Finite Group Databasebeta

← C22×#11598 #72287434

Subgroups — C22×#11598 #72287434

Order 1024.

7797 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#135C211yesmaximal, characteristic
2C22×C4×M32C211yesmaximal, characteristic, complemented
3C23×#334C211yesmaximal, characteristic, complemented
4C2×#11598C2124yesmaximal, direct product factor
5C22×#1334C212yesmaximal, complemented
6C22×#1333C212yesmaximal, complemented
7C23×#135C411yescharacteristic
8C22×C4×#41—21no—
9C23×M32—21no—
10C2×C4×#135—26no—
11C2×C4×#135—26no—
12C2×C4×#135C4112yes—
13C2×C4×#135C22112yes—
14C22×C4×Q16C411yescharacteristic
15C22×C4×Q16C2211yescharacteristic
16C22×C4×SD16—21no—
17C22×C4×SD16—21no—
18C22×C4×D8C411yescharacteristic
19C22×C4×D8C2211yescharacteristic
20C22×#427C411yescharacteristic
21C23×C4⋊C8—21no—
22C22×#416—21no—
23C23×C4×C8C2211yescharacteristic
24C2×C4×M32—26no—
25C2×C4×M32C22112yescomplemented
26C22×#346C2212yes—
27C22×C4×C16—21no—
28C22×#334—26no—
29C22×#334—21no—
30C22×#334C2212yes—
31C22×#334C22112yescomplemented
32#11598C22164yesdirect product factor
33C2×#1334C22124yescomplemented
34C2×#1333C22124yescomplemented
35C23×#41—21no—
36C22×#135—26no—
37C22×#135—28no—
38C22×#135C8116yescomplemented
39C22×#135C2×C4112yes—
40C23×Q16C2×C411yescharacteristic
41C23×SD16—21no—
42C23×D8C2×C411yescharacteristic
43C24×C8—41no—
44C24×C8C2×C411yescharacteristic
45C24×C8D411yescharacteristic
46C2×C4×#41—212no—
47C2×C4×#41—212no—
48C2×C4×#41—46no—
49C2×C4×#41—46no—
50C22×C4×Q8—21no—
51C22×C4×Q8—21no—
52C22×C4×D4—21no—
53C22×C4×D4—21no—
54C22×C4×D4—41no—
55C22×C4×D4—41no—
56C22×#117—21no—
57C22×#117—41no—
58C23×C42—41no—
59C23×C42D411yescharacteristic
60C22×M32—216no—
61C22×M32—212no—
62C23×C16—22no—
63C4×#135—248no—
64C4×#135—224no—
65C4×#135—224no—
66C4×#135C2×C4148yes—
67C4×#135C23116yes—
68C2×C4×Q16—26no—
69C2×C4×Q16—23no—
70C2×C4×Q16—23no—
71C2×C4×Q16C2×C4112yes—
72C2×C4×Q16C2×C416yes—
73C2×C4×Q16C2316yes—
74C2×C4×SD16—212no—
75C2×C4×SD16—212no—
76C2×C4×SD16—212no—
77C2×C4×SD16—212no—
78C2×C4×D8—26no—
79C2×C4×D8—23no—
80C2×C4×D8—23no—
81C2×C4×D8C2×C4112yes—
82C2×C4×D8C2×C416yes—
83C2×C4×D8C2316yes—
84C2×#427—26no—
85C2×#427C2×C4112yes—
86C22×#107C2×C411yescharacteristic
87C22×#107C2×C411yescharacteristic
88C22×#106—21no—
89C22×C4⋊C8—22no—
90C22×C4⋊C8—212no—
91C22×C4⋊C8—41no—
92C22×C4⋊C8—46no—
93C2×#416—212no—
94C2×#416—212no—
95C22×#103—21no—
96C22×#103—21no—
