Finite Group Databasebeta

← C2×#26036 #6449293

Subgroups — C2×#26036 #6449293

Order 512.

2104 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2357C211yesmaximal, characteristic
3#26036C2116yesmaximal, direct product factor
4C2×#2067C211yesmaximal, characteristic
5C2×#2065C211yesmaximal, characteristic
6C2×#2023C211yesmaximal, characteristic
7C2×#2023C211yesmaximal, characteristic
8C2×#1988C211yesmaximal, characteristic, complemented
9C2×#1987C211yesmaximal, characteristic, complemented
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1940C211yesmaximal, characteristic, complemented
12C2×#1921C211yesmaximal, characteristic, complemented
13C2×#1919C211yesmaximal, characteristic, complemented
14C2×#1836C211yesmaximal, characteristic, complemented
15C2×#1802C211yesmaximal, characteristic
16C2×#1762C211yesmaximal, characteristic
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2551—24no—
32#2551C2218yescomplemented
33C2×#395—21no—
34C2×#395—21no—
35#2357—24no—
36#2357C2218yes—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389—21no—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#381—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52#2067C2218yes—
53#2065C2218yes—
54#2023C2218yes—
55#2023C2218yes—
56#1988C2218yescomplemented
57#1987C2218yescomplemented
58#1949C2218yescomplemented
59#1940C2218yescomplemented
60#1921C2218yescomplemented
61#1919C2218yescomplemented
62#1836C2218yescomplemented
63#1802C2218yes—
64#1762C2218yes—
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×Q8—21no—
78C23×Q8—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—42no—
91C2×C4⋊Q8—42no—
92C2×C4⋊Q8—48no—
93C2×C4⋊Q8C2314yes—
94C2×C4⋊Q8C2314yes—
95C2×C4×D4—22no—
96C2×C4×D4—24no—
97C2×C4×D4Q818yescomplemented
98C2×C4×D4C2314yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C2×#107—24no—
126C2×#107C2314yes—
127C2×#106—24no—
128C2×#106C2314yes—
129C2×C4⋊C8—24no—
130C2×C4⋊C8—42no—
131C2×C4⋊C8—42no—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102—42no—
138C2×#102—42no—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#97—24no—
144C2×#97C2314yes—
145#395—28no—
146#395—28no—
147#390—22no—
148#390—22no—
149#390C2314yes—
150#390C2314yes—
151#389—28no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#388—28no—
157#388—28no—
158#384—22no—
159#384C2314yes—
160#382—22no—
161#382C2314yes—
162#381—28no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4D414yes—
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2314yes—
170C4×C4⋊C4C2314yes—
171C4×C22⋊C4—28no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95D411yescharacteristic
180C2×#95D411yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193C25—21no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—82no—
201C22×Q8—82no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4C2×Q814yes—
209C22×D4C2412yes—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—82no—
220C23×C4—82no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2411yescharacteristic
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C4⋊Q8—28no—
245C4⋊Q8—28no—
246C4⋊Q8—44no—
247C4⋊Q8—44no—
248C4⋊Q8—48no—
249C4⋊Q8—48no—
250C4⋊Q8—432no—
251C4×D4—216no—
252C4×D4—28no—
253C4×D4C2×Q8116yescomplemented
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×Q814yes—
289C2×C4⋊C4#4114yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—82no—
307C2×C22⋊C4—82no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—82no—
317C2×C42—82no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×Q814yes—
322C2×C42#4114yes—
323C2×C42C2412yes—
324C2×C42C2412yes—
325C2×C42C2412yes—
326#107—28no—
327#106—28no—
328C4⋊C8—28no—
329C4⋊C8—44no—
330C4⋊C8—44no—
