8—28no—
155C2×C71⋊C8—22no—
156C2×C71⋊C8—22no—
157C2×C71⋊C8C2×D414yes—
158C4⋊Dic71—216no—
159C4⋊Dic71—24no—
160C4⋊Dic71—22no—
161C4⋊Dic71C2×D414yes—
162Dic71⋊C4—216no—
163(C2×D71)⋊C4—216no—
164(C2×D71)⋊C4—216no—
165C24×C71C2×D411yescharacteristic
166C2×Q8×C71—24no—
167C2×Q8×C71—22no—
168C2×Q8×C71—28no—
169C2×Q8×C71—28no—
170C2×Q8×C71—22no—
171C2×Q8×C71—22no—
172C2×Q8×C71—22no—
173C2×Q8×C71—22no—
174C2×Q8×C71—42no—
175C2×Q8×C71—42no—
176C2×Q8×C71—44no—
177C2×Q8×C71—44no—
178C2×Q8×C71C2×D414yes—
179C22×C4×C71—24no—
180C22×C4×C71—24no—
181C22×C4×C71—22no—
182C22×C4×C71—22no—
183C22×C4×C71—42no—
184C22×C4×C71—42no—
185C22×C4×C71—42no—
186C22×C4×C71—42no—
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D412yes—
189C22×C4×C71C2×D412yes—
190C22×C4×C71C2×D412yes—
191C22×C4×C71C2×D412yes—
192C22×C4×C71C2×D412yes—
193C22×C4×C71#4112yes—
194C22×C4×C71C2411yescharacteristic
195C2×C8×C71—22no—
196C2×C8×C71—22no—
197C2×C8×C71—22no—
198C2×C8×C71—44no—
199C4⋊C4×C71—24no—
200C4⋊C4×C71—22no—
201C4⋊C4×C71C2×D414yes—
202C23×C71#12012yes—
203C23×C71#12012yes—
204C23×C71#12112yes—
205C23×C71C2×Q1612yes—
206C23×C71#13512yes—
207C23×C71#13712yes—
208C23×C71#13712yes—
209C23×C71C22×D411yescharacteristic
210C22×D71—22no—
211C22×D71—22no—
212C22×D71—22no—
213C22×D71—22no—
214C22×D71—22no—
215C22×D71—22no—
216C22×D71—28no—
217C22×D71—28no—
218C22×D71—28no—
219C22×D71—44no—
220C22×D71—44no—
221C22×D71—44no—
222C22×D71—44no—
223C22×D71C2×Q1614yescomplemented
224Q8×C71—24no—
225Q8×C71—24no—
226Q8×C71—42no—
227Q8×C71—42no—
228Q8×C71—48no—
229C2×C4×C71—24no—
230C2×C4×C71—24no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71—42no—
234C2×C4×C71—42no—
235C2×C4×C71—48no—
236C2×C4×C71—48no—
237C2×C4×C71#12012yes—
238C2×C4×C71#12014yes—
239C2×C4×C71#12012yes—
240C2×C4×C71#12114yes—
241C2×C4×C71#12112yes—
242C2×C4×C71#12114yes—
243C2×C4×C71#12714yes—
244C2×C4×C71C22×D411yescharacteristic
245C2×C4×C71C22×D411yescharacteristic
246C2×C4×C71C22×D411yescharacteristic
247C2×C4×C71C22×D411yescharacteristic
248C2×C4×C71C22×D411yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—22no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—42no—
258C2×Dic71—42no—
259C2×Dic71—44no—
260C2×Dic71—48no—
261C2×Dic71—48no—
262C4×D71—42no—
263C4×D71—42no—
264C4×D71—42no—
265C4×D71—42no—
266C4×D71—48no—
267Dic142—24no—
268Dic142—24no—
269Dic142—42no—
270Dic142—42no—
271Dic142—48no—
272C8×C71—42no—
273C71⋊C8—24no—
274C22×C71#39214yes—
275C22×C71#45114yes—
276C22×C71#45114yes—
277C22×C71#45214yes—
278C22×C71#45214yes—
279C22×C71#46714yes—
280C22×C71#47014yes—
281C22×C71C2×#12011yescharacteristic
282C22×C71C2×#12011yescharacteristic
283C22×C71C2×#12111yescharacteristic
284C22×C71C22×Q1611yescharacteristic
285C22×C71C2×#13511yescharacteristic
286C22×C71C2×#13711yescharacteristic
287C22×C71C2×#13711yescharacteristic
288C2×D71—24no—
289C2×D71—24no—
290C2×D71—24no—
291C2×D71—42no—
292C2×D71—42no—
293C2×D71—42no—
294C2×D71—416no—
295C2×D71—42no—
296C4×C71—44no—
297C4×C71—44no—
298C4×C71#39214yes—
299C4×C71C2×#12011yescharacteristic
300C4×C71C2×#12111yescharacteristic
301C4×C71C2×#12111yescharacteristic
302C4×C71C2×#12711yescharacteristic
303Dic71—44no—
304Dic71—42no—
305Dic71—44no—
306C2×#2001—711nomaximal, Sylow
307C2×C71#200118yescomplemented
308C2×C71C2×#39211yescharacteristic
309C2×C71C2×#45111yescharacteristic
310C2×C71C2×#45111yescharacteristic
311C2×C71C2×#45211yescharacteristic
312C2×C71C2×#45211yescharacteristic
