Finite Group Databasebeta

← C2×#27941 #6457328

Subgroups — C2×#27941 #6457328

Order 512.

2934 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic, complemented
2C22×#482C211yesmaximal, characteristic, complemented
3C22×#469C211yesmaximal, characteristic
4C22×#428C211yesmaximal, characteristic, complemented
5C2×#2500C211yesmaximal, characteristic, complemented
6C2×#2501C211yesmaximal, characteristic, complemented
7#27941C2116yesmaximal, direct product factor
8C2×#2094C211yesmaximal, characteristic, complemented
9C2×#2094C211yesmaximal, characteristic, complemented
10C2×#2093C211yesmaximal, characteristic, complemented
11C2×#2092C211yesmaximal, characteristic, complemented
12C2×#1972C211yesmaximal, characteristic, complemented
13C2×#1970C211yesmaximal, characteristic, complemented
14C2×#1847C211yesmaximal, characteristic, complemented
15C2×#1844C211yesmaximal, characteristic, complemented
16C2×#1753C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C23×M16C2211yescharacteristic
19C22×#123—22no—
20C22×#121C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C2×#481—28no—
25C2×#481C2218yescomplemented
26C2×#482—28no—
27C2×#482C2218yescomplemented
28C2×#469—28no—
29C2×#469C2218yes—
30C2×#428—28no—
31C2×#428C2218yescomplemented
32C22×#107C2211yescharacteristic
33C22×#107C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C22×#106C2211yescharacteristic
39C22×#106C2211yescharacteristic
40C22×#106C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C22×#102C2211yescharacteristic
47C22×C22⋊C8C2211yescharacteristic
48C22×#97C2211yescharacteristic
49C22×C4×C8C2211yescharacteristic
50#2501—24no—
51#2501C2218yescomplemented
52#2500—24no—
53#2500C2218yescomplemented
54C2×#399—22no—
55C2×#394—22no—
56C2×#390C2211yescharacteristic
57C2×#390C2211yescharacteristic
58C2×#390C2211yescharacteristic
59C2×#390C2211yescharacteristic
60C2×#388—22no—
61C2×#384C2211yescharacteristic
62C2×#384C2211yescharacteristic
63C2×#384C2211yescharacteristic
64C2×#382C2211yescharacteristic
65C2×C4×C22⋊C4C2211yescharacteristic
66#2094C2218yescomplemented
67#2094C2218yescomplemented
68#2093C2218yescomplemented
69#2092C2218yescomplemented
70#1972C2218yescomplemented
71#1970C2218yescomplemented
72#1847C2218yescomplemented
73#1844C2218yescomplemented
74#1753C2218yescomplemented
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C22×D8—24no—
84C22×D8—24no—
85C22×D8—24no—
86C22×D8C2314yes—
87C22×M16—24no—
88C22×M16—24no—
89C22×M16D418yes—
90C22×M16C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×#123—216no—
96C2×#123—416no—
97C2×#121—22no—
98C2×#121—22no—
99C2×#121—24no—
100C2×#121—24no—
101C2×#121—24no—
102C2×#121—24no—
103C2×#121C2314yes—
104C2×#121C2314yes—
105C2×#117—22no—
106C2×#117—24no—
107C2×#117—24no—
108C2×#117C2314yes—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—22no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4D412yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#482—232no—
135#481—232no—
136#469—232no—
137#428—232no—
138C2×#107—24no—
139C2×#107—24no—
140C2×#107—24no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106—24no—
145C2×#106—24no—
146C2×#106—24no—
147C2×#106—24no—
148C2×#106—24no—
149C2×#106—24no—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×#106C2314yes—
155C2×#106C2314yes—
156C2×#102—24no—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102—24no—
160C2×#102—24no—
161C2×#102—24no—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×C22⋊C8D418yes—
169C2×C22⋊C8C2314yes—
170C2×#97—24no—
171C2×#97C2314yes—
172C2×C4×C8—24no—
173C2×C4×C8C2314yes—
174#399—216no—
175#394—216no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390C2314yes—
181#390C2314yes—
182#390C2314yes—
183#390C2314yes—
184#388—216no—
185#384—22no—
186#384—22no—
187#384—22no—
188#384C2314yes—
189#384C2314yes—
190#384C2314yes—
191#382—22no—
192#382C2314yes—
193C4×C22⋊C4D414yes—
194C4×C22⋊C4C2314yes—
195C2×#95—22no—
196C2×#95D412yes—
197#336C2314yes—
198#336C2314yes—
199C25—21no—
200C25—21no—
201C25C2×D411yescharacteristic
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—48no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4—84no—
222C22×D4C2412yes—
223C22×D4C2412yes—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—48no—
236C23×C4—44no—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C2×D8—28no—
246C2×D8—28no—
247C2×D8—28no—
248C2×D8—416no—
249C2×M16—232no—
250C2×M16—28no—
251C2×M16—28no—
252C2×M16C2×D4116yes—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8C2×D414yes—
260C22×C8C2×D414yes—
261C22×C8C2×D414yes—
262C22×C8C2×D414yes—
263C22×C8#4114yes—
