Finite Group Databasebeta

← C2×#36290 #7629941

Subgroups — C2×#36290 #7629941

Order 512.

6602 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C2×#2851C212yesmaximal, complemented
3#36290C2132yesmaximal, direct product factor
4C2×#2571C211yesmaximal, characteristic, complemented
5C2×#2552C211yesmaximal, characteristic, complemented
6C2×#2509C212yesmaximal, complemented
7C2×#2507C212yesmaximal, complemented
8C2×#2505C214yesmaximal, complemented
9C2×#2442C212yesmaximal, complemented
10C2×#2442C212yesmaximal, complemented
11C2×#2473C214yesmaximal, complemented
12C2×#2349C212yesmaximal
13C2×D4×C4⋊C4C211yesmaximal, characteristic
14C2×D4×C4⋊C4C212yesmaximal
15C2×D4×C4⋊C4C211yesmaximal, characteristic
16C2×C4×#125C211yesmaximal, characteristic, complemented
17C22×#389C212yesmaximal, complemented
18C2×C42×D4C211yesmaximal, characteristic
19C22×#125C2211yescharacteristic
20C22×#125C2212yes—
21C22×#125C2212yes—
22C22×C4×D4C2212yes—
23C22×C4×D4C2212yes—
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2214yes—
29C23×C4⋊C4C2212yes—
30C23×C42C2212yes—
31#2859C22116yescomplemented
32#2851C22132yescomplemented
33#2571C22116yescomplemented
34#2552C22116yescomplemented
35#2509C22132yescomplemented
36#2507C22132yescomplemented
37#2505C22164yescomplemented
38#2473C22164yescomplemented
39#2442C22132yescomplemented
40#2442C22132yescomplemented
41C2×#400C2214yes—
42C2×#400C2214yes—
43C2×#400C2214yes—
44C2×#398C2214yes—
45C2×#398C2214yes—
46C2×#398C2214yes—
47#2349C22132yes—
48D4×C4⋊C4C22116yes—
49D4×C4⋊C4C22116yes—
50D4×C4⋊C4C22132yes—
51C4×#125C22116yescomplemented
52C2×#389C2211yescharacteristic
53C2×#389C2212yes—
54C2×#389C2212yes—
55C2×#389C2212yes—
56C2×#389C2212yes—
57C2×#389C2212yes—
58C2×#389C2212yes—
59C2×#389C2212yes—
60C2×#389C2212yes—
61C2×#389C2214yes—
62C2×#389C2212yes—
63C2×#389C2212yes—
64C2×#389C2214yes—
65C2×#389C2211yescharacteristic
66C2×#389C22132yescomplemented
67C2×#387C2214yes—
68C2×#387C2214yes—
69C2×#387C2212yes—
70C2×#387C2212yes—
71C2×#385C2214yes—
72C2×#385C2214yes—
73C2×#385C2214yes—
74C2×#385C2214yes—
75C2×#385C2214yes—
76C2×#385C2214yes—
77C2×#384C2211yescharacteristic
78C2×#384C2211yescharacteristic
79C2×#384C2212yes—
80C2×#382C2212yes—
81C2×#382C2212yes—
82C2×#382C2212yes—
83C2×#382C2211yescharacteristic
84C2×#380C2212yes—
85C2×#380C2212yes—
86C2×#380C2214yes—
87C2×#380C2214yes—
88C42×D4C22116yes—
89C2×C4×C4⋊C4C2211yescharacteristic
90C2×C4×C4⋊C4C2211yescharacteristic
91C2×C4×C4⋊C4C2212yes—
92C2×C4×C4⋊C4C2212yes—
93C2×C4×C4⋊C4C2211yescharacteristic
94C2×C4×C4⋊C4C2211yescharacteristic
95C2×C4×C4⋊C4C2211yescharacteristic
96C2×C4×C22⋊C4C2212yes—
97C2×C4×C22⋊C4C2212yes—
98C2×C4×C22⋊C4C2212yes—
99C22×#95C2214yes—
100C2×C43C2211yescharacteristic
101C23×D4C2311yescharacteristic
102C24×C4C2312yes—
103C24×C4C2312yes—
104C24×C4C2314yes—
105C24×C4C2312yes—
106C24×C4C2314yes—
107C2×#125—28no—
108C2×#125—48no—
109C2×#125—48no—
110C2×#125C2318yes—
111C2×#125C23116yes—
112C2×#125C23116yes—
113C2×C4×D4—216no—
114C2×C4×D4—28no—
115C2×C4×D4—28no—
116C2×C4×D4—216no—
117C2×C4×D4Q8116yes—
118C2×C4×D4C23116yes—
119C2×C4×D4C23116yes—
120C2×C4×D4C2318yes—
121C2×C4×D4C2318yes—
122C2×C4×D4C2318yes—
123C22×C4⋊C4—216no—