97C22×#102—21no—
98C22×#102—21no—
99C22×C4×C8C2×C411yescharacteristic
100C22×C4×C8C2×C416yes—
101C22×C4×C8D411yescharacteristic
102C22×C4×C8D416yes—
103C22×C4×C8D411yescharacteristic
104C22×C4×C8D416yes—
105C22×C4×C8C2311yescharacteristic
106C22×C4×C8C2316yes—
107C4×M32—224no—
108C4×M32C23116yescomplemented
109C2×#346—26no—
110C2×#346C23112yes—
111C2×C4×C16—212no—
112C2×#334—26no—
113C2×#334—224no—
114C2×#334—212no—
115C2×#334C23112yes—
116C2×#334C23116yescomplemented
117#1334C23132yescomplemented
118#1333C23132yescomplemented
119C22×#41—216no—
120C22×#41—212no—
121C22×#41—46no—
122C22×#41—48no—
123C23×Q8—21no—
124C23×D4—21no—
125C23×D4—41no—
126C24×C4—41no—
127C24×C4—41no—
128C24×C4—41no—
129C24×C4C22⋊C411yescharacteristic
130C24×C4D811yescharacteristic
131C24×C4SD1611yescharacteristic
132C2×#135—296no—
133C2×#135—224no—
134C2×#135—248no—
135C2×#135C2×C8196yescomplemented
136C2×#135C22×C4116yes—
137C22×Q16—23no—
138C22×Q16—24no—
139C22×Q16C2×C818yescomplemented
140C22×Q16C22×C416yes—
141C22×SD16—216no—
142C22×SD16—212no—
143C22×D8—23no—
144C22×D8—24no—
145C22×D8C2×C818yescomplemented
146C22×D8C22×C416yes—
147C23×C8—28no—
148C23×C8—42no—
149C23×C8—412no—
150C23×C8—88no—
151C23×C8C22⋊C411yescharacteristic
152C23×C8C22⋊C416yes—
153C23×C8C22⋊C411yescharacteristic
154C23×C8C22⋊C416yes—
155C23×C8C2×C818yes—
156C23×C8M1618yes—
157C23×C8C22×C411yescharacteristic
158C23×C8C22×C416yes—
159C23×C8C2×D411yescharacteristic
160C23×C8C2×D416yes—
161C4×#41—248no—
162C4×#41—216no—
163C4×#41—448no—
164C4×#41—424no—
165C4×#41—424no—
166C2×C4×Q8—212no—
167C2×C4×Q8—26no—
168C2×C4×Q8—26no—
169C2×C4×Q8—46no—
170C2×C4×Q8—43no—
171C2×C4×Q8—43no—
172C2×C4×D4—212no—
173C2×C4×D4—26no—
174C2×C4×D4—26no—
175C2×C4×D4—412no—
176C2×C4×D4—412no—
177C2×C4×D4—412no—
178C2×C4×D4—412no—
179C2×C4×D4—46no—
180C2×C4×D4—43no—
181C2×C4×D4—43no—
182C2×#117—212no—
183C2×#117—46no—
184C2×#117—412no—
185C2×#117—412no—
186C22×C4⋊C4—21no—
187C22×C4⋊C4—21no—
188C22×C4⋊C4—41no—
189C22×C4⋊C4—41no—
190C22×C22⋊C4—41no—
191C22×C22⋊C4—41no—
192C22×C22⋊C4—41no—
193C22×C42—41no—
194C22×C42—41no—
195C22×C42—41no—
196C22×C42—412no—
197C22×C42—412no—
198C22×C42C22⋊C416yes—
199C22×C42C22⋊C411yescharacteristic
200C22×C42D811yescharacteristic
201C22×C42D816yes—
202C22×C42SD1616yes—
203C22×C42SD1611yescharacteristic
204C22×C42C2×D411yescharacteristic
205C22×C42C2×D416yes—
206C2×M32—296no—
207C2×M32—216no—
