Finite Group Databasebeta

← C2×#15133 #608365

Subgroups — C2×#15133 #608365

Order 512.

4486 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C212yesmaximal
2C22×#389C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4C22×#384C211yesmaximal, characteristic
5C22×C4×C4⋊C4C211yesmaximal, characteristic
6#15133C2116yesmaximal, direct product factor
7C2×#1441C212yesmaximal
8C2×#1441C212yesmaximal
9C2×#1441C212yesmaximal
10C2×C4⋊#95C211yesmaximal, characteristic
11C2×C4⋊#95C211yesmaximal, characteristic
12C2×C4⋊#95C211yesmaximal, characteristic
13C23×C4⋊C4C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2212yes—
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C23×C42C2211yescharacteristic
26C2×#398—216no—
27C2×#398C22116yes—
28C2×#389—28no—
29C2×#389—28no—
30C2×#389C2218yes—
31C2×#389C2218yes—
32C2×#384—28no—
33C2×#384C2218yes—
34C2×C4×C4⋊C4—28no—
35C2×C4×C4⋊C4C2218yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1441C22116yes—
47#1441C22116yes—
48#1441C22116yes—
49C4⋊#95C2218yes—
50C4⋊#95C2218yes—
51C4⋊#95C2218yes—
52C24×C4C2312yes—
53C24×C4C2312yes—
54C24×C4C2311yescharacteristic
55C24×C4C2312yes—
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—28no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—28no—
74C22×C4⋊C4—28no—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—28no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—28no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4D418yes—
90C22×C4⋊C4D418yes—
91C22×C4⋊C4Q818yes—
92C22×C4⋊C4Q818yes—
93C22×C4⋊C4C2318yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2318yes—
97C22×C4⋊C4C2318yes—
98C22×C4⋊C4C2318yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C42—24no—
104C22×C42—24no—
105C22×C42—24no—
106C22×C42—24no—
107C22×C42—24no—
108C22×C42—24no—
109C22×C42—24no—
110C22×C42D418yes—
111C22×C42Q818yes—
112C22×C42C2314yes—
113C22×C42C2314yes—
114C22×C42C2314yes—
115#398—264no—
116#389—232no—
117#389—232no—
118#384—232no—
119C4×C4⋊C4—232no—
120C2×#95—28no—
121C2×#95—28no—
122C2×#95—28no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95—44no—
128C2×#95—44no—
129C2×#95—44no—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C2×#95C2318yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C25C2411yescharacteristic
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4C2×D418yes—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×Q818yes—
206C23×C4C2×Q814yes—
207C23×C4C2×Q814yes—
208C23×C4C2×Q814yes—
209C23×C4C2×Q814yes—
210C23×C4C2×Q814yes—
211C23×C4C2×Q814yes—
212C23×C4C2×Q814yes—
213C23×C4#4114yes—
214C23×C4#4114yes—
215C23×C4#4114yes—
216C23×C4#4114yes—
217C23×C4#4114yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C2×C4⋊C4—232no—
230C2×C4⋊C4—232no—
231C2×C4⋊C4—232no—
232C2×C4⋊C4—232no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—432no—
260C2×C4⋊C4—432no—
261C2×C4⋊C4—432no—
262C2×C4⋊C4—432no—
263C2×C4⋊C4—416no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4C2×D4116yes—
266C2×C4⋊C4C2×D4116yes—
267C2×C4⋊C4C2×Q8116yes—
268C2×C4⋊C4C2×Q8116yes—
269C2×C42—232no—
270C2×C42—232no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—28no—
274C2×C42—28no—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—416no—
279C2×C42C2×D4116yes—
280C2×C42C2×Q8116yes—
281#95—28no—
282#95—28no—
283#95—216no—
284#95—216no—
285#95—216no—
286#95—216no—
287#95—216no—
288#95—48no—
289#95—48no—
290#95—48no—
291C24C22×D412yes—
292C24C22×D412yes—
293C24C22×D412yes—
294C24C22×Q812yes—
295C24C22×Q812yes—
296C24C22×Q812yes—
297C24C2×#4112yes—
298C24C2×#4112yes—
299C24C2×#4112yes—
300C24C2×#4112yes—
301C24C2×#4112yes—
302C24H(5,𝔽2)12yes—
303C24H(5,𝔽2)12yes—
304C242−1+412yes—
305C242−1+412yes—
306C24C2511yescharacteristic
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—216no—
350C22×C4—28no—
351C22×C4—216no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4#121116yes—
431C22×C4#121116yes—
432C22×C4C22⋊Q8116yes—
433C22×C4C22⋊Q8116yes—
434C22×C4C22⋊Q8116yes—
435C22×C4#123116yes—
436C22×C4#125116yes—
437C22×C4#125116yes—
438C22×C4C4⋊Q8116yes—
439C22×C4C4⋊Q8116yes—
440C22×C4C22×D414yes—
441C22×C4C22×D418yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×Q814yes—
449C22×C4C22×Q818yes—
450C22×C4C22×Q814yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q814yes—
455C22×C4C22×Q814yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4114yes—