331#102—28no—
332#102—28no—
333#102—44no—
334#102—44no—
335#97—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95C2×D414yes—
344#95C2×D414yes—
345C24—24no—
346C24—22no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24C22×D411yescharacteristic
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×Q8—816no—
361C2×Q8—816no—
362C2×D4—24no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4C22×Q814yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—816no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4C22⋊Q814yes—
422C22×C4C22⋊Q814yes—
423C22×C4AGL(1,ℤ/8)14yes—
424C22×C4#13714yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×Q812yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8—44no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—84no—
481C4⋊C4C22×Q814yes—
482C4⋊C4C2×#4114yes—
483C4⋊C4C2×#4114yes—
484C22⋊C4—216no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C42—24no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—48no—
496C42—84no—
497C42—84no—
498C42C22⋊Q818yes—
499C42C22×Q814yes—
500C42C2×#4114yes—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23C2×C22⋊Q812yes—
516C23#54612yes—
517C23#54812yes—
518C23C2×AGL(1,ℤ/8)12yes—
519C23C2×#13712yes—
520C23#57612yes—
521C23#57712yes—
522C23C23×D411yescharacteristic
523Q8—816no—
524Q8—816no—
525D4—24no—
526D4—24no—
527D4—48no—
528D4—42no—
529D4—42no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—816no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4C2×C22⋊Q814yes—
580C2×C4C2×C22⋊Q814yes—
581C2×C4C2×C22⋊Q812yes—
582C2×C4#53314yes—
583C2×C4#54414yes—
584C2×C4#54612yes—
585C2×C4#54812yes—
586C2×C4#55514yes—
587C2×C4#55614yes—
588C2×C4C2×AGL(1,ℤ/8)14yes—
589C2×C4C2×#13714yes—
590C2×C4C23×Q811yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C22×#4111yescharacteristic
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C2×C4C2×2−1+411yescharacteristic
596C8—42no—
597C8—42no—
598C8—42no—
599C8—42no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—82no—
604C22—82no—
605C22—816no—
606C22—82no—
607C22—82no—
608C22#243914yes—
609C22#307914yes—
610C22#308114yes—
611C22#331814yes—
612C22#332014yes—
613C22#337314yes—
614C22#337514yes—
615C22C22×C22⋊Q811yescharacteristic
616C22C2×#54611yescharacteristic
617C22C2×#54811yescharacteristic
618C22C22×AGL(1,ℤ/8)11yescharacteristic
619C22C22×#13711yescharacteristic
620C22C2×#57611yescharacteristic
621C22C2×#57711yescharacteristic
622C4—24no—
623C4—24no—
624C4—24no—
625C4—24no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4#243914yes—
636C4C2×#53311yescharacteristic
637C4C2×#54411yescharacteristic
638C4C2×#55511yescharacteristic
639C4C2×#55611yescharacteristic
640C2—84no—
641C2#2603618yesdirect product factor
642C2C2×#243911yescharacteristic
643C2C2×#307911yescharacteristic
644C2C2×#308111yescharacteristic
645C2C2×#331811yescharacteristic
646C2C2×#332011yescharacteristic
647C2C2×#337311yescharacteristic
648C2C2×#337511yescharacteristic
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. C2×#2551C2semidirect
3. #26036C2direct (N × H)
8. C2×#1988C2semidirect
9. C2×#1987C2semidirect
10. C2×#1949C2semidirect
11. C2×#1940C2semidirect
12. C2×#1921C2semidirect
13. C2×#1919C2semidirect
14. C2×#1836C2semidirect
32. #2551C22semidirect
56. #1988C22semidirect
57. #1987C22semidirect
58. #1949C22semidirect
59. #1940C22semidirect
60. #1921C22semidirect
61. #1919C22semidirect
62. #1836C22semidirect
97. C2×C4×D4Q8semidirect
253. C4×D4C2×Q8semidirect
641. C2#26036direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#23572
#260362
C2×#20672
C2×#20652
C2×#20232
C2×#19882
C2×#19872
C2×#19492
C2×#19402
C2×#19212
C2×#19192
C2×#18362
C2×#18022
C2×#17622