Finite Group Databasebeta

← C2×#59679 #7666224

Subgroups — C2×#59679 #7666224

Order 512.

6142 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59679C2132yesmaximal, direct product factor
2C2×#2800C214yesmaximal, complemented
3C2×#2548C211yesmaximal, characteristic
4C2×#2548C211yesmaximal, characteristic
5C2×#2545C212yesmaximal, complemented
6C2×#2566C212yesmaximal, complemented
7C2×#2513C212yesmaximal, complemented
8C2×#2512C212yesmaximal, complemented
9C2×#2512C212yesmaximal, complemented
10C2×#2447C212yesmaximal, complemented
11C2×#2466C214yesmaximal, complemented
12C2×#2367C212yesmaximal, complemented
13C2×#2357C212yesmaximal, complemented
14C2×#2355C211yesmaximal, characteristic, complemented
15C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
16C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
17C2×C4×#127C211yesmaximal, characteristic
18C2×C4×#125C211yesmaximal, characteristic, complemented
19C22×C4⋊Q8C2211yescharacteristic
20C22×#127C2211yescharacteristic
21C22×#126C2214yes—
22C22×#125C2211yescharacteristic
23C22×#124C2212yes—
24C22×#124C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2212yes—
28C22×C4×D4C2212yes—
29C22×C4×D4C2212yes—
30C22×C4×D4C2212yes—
31#2800C22164yescomplemented
32#2566C22132yescomplemented
33#2548C22116yes—
34#2548C22116yes—
35#2545C22132yescomplemented
36#2513C22132yescomplemented
37#2512C22132yescomplemented
38#2512C22132yescomplemented
39#2466C22164yescomplemented
40#2447C22132yescomplemented
41C2×#397C2214yes—
42C2×#397C2214yes—
43C2×#397C2214yes—
44C2×#396C2214yes—
45C2×#396C2214yes—
46C2×#396C2214yes—
47C2×#394C2214yes—
48C2×#394C2214yes—
49#2367C22132yescomplemented
50#2357C22132yescomplemented
51#2355C22116yescomplemented
52Q8×C4⋊C4C22116yescomplemented
53C4×C4⋊Q8C22116yescomplemented
54C4×#127C22116yes—
55C4×#125C22116yescomplemented
56C2×#391C2212yes—
57C2×#391C2211yescharacteristic
58C2×#391C2211yescharacteristic
59C2×#390C2212yes—
60C2×#390C2212yes—
61C2×#390C2212yes—
62C2×#390C2212yes—
63C2×#390C2212yes—
64C2×#390C2212yes—
65C2×#390C2212yes—
66C2×#390C2212yes—
67C2×#390C2212yes—
68C2×#389C2212yes—
69C2×#389C2214yes—
70C2×#389C2212yes—
71C2×#389C2212yes—
72C2×#389C2211yescharacteristic
73C2×#389C2211yescharacteristic
74C2×#388C2214yes—
75C2×#388C2214yes—
76C2×#388C2214yes—
77C2×#388C2214yes—
78C2×#388C2214yes—
79C2×#387C2212yes—
80C2×#387C2214yes—
81C2×#387C2212yes—
82C2×#385C2214yes—
83C2×#385C2214yes—
84C2×#384C2211yescharacteristic
85C2×#384C2212yes—
86C2×#384C2211yescharacteristic
87C2×#382C2212yes—
88C2×#382C2211yescharacteristic
89C2×#381C2214yes—
90C2×C4×C4⋊C4C2212yes—
91C2×C4×C4⋊C4C2211yescharacteristic
92C2×C4×C4⋊C4C2212yes—
93C2×C4×C4⋊C4C2212yes—
94C2×C4×C4⋊C4C2211yescharacteristic
95C2×C4×C4⋊C4C2212yes—
96C2×C4×C4⋊C4C2212yes—
97C2×C4×C4⋊C4C2212yes—
98C2×C4×C4⋊C4C2212yes—
99C2×C4×C22⋊C4C2214yes—
100C2×C43C2211yescharacteristic
101C23×Q8C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C23×D4C2312yes—
104C23×D4C2312yes—
105C23×D4C2312yes—
106C24×C4C2314yes—
107C2×C4⋊Q8—28no—
108C2×C4⋊Q8C2318yes—
109C2×#127—28no—
110C2×#127C2318yes—