124C22×C4⋊C4—48no—
125C22×C4⋊C4—416no—
126C22×C4⋊C4C23116yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C23116yes—
143C22×C4⋊C4C23132yes—
144C22×C4⋊C4C2314yes—
145C22×C4⋊C4C2314yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2314yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C2314yes—
151C22×C4⋊C4C2312yes—
152C22×C4⋊C4C2311yescharacteristic
153C22×C4⋊C4C2311yescharacteristic
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2314yes—
158C22×C22⋊C4C2312yes—
159C22×C42—48no—
160C22×C42C2311yescharacteristic
161C22×C42C2312yes—
162C22×C42C2311yescharacteristic
163C22×C42C2311yescharacteristic
164C22×C42C2312yes—
165C22×C42C2312yes—
166C22×C42C2311yescharacteristic
167C22×C42C2311yescharacteristic
168C22×C42C2311yescharacteristic
169C22×C42C23116yes—
170C22×C42C2312yes—
171C22×C42C2312yes—
172C22×C42C2311yescharacteristic
173#400C23132yes—
174#400C23132yes—
175#400C23132yes—
176#398C23132yes—
177#398C23132yes—
178#398C23132yes—
179#389C2318yes—
180#389C2318yes—
181#389C23116yes—
182#389C23116yes—
183#389C23116yes—
184#389C23116yes—
185#389C23116yes—
186#389C23116yes—
187#389C23116yes—
188#389C23116yes—
189#389C23116yes—
190#389C23116yes—
191#389C23132yes—
192#389C23132yes—
193#387C23116yes—
194#387C23116yes—
195#387C23132yes—
196#387C23132yes—
197#385C23132yes—
198#385C23132yes—
199#385C23132yes—
200#385C23132yes—
201#385C23132yes—
202#385C23132yes—
203#384C2318yes—
204#384C2318yes—
205#384C23116yes—
206#382C2318yes—
207#382C23116yes—
208#382C23116yes—
209#382C23116yes—
210#380C23116yes—
211#380C23116yes—
212#380C23132yes—
213#380C23132yes—
214C4×C4⋊C4C2318yes—
215C4×C4⋊C4C23116yes—
216C4×C4⋊C4C2318yes—
217C4×C4⋊C4C23116yes—
218C4×C4⋊C4C2318yes—
219C4×C4⋊C4C2318yes—
220C4×C4⋊C4C2318yes—
221C4×C22⋊C4C23116yes—
222C4×C22⋊C4C23116yes—
223C4×C22⋊C4C23116yes—
224C2×#95C23132yes—
225C2×#95C2312yes—
226C2×#95C2312yes—
227C2×#95C2312yes—
228C2×#95C2312yes—
229C2×#95C2314yes—
230C2×#95C2314yes—
231C2×#95C2314yes—
232C2×#95C2314yes—
233C2×#95C2312yes—
234C2×#95C2314yes—
235C2×#95C2312yes—
236C2×#95C2312yes—
237C2×#95C2314yes—
238C2×#95C2314yes—
239C2×#95C2312yes—
240C2×#95C2314yes—
241C2×#95C2312yes—
242C43C2318yes—
243C25C2412yes—
244C22×D4C2×Q818yes—
245C22×D4#4118yes—
246C22×D4#4118yes—
247C22×D4C2414yes—
248C23×C4—28no—
249C23×C4—28no—
250C23×C4—28no—
251C23×C4—28no—
252C23×C4—216no—
253C23×C4—216no—
254C23×C4—216no—
255C23×C4—28no—
256C23×C4—28no—
257C23×C4—48no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—48no—
261C23×C4—48no—
262C23×C4C2×Q8116yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C23×C4C2414yes—
268C23×C4C2414yes—
269C23×C4C2412yes—
270C23×C4C2412yes—
271C23×C4C2412yes—
272C23×C4C24116yes—
273C23×C4C2418yes—
274C23×C4C2418yes—
275C23×C4C2418yes—
276C23×C4C2411yescharacteristic
277C23×C4C2412yes—
278C23×C4C2412yes—
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C23×C4C24116yes—
282C23×C4C2411yescharacteristic
283C23×C4C2411yescharacteristic
284#125—232no—
285#125—432no—
286#125—432no—
287C4×D4—264no—
288C4×D4—232no—
289C4×D4—232no—
290C4×D4—264no—
291C4×D4C2×Q8164yes—
292C2×C4⋊C4—216no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—216no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—264no—
