Finite Group Databasebeta

← C2×#59321 #7666219

Subgroups — C2×#59321 #7666219

Order 512.

5930 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59321C2132yesmaximal, direct product factor
2C2×#2791C214yesmaximal, complemented
3C2×#2582C211yesmaximal, characteristic, complemented
4C2×#2552C211yesmaximal, characteristic, complemented
5C2×#2552C212yesmaximal, complemented
6C2×#2565C212yesmaximal, complemented
7C2×#2511C212yesmaximal, complemented
8C2×#2502C212yesmaximal
9C2×#2506C212yesmaximal, complemented
10C2×#2442C212yesmaximal
11C2×#2473C214yesmaximal
12C2×#2360C212yesmaximal
13C2×#2355C211yesmaximal, characteristic, complemented
14C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
15C2×#2345C212yesmaximal
16C2×C4×C22⋊Q8C211yesmaximal, characteristic
17C2×C4×C22⋊Q8C211yesmaximal, characteristic
18C22×#384C211yesmaximal, characteristic
19C22×#125C2212yes—
20C22×#125C2214yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2212yes—
26C23×C4⋊C4C2212yes—
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2212yes—
29C23×C42C2211yescharacteristic
30#2791C22164yescomplemented
31#2582C22116yescomplemented
32#2565C22132yescomplemented
33#2552C22116yescomplemented
34#2552C22132yescomplemented
35#2511C22132yescomplemented
36#2506C22132yescomplemented
37#2502C22132yes—
38#2473C22164yes—
39#2442C22132yes—
40C2×#400C2214yes—
41C2×#400C2214yes—
42C2×#398C2214yes—
43C2×#398C2214yes—
44C2×#398C2214yes—
45C2×#398C2214yes—
46C2×#398C2214yes—
47C2×#398C2214yes—
48#2360C22132yes—
49#2355C22116yescomplemented
50Q8×C4⋊C4C22116yescomplemented
51#2345C22132yes—
52C4×C22⋊Q8C22116yes—
53C4×C22⋊Q8C22116yes—
54C2×#391C2212yes—
55C2×#391C2211yescharacteristic
56C2×#391C2212yes—
57C2×#391C2212yes—
58C2×#391C2211yescharacteristic
59C2×#389C2214yes—
60C2×#389C2214yes—
61C2×#389C2212yes—
62C2×#389C2212yes—
63C2×#389C2212yes—
64C2×#389C2212yes—
65C2×#389C2211yescharacteristic
66C2×#389C2212yes—
67C2×#389C2211yescharacteristic
68C2×#387C2212yes—
69C2×#387C2212yes—
70C2×#387C2214yes—
71C2×#387C2214yes—
72C2×#387C2214yes—
73C2×#387C2214yes—
74C2×#387C2214yes—
75C2×#385C2214yes—
76C2×#385C2212yes—
77C2×#385C2212yes—
78C2×#385C2214yes—
79C2×#384C2212yes—
80C2×#384C2211yescharacteristic
81C2×#384C2212yes—
82C2×#384C2212yes—
83C2×#384C2212yes—
84C2×#384C2211yescharacteristic
85C2×#384C2212yes—
86C2×#384C22116yes—
87C2×#382C2212yes—
88C2×#380C2211yescharacteristic
89C2×#380C2214yes—
90C2×#380C2211yescharacteristic
91C2×C4×C4⋊C4C2212yes—
92C2×C4×C4⋊C4C2212yes—
93C2×C4×C4⋊C4C2211yescharacteristic
94C2×C4×C4⋊C4C2212yes—
95C2×C4×C4⋊C4C2211yescharacteristic
96C2×C4×C4⋊C4C2212yes—
97C2×C4×C4⋊C4C2212yes—
98C2×C4×C22⋊C4C2214yes—
99C2×C4×C22⋊C4C2212yes—
100C2×C4×C22⋊C4C2212yes—
101C23×Q8C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C24×C4C2311yescharacteristic
104C24×C4C2311yescharacteristic
105C24×C4C2311yescharacteristic
106C24×C4C2314yes—
107C2×#125—48no—
108C2×#125—416no—
109C2×#125C23116yes—
110C2×#125C23132yes—
111C2×C22⋊Q8—28no—
112C2×C22⋊Q8Q8116yescomplemented
113C2×C22⋊Q8C2318yes—
114C2×C22⋊Q8C2318yes—
115C2×C4×Q8—28no—
116C2×C4×Q8—28no—