111C2×#126—416no—
112C2×#126C23132yes—
113C2×#125—28no—
114C2×#125C2318yes—
115C2×#124—216no—
116C2×#124—48no—
117C2×#124C23116yes—
118C2×#124C23116yes—
119C2×C4×Q8—28no—
120C2×C4×Q8—28no—
121C2×C4×Q8—48no—
122C2×C4×Q8C2318yes—
123C2×C4×Q8C2318yes—
124C2×C4×Q8C23116yes—
125C2×C4×D4—216no—
126C2×C4×D4—48no—
127C2×C4×D4—48no—
128C2×C4×D4C23116yes—
129C2×C4×D4C23116yes—
130C2×C4×D4C23116yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2312yes—
152C22×C22⋊C4C2314yes—
153C22×C22⋊C4C2314yes—
154C22×C22⋊C4C2314yes—
155C22×C22⋊C4C2314yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2314yes—
158C22×C22⋊C4C2314yes—
159C22×C22⋊C4C2314yes—
160C22×C22⋊C4C2314yes—
161C22×C22⋊C4C2314yes—
162C22×C22⋊C4C2314yes—
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165C22×C42C2312yes—
166C22×C42C2312yes—
167C22×C42C2312yes—
168C22×C42C2312yes—
169C22×C42C2311yescharacteristic
170C22×C42C2311yescharacteristic
171C22×C42C2314yes—
172C22×C42C2314yes—
173C22×C42C2311yescharacteristic
174C22×C42C2311yescharacteristic
175C22×C42C2311yescharacteristic
176#397C23132yes—
177#397C23132yes—
178#397C23132yes—
179#396C23132yes—
180#396C23132yes—
181#396C23132yes—
182#394C23132yes—
183#394C23132yes—
184#391C2318yes—
185#391C2318yes—
186#391C23116yes—
187#390C23116yes—
188#390C23116yes—
189#390C23116yes—
190#390C23116yes—
191#390C23116yes—
192#390C23116yes—
193#390C23116yes—
194#390C23116yes—
195#390C23116yes—
196#389C2318yes—
197#389C2318yes—
198#389C23116yes—
199#389C23116yes—
200#389C23116yes—
201#389C23132yes—
202#388C23132yes—
203#388C23132yes—
204#388C23132yes—
205#388C23132yes—
206#388C23132yes—
207#387C23116yes—
208#387C23116yes—
209#387C23132yes—
210#385C23132yes—
211#385C23132yes—
212#384C2318yes—
213#384C2318yes—
214#384C23116yes—
215#382C2318yes—
216#382C23116yes—
217#381C23132yes—
218C4×C4⋊C4C2318yes—
219C4×C4⋊C4C2318yes—
220C4×C4⋊C4C23116yes—
221C4×C4⋊C4C23116yes—
222C4×C4⋊C4C23116yes—
223C4×C4⋊C4C23116yes—
224C4×C4⋊C4C23116yes—
225C4×C4⋊C4C23116yes—
226C4×C4⋊C4C23116yes—
227C4×C22⋊C4C23132yes—
228C2×#95C2312yes—
229C2×#95C2312yes—
230C2×#95C2314yes—
231C2×#95C2314yes—
232C2×#95C2314yes—
233C2×#95C2312yes—
234C2×#95C2314yes—
235C2×#95C2312yes—
236C2×#95C2312yes—
237C2×#95C2312yes—
238C2×#95C2312yes—
239C2×#95C2312yes—
240C2×#95C2312yes—
241C2×#95C2312yes—
242C43C2318yes—
243C25C2414yes—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8#4118yes—
250C22×Q8C2414yes—
251C22×Q8C2414yes—
252C22×D4—28no—
253C22×D4—28no—
254C22×D4—28no—
255C22×D4—28no—
256C22×D4—28no—
257C22×D4—28no—
258C22×D4—28no—
259C22×D4—44no—
260C22×D4—44no—
261C22×D4C2418yes—
262C22×D4C2418yes—
263C22×D4C2418yes—
264C23×C4—216no—
265C23×C4—48no—
266C23×C4—48no—
267C23×C4C2412yes—
268C23×C4C2412yes—
269C23×C4C2412yes—
270C23×C4C2412yes—
271C23×C4C2412yes—
272C23×C4C2412yes—
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C23×C4C2414yes—
276C23×C4C2411yescharacteristic
277C23×C4C2412yes—
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C23×C4C24116yes—
282C23×C4C2411yescharacteristic
283C23×C4C2411yescharacteristic
284C4⋊Q8—232no—
285#127—232no—