308C2×C4⋊C4—432no—
309C2×C4⋊C4—464no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×Q818yes—
319C2×C4⋊C4#4118yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2418yes—
322C2×C4⋊C4C2418yes—
323C2×C4⋊C4C24116yes—
324C2×C4⋊C4C2418yes—
325C2×C4⋊C4C24116yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2418yes—
329C2×C4⋊C4C2418yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2418yes—
333C2×C4⋊C4C2418yes—
334C2×C4⋊C4C2418yes—
335C2×C4⋊C4C24116yes—
336C2×C4⋊C4C24116yes—
337C2×C4⋊C4C24116yes—
338C2×C4⋊C4C24116yes—
339C2×C4⋊C4C24116yes—
340C2×C4⋊C4C2418yes—
341C2×C4⋊C4C2418yes—
342C2×C4⋊C4C24116yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2414yes—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—216no—
348C2×C22⋊C4—216no—
349C2×C22⋊C4C2×Q8116yes—
350C2×C22⋊C4C2418yes—
351C2×C22⋊C4C2418yes—
352C2×C22⋊C4C24116yes—
353C2×C22⋊C4C24116yes—
354C2×C22⋊C4C2418yes—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—28no—
358C2×C42—28no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—28no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—432no—
365C2×C42—44no—
366C2×C42—44no—
367C2×C42C2×Q818yes—
368C2×C42C2414yes—
369C2×C42C2414yes—
370C2×C42C2418yes—
371C2×C42C2418yes—
372C2×C42C2414yes—
373C2×C42C2414yes—
374C2×C42C2418yes—
375C2×C42C2414yes—
376C2×C42C2414yes—
377C2×C42C2418yes—
378C2×C42C2418yes—
379C2×C42C2414yes—
380#95C2418yes—
381#95C24116yes—
382#95C24116yes—
383#95C24116yes—
384#95C2418yes—
385#95C2418yes—
386#95C2418yes—
387#95C2418yes—
388#95C2418yes—
389#95C2418yes—
390#95C2418yes—
391#95C2418yes—
392#95C24116yes—
393#95C24116yes—
394#95C24116yes—
395#95C24116yes—
396#95C24116yes—
397C24—24no—
398C24—24no—
399C24—24no—
400C24—24no—
401C24C22×Q818yes—
402C24C2×#4118yes—
403C24C2×#4118yes—
404C24C2514yes—
405C24C2511yescharacteristic
406C2×D4#125164yescomplemented
407C2×D4C22×Q8116yes—
408C2×D4C2×#41116yes—
409C2×D4C2×#41116yes—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—264no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—232no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—232no—
454C22×C4—232no—
455C22×C4—216no—
456C22×C4—216no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—464no—
463C22×C4—432no—
464C22×C4—432no—
465C22×C4—832no—
466C22×C4C22×Q814yes—
467C22×C4C22×Q818yes—
468C22×C4C22×Q8132yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q814yes—
471C22×C4C2×#4118yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C42−1+414yes—
478C22×C4C2512yes—
479C22×C4C2514yes—
480C22×C4C2514yes—
481C22×C4C2514yes—
482C22×C4C2514yes—
483C22×C4C2518yes—
484C22×C4C2512yes—
485C22×C4C2514yes—
486C22×C4C2518yes—
487C22×C4C2512yes—
488C22×C4C2514yes—
489C22×C4C2514yes—
490C22×C4C2512yes—
491C22×C4C2512yes—
492C22×C4C2512yes—
493C22×C4C2512yes—
494C4⋊C4—232no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—28no—
503C4⋊C4—216no—
504C4⋊C4—28no—
505C4⋊C4—232no—
506C4⋊C4—216no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—416no—
510C4⋊C4—416no—
511C4⋊C4—416no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—416no—
515C4⋊C4—416no—
516C4⋊C4—416no—
517C4⋊C4C22×Q8116yes—
518C4⋊C4C2×#41116yes—
519C22⋊C4—216no—
520C22⋊C4—216no—