313C2×C71C2×#46711yescharacteristic
314C2×C71C2×#47011yescharacteristic
315D71—44no—
316C23×Q16—711no—
317C22×C22⋊Q8—711no—
318C22×C22⋊Q8—711no—
319C22×#103—711no—
320C22×#103D7111yescharacteristic, complemented
321C22×C22⋊C8—711no—
322#2001—718no—
323C2×#340—711no—
324C71C2×#200111yescharacteristic, complemented, Sylow
325C23×Q8—711no—
326C23×Q8C2×D7111yescharacteristic
327C24×C4—711no—
328C22×Q16—714no—
329C22×Q16—1424no—
330C22×Q16—1424no—
331C22×Q16—1424no—
332C23×C8—711no—
333C23×C8C2×D7111yescharacteristic
334C2×C22⋊Q8—714no—
335C2×C22⋊Q8—714no—
336C2×C22⋊Q8—1422no—
337C2×C22⋊Q8—1424no—
338C2×C22⋊Q8—1424no—
339C2×C22⋊Q8—1422no—
340C2×C22⋊Q8—1424no—
341C2×C22⋊Q8—1424no—
342C22×C4⋊C4—711no—
343C22×C4⋊C4—1421no—
344C22×C4⋊C4—1421no—
345C22×C4⋊C4C2×D7111yescharacteristic
346C22×C22⋊C4—1421no—
347C22×C22⋊C4—1421no—
348C2×#103—24no—
349C2×#103—714no—
350C2×#103—1424no—
351C2×#103C2×D7114yescomplemented
352C2×C22⋊C8—714no—
353C2×C22⋊C8—1424no—
354#340—714no—
355C25—711no—
356C22×Q8—21no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—714no—
363C22×Q8—712no—
364C22×Q8—1422no—
365C22×Q8—1422no—
366C22×Q8—1422no—
367C22×Q8—1422no—
368C22×Q8—1421no—
369C22×Q8—1422no—
370C22×Q8D28414yes—
371C22×Q8C22×D7112yes—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—711no—
375C23×C4—712no—
376C23×C4—712no—
377C23×C4—1421no—
378C23×C4—1421no—
379C23×C4—1422no—
380C23×C4—1422no—
381C23×C4—1422no—
382C23×C4—1422no—
383C23×C4—1422no—
384C23×C4—1422no—
385C23×C4C22×D7111yescharacteristic
386C2×Q16—1428no—
387C2×Q16—1428no—
388C2×Q16—14232no—
389C2×Q16—1428no—
390C22×C8—22no—
391C22×C8—22no—
392C22×C8—22no—
393C22×C8—714no—
394C22×C8—712no—
395C22×C8—1422no—
396C22×C8—1422no—
397C22×C8C22×D7112yes—
398C22⋊Q8—1428no—
399C22⋊Q8—14216no—
400C22⋊Q8—1428no—
401C22⋊Q8—14216no—
402C22⋊Q8—1428no—
403C22⋊Q8—1428no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—714no—
407C2×C4⋊C4—712no—
408C2×C4⋊C4—1424no—
409C2×C4⋊C4—1424no—
410C2×C4⋊C4—1428no—
411C2×C4⋊C4—1428no—
412C2×C4⋊C4—1421no—
413C2×C4⋊C4—1422no—
414C2×C4⋊C4C22⋊D7114yes—
415C2×C4⋊C4C22×D7112yes—
416C2×C22⋊C4—1424no—
417C2×C22⋊C4—1424no—
418C2×C22⋊C4—1428no—
419C2×C22⋊C4—1428no—
420#103—28no—
421#103—1428no—
422C22⋊C8—1428no—
423C24—712no—
424C24—714no—
425C24—1422no—
426C24—1422no—
427C24—1422no—
428C24—1422no—
429C24—1422no—
430C24—1422no—
431C24D4×D7111yescharacteristic
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—28no—
439C2×Q8—28no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—2842no—
445C2×Q8—2842no—
446C2×Q8—2844no—
447C2×Q8—2844no—
448C2×Q8—714no—
449C2×Q8—1422no—
450C2×Q8—1422no—
451C2×Q8—1422no—
452C2×Q8—1422no—
453C2×Q8—1428no—
454C2×Q8—1428no—
455C2×Q8—1422no—
456C2×Q8—1424no—
457C2×Q8C2×D28414yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—2842no—
467C22×C4—2842no—
468C22×C4—2842no—
469C22×C4—2842no—
470C22×C4—2844no—
471C22×C4—2844no—
472C22×C4—2844no—
473C22×C4—2844no—
474C22×C4—712no—
475C22×C4—1422no—
476C22×C4—1422no—
477C22×C4—1424no—
478C22×C4—1422no—
479C22×C4—1424no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1424no—
489C22×C4—1424no—
490C22×C4—1424no—
491C22×C4D4×D7112yes—
492C22×C4D4×D7112yes—
493C22×C4D4×D7112yes—
494C22×C4D4×D7111yescharacteristic
495C22×C4C2×C22⋊D7112yes—
496C22×C4C2×D28412yes—
497C22×C4#1176586512yes—
498C22×C4C23×D7111yescharacteristic
499Q16—14232no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—44no—
504C2×C8—714no—
505C2×C8—1422no—
506C2×C8—1422no—
