Finite Group Databasebeta

← C2×#26437 #6441923

Subgroups — C2×#26437 #6441923

Order 512.

2288 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
1C2×#2442C211yesmaximal, characteristic, complemented
2C2×#2363C211yesmaximal, characteristic
3#26437C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2086C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2025C211yesmaximal, characteristic, complemented
8C2×#1989C211yesmaximal, characteristic, complemented
9C2×#1920C211yesmaximal, characteristic, complemented
10C2×#1895C211yesmaximal, characteristic
11C2×#1874C211yesmaximal, characteristic
12C2×#1864C211yesmaximal, characteristic, complemented
13C2×#1854C211yesmaximal, characteristic, complemented
14C2×#1851C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic, complemented
16C2×#1736C211yesmaximal, characteristic, complemented
17C23×M16C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32#2442—24no—
33#2442C2218yescomplemented
34C2×#400—21no—
35C2×#398—21no—
36#2363—24no—
37#2363C2218yes—
38C2×#390—21no—
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#386—21no—
45C2×#385—21no—
46C2×#382C2211yescharacteristic
47C2×#380—21no—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4C2211yescharacteristic
52#2101C2218yescomplemented
53#2086C2218yescomplemented
54#2067C2218yescomplemented
55#2025C2218yescomplemented
56#1989C2218yescomplemented
57#1920C2218yescomplemented
58#1895C2218yes—
59#1874C2218yes—
60#1864C2218yescomplemented
61#1854C2218yescomplemented
62#1851C2218yescomplemented
63#1804C2218yescomplemented
64#1736C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C22×M16—24no—
83C22×M16—42no—
84C22×M16—42no—
85C22×M16C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×#125—24no—
91C2×#125—22no—
92C2×#125—42no—
93C2×#125C2314yes—
94C2×#121—22no—
95C2×#121—24no—
96C2×#121Q818yes—
97C2×#121C2314yes—
98C2×C4×D4—24no—
99C2×C4×D4—22no—
100C2×C4×D4—42no—
101C2×C4×D4C2314yes—
102C2×#117—22no—
103C2×#117—24no—
104C2×#117—42no—
105C2×#117C2314yes—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—42no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
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×C22⋊C4—21no—
129C22×C22⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42—21no—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C2×#107—42no—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106C2314yes—
139C2×C4⋊C8—42no—
140C2×C4⋊C8—42no—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×#102—24no—
144C2×#102—42no—
145C2×#102—42no—
146C2×#102—42no—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8—24no—
152C2×C22⋊C8C2314yes—
153#400—28no—
154#398—28no—
155#390—28no—
156#390—22no—
157#390C2314yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#388—28no—
165#386—28no—
166#385—28no—
167#382—22no—
168#382C2314yes—
169#380—22no—
170#380—28no—
171#380C2314yes—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2314yes—
174C4×C22⋊C4—28no—
175C4×C22⋊C4D414yes—
176C4×C22⋊C4C2314yes—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95D411yescharacteristic
185C2×#95D411yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198C25—21no—
199C25C2×D411yescharacteristic
200C22×D4—28no—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—82no—
209C22×D4—82no—
210C22×D4C2×Q814yes—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—82no—
230C23×C4—82no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×Q814yes—
238C23×C4#4114yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C2×M16—28no—
242C2×M16—44no—
243C2×M16—44no—
244C2×M16—416no—
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×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261#125—28no—
262#125—44no—
263#125—48no—
264#121—216no—
265#121—28no—
266#121C2×Q8116yes—
267C4×D4—28no—
268C4×D4—44no—
269C4×D4—48no—
270#117—28no—
271#117—44no—
272#117—48no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
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—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×Q814yes—
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4#4114yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4C2×D414yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42—82no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—44no—
345#106—28no—
346C4⋊C8—44no—
347C4⋊C8—44no—
348#102—28no—
349#102—44no—
350#102—44no—
351#102—44no—
352C22⋊C8—28no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—22no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24C22⋊Q814yes—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379C2×D4—216no—
380C2×D4—24no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—816no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4C22×Q814yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
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—22no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
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—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—816no—
455C22×C4—816no—
456C22×C4C22⋊Q814yes—
457C22×C4C22⋊Q818yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×Q814yes—
465C22×C4C22×Q812yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4C2511yescharacteristic
473M16—416no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—44no—
487C2×C8—44no—
488C2×C8—44no—
489C2×C8—44no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C22×Q814yes—
515C4⋊C4C2×#4114yes—
516C4⋊C4C2×#4114yes—
517C22⋊C4—216no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—84no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—48no—
532C42—84no—
533C23—28no—
534C23—28no—
535C23—22no—
536C23—22no—
537C23—28no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—48no—
548C23—48no—
549C23—42no—
550C23—42no—
551C23—48no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23C2×C22⋊Q814yes—
561C23C2×C22⋊Q812yes—
562C23#54612yes—
563C23#54612yes—
564C23#57512yes—
565C23#57512yes—
566C23#57612yes—
567C23#57712yes—
568C23C23×D411yescharacteristic
569D4—24no—
570D4—24no—
571D4—42no—
572D4—42no—
573D4—48no—
574D4—816no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—216no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
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—816no—
624C2×C4—816no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4C2×C22⋊Q814yes—
630C2×C4C2×C22⋊Q812yes—
631C2×C4#52914yes—
632C2×C4#53814yes—
633C2×C4#54612yes—
634C2×C4#54612yes—
635C2×C4#55414yes—
636C2×C4#55414yes—
637C2×C4C23×Q811yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C8—42no—
644C8—42no—
645C8—42no—
646C8—42no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—416no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—816no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22#245014yes—
664C22#308014yes—
665C22#308114yes—
666C22#330914yes—
667C22#331914yes—
668C22#332314yes—
669C22#332614yes—
670C22C22×C22⋊Q811yescharacteristic
671C22C2×#54611yescharacteristic
672C22C2×#54611yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57511yescharacteristic
675C22C2×#57611yescharacteristic
676C22C2×#57711yescharacteristic
677C4—24no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#245014yes—
690C4C2×#52911yescharacteristic
691C4C2×#53811yescharacteristic
692C4C2×#55411yescharacteristic
693C4C2×#55411yescharacteristic
694C2—44no—
695C2—84no—
696C2#2643718yesdirect product factor
697C2C2×#245011yescharacteristic
698C2C2×#308011yescharacteristic
699C2C2×#308111yescharacteristic
700C2C2×#331911yescharacteristic
701C2C2×#330911yescharacteristic
702C2C2×#332311yescharacteristic
703C2C2×#332611yescharacteristic
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. C2×#2442C2semidirect
3. #26437C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2086C2semidirect
6. C2×#2067C2semidirect
7. C2×#2025C2semidirect
8. C2×#1989C2semidirect
9. C2×#1920C2semidirect
12. C2×#1864C2semidirect
13. C2×#1854C2semidirect
14. C2×#1851C2semidirect
15. C2×#1804C2semidirect
16. C2×#1736C2semidirect
33. #2442C22semidirect
52. #2101C22semidirect
53. #2086C22semidirect
54. #2067C22semidirect
55. #2025C22semidirect
56. #1989C22semidirect
57. #1920C22semidirect
60. #1864C22semidirect
61. #1854C22semidirect
62. #1851C22semidirect
63. #1804C22semidirect
64. #1736C22semidirect
696. C2#26437direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24422
C2×#23632
#264372
C2×#21012
C2×#20862
C2×#20672
C2×#20252
C2×#19892
C2×#19202
C2×#18952
C2×#18742
C2×#18642
C2×#18542
C2×#18512
C2×#18042
C2×#17362