Finite Group Databasebeta

← C2×#26840 #6443874

Subgroups — C2×#26840 #6443874

Order 512.

2352 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C22×#440C211yesmaximal, characteristic
3C2×#2447C211yesmaximal, characteristic
4C2×#2451C211yesmaximal, characteristic, complemented
5#26840C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2057C211yesmaximal, characteristic, complemented
8C2×#2055C211yesmaximal, characteristic
9C2×#2015C211yesmaximal, characteristic
10C2×#1969C211yesmaximal, characteristic, complemented
11C2×#1935C211yesmaximal, characteristic, complemented
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1934C211yesmaximal, characteristic
14C2×#1878C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic
16C2×#1770C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#489—28no—
24C2×#489C2218yes—
25C2×#440—28no—
26C2×#440C2218yes—
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#97C2211yescharacteristic
39C22×#97C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#2451—24no—
42#2451C2218yescomplemented
43#2447—24no—
44#2447C2218yes—
45C2×#398—21no—
46C2×#397—21no—
47C2×#397—21no—
48C2×#395—21no—
49C2×#391—21no—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#387—21no—
59C2×#385—21no—
60C2×#381—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2068C2218yescomplemented
63#2057C2218yescomplemented
64#2055C2218yes—
65#2015C2218yes—
66#1969C2218yescomplemented
67#1935C2218yescomplemented
68#1935C2218yescomplemented
69#1934C2218yes—
70#1878C2218yescomplemented
71#1862C2218yes—
72#1770C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C22×Q16—24no—
85C22×Q16—24no—
86C22×Q16—42no—
87C22×Q16C2314yes—
88C22×SD16—24no—
89C22×SD16—24no—
90C22×SD16—24no—
91C22×SD16C2314yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8C2314yes—
100C2×#124—24no—
101C2×#124—22no—
102C2×#124—42no—
103C2×#124C2314yes—
104C2×C4×Q8—22no—
105C2×C4×Q8—24no—
106C2×C4×Q8—42no—
107C2×C4×Q8C2314yes—
108C2×C4×D4—24no—
109C2×C4×D4—22no—
110C2×C4×D4—42no—
111C2×C4×D4C2314yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#489—232no—
134#440—232no—
135C2×#107—24no—
136C2×#107—42no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×C4⋊C8—24no—
140C2×C4⋊C8C2314yes—
141C2×#103—24no—
142C2×#103—24no—
143C2×#103—24no—
144C2×#103—24no—
145C2×#103—24no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102—24no—
152C2×#102—24no—
153C2×#102D418yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#97—24no—
158C2×#97—24no—
159C2×#97C2314yes—
160C2×#97C2314yes—
161C2×C4×C8—24no—
162C2×C4×C8C2314yes—
163#398—28no—
164#397—28no—
165#397—28no—
166#395—28no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—28no—
172#391C2314yes—
173#391C2314yes—
174#391C2314yes—
175#391C2314yes—
176#390—22no—
177#390C2314yes—
178#389—22no—
179#389—22no—
180#389—22no—
181#389C2314yes—
182#389C2314yes—
183#389C2314yes—
184#387—28no—
185#385—28no—
186#381—28no—
187C4×C4⋊C4D414yes—
188C4×C4⋊C4C2314yes—
189C2×#95—21no—
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×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—82no—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C22×D4—21no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4C2×D414yes—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×Q16—28no—
256C2×Q16—28no—
257C2×Q16—44no—
258C2×Q16—416no—
259C2×SD16—232no—
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16—28no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8C2×D414yes—
273C22×C8#4114yes—
274C22×C8#4114yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C4⋊Q8—28no—
280C4⋊Q8—44no—
281C4⋊Q8—48no—
282#124—28no—
283#124—44no—
284#124—48no—
285C4×Q8—28no—
286C4×Q8—44no—
287C4×Q8—48no—
288C4×D4—28no—
289C4×D4—44no—
290C4×D4—48no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
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—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