117C2×C4×Q8—48no—
118C2×C4×Q8C2318yes—
119C2×C4×Q8C2318yes—
120C2×C4×Q8C23116yes—
121C22×C4⋊C4—216no—
122C22×C4⋊C4—48no—
123C22×C4⋊C4—48no—
124C22×C4⋊C4C23116yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C23116yes—
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2314yes—
144C22×C4⋊C4C2314yes—
145C22×C4⋊C4C2314yes—
146C22×C4⋊C4C2314yes—
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2314yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C23116yes—
151C22×C4⋊C4C2312yes—
152C22×C4⋊C4C2311yescharacteristic
153C22×C4⋊C4C2312yes—
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2312yes—
157C22×C22⋊C4C2312yes—
158C22×C42—28no—
159C22×C42C2311yescharacteristic
160C22×C42C2311yescharacteristic
161C22×C42C2311yescharacteristic
162C22×C42C2318yes—
163C22×C42C2312yes—
164C22×C42C2312yes—
165C22×C42C2312yes—
166C22×C42C2312yes—
167C22×C42C2314yes—
168C22×C42C2314yes—
169C22×C42C2311yescharacteristic
170#400C23132yes—
171#400C23132yes—
172#398C23132yes—
173#398C23132yes—
174#398C23132yes—
175#398C23132yes—
176#398C23132yes—
177#398C23132yes—
178#391C2318yes—
179#391C23116yes—
180#391C23116yes—
181#391C23116yes—
182#391C2318yes—
183#389C23116yes—
184#389C23116yes—
185#389C23116yes—
186#389C23116yes—
187#389C23132yes—
188#389C23116yes—
189#389C23132yes—
190#389C2318yes—
191#389C2318yes—
192#387C23132yes—
193#387C23132yes—
194#387C23132yes—
195#387C23132yes—
196#387C23132yes—
197#387C23116yes—
198#387C23116yes—
199#385C23132yes—
200#385C23132yes—
201#385C23116yes—
202#385C23116yes—
203#384C23116yes—
204#384C2318yes—
205#384C2318yes—
206#384C23116yes—
207#384C23116yes—
208#384C23116yes—
209#384C23116yes—
210#382C23116yes—
211#380C2318yes—
212#380C23132yes—
213#380C2318yes—
214C4×C4⋊C4C23116yes—
215C4×C4⋊C4C23116yes—
216C4×C4⋊C4C23116yes—
217C4×C4⋊C4C2318yes—
218C4×C4⋊C4C23116yes—
219C4×C4⋊C4C2318yes—
220C4×C4⋊C4C23116yes—
221C4×C22⋊C4C23132yes—
222C4×C22⋊C4C23116yes—
223C4×C22⋊C4C23116yes—
224C2×#95C2312yes—
225C2×#95C2314yes—
226C2×#95C2312yes—
227C2×#95C2312yes—
228C2×#95C2312yes—
229C2×#95C2314yes—
230C2×#95C2314yes—
231C2×#95C2312yes—
232C2×#95C2312yes—
233C2×#95C2312yes—
234C2×#95C2314yes—
235C2×#95C2314yes—
236C2×#95C2314yes—
237C2×#95C2314yes—
238C2×#95C2312yes—
239C2×#95C2314yes—
240C2×#95C2312yes—
241C2×#95C2312yes—
242C25C2411yescharacteristic
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8C2×Q818yes—
249C22×Q8C2414yes—
250C22×Q8C2414yes—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4—24no—
257C23×C4—24no—
258C23×C4—216no—
259C23×C4—48no—
260C23×C4—48no—
261C23×C4C2×Q818yes—
262C23×C4#4118yes—
263C23×C4C2414yes—
264C23×C4C2412yes—
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2412yes—
268C23×C4C2412yes—
269C23×C4C2412yes—
270C23×C4C2414yes—
271C23×C4C2414yes—
272C23×C4C2414yes—
273C23×C4C2412yes—
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C23×C4C2414yes—
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C23×C4C24116yes—
281C23×C4C2411yescharacteristic
282C23×C4C2411yescharacteristic
283#125—432no—
284#125—464no—
285C22⋊Q8—232no—
286C22⋊Q8C2×Q8164yescomplemented
