Finite Group Databasebeta

← C2×#27801 #6457589

Subgroups — C2×#27801 #6457589

Order 512.

2294 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#499C211yesmaximal, characteristic, complemented
2C22×#468C211yesmaximal, characteristic
3C2×#2490C211yesmaximal, characteristic
4C2×#2489C211yesmaximal, characteristic, complemented
5#27801C2116yesmaximal, direct product factor
6C2×#2034C211yesmaximal, characteristic
7C2×#2035C211yesmaximal, characteristic, complemented
8C2×#2026C211yesmaximal, characteristic, complemented
9C2×#2025C211yesmaximal, characteristic, complemented
10C2×#1973C211yesmaximal, characteristic, complemented
11C2×#1968C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1809C211yesmaximal, characteristic, complemented
15C2×#1776C211yesmaximal, characteristic, complemented
16C2×#1753C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×M16C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×#121C2211yescharacteristic
23C2×#499—28no—
24C2×#499C2218yescomplemented
25C2×#468—28no—
26C2×#468C2218yes—
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39C22×#97C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#2490—24no—
42#2490C2218yes—
43#2489—24no—
44#2489C2218yescomplemented
45C2×#400—21no—
46C2×#397—21no—
47C2×#397—21no—
48C2×#394—21no—
49C2×#391C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#390—21no—
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#388—21no—
55C2×#388—21no—
56C2×#382—21no—
57C2×#380C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C22⋊C4C2211yescharacteristic
62#2035C2218yescomplemented
63#2034C2218yes—
64#2026C2218yescomplemented
65#2025C2218yescomplemented
66#1973C2218yescomplemented
67#1968C2218yescomplemented
68#1927C2218yescomplemented
69#1926C2218yescomplemented
70#1809C2218yescomplemented
71#1776C2218yescomplemented
72#1753C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16C2314yes—
87C22×M16—24no—
88C22×M16—24no—
89C22×M16—24no—
90C22×M16C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×#125—24no—
96C2×#125—24no—
97C2×#125—22no—
98C2×#125—22no—
99C2×#125—42no—
100C2×#125—42no—
101C2×#125C2314yes—
102C2×#125C2314yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—42no—
106C2×C22⋊Q8C2314yes—
107C2×#121—22no—
108C2×#121—24no—
109C2×#121—42no—
110C2×#121C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4D411yescharacteristic
131C22×C22⋊C4D411yescharacteristic
132C22×C42—21no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#499—232no—
137#468—232no—
138C2×#107—24no—
139C2×#107D418yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106—24no—
143C2×#106—24no—
144C2×#106—24no—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103—42no—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102—24no—
155C2×#102—24no—
156C2×#102—42no—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8—24no—
161C2×C22⋊C8C2314yes—
162C2×#97—24no—
163C2×#97C2314yes—
164C2×C4×C8—24no—
165C2×C4×C8C2314yes—
166#400—28no—
167#397—28no—
168#397—28no—
169#394—28no—
170#391—22no—
171#391—22no—
172#391C2314yes—
173#391C2314yes—
174#390—28no—
175#390—22no—
176#390—22no—
177#390C2314yes—
178#390C2314yes—
179#388—28no—
180#388—28no—
181#382—28no—
182#380—22no—
183#380—22no—
184#380C2314yes—
185#380C2314yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2314yes—
190C4×C22⋊C4D414yes—
191C4×C22⋊C4C2314yes—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C25C2×D411yescharacteristic
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2412yes—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—82no—
229C22×D4C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—28no—
254C2×SD16—416no—
255C2×M16—28no—
256C2×M16—28no—
257C2×M16—28no—
258C2×M16—416no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8C2×D414yes—
268C22×C8#4114yes—
269C22×C8#4114yes—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#125—28no—
276#125—28no—
277#125—44no—
278#125—44no—
