Finite Group Databasebeta

← C2×#25989 #6452287

Subgroups — C2×#25989 #6452287

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C211yesmaximal, characteristic
2C2×#2347C211yesmaximal, characteristic
3#25989C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2087C211yesmaximal, characteristic, complemented
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2057C211yesmaximal, characteristic, complemented
8C2×#1988C211yesmaximal, characteristic, complemented
9C2×#1949C211yesmaximal, characteristic, complemented
10C2×#1921C211yesmaximal, characteristic, complemented
11C2×#1918C211yesmaximal, characteristic, complemented
12C2×#1854C211yesmaximal, characteristic
13C2×#1850C211yesmaximal, characteristic
14C2×#1825C211yesmaximal, characteristic
15C2×#1808C211yesmaximal, characteristic
16C2×#1735C211yesmaximal, characteristic
17C24×C8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28#2439—24no—
29#2439C2218yes—
30C2×#396—21no—
31C2×#393—21no—
32#2347—24no—
33#2347C2218yes—
34C2×#391—21no—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#388—21no—
40C2×#388—21no—
41C2×#387—21no—
42C2×#383—21no—
43C2×#382C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2218yescomplemented
51#2088C2218yescomplemented
52#2087C2218yescomplemented
53#2057C2218yescomplemented
54#1988C2218yescomplemented
55#1949C2218yescomplemented
56#1921C2218yescomplemented
57#1918C2218yescomplemented
58#1854C2218yes—
59#1850C2218yes—
60#1825C2218yes—
61#1808C2218yes—
62#1735C2218yes—
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
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—
78C24×C4C2311yescharacteristic
79C23×C8—24no—
80C23×C8—24no—
81C23×C8—42no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2314yes—
86C23×C8C2311yescharacteristic
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8C2314yes—
91C2×#117—22no—
92C2×#117—24no—
93C2×#117—22no—
94C2×#117—24no—
95C2×#117—42no—
96C2×#117Q818yescomplemented
97C2×#117C2314yes—
98C2×#117C2314yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4D411yescharacteristic
117C22×C22⋊C4D411yescharacteristic
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C2×#107—24no—
125C2×#107—24no—
126C2×#107—42no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106—42no—
131C2×#106C2314yes—
132C2×C4⋊C8—24no—
133C2×C4⋊C8—42no—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C22⋊C8—24no—
137C2×C22⋊C8C2314yes—
138#396—28no—
139#393—28no—
140#391—28no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389C2314yes—
146#389C2314yes—
147#389C2314yes—
148#389C2314yes—
149#388—28no—
150#388—28no—
151#387—28no—
152#383—28no—
153#382—22no—
154#382C2314yes—
155#380—22no—
156#380—22no—
157#380C2314yes—
158#380C2314yes—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4C2314yes—
163C4×C4⋊C4C2314yes—
164C4×C22⋊C4D414yes—
165C4×C22⋊C4C2314yes—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95D411yescharacteristic
177C2×#95D411yescharacteristic
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192C25C2×D411yescharacteristic
193C22×Q8—24no—
194C22×Q8—44no—
195C22×Q8—44no—
196C22×Q8—82no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×Q814yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—28no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—28no—
232C22×C8—44no—
233C22×C8—416no—
234C22×C8#4114yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C4⋊Q8—28no—
240C4⋊Q8—44no—
241C4⋊Q8—48no—
242#117—216no—
243#117—28no—
244#117—28no—
245#117—44no—
246#117—48no—
247#117C2×Q8116yescomplemented
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4C2×Q814yes—
281C2×C4⋊C4#4114yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—82no—
299C2×C22⋊C4—82no—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—44no—
312C2×C42—82no—
313C2×C42—82no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2412yes—