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—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—82no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349#107—28no—
350#107—44no—
351C4⋊C8—28no—
352#103—28no—
353#103—28no—
354#103—28no—
355#103—28no—
356#103—28no—
357#102—28no—
358#102—28no—
359#102C2×D4116yes—
360#97—28no—
361#97—28no—
362C4×C8—28no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95C2×D414yes—
375#95C2×D414yes—
376C24—24no—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—84no—
413C2×Q8—816no—
414C2×Q8C2×#4114yes—
415C2×D4—28no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4C22×D414yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
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—48no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
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—84no—
476C22×C4—816no—
477C22×C4#13714yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C4C2511yescharacteristic
492Q16—416no—
493SD16—232no—
494C2×C8—28no—
495C2×C8—28no—
496C2×C8—28no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—44no—
507C2×C8C22×D414yes—
508C2×C8C2×#4114yes—
509C2×C8C2×#4114yes—
510C4⋊C4—24no—
511C4⋊C4—28no—
512C4⋊C4—22no—
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—44no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4C22×D414yes—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—84no—
537C42—24no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—48no—
547C23—24no—
548C23—42no—
549C23—42no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23#53212yes—
562C23D4212yes—
563C23#54812yes—
564C23C2×#13712yes—
565C23#57512yes—
566C23#57612yes—
567C23#57812yes—
568C23C23×D411yescharacteristic
569Q8—24no—
570Q8—24no—
571Q8—42no—
572Q8—42no—
573Q8—48no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—816no—
579Q8—84no—
580D4—24no—
581D4—24no—
582D4—42no—
583D4—42no—
584D4—48no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
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—816no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4#52514yes—
638C2×C4#53212yes—
639C2×C4#54214yes—
640C2×C4D4212yes—
641C2×C4#54714yes—
642C2×C4#54812yes—
643C2×C4#55114yes—
644C2×C4C2×#13714yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C8—24no—
652C8—24no—
653C8—24no—
654C8—42no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—816no—
663C22#246114yes—
664C22#317514yes—
665C22#318114yes—
666C22#327314yes—
667C22#327614yes—
668C22#337514yes—
669C22#337714yes—
670C22C2×#53211yescharacteristic
671C22C2×D4211yescharacteristic
672C22C2×#54811yescharacteristic
673C22C22×#13711yescharacteristic
674C22C2×#57511yescharacteristic
675C22C2×#57611yescharacteristic
676C22C2×#57811yescharacteristic
677C4—24no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4#246114yes—
691C4C2×#52511yescharacteristic
692C4C2×#54211yescharacteristic
693C4C2×#54711yescharacteristic
694C4C2×#55111yescharacteristic
695C2—84no—
696C2#2684018yesdirect product factor
697C2C2×#246111yescharacteristic
698C2C2×#317511yescharacteristic
699C2C2×#318111yescharacteristic
700C2C2×#327311yescharacteristic
701C2C2×#327611yescharacteristic
702C2C2×#337511yescharacteristic
703C2C2×#337711yescharacteristic
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
4. C2×#2451C2semidirect
5. #26840C2direct (N × H)
6. C2×#2068C2semidirect
7. C2×#2057C2semidirect
10. C2×#1969C2semidirect
11. C2×#1935C2semidirect
12. C2×#1935C2semidirect
14. C2×#1878C2semidirect
16. C2×#1770C2semidirect
42. #2451C22semidirect
62. #2068C22semidirect
63. #2057C22semidirect
66. #1969C22semidirect
67. #1935C22semidirect
68. #1935C22semidirect
70. #1878C22semidirect
72. #1770C22semidirect
696. C2#26840direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4892
C22×#4402
C2×#24472
C2×#24512
#268402
C2×#20682
C2×#20572
C2×#20552
C2×#20152
C2×#19692
C2×#19352
C2×#19342
C2×#18782
C2×#18622
C2×#17702