208C22×C16—216no—
209C22×C16—212no—
210C4×Q16—224no—
211C4×Q16—212no—
212C4×Q16—26no—
213C4×Q16—26no—
214C4×Q16C22×C4116yes—
215C4×Q16C22×C4112yes—
216C4×Q16C2414yes—
217C4×SD16—248no—
218C4×SD16—248no—
219C4×SD16—216no—
220C4×SD16—216no—
221C4×D8—224no—
222C4×D8—212no—
223C4×D8—26no—
224C4×D8—26no—
225C4×D8C22×C4116yes—
226C4×D8C22×C4112yes—
227C4×D8C2414yes—
228#427—224no—
229#427C22×C4116yes—
230C2×#107—23no—
231C2×#107—23no—
232C2×#107C22×C416yes—
233C2×#107C22×C416yes—
234C2×#106—212no—
235C2×C4⋊C8—212no—
236C2×C4⋊C8—216no—
237C2×C4⋊C8—412no—
238C2×C4⋊C8—46no—
239C2×C4⋊C8—424no—
240#416—248no—
241#416—216no—
242C2×#103—212no—
243C2×#103—26no—
244C2×#103—26no—
245C2×#102—212no—
246C2×#102—26no—
247C2×#102—26no—
248C2×C4×C8C22⋊C416yes—
249C2×C4×C8C22⋊C416yes—
250C2×C4×C8C22⋊C416yes—
251C2×C4×C8C22⋊C4124yes—
252C2×C4×C8C22×C416yes—
253C2×C4×C8C22×C413yes—
254C2×C4×C8C22×C4112yes—
255C2×C4×C8C2×D416yes—
256C2×C4×C8C2×D413yes—
257C2×C4×C8C2×D4112yes—
258C2×C4×C8C2×D413yes—
259C2×C4×C8C2×D416yes—
260C2×C4×C8C2×D4112yes—
261C2×C4×C8C2413yes—
262C2×C4×C8C2414yes—
263#346—212no—
264#346C2418yes—
265C4×C16—216no—
266#334—212no—
267#334—216no—
268#334C2418yes—
269C25—41no—
270C25#10211yescharacteristic
271C2×#41—216no—
272C2×#41—296no—
273C2×#41—496no—
274C2×#41—424no—
275C2×#41—448no—
276C22×Q8—26no—
277C22×Q8—28no—
278C22×Q8—43no—
279C22×Q8—44no—
280C22×D4—26no—
281C22×D4—28no—
282C22×D4—416no—
283C22×D4—412no—
284C22×D4—43no—
285C22×D4—44no—
286C23×C4—28no—
287C23×C4—44no—
288C23×C4—41no—
289C23×C4—416no—
290C23×C4—412no—
291C23×C4—41no—
292C23×C4—41no—
293C23×C4—46no—
294C23×C4—46no—
295C23×C4—412no—
296C23×C4—88no—
297C23×C4—88no—
298C23×C4C22⋊C818yes—
299C23×C4#10211yescharacteristic
300C23×C4#10216yes—
301C23×C4#10216yes—
302C23×C4#10216yes—
303C23×C4#10211yescharacteristic
304C23×C4#10211yescharacteristic
305C23×C4#10418yes—
306C23×C4C2×C22⋊C411yescharacteristic
307C23×C4C2×C22⋊C416yes—
308C23×C4C2×D811yescharacteristic
309C23×C4C2×D816yes—
310C23×C4C2×SD1611yescharacteristic
311C23×C4C2×SD1616yes—
312#135—232no—
313#135—2192no—
314#135C22×C8164yescomplemented
315C2×Q16—26no—
316C2×Q16—212no—
317C2×Q16—224no—
318C2×Q16C22×C8124yescomplemented
319C2×Q16C23×C414yes—
320C2×SD16—216no—
321C2×SD16—296no—
322C2×D8—26no—
323C2×D8—212no—
324C2×D8—224no—
325C2×D8C22×C8124yescomplemented
326C2×D8C23×C414yes—
327C22×C8—24no—
328C22×C8—224no—