521C22⋊C4—232no—
522C22⋊C4—232no—
523C22⋊C4C22×Q8132yes—
524C42—28no—
525C42—28no—
526C42—216no—
527C42—216no—
528C42—28no—
529C42—28no—
530C42—216no—
531C42—28no—
532C42—28no—
533C42—48no—
534C42—48no—
535C42C22×Q8116yes—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—216no—
541C23—216no—
542C23—216no—
543C23—216no—
544C23—216no—
545C23—216no—
546C23C2×#125132yes—
547C23C23×Q818yes—
548C23C23×Q812yes—
549C23C22×#4118yes—
550C23C22×#4118yes—
551C23C22×#4112yes—
552C23C22×#4112yes—
553C23C22×#4112yes—
554C23C2×H(5,𝔽2)12yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×2−1+412yes—
557C23C2611yescharacteristic
558D4C2×#125164yescomplemented
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—216no—
570C2×C4—216no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—264no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—432no—
603C2×C4—432no—
604C2×C4—464no—
605C2×C4—44no—
606C2×C4—416no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—416no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—832no—
615C2×C4C2×#125116yes—
616C2×C4#554116yes—
617C2×C4C23×Q814yes—
618C2×C4C23×Q818yes—
619C2×C4C23×Q814yes—
620C2×C4C23×Q814yes—
621C2×C4C22×#4118yes—
622C2×C4C22×#4114yes—
623C2×C4C22×#4114yes—
624C2×C4C22×#4114yes—
625C2×C4C2×H(5,𝔽2)14yes—
626C2×C4C2×H(5,𝔽2)14yes—
627C2×C4C2×2−1+414yes—
628C22—28no—
629C22—28no—
630C22—28no—
631C22—264no—
632C22—28no—
633C22—28no—
634C22—28no—
635C22C22×#125116yes—
636C22C22×#12514yes—
637C22C2×#55414yes—
638C22#348114yes—
639C22#348214yes—
640C22#350314yes—
641C22#351914yes—
642C22#352014yes—
643C22C24×Q811yescharacteristic
644C22C23×#4111yescharacteristic
645C22C23×#4111yescharacteristic
646C22C23×#4111yescharacteristic
647C22C22×H(5,𝔽2)11yescharacteristic
648C22C22×H(5,𝔽2)11yescharacteristic
649C22C22×2−1+411yescharacteristic
650C4—24no—
651C4—24no—
652C4—24no—
653C4—28no—
654C4—28no—
655C4—44no—
656C4—48no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—48no—
662C4—42no—
663C4—44no—
664C4C22×#12518yes—
665C4C2×#55418yes—
666C2—216no—
667C2#3629018yesdirect product factor
668C2C23×#12511yescharacteristic
669C2C22×#55411yescharacteristic
670C2C2×#348111yescharacteristic
671C2C2×#348211yescharacteristic
672C2C2×#351911yescharacteristic
673C2C2×#352011yescharacteristic
674C2C2×#350311yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2859C2semidirect
2. C2×#2851C2semidirect
3. #36290C2direct (N × H)
4. C2×#2571C2semidirect
5. C2×#2552C2semidirect
6. C2×#2509C2semidirect
7. C2×#2507C2semidirect
8. C2×#2505C2semidirect
9. C2×#2442C2semidirect
10. C2×#2442C2semidirect
11. C2×#2473C2semidirect
16. C2×C4×#125C2semidirect
17. C22×#389C2semidirect
31. #2859C22semidirect
32. #2851C22semidirect
33. #2571C22semidirect
34. #2552C22semidirect
35. #2509C22semidirect
36. #2507C22semidirect
37. #2505C22semidirect
38. #2473C22semidirect
39. #2442C22semidirect
40. #2442C22semidirect
51. C4×#125C22semidirect
66. C2×#389C22semidirect
406. C2×D4#125semidirect
558. D4C2×#125semidirect
667. C2#36290direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#28512
#362902
C2×#25712
C2×#25522
C2×#25092
C2×#25072
C2×#25052
C2×#24422
C2×#24732
C2×#23492
C2×D4×C4⋊C42
C2×C4×#1252
C22×#3892
C2×C42×D42