461C4⋊C4—232no—
462C4⋊C4—232no—
463C4⋊C4—232no—
464C4⋊C4—232no—
465C4⋊C4—432no—
466C4⋊C4—432no—
467C4⋊C4—432no—
468C4⋊C4—432no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C42—232no—
472C42—232no—
473C42—416no—
474C23C2×#12114yes—
475C23C2×#12114yes—
476C23C2×C22⋊Q814yes—
477C23C2×C22⋊Q814yes—
478C23C2×C22⋊Q814yes—
479C23C2×#12314yes—
480C23C2×#12514yes—
481C23C2×#12514yes—
482C23C2×C4⋊Q814yes—
483C23C2×C4⋊Q814yes—
484C23#53214yes—
485C23#53314yes—
486C23#53314yes—
487C23#53714yes—
488C23#53714yes—
489C23#54014yes—
490C23#54114yes—
491C23#54414yes—
492C23#54414yes—
493C23D4214yes—
494C23#54614yes—
495C23#54714yes—
496C23#54814yes—
497C23D4×Q814yes—
498C23D4×Q814yes—
499C23#55014yes—
500C23#55214yes—
501C23#55314yes—
502C23#55414yes—
503C23#55414yes—
504C23#55614yes—
505C23#55714yes—
506C23#55714yes—
507C23Q8214yes—
508C23#55914yes—
509C23C23×D411yescharacteristic
510C23C23×D411yescharacteristic
511C23C23×D411yescharacteristic
512C23C23×Q811yescharacteristic
513C23C23×Q811yescharacteristic
514C23C23×Q811yescharacteristic
515C23C22×#4111yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4111yescharacteristic
519C23C22×#4111yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23C2×2−1+411yescharacteristic
524C2×C4—232no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—232no—
531C2×C4—232no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—416no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—416no—
565C2×C4—416no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—432no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—432no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—816no—
600C2×C4—816no—
601C2×C4#398164yescomplemented
602C2×C4C2×#12118yes—
603C2×C4C2×#12118yes—
604C2×C4C2×C22⋊Q818yes—
605C2×C4C2×C22⋊Q818yes—
606C2×C4C2×C22⋊Q818yes—
607C2×C4C2×#12318yes—
608C2×C4C2×#12518yes—
609C2×C4C2×#12518yes—
610C2×C4C2×C4⋊Q818yes—
611C2×C4C2×C4⋊Q818yes—
612C22C2×#39818yes—
613C22#241618yes—
614C22#241718yes—
615C22#244518yes—
616C22#244618yes—
617C22#250618yes—
618C22#250718yes—
619C22#253818yes—
620C22#255018yes—
621C22#255118yes—
622C22#255218yes—
623C22#257018yes—
624C22#258218yes—
625C22#262918yes—
626C22#285718yes—
627C22C22×#12111yescharacteristic
628C22C22×#12111yescharacteristic
629C22C22×C22⋊Q811yescharacteristic
630C22C22×C22⋊Q811yescharacteristic
631C22C22×C22⋊Q811yescharacteristic
632C22C22×#12311yescharacteristic
633C22C22×#12511yescharacteristic
634C22C22×#12511yescharacteristic
635C22C22×C4⋊Q811yescharacteristic
636C22C22×C4⋊Q811yescharacteristic
637C22C2×#53211yescharacteristic
638C22C2×#53311yescharacteristic
639C22C2×#53311yescharacteristic
640C22C2×#53711yescharacteristic
641C22C2×#53711yescharacteristic
642C22C2×#54011yescharacteristic
643C22C2×#54111yescharacteristic
644C22C2×#54411yescharacteristic
645C22C2×#54411yescharacteristic
646C22C2×D4211yescharacteristic
647C22C2×#54611yescharacteristic
648C22C2×#54711yescharacteristic
649C22C2×#54811yescharacteristic
650C22C2×D4×Q811yescharacteristic
651C22C2×D4×Q811yescharacteristic
652C22C2×#55011yescharacteristic
653C22C2×#55211yescharacteristic
654C22C2×#55311yescharacteristic
655C22C2×#55411yescharacteristic
656C22C2×#55411yescharacteristic
657C22C2×#55611yescharacteristic
658C22C2×#55711yescharacteristic
659C22C2×#55711yescharacteristic
660C22C2×Q8211yescharacteristic
661C22C2×#55911yescharacteristic
662C4—28no—
663C4—28no—
664C4—28no—
665C4—44no—
666C4—48no—
667C4—44no—
668C4—48no—
669C4—44no—
670C4—84no—
671C4—84no—
672C4C2×#398116yescomplemented
673C2#15133116yesdirect product factor
674C2C22×#39811yescharacteristic
675C2C2×#262911yescharacteristic
676C2C2×#241611yescharacteristic
677C2C2×#244511yescharacteristic
678C2C2×#241711yescharacteristic
679C2C2×#244611yescharacteristic
680C2C2×#250611yescharacteristic
681C2C2×#250711yescharacteristic
682C2C2×#255011yescharacteristic
683C2C2×#257011yescharacteristic
684C2C2×#255111yescharacteristic
685C2C2×#253811yescharacteristic
686C2C2×#255211yescharacteristic
687C2C2×#258211yescharacteristic
688C2C2×#285711yescharacteristic
subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #15133C2direct (N × H)
601. C2×C4#398semidirect
672. C4C2×#398semidirect
673. C2#15133direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3982
C22×#3892
C22×#3842
C22×C4×C4⋊C42
#151332
C2×#14412
C2×C4⋊#952