264C22×C8#4114yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269#123—464no—
270#121—28no—
271#121—28no—
272#121—28no—
273#121—28no—
274#121—48no—
275#121—48no—
276#117—28no—
277#117—28no—
278#117—48no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—84no—
329C2×C22⋊C4C2×D418yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—44no—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340#107—28no—
341#107—28no—
342#107—28no—
343#106—28no—
344#106—28no—
345#106—28no—
346#106—28no—
347#106—28no—
348#106—28no—
349#102—28no—
350#102—28no—
351#102—28no—
352#102—28no—
353#102—28no—
354#102—28no—
355C22⋊C8C2×D4116yes—
356#97—28no—
357C4×C8—28no—
358#95—28no—
359#95C2×D418yes—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24—44no—
376C24#12714yes—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—416no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—88no—
409C2×D4—816no—
410C2×D4—816no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—416no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—816no—
449C22×C4#12714yes—
450C22×C4#12718yes—
451C22×C4AGL(1,ℤ/8)18yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C22×C4C2511yescharacteristic
465D8—416no—
466M16—232no—
467C2×C8—28no—
468C2×C8—28no—
469C2×C8—28no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8#123116yescomplemented
477C2×C8C22×D414yes—
478C2×C8C22×D414yes—
479C2×C8C22×D414yes—
480C2×C8C22×D414yes—
481C2×C8C2×#4114yes—
482C2×C8C2×#4114yes—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—416no—
504C4⋊C4—88no—
505C22⋊C4—216no—
506C22⋊C4—416no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—88no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—48no—
520C23—28no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—24no—
526C23—24no—
527C23—48no—
528C23—48no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—88no—
540C23—88no—
541C23—88no—
542C23—88no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23C2×#12714yes—
550C23C2×#12712yes—
551C23#54712yes—
552C23#54712yes—
553C23C2×AGL(1,ℤ/8)14yes—
554C23#57714yes—
555C23C23×D411yescharacteristic
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—816no—
565D4—816no—
566D4—84no—
567D4—84no—
568C2×C4—216no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—216no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—816no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4C2×#12314yes—
603C2×C4C2×#12714yes—
604C2×C4C2×#12712yes—
605C2×C4#53614yes—
606C2×C4#54714yes—
607C2×C4#54714yes—
608C2×C4#54712yes—
609C2×C4#54712yes—
610C2×C4C2×AGL(1,ℤ/8)18yes—
611C2×C4C23×D411yescharacteristic
612C2×C4C23×D411yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C8—24no—
618C8—24no—
619C8—24no—
620C8C2×#12318yescomplemented
621C22—24no—
622C22—24no—
623C22—24no—
624C22—416no—
625C22—44no—
626C22—44no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—168no—
638C22—168no—
639C22#250114yes—
640C22#315114yes—
641C22#315414yes—
642C22#334518yes—
643C22#335418yes—
644C22C22×#12711yescharacteristic
645C22C2×#54711yescharacteristic
646C22C2×#54711yescharacteristic
647C22C22×AGL(1,ℤ/8)12yes—
648C22C2×#57712yes—
649C4—28no—
650C4—24no—
651C4—44no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#250114yes—
659C4C22×#12311yescharacteristic
660C4C2×#53611yescharacteristic
661C4C2×#54711yescharacteristic
662C4C2×#54711yescharacteristic
663C2—44no—
664C2—162no—
665C2—162no—
666C2#2794118yesdirect product factor
667C2C2×#250111yescharacteristic
668C2C2×#315111yescharacteristic
669C2C2×#315411yescharacteristic
670C2C2×#334512yes—
671C2C2×#335412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#469C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#481C2semidirect
2. C22×#482C2semidirect
4. C22×#428C2semidirect
5. C2×#2500C2semidirect
6. C2×#2501C2semidirect
7. #27941C2direct (N × H)
8. C2×#2094C2semidirect
9. C2×#2094C2semidirect
10. C2×#2093C2semidirect
11. C2×#2092C2semidirect
12. C2×#1972C2semidirect
13. C2×#1970C2semidirect
14. C2×#1847C2semidirect
15. C2×#1844C2semidirect
16. C2×#1753C2semidirect
25. C2×#481C22semidirect
27. C2×#482C22semidirect
31. C2×#428C22semidirect
51. #2501C22semidirect
53. #2500C22semidirect
66. #2094C22semidirect
67. #2094C22semidirect
68. #2093C22semidirect
69. #2092C22semidirect
70. #1972C22semidirect
71. #1970C22semidirect
72. #1847C22semidirect
73. #1844C22semidirect
74. #1753C22semidirect
476. C2×C8#123semidirect
620. C8C2×#123semidirect
666. C2#27941direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4812
C22×#4822
C22×#4692
C22×#4282
C2×#25002
C2×#25012
#279412
C2×#20942
C2×#20932
C2×#20922
C2×#19722
C2×#19702
C2×#18472
C2×#18442
C2×#17532