507C2×C8—1428no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—714no—
511C4⋊C4—14216no—
512C4⋊C4—14216no—
513C4⋊C4—1422no—
514C4⋊C4—1424no—
515C4⋊C4C2×C22⋊D7114yes—
516C22⋊C4—14216no—
517C22⋊C4—14216no—
518C23—2844no—
519C23—2844no—
520C23—2844no—
521C23—2844no—
522C23—714no—
523C23—1428no—
524C23—1428no—
525C23—1422no—
526C23—1422no—
527C23—1422no—
528C23—1428no—
529C23—1422no—
530C23—1422no—
531C23—1422no—
532C23#1455609612yes—
533C23#1455609812yes—
534C23#1455610112yes—
535C23Q16×D7112yes—
536C23#1455614912yes—
537C23#1455615712yes—
538C23#1455616012yes—
539C23C2×D4×D7111yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—48no—
543Q8—42no—
544Q8—42no—
545Q8—2848no—
546Q8—2842no—
547Q8—2842no—
548Q8—1424no—
549Q8—1424no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—2844no—
559C2×C4—2848no—
560C2×C4—2848no—
561C2×C4—2842no—
562C2×C4—2842no—
563C2×C4—2842no—
564C2×C4—2842no—
565C2×C4—2848no—
566C2×C4—2842no—
567C2×C4—2842no—
568C2×C4—2842no—
569C2×C4—2842no—
570C2×C4—1424no—
571C2×C4—1422no—
572C2×C4—1424no—
573C2×C4—1422no—
574C2×C4—1422no—
575C2×C4#1455609612yes—
576C2×C4#1455609812yes—
577C2×C4#1455609814yes—
578C2×C4#1455610112yes—
579C2×C4#1455610114yes—
580C2×C4#1455610214yes—
581C2×C4#1455612414yes—
582C2×C4C2×D4×D7111yescharacteristic
583C2×C4C2×D4×D7111yescharacteristic
584C2×C4C2×D4×D7111yescharacteristic
585C2×C4C22×C22⋊D7111yescharacteristic
586C2×C4C22×D28411yescharacteristic
587C2×C4C2×#1176586511yescharacteristic
588C8—42no—
589C8—1424no—
590C22—2842no—
591C22—2842no—
592C22—28416no—
593C22—2842no—
594C22—2842no—
595C22—1424no—
596C22—1424no—
597C22—1424no—
598C22#1798590814yes—
599C22#1798615314yes—
600C22#1798615614yes—
601C22#1798615814yes—
602C22#1798616414yes—
603C22#1798623914yes—
604C22#1798625614yes—
605C22C2×#1455609611yescharacteristic
606C22C2×#1455609811yescharacteristic
607C22C2×#1455610111yescharacteristic
608C22C2×Q16×D7111yescharacteristic
609C22C2×#1455614911yescharacteristic
610C22C2×#1455615711yescharacteristic
611C22C2×#1455616011yescharacteristic
612C4—44no—
613C4—44no—
614C4—2842no—
615C4—2844no—
616C4—2844no—
617C4#1798590814yes—
618C4C2×#1455609811yescharacteristic
619C4C2×#1455610111yescharacteristic
620C4C2×#1455610211yescharacteristic
621C4C2×#1455612411yescharacteristic
622C2—2844no—
623C2#2258934918yesdirect product factor
624C2C2×#1798590811yescharacteristic
625C2C2×#1798615311yescharacteristic
626C2C2×#1798615611yescharacteristic
627C2C2×#1798615811yescharacteristic
628C2C2×#1798616411yescharacteristic
629C2C2×#1798623911yescharacteristic
630C2C2×#1798625611yescharacteristic
subgroupG/H
centerC24D4×D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC22×#103×C71C2
socleC24×C71C2×D4
prime radicalC24×D71D4
hypercenterC22×#103D71
hypocenterC71C2×#2001
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
1. C23×#11765855C2semidirect
4. C22×#14556053C2semidirect
6. #22589349C2direct (N × H)
7. C2×#17985761C2semidirect
8. C22×#103×C71C2semidirect
14. C22×#11765855C22semidirect
30. C2×#14556053C22semidirect
33. #17985761C22semidirect
38. C2×#103×C71C22semidirect
113. C23×D71Q16semidirect
223. C22×D71C2×Q16semidirect
307. C2×C71#2001semidirect
320. C22×#103D71semidirect
324. C71C2×#2001semidirect
351. C2×#103C2×D71semidirect
623. C2#22589349direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658552
C22×#145561082
C22×#145561042
C22×#145560532
C22×(C2×D71)⋊C82
#225893492
C2×#179857612
C22×#103×C712
C2×#200171