287C4×Q8—232no—
288C4×Q8—232no—
289C4×Q8—432no—
290C2×C4⋊C4—216no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—264no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—432no—
308C2×C4⋊C4—432no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×Q818yes—
319C2×C4⋊C4C2×Q8116yes—
320C2×C4⋊C4C2418yes—
321C2×C4⋊C4C24116yes—
322C2×C4⋊C4C2418yes—
323C2×C4⋊C4C2418yes—
324C2×C4⋊C4C2418yes—
325C2×C4⋊C4C24116yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2418yes—
329C2×C4⋊C4C2418yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2418yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2418yes—
334C2×C4⋊C4C2418yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2418yes—
337C2×C4⋊C4C24116yes—
338C2×C4⋊C4C2418yes—
339C2×C4⋊C4C24116yes—
340C2×C4⋊C4C24116yes—
341C2×C4⋊C4C24116yes—
342C2×C4⋊C4C24116yes—
343C2×C4⋊C4C24116yes—
344C2×C4⋊C4C24116yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2418yes—
347C2×C22⋊C4—28no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—28no—
350C2×C22⋊C4C2×Q8116yes—
351C2×C22⋊C4C2418yes—
352C2×C22⋊C4C2418yes—
353C2×C22⋊C4C2418yes—
354C2×C22⋊C4C2418yes—
355C2×C42—28no—
356C2×C42—28no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—28no—
361C2×C42—28no—
362C2×C42—232no—
363C2×C42—48no—
364C2×C42—48no—
365C2×C42#4118yes—
366C2×C42C2418yes—
367C2×C42C2418yes—
368C2×C42C2418yes—
369C2×C42C2418yes—
370C2×C42C2414yes—
371C2×C42C2414yes—
372C2×C42C2414yes—
373C2×C42C24116yes—
374C2×C42C24116yes—
375C2×C42C2414yes—
376#95C2418yes—
377#95C2418yes—
378#95C2418yes—
379#95C24116yes—
380#95C2418yes—
381#95C24116yes—
382#95C24116yes—
383#95C2418yes—
384#95C24116yes—
385#95C2418yes—
386#95C2418yes—
387#95C2418yes—
388#95C24116yes—
389#95C2418yes—
390#95C2418yes—
391#95C24116yes—
392#95C24116yes—
393#95C24116yes—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24C22×Q814yes—
399C24C2×#4114yes—
400C242−1+414yes—
401C24C2512yes—
402C24C2511yescharacteristic
403C2×Q8—232no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—28no—
409C2×Q8—416no—
410C2×Q8C22×Q8116yes—
411C22×C4—232no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—232no—
426C22×C4—232no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—232no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—832no—
461C22×C4C22×Q818yes—
462C22×C4C22×Q818yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q8116yes—
465C22×C4C22×Q814yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#41116yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C4H(5,𝔽2)14yes—
474C22×C42−1+414yes—
475C22×C4C2514yes—
476C22×C4C2514yes—
477C22×C4C2514yes—
478C22×C4C2514yes—
479C22×C4C2514yes—
480C22×C4C2518yes—
481C22×C4C2518yes—
482C22×C4C2514yes—
483C22×C4C2514yes—
484C22×C4C2514yes—
485C22×C4C2512yes—
486C22×C4C2512yes—
487C22×C4C2512yes—
488C22×C4C2512yes—
489C22×C4C2512yes—
490C22×C4C2512yes—
491C4⋊C4—232no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—216no—
498C4⋊C4—28no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—232no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—48no—
