Finite Group Databasebeta

← C2×#26481 #6444647

Subgroups — C2×#26481 #6444647

Order 512.

2282 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#499C211yesmaximal, characteristic, complemented
2C22×C4×SD16C211yesmaximal, characteristic
3C2×#2453C211yesmaximal, characteristic, complemented
4C2×#2454C211yesmaximal, characteristic
5#26481C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic
8C2×#2026C211yesmaximal, characteristic, complemented
9C2×#2025C211yesmaximal, characteristic
10C2×#1968C211yesmaximal, characteristic
11C2×#1968C211yesmaximal, characteristic
12C2×#1941C211yesmaximal, characteristic, complemented
13C2×#1941C211yesmaximal, characteristic, complemented
14C2×#1863C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic
16C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#499—28no—
23C2×#499C2218yescomplemented
24C2×C4×SD16—28no—
25C2×C4×SD16C2218yes—
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#2454—24no—
42#2454C2218yes—
43#2453—24no—
44#2453C2218yescomplemented
45C2×#400—21no—
46C2×#399—21no—
47C2×#398—21no—
48C2×#396—21no—
49C2×#391C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#389—21no—
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#387—21no—
59C2×#385—21no—
60C2×#380—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2068C2218yescomplemented
63#2067C2218yes—
64#2026C2218yescomplemented
65#2025C2218yes—
66#1968C2218yes—
67#1968C2218yes—
68#1941C2218yescomplemented
69#1941C2218yescomplemented
70#1863C2218yescomplemented
71#1862C2218yes—
72C8×C4⋊C4C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C22×SD16—24no—
83C22×SD16—24no—
84C22×SD16Q818yes—
85C22×SD16C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×#125—24no—
91C2×#125—24no—
92C2×#125—22no—
93C2×#125—22no—
94C2×#125—42no—
95C2×#125—42no—
96C2×#125C2314yes—
97C2×#125C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—24no—
100C2×C4×Q8—42no—
101C2×C4×Q8C2314yes—
102C2×C4×D4—24no—
103C2×C4×D4—22no—
104C2×C4×D4—42no—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
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—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#499—232no—
133C4×SD16—232no—
134C2×#107—24no—
135C2×#107—24no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106—24no—
139C2×#106—24no—
140C2×#106D418yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8—24no—
145C2×C4⋊C8C2314yes—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103—42no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102—24no—
153C2×#102—24no—
154C2×#102—42no—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C4×C8—24no—
159C2×C4×C8—24no—
160C2×C4×C8—24no—
161C2×C4×C8C2314yes—
162C2×C4×C8C2314yes—
163C2×C4×C8C2314yes—
164#400—28no—
165#399—28no—
166#398—28no—
167#396—28no—
168#391—22no—
169#391—22no—
170#391C2314yes—
171#391C2314yes—
172#390—22no—
173#390—22no—
174#390C2314yes—
175#390C2314yes—
176#389—28no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389C2314yes—
182#389C2314yes—
183#389C2314yes—
184#389C2314yes—
185#387—28no—
186#385—28no—
187#380—28no—
188C4×C4⋊C4D414yes—
189C4×C4⋊C4C2314yes—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25—21no—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8C2×Q814yes—
214C22×Q8C2412yes—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2×Q814yes—
222C22×D4C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—82no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×SD16—232no—
242C2×SD16—28no—
243C2×SD16—28no—
244C2×SD16C2×Q8116yes—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8C2×D414yes—
254C22×C8C2×Q814yes—
255C22×C8#4114yes—
256C22×C8#4114yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261#125—28no—
262#125—28no—
263#125—44no—
264#125—44no—
265#125—48no—
266#125—48no—
267C4×Q8—28no—
268C4×Q8—44no—
269C4×Q8—48no—
270C4×D4—28no—
