Finite Group Databasebeta

← C2×#29083 #6460595

Subgroups — C2×#29083 #6460595

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2514C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic
3#29083C2116yesmaximal, direct product factor
4C2×#2086C211yesmaximal, characteristic, complemented
5C2×#2073C211yesmaximal, characteristic
6C2×#2055C211yesmaximal, characteristic
7C2×#2024C211yesmaximal, characteristic, complemented
8C2×#1993C211yesmaximal, characteristic, complemented
9C2×#1973C211yesmaximal, characteristic
10C2×#1935C211yesmaximal, characteristic, complemented
11C2×#1934C211yesmaximal, characteristic
12C2×#1933C211yesmaximal, characteristic, complemented
13C2×#1921C211yesmaximal, characteristic, complemented
14C2×#1877C211yesmaximal, characteristic
15C2×#1770C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2514—24no—
33#2514C2218yescomplemented
34#2452—24no—
35#2452C2218yes—
36C2×#401—21no—
37C2×#400—21no—
38C2×#396—21no—
39C2×#393—21no—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#387—21no—
49C2×#385—21no—
50C2×#383—21no—
51C2×#382C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2086C2218yescomplemented
55#2073C2218yes—
56#2055C2218yes—
57#2024C2218yescomplemented
58#1993C2218yescomplemented
59#1973C2218yes—
60#1935C2218yescomplemented
61#1934C2218yes—
62#1933C2218yescomplemented
63#1921C2218yescomplemented
64#1877C2218yes—
65#1770C2218yescomplemented
66#1762C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×SD16—24no—
82C22×SD16—24no—
83C22×SD16—42no—
84C22×SD16C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—42no—
92C2×C4⋊Q8C2314yes—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—48no—
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4—42no—
98C2×C4×D4C2314yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C2×#107—24no—
121C2×#107C2314yes—
122C2×C4⋊C8—24no—
123C2×C4⋊C8C2314yes—
124C2×#103—24no—
125C2×#103—24no—
126C2×#103—42no—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102—24no—
131C2×#102—24no—
132C2×#102—24no—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#97—42no—
137C2×#97—42no—
138C2×#97C2314yes—
139C2×#97C2314yes—
140C2×C4×C8—24no—
141C2×C4×C8C2314yes—
142#401—28no—
143#400—28no—
144#396—28no—
145#393—28no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391C2314yes—
150#391C2314yes—
151#391C2314yes—
152#390—22no—
153#390—22no—
154#390C2314yes—
155#390C2314yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#387—28no—
163#385—28no—
164#383—28no—
165#382—22no—
166#382C2314yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4D414yes—
169C4×C4⋊C4C2314yes—
170C4×C4⋊C4C2314yes—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C22×Q8—21no—
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—82no—
207C22×Q8C2412yes—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4#4114yes—
215C22×D4C2412yes—
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—44no—
225C23×C4—44no—
226C23×C4—82no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C2×SD16—28no—
235C2×SD16—28no—
236C2×SD16—44no—
237C2×SD16—416no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8#4114yes—
249C22×C8#4114yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—28no—
255C4⋊Q8—44no—
256C4⋊Q8—48no—
257C22⋊Q8—432no—
258C4×D4—28no—
259C4×D4—44no—
260C4×D4—48no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
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—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—82no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—44no—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2412yes—
319C2×C42C2412yes—
320C2×C42C2412yes—
321#107—28no—
322C4⋊C8—28no—
323#103—28no—
324#103—28no—
325#103—44no—
326#102—28no—
327#102—28no—
328#102—28no—
329#97—44no—
330#97—44no—
331C4×C8—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95C2×D414yes—
345#95C2×D414yes—
346C24—24no—
347C24—22no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24C22×D411yescharacteristic
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—84no—
371C2×Q8—816no—
372C2×D4—28no—
373C2×D4—24no—
374C2×D4—28no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4C2×#4114yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—84no—
433C22×C4—816no—
434C22×C4AGL(1,ℤ/8)14yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+412yes—
448C22×C4C2511yescharacteristic
449SD16—416no—
450C2×C8—28no—
451C2×C8—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—44no—
463C2×C8—44no—
464C2×C8C2×#4114yes—
465C2×C8C2×#4114yes—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4C2×#4114yes—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C42—24no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—48no—
505C23—24no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—84no—
516C23—84no—
517C23—84no—
518C23—84no—
519C23#53212yes—
520C23#54612yes—
521C23#54812yes—
522C23C2×AGL(1,ℤ/8)12yes—
523C23#57512yes—
524C23#57712yes—
525C23#57812yes—
526C23C23×D411yescharacteristic
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—816no—
532Q8—84no—
533D4—24no—
534D4—24no—
535D4—48no—
536D4—42no—
537D4—42no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
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—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—816no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4#52914yes—
591C2×C4#53212yes—
592C2×C4#53914yes—
593C2×C4#54612yes—
594C2×C4#54812yes—
595C2×C4#55114yes—
596C2×C4#55614yes—
597C2×C4C2×AGL(1,ℤ/8)14yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×2−1+411yescharacteristic
604C8—24no—
605C8—24no—
606C8—42no—
607C8—42no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—82no—
612C22—82no—
613C22—816no—
614C22—82no—
615C22—82no—
616C22#256014yes—
617C22#317814yes—
618C22#318014yes—
619C22#332414yes—
620C22#332714yes—
621C22#336714yes—
622C22#337614yes—
623C22C2×#53211yescharacteristic
624C22C2×#54611yescharacteristic
625C22C2×#54811yescharacteristic
626C22C22×AGL(1,ℤ/8)11yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57711yescharacteristic
629C22C2×#57811yescharacteristic
630C4—24no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4#256014yes—
644C4C2×#52911yescharacteristic
645C4C2×#53911yescharacteristic
646C4C2×#55111yescharacteristic
647C4C2×#55611yescharacteristic
648C2—84no—
649C2#2908318yesdirect product factor
650C2C2×#256011yescharacteristic
651C2C2×#317811yescharacteristic
652C2C2×#318011yescharacteristic
653C2C2×#332411yescharacteristic
654C2C2×#332711yescharacteristic
655C2C2×#336711yescharacteristic
656C2C2×#337611yescharacteristic
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×#2514C2semidirect
3. #29083C2direct (N × H)
4. C2×#2086C2semidirect
7. C2×#2024C2semidirect
8. C2×#1993C2semidirect
10. C2×#1935C2semidirect
12. C2×#1933C2semidirect
13. C2×#1921C2semidirect
15. C2×#1770C2semidirect
33. #2514C22semidirect
54. #2086C22semidirect
57. #2024C22semidirect
58. #1993C22semidirect
60. #1935C22semidirect
62. #1933C22semidirect
63. #1921C22semidirect
65. #1770C22semidirect
649. C2#29083direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25142
C2×#24522
#290832
C2×#20862
C2×#20732
C2×#20552
C2×#20242
C2×#19932
C2×#19732
C2×#19352
C2×#19342
C2×#19332
C2×#19212
C2×#18772
C2×#17702
C2×#17622