508C4⋊C4—416no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—416no—
512C4⋊C4—416no—
513C4⋊C4—416no—
514C4⋊C4—416no—
515C4⋊C4—416no—
516C4⋊C4C22×Q8116yes—
517C4⋊C4C22×Q8132yes—
518C22⋊C4—216no—
519C22⋊C4—216no—
520C22⋊C4—216no—
521C22⋊C4C22×Q8132yes—
522C42—28no—
523C42—28no—
524C42—28no—
525C42—216no—
526C42—216no—
527C42—216no—
528C42—216no—
529C42—416no—
530C42—416no—
531C42C2×#41116yes—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23#557116yes—
543C23C23×Q814yes—
544C23C23×Q812yes—
545C23C22×#4114yes—
546C23C22×#4112yes—
547C23C22×#4112yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+414yes—
551C23C2×2−1+412yes—
552C23C2×2−1+412yes—
553C23C2611yescharacteristic
554Q8—232no—
555Q8—416no—
556C2×C4—232no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—216no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—216no—
575C2×C4—232no—
576C2×C4—232no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—416no—
606C2×C4—416no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—832no—
610C2×C4#552116yes—
611C2×C4#554116yes—
612C2×C4C23×Q818yes—
613C2×C4C23×Q818yes—
614C2×C4C23×Q814yes—
615C2×C4C23×Q814yes—
616C2×C4C22×#4114yes—
617C2×C4C22×#4114yes—
618C2×C4C22×#4114yes—
619C2×C4C22×#4114yes—
620C2×C4C22×#4114yes—
621C2×C4C2×H(5,𝔽2)14yes—
622C2×C4C2×H(5,𝔽2)14yes—
623C2×C4C2×2−1+414yes—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—232no—
629C22—24no—
630C22—24no—
631C22C2×#55214yes—
632C22C2×#55414yes—
633C22C2×#55718yes—
634C22C2×#55714yes—
635C22#349214yes—
636C22#350314yes—
637C22#350814yes—
638C22#355214yes—
639C22C24×Q811yescharacteristic
640C22C23×#4111yescharacteristic
641C22C23×#4111yescharacteristic
642C22C22×H(5,𝔽2)11yescharacteristic
643C22C22×H(5,𝔽2)11yescharacteristic
644C22C22×2−1+411yescharacteristic
645C22C22×2−1+411yescharacteristic
646C4—24no—
647C4—24no—
648C4—28no—
649C4—24no—
650C4—28no—
651C4—24no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—48no—
658C4—48no—
659C4—44no—
660C4C2×#55218yes—
661C4C2×#55418yes—
662C2—28no—
663C2#5932118yesdirect product factor
664C2C22×#55411yescharacteristic
665C2C22×#55211yescharacteristic
666C2C22×#55711yescharacteristic
667C2C2×#355211yescharacteristic
668C2C2×#349211yescharacteristic
669C2C2×#350311yescharacteristic
670C2C2×#350811yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59321C2direct (N × H)
2. C2×#2791C2semidirect
3. C2×#2582C2semidirect
4. C2×#2552C2semidirect
5. C2×#2552C2semidirect
6. C2×#2565C2semidirect
7. C2×#2511C2semidirect
9. C2×#2506C2semidirect
13. C2×#2355C2semidirect
14. C2×Q8×C4⋊C4C2semidirect
30. #2791C22semidirect
31. #2582C22semidirect
32. #2565C22semidirect
33. #2552C22semidirect
34. #2552C22semidirect
35. #2511C22semidirect
36. #2506C22semidirect
49. #2355C22semidirect
50. Q8×C4⋊C4C22semidirect
112. C2×C22⋊Q8Q8semidirect
286. C22⋊Q8C2×Q8semidirect
663. C2#59321direct (N × H)

Maximal subgroups

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

subgroupindex
#593212
C2×#27912
C2×#25822
C2×#25522
C2×#25652
C2×#25112
C2×#25022
C2×#25062
C2×#24422
C2×#24732
C2×#23602
C2×#23552
C2×Q8×C4⋊C42
C2×#23452
C2×C4×C22⋊Q82
C22×#3842