329C22×C8—24no—
330C22×C8—224no—
331C22×C8—412no—
332C22×C8—416no—
333C22×C8—88no—
334C22×C8—848no—
335C22×C8C22⋊C818yes—
336C22×C8C22⋊C8148yes—
337C22×C8C2×C22⋊C416yes—
338C22×C8C2×C22⋊C413yes—
339C22×C8C2×C22⋊C4112yes—
340C22×C8C2×C22⋊C413yes—
341C22×C8C2×C22⋊C416yes—
342C22×C8C2×C22⋊C4112yes—
343C22×C8C22×C814yes—
344C22×C8C22×C8124yes—
345C22×C8C2×M1614yes—
346C22×C8C2×M16124yes—
347C22×C8C23×C413yes—
348C22×C8C23×C414yes—
349C22×C8C22×D413yes—
350C22×C8C22×D414yes—
351C4×Q8—216no—
352C4×Q8—212no—
353C4×Q8—24no—
354C4×Q8—46no—
355C4×Q8—46no—
356C4×Q8—424no—
357C4×Q8—412no—
358C4×D4—216no—
359C4×D4—212no—
360C4×D4—24no—
361C4×D4—416no—
362C4×D4—416no—
363C4×D4—448no—
364C4×D4—448no—
365C4×D4—46no—
366C4×D4—46no—
367C4×D4—424no—
368C4×D4—412no—
369#117—216no—
370#117—424no—
371#117—416no—
372#117—448no—
373C2×C4⋊C4—26no—
374C2×C4⋊C4—26no—
375C2×C4⋊C4—46no—
376C2×C4⋊C4—43no—
377C2×C4⋊C4—43no—
378C2×C4⋊C4—412no—
379C2×C4⋊C4—46no—
380C2×C22⋊C4—412no—
381C2×C22⋊C4—46no—
382C2×C22⋊C4—412no—
383C2×C22⋊C4—46no—
384C2×C42—46no—
385C2×C42—412no—
386C2×C42—46no—
387C2×C42—412no—
388C2×C42—416no—
389C2×C42—448no—
390C2×C42#10216yes—
391C2×C42#10216yes—
392C2×C42#102124yes—
393C2×C42#10216yes—
394C2×C42C2×C22⋊C413yes—
395C2×C42C2×C22⋊C416yes—
396C2×C42C2×C22⋊C4112yes—
397C2×C42C2×D816yes—
398C2×C42C2×D813yes—
399C2×C42C2×D8112yes—
400C2×C42C2×SD1613yes—
401C2×C42C2×SD16112yes—
402C2×C42C2×SD1616yes—
403C2×C42C22×D413yes—
404C2×C42C22×D414yes—
405M32—264no—
406C2×C16—248no—
407C2×C16—28no—
408#107—26no—
409#107—26no—
410#107C23×C414yes—
411#107C23×C414yes—
412#106—216no—
413C4⋊C8—28no—
414C4⋊C8—416no—
415C4⋊C8—412no—
416#103—216no—
417#103—212no—
418#103—24no—
419#102—216no—
420#102—212no—
421#102—24no—
422C4×C8C2×C22⋊C4112yes—
423C4×C8C2×C22⋊C416yes—
424C4×C8C2×C22⋊C4112yes—
425C4×C8C2×C22⋊C4112yes—
426C4×C8C23×C414yes—
427C4×C8C23×C413yes—
428C4×C8C22×D414yes—
429C4×C8C22×D413yes—
430C4×C8C22×D413yes—
431C4×C8C22×D414yes—
432C4×C8C2511yescharacteristic
433C24—24no—
434C24—24no—
435C24—46no—
436C24—48no—
437C24—84no—
438C24—84no—
439C24D4⋊C818yes—
440C24C2×#10216yes—
441C24C2×#10211yescharacteristic
442#41—264no—
443#41—432no—
444#41—4192no—
445C2×Q8—224no—
446C2×Q8—24no—
447C2×Q8—46no—
448C2×Q8—412no—
449C2×Q8—424no—
450C2×D4—224no—
451C2×D4—24no—
452C2×D4—416no—
453C2×D4—496no—
454C2×D4—46no—
455C2×D4—412no—
456C2×D4—424no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—224no—