317C2×C42C2412yes—
318#107—28no—
319#107—28no—
320#107—44no—
321#106—44no—
322C4⋊C8—28no—
323C4⋊C8—44no—
324C22⋊C8—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95C2×D414yes—
336#95C2×D414yes—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24C22⋊Q814yes—
344C24C22×D412yes—
345C24C22×D411yescharacteristic
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—84no—
349C2×Q8—816no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
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—22no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4C22⋊Q814yes—
406C22×C4C22⋊Q818yes—
407C22×C4#13514yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×Q814yes—
415C22×C4C22×Q812yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+412yes—
422C22×C4C2511yescharacteristic
423C2×C8—28no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—416no—
436C2×C8—44no—
437C2×C8—44no—
438C2×C8—44no—
439C2×C8C2×#4114yes—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—48no—
465C4⋊C4—84no—
466C4⋊C4C22×Q814yes—
467C4⋊C4C22×Q814yes—
468C4⋊C4C2×#4114yes—
469C22⋊C4—216no—
470C22⋊C4—28no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C42—216no—
476C42—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—48no—
483C42—84no—
484C42—84no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23C2×C22⋊Q814yes—
499C23C2×C22⋊Q812yes—
500C23#54612yes—
501C23#54712yes—
502C23C2×#13512yes—
503C23#57512yes—
504C23#57612yes—
505C23#57812yes—
506C23C23×D411yescharacteristic
507Q8—816no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—216no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4C2×C22⋊Q814yes—
564C2×C4C2×C22⋊Q812yes—
565C2×C4#53314yes—
566C2×C4#54414yes—
567C2×C4#54612yes—
568C2×C4#54712yes—
569C2×C4#55114yes—
570C2×C4#55214yes—
571C2×C4C2×#13514yes—
572C2×C4C23×Q811yescharacteristic
573C2×C4C22×#4111yescharacteristic
574C2×C4C22×#4111yescharacteristic
575C2×C4C22×#4111yescharacteristic
576C2×C4C2×H(5,𝔽2)11yescharacteristic
577C2×C4C2×2−1+411yescharacteristic
578C8—24no—
579C8—42no—
580C8—42no—
581C8—42no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—416no—
588C22—42no—
589C22—42no—
590C22#243814yes—
591C22#307614yes—
592C22#308314yes—
593C22#329714yes—
594C22#331314yes—
595C22#334214yes—
596C22#334714yes—
597C22C22×C22⋊Q811yescharacteristic
598C22C2×#54611yescharacteristic
599C22C2×#54711yescharacteristic
600C22C22×#13511yescharacteristic
601C22C2×#57511yescharacteristic
602C22C2×#57611yescharacteristic
603C22C2×#57811yescharacteristic
604C4—24no—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—44no—
609C4—44no—
610C4—44no—
611C4—42no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4#243814yes—
618C4C2×#53311yescharacteristic
619C4C2×#54411yescharacteristic
620C4C2×#55111yescharacteristic
621C4C2×#55211yescharacteristic
622C2—44no—
623C2#2598918yesdirect product factor
624C2C2×#243811yescharacteristic
625C2C2×#307611yescharacteristic
626C2C2×#308311yescharacteristic
627C2C2×#329711yescharacteristic
628C2C2×#331311yescharacteristic
629C2C2×#334211yescharacteristic
630C2C2×#334711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #25989C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2087C2semidirect
6. C2×#2088C2semidirect
7. C2×#2057C2semidirect
8. C2×#1988C2semidirect
9. C2×#1949C2semidirect
10. C2×#1921C2semidirect
11. C2×#1918C2semidirect
50. #2101C22semidirect
51. #2088C22semidirect
52. #2087C22semidirect
53. #2057C22semidirect
54. #1988C22semidirect
55. #1949C22semidirect
56. #1921C22semidirect
57. #1918C22semidirect
96. C2×#117Q8semidirect
247. #117C2×Q8semidirect
623. C2#25989direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#23472
#259892
C2×#21012
C2×#20872
C2×#20882
C2×#20572
C2×#19882
C2×#19492
C2×#19212
C2×#19182
C2×#18542
C2×#18502
C2×#18252
C2×#18082
C2×#17352