279#125—48no—
280#125—48no—
281C22⋊Q8—28no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284#121—28no—
285#121—44no—
286#121—48no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4C2×D414yes—
340C2×C22⋊C4C2×D414yes—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—44no—
348C2×C42—82no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353#107—28no—
354#107C2×D4116yes—
355#106—28no—
356#106—28no—
357#106—28no—
358#103—28no—
359#103—28no—
360#103—44no—
361#102—28no—
362#102—28no—
363#102—44no—
364C22⋊C8—28no—
365#97—28no—
366C4×C8—28no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95C2×D414yes—
376#95C2×D414yes—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24C22×D412yes—
393C24C22×D411yescharacteristic
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—84no—
423C2×D4—816no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C42−1+412yes—
492C22×C4C2511yescharacteristic
493SD16—416no—
494M16—416no—
495C2×C8—28no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—44no—
505C2×C8—44no—
506C2×C8#124116yes—
507C2×C8C22×D414yes—
508C2×C8C2×#4114yes—
509C2×C8C2×#4114yes—
510C2×C8C2×#4114yes—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4C22×D414yes—
533C4⋊C4C22×D414yes—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—48no—
551C42—84no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23#53212yes—
581C23D4212yes—
582C23#54712yes—
583C23#57512yes—
584C23#57512yes—
585C23#57712yes—
586C23#57712yes—
587C23C23×D411yescharacteristic
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—84no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—816no—
598D4—84no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4C2×#12414yes—
653C2×C4#53212yes—
654C2×C4#54214yes—
655C2×C4D4212yes—
656C2×C4#54712yes—
657C2×C4#54814yes—
658C2×C4#55514yes—
659C2×C4C23×D411yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×2−1+411yescharacteristic
665C8—24no—
666C8—42no—
667C8—42no—
668C8C2×#12418yes—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—88no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—168no—
685C22#249114yes—
686C22#317414yes—
687C22#318314yes—
688C22#326714yes—
689C22#327114yes—
690C22#335114yes—
691C22#335114yes—
692C22C2×#53211yescharacteristic
693C22C2×D4211yescharacteristic
694C22C2×#54711yescharacteristic
695C22C2×#57511yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57711yescharacteristic
698C22C2×#57711yescharacteristic
699C4—42no—
700C4—42no—
701C4—42no—
702C4—44no—
703C4—44no—
704C4—42no—
705C4—42no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#249114yes—
712C4C22×#12411yescharacteristic
713C4C2×#54211yescharacteristic
714C4C2×#54811yescharacteristic
715C4C2×#55511yescharacteristic
716C2—82no—
717C2—162no—
718C2#2780118yesdirect product factor
719C2C2×#249111yescharacteristic
720C2C2×#317411yescharacteristic
721C2C2×#318311yescharacteristic
722C2C2×#326711yescharacteristic
723C2C2×#327111yescharacteristic
724C2C2×#335111yescharacteristic
725C2C2×#335111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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×#499C2semidirect
4. C2×#2489C2semidirect
5. #27801C2direct (N × H)
7. C2×#2035C2semidirect
8. C2×#2026C2semidirect
9. C2×#2025C2semidirect
10. C2×#1973C2semidirect
11. C2×#1968C2semidirect
12. C2×#1926C2semidirect
13. C2×#1927C2semidirect
14. C2×#1809C2semidirect
15. C2×#1776C2semidirect
16. C2×#1753C2semidirect
24. C2×#499C22semidirect
44. #2489C22semidirect
62. #2035C22semidirect
64. #2026C22semidirect
65. #2025C22semidirect
66. #1973C22semidirect
67. #1968C22semidirect
68. #1927C22semidirect
69. #1926C22semidirect
70. #1809C22semidirect
71. #1776C22semidirect
72. #1753C22semidirect
718. C2#27801direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4992
C22×#4682
C2×#24902
C2×#24892
#278012
C2×#20342
C2×#20352
C2×#20262
C2×#20252
C2×#19732
C2×#19682
C2×#19262
C2×#19272
C2×#18092
C2×#17762
C2×#17532