271C4×D4—44no—
272C4×D4—48no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107—28no—
344#107—28no—
345#106—28no—
346#106—28no—
347#106C2×D4116yes—
348C4⋊C8—28no—
349#103—28no—
350#103—28no—
351#103—44no—
352#102—28no—
353#102—28no—
354#102—44no—
355C4×C8—28no—
356C4×C8—28no—
357C4×C8—28no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95C2×D414yes—
367#95C2×D414yes—
368C24—24no—
369C24—22no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24C22×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8C22×Q814yes—
390C2×D4—28no—
391C2×D4—24no—
392C2×D4—28no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4C22×Q814yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—48no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—48no—
449C22×C4—84no—
450C22×C4—816no—
451C22×C4#13514yes—
452C22×C4#13514yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×Q812yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C4C2511yescharacteristic
467SD16—232no—
468C2×C8—28no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8C22⋊Q8116yes—
480C2×C8C22×D414yes—
481C2×C8C22×Q814yes—
482C2×C8C2×#4114yes—
483C2×C8C2×#4114yes—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4C22×D414yes—
512C4⋊C4C22×D414yes—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—24no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—48no—
528C23—24no—
529C23—48no—
530C23—48no—
531C23—42no—
532C23—42no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23#53212yes—
543C23D4212yes—
544C23D4×Q812yes—
545C23C2×#13512yes—
546C23C2×#13512yes—
547C23#57712yes—
548C23#57712yes—
549C23C23×D411yescharacteristic
550Q8—24no—
551Q8—24no—
552Q8—42no—
553Q8—42no—
554Q8—48no—
555D4—24no—
556D4—24no—
557D4—42no—
558D4—42no—
559D4—48no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—816no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4C2×C22⋊Q814yes—
612C2×C4#53212yes—
613C2×C4#54214yes—
614C2×C4D4212yes—
615C2×C4#54814yes—
616C2×C4D4×Q812yes—
617C2×C4#55414yes—
618C2×C4C2×#13514yes—
619C2×C4C2×#13514yes—
620C2×C4C23×D411yescharacteristic
621C2×C4C23×Q811yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C8—24no—
627C8—42no—
628C8—42no—
629C8C2×C22⋊Q818yes—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—82no—
634C22—816no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22#245214yes—
639C22#316814yes—
640C22#317414yes—
641C22#328214yes—
642C22#328414yes—
643C22#338114yes—
644C22#338114yes—
645C22C2×#53211yescharacteristic
646C22C2×D4211yescharacteristic
647C22C2×D4×Q811yescharacteristic
648C22C22×#13511yescharacteristic
649C22C22×#13511yescharacteristic
650C22C2×#57711yescharacteristic
651C22C2×#57711yescharacteristic
652C4—24no—
653C4—24no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4#245214yes—
666C4C22×C22⋊Q811yescharacteristic
667C4C2×#54211yescharacteristic
668C4C2×#54811yescharacteristic
669C4C2×#55411yescharacteristic
670C2—84no—
671C2#2648118yesdirect product factor
672C2C2×#245211yescharacteristic
673C2C2×#316811yescharacteristic
674C2C2×#317411yescharacteristic
675C2C2×#328211yescharacteristic
676C2C2×#328411yescharacteristic
677C2C2×#338111yescharacteristic
678C2C2×#338111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#499C2semidirect
3. C2×#2453C2semidirect
5. #26481C2direct (N × H)
6. C2×#2068C2semidirect
8. C2×#2026C2semidirect
12. C2×#1941C2semidirect
13. C2×#1941C2semidirect
14. C2×#1863C2semidirect
16. C2×C8×C4⋊C4C2semidirect
23. C2×#499C22semidirect
44. #2453C22semidirect
62. #2068C22semidirect
64. #2026C22semidirect
68. #1941C22semidirect
69. #1941C22semidirect
70. #1863C22semidirect
72. C8×C4⋊C4C22semidirect
671. C2#26481direct (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×C4×SD162
C2×#24532
C2×#24542
#264812
C2×#20682
C2×#20672
C2×#20262
C2×#20252
C2×#19682
C2×#19412
C2×#18632
C2×#18622
C2×C8×C4⋊C42