461C22×C4—224no—
462C22×C4—412no—
463C22×C4—412no—
464C22×C4—46no—
465C22×C4—48no—
466C22×C4—416no—
467C22×C4—496no—
468C22×C4—46no—
469C22×C4—46no—
470C22×C4—412no—
471C22×C4—416no—
472C22×C4—44no—
473C22×C4—824no—
474C22×C4—824no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—848no—
478C22×C4D4⋊C818yes—
479C22×C4D4⋊C8148yes—
480C22×C4#36018yes—
481C22×C4C2×C22⋊C814yes—
482C22×C4C2×C22⋊C8124yes—
483C22×C4C2×#10213yes—
484C22×C4C2×#10216yes—
485C22×C4C2×#10213yes—
486C22×C4C2×#102112yes—
487C22×C4C2×#102112yes—
488C22×C4C2×#10216yes—
489C22×C4C2×#10216yes—
490C22×C4C2×#10213yes—
491C22×C4C2×#102112yes—
492C22×C4C2×#10414yes—
493C22×C4C2×#104124yes—
494C22×C4C22×C22⋊C413yes—
495C22×C4C22×C22⋊C414yes—
496C22×C4C22×D813yes—
497C22×C4C22×D814yes—
498C22×C4C22×SD1613yes—
499C22×C4C22×SD1614yes—
500Q16—224no—
501Q16—24no—
502Q16C23×C818yescomplemented
503SD16—264no—
504D8—224no—
505D8—24no—
506D8C23×C818yescomplemented
507C2×C8—26no—
508C2×C8—28no—
509C2×C8—26no—
510C2×C8—212no—
511C2×C8—224no—
512C2×C8—48no—
513C2×C8—824no—
514C2×C8—832no—
515C2×C8C2×C22⋊C8112yes—
516C2×C8C2×C22⋊C8124yes—
517C2×C8C2×C22⋊C8148yes—
518C2×C8C22×C22⋊C414yes—
519C2×C8C22×C22⋊C413yes—
520C2×C8C22×C22⋊C414yes—
521C2×C8C22×C22⋊C413yes—
522C2×C8C23×C816yes—
523C2×C8C23×C818yes—
524C2×C8C22×M1616yes—
525C2×C8C22×M1618yes—
526C2×C8C24×C411yescharacteristic
527C2×C8C23×D411yescharacteristic
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—46no—
531C4⋊C4—46no—
532C4⋊C4—416no—
533C4⋊C4—412no—
534C4⋊C4—44no—
535C22⋊C4—416no—
536C22⋊C4—416no—
537C22⋊C4—412no—
538C22⋊C4—44no—
539C42—416no—
540C42—412no—
541C42—416no—
542C42—44no—
543C42C2×#102112yes—
544C42C2×#102112yes—
545C42C2×#10216yes—
546C42C2×#102112yes—
547C42C22×C22⋊C413yes—
548C42C22×C22⋊C414yes—
549C42C22×D814yes—
550C42C22×D813yes—
551C42C22×SD1614yes—
552C42C22×SD1613yes—
553C42C23×D411yescharacteristic
554C16—216no—
555C23—232no—
556C23—212no—
557C23—212no—
558C23—424no—
559C23—44no—
560C23—832no—
561C23—812no—
562C23—812no—
563C23C2×D4⋊C8124yes—
564C23C2×D4⋊C814yes—
565C23C2×#36014yes—
566C23#214814yes—
567C23C22×#10214yes—
568C23C22×#10213yes—
569Q8—28no—
570Q8—424no—
571Q8—44no—
572D4—28no—
573D4—464no—
574D4—424no—
575D4—44no—
576C2×C4—26no—
577C2×C4—26no—
578C2×C4—26no—
579C2×C4—26no—
580C2×C4—212no—
581C2×C4—28no—
582C2×C4—224no—
583C2×C4—44no—
584C2×C4—412no—