286#126—464no—
287#125—232no—
288#124—264no—
289#124—432no—
290C4×Q8—232no—
291C4×Q8—232no—
292C4×Q8—432no—
293C4×D4—264no—
294C4×D4—432no—
295C4×D4—432no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4C2418yes—
318C2×C4⋊C4C2418yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2418yes—
322C2×C4⋊C4C2418yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2418yes—
326C2×C4⋊C4C2418yes—
327C2×C4⋊C4C2418yes—
328C2×C4⋊C4C2418yes—
329C2×C4⋊C4C2418yes—
330C2×C4⋊C4C2418yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2418yes—
334C2×C4⋊C4C24116yes—
335C2×C4⋊C4C24116yes—
336C2×C4⋊C4C2418yes—
337C2×C4⋊C4C2418yes—
338C2×C22⋊C4—216no—
339C2×C22⋊C4—216no—
340C2×C22⋊C4—216no—
341C2×C22⋊C4—216no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4—48no—
344C2×C22⋊C4—48no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4—48no—
349C2×C22⋊C4C24116yes—
350C2×C22⋊C4C24116yes—
351C2×C22⋊C4C24116yes—
352C2×C22⋊C4C24116yes—
353C2×C22⋊C4C24116yes—
354C2×C22⋊C4C24116yes—
355C2×C22⋊C4C24116yes—
356C2×C22⋊C4C24116yes—
357C2×C22⋊C4C24116yes—
358C2×C22⋊C4C24116yes—
359C2×C22⋊C4C24116yes—
360C2×C42—28no—
361C2×C42—28no—
362C2×C42—28no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—28no—
367C2×C42—24no—
368C2×C42—48no—
369C2×C42—48no—
370C2×C42—42no—
371C2×C42#4118yes—
372C2×C42#4118yes—
373C2×C42C2414yes—
374C2×C42C2414yes—
375C2×C42C2418yes—
376C2×C42C2418yes—
377C2×C42C2418yes—
378C2×C42C2418yes—
379C2×C42C2414yes—
380C2×C42C2414yes—
381C2×C42C24116yes—
382C2×C42C24116yes—
383C2×C42C2414yes—
384C2×C42C2414yes—
385C2×C42C2414yes—
386#95C2418yes—
387#95C2418yes—
388#95C2418yes—
389#95C24116yes—
390#95C24116yes—
391#95C24116yes—
392#95C2418yes—
393#95C2418yes—
394#95C2418yes—
395#95C2418yes—
396#95C2418yes—
397#95C2418yes—
398#95C2418yes—
399#95C24116yes—
400C24—28no—
401C24—28no—
402C24—28no—
403C24—28no—
404C24—28no—
405C24—28no—
406C24—28no—
407C24C2518yes—
408C24C2511yescharacteristic
409C2×Q8—232no—
410C2×Q8—28no—
411C2×Q8—28no—
412C2×Q8—28no—
413C2×Q8—28no—
414C2×Q8—28no—
415C2×Q8—416no—
416C2×Q8C2×#41116yes—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—216no—
420C2×D4—216no—
421C2×D4—216no—
422C2×D4—216no—
423C2×D4—216no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—432no—
427C2×D4—432no—
428C2×D4—432no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—232no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—832no—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4118yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4114yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C42−1+414yes—
481C22×C4C2514yes—
482C22×C4C2514yes—
483C22×C4C2512yes—
484C22×C4C2514yes—
485C22×C4C2514yes—
486C22×C4C2514yes—
487C22×C4C2518yes—
488C22×C4C2514yes—
489C22×C4C2514yes—
490C22×C4C2512yes—
491C22×C4C2512yes—
492C22×C4C2512yes—
493C22×C4C2514yes—
494C22×C4C2512yes—
495C22×C4C2512yes—
496C22×C4C2512yes—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—416no—
517C4⋊C4—416no—
518C22⋊C4—232no—
519C22⋊C4—232no—
520C22⋊C4—232no—
521C22⋊C4—232no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—416no—