585C2×C4—424no—
586C2×C4—44no—
587C2×C4—464no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—824no—
591C2×C4—88no—
592C2×C4—812no—
593C2×C4—812no—
594C2×C4—832no—
595C2×C4C2×D4⋊C8124yes—
596C2×C4C2×D4⋊C8112yes—
597C2×C4C2×D4⋊C8148yes—
598C2×C4C2×#360124yes—
599C2×C4C22×C22⋊C816yes—
600C2×C4C22×C22⋊C818yes—
601C2×C4C22×#10213yes—
602C2×C4C22×#10214yes—
603C2×C4C22×#10213yes—
604C2×C4C22×#10214yes—
605C2×C4C22×#10213yes—
606C2×C4C22×#10214yes—
607C2×C4C22×#10416yes—
608C2×C4C22×#10418yes—
609C2×C4C23×C22⋊C411yescharacteristic
610C2×C4C23×D811yescharacteristic
611C2×C4C23×SD1611yescharacteristic
612C8—21no—
613C8—23no—
614C8—24no—
615C8—88no—
616C8C22×C22⋊C816yes—
617C8C22×C22⋊C818yes—
618C8C24×C811yescharacteristic
619C8C23×M1611yescharacteristic
620C22—248no—
621C22—24no—
622C22—24no—
623C22—48no—
624C22—848no—
625C22—84no—
626C22—84no—
627C22#11598116yesdirect product factor
628C22C22×D4⋊C818yes—
629C22C22×D4⋊C816yes—
630C22C22×#36016yes—
631C22C2×#214816yes—
632C22C23×#10211yescharacteristic
633C4—21no—
634C4—23no—
635C4—23no—
636C4—21no—
637C4—24no—
638C4—48no—
639C4—84no—
640C4—84no—
641C4C22×D4⋊C818yes—
642C4C22×D4⋊C816yes—
643C4C22×#36018yes—
644C4C23×C22⋊C811yescharacteristic
645C4C23×#10411yescharacteristic
646C2—28no—
647C2—88no—
648C2C2×#11598112yesdirect product factor
649C2C23×D4⋊C811yescharacteristic
650C2C23×#36011yescharacteristic
651C2C22×#214811yescharacteristic
subgroupG/H
centerC23×C4#102
commutatorC8C24×C8
Frattini, cosocleC4×C8C25
socleC24C2×#102
prime radicalC23×#135C4
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ C23×C4×C8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×M32C2semidirect
3. C23×#334C2semidirect
4. C2×#11598C2direct (N × H)
5. C22×#1334C2semidirect
6. C22×#1333C2semidirect
25. C2×C4×M32C22semidirect
31. C22×#334C22semidirect
32. #11598C22direct (N × H)
33. C2×#1334C22semidirect
34. C2×#1333C22semidirect
38. C22×#135C8semidirect
108. C4×M32C23semidirect
116. C2×#334C23semidirect
117. #1334C23semidirect
118. #1333C23semidirect
135. C2×#135C2×C8semidirect
139. C22×Q16C2×C8semidirect
145. C22×D8C2×C8semidirect
314. #135C22×C8semidirect
318. C2×Q16C22×C8semidirect
325. C2×D8C22×C8semidirect
502. Q16C23×C8semidirect
506. D8C23×C8semidirect
627. C22#11598direct (N × H)
648. C2C2×#11598direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#1352
C22×C4×M322
C23×#3342
C2×#115982
C22×#13342
C22×#13332