528C22⋊C4—416no—
529C42—216no—
530C42—216no—
531C42—216no—
532C42—28no—
533C42—28no—
534C42—28no—
535C42—216no—
536C42—28no—
537C42—44no—
538C42—416no—
539C42—416no—
540C42C2×#41116yes—
541C42C2×#41116yes—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—416no—
550C23—416no—
551C23—416no—
552C23—416no—
553C23—416no—
554C23—416no—
555C23—416no—
556C23C22×#4112yes—
557C23C22×#4112yes—
558C23C22×#4112yes—
559C23C2×H(5,𝔽2)12yes—
560C23C2×H(5,𝔽2)12yes—
561C23C2×2−1+412yes—
562C23#58512yes—
563C23C2611yescharacteristic
564Q8—232no—
565Q8—416no—
566D4—432no—
567D4—432no—
568D4—432no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—216no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—416no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—44no—
620C2×C4—832no—
621C2×C4#553116yes—
622C2×C4#556116yes—
623C2×C4C22×#4114yes—
624C2×C4C22×#4114yes—
625C2×C4C22×#4114yes—
626C2×C4C22×#4118yes—
627C2×C4C22×#4114yes—
628C2×C4C22×#4114yes—
629C2×C4C22×#4114yes—
630C2×C4C22×#4114yes—
631C2×C4C2×H(5,𝔽2)14yes—
632C2×C4C2×H(5,𝔽2)14yes—
633C2×C4C2×2−1+414yes—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—832no—
642C22C2×#55314yes—
643C22C2×#55614yes—
644C22#348014yes—
645C22#350614yes—
646C22#350814yes—
647C22#352214yes—
648C22#355414yes—
649C22C23×#4111yescharacteristic
650C22C23×#4111yescharacteristic
651C22C23×#4111yescharacteristic
652C22C22×H(5,𝔽2)11yescharacteristic
653C22C22×H(5,𝔽2)11yescharacteristic
654C22C22×2−1+411yescharacteristic
655C22C2×#58511yescharacteristic
656C4—24no—
657C4—24no—
658C4—24no—
659C4—28no—
660C4—24no—
661C4—44no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—48no—
668C4—44no—
669C4—44no—
670C4C2×#55318yes—
671C4C2×#55618yes—
672C2—88no—
673C2#5967918yesdirect product factor
674C2C22×#55311yescharacteristic
675C2C22×#55611yescharacteristic
676C2C2×#348011yescharacteristic
677C2C2×#355411yescharacteristic
678C2C2×#350611yescharacteristic
679C2C2×#352211yescharacteristic
680C2C2×#350811yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC22×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59679C2direct (N × H)
2. C2×#2800C2semidirect
5. C2×#2545C2semidirect
6. C2×#2566C2semidirect
7. C2×#2513C2semidirect
8. C2×#2512C2semidirect
9. C2×#2512C2semidirect
10. C2×#2447C2semidirect
11. C2×#2466C2semidirect
12. C2×#2367C2semidirect
13. C2×#2357C2semidirect
14. C2×#2355C2semidirect
15. C2×Q8×C4⋊C4C2semidirect
16. C2×C4×C4⋊Q8C2semidirect
18. C2×C4×#125C2semidirect
31. #2800C22semidirect
32. #2566C22semidirect
35. #2545C22semidirect
36. #2513C22semidirect
37. #2512C22semidirect
38. #2512C22semidirect
39. #2466C22semidirect
40. #2447C22semidirect
49. #2367C22semidirect
50. #2357C22semidirect
51. #2355C22semidirect
52. Q8×C4⋊C4C22semidirect
53. C4×C4⋊Q8C22semidirect
55. C4×#125C22semidirect
673. C2#59679direct (N × H)

Maximal subgroups

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

subgroupindex
#596792
C2×#28002
C2×#25482
C2×#25452
C2×#25662
C2×#25132
C2×#25122
C2×#24472
C2×#24662
C2×#23672
C2×#23572
C2×#23552
C2×Q8×C4⋊C42
C2×C4×C4⋊Q82
C2×C4×#1272
C2×C4×#1252