Finite Group Databasebeta

← C2×#25994 #6452293

Subgroups — C2×#25994 #6452293

Order 512.

2218 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×#2440C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic
3#25994C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2087C211yesmaximal, characteristic, complemented
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2057C211yesmaximal, characteristic, complemented
8C2×#1987C211yesmaximal, characteristic, complemented
9C2×#1947C211yesmaximal, characteristic, complemented
10C2×#1921C211yesmaximal, characteristic, complemented
11C2×#1921C211yesmaximal, characteristic, complemented
12C2×#1852C211yesmaximal, characteristic
13C2×#1850C211yesmaximal, characteristic
14C2×#1824C211yesmaximal, characteristic
15C2×#1808C211yesmaximal, characteristic
16C22×#340C211yesmaximal, characteristic
17C24×C8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2440—24no—
30#2440C2218yes—
31C2×#398—21no—
32C2×#395—21no—
33#2345—24no—
34#2345C2218yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#387—21no—
40C2×#387—21no—
41C2×#385—21no—
42C2×#384C2211yescharacteristic
43C2×#380—21no—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2100C2218yescomplemented
51#2087C2218yescomplemented
52#2087C2218yescomplemented
53#2057C2218yescomplemented
54#1987C2218yescomplemented
55#1947C2218yescomplemented
56#1921C2218yescomplemented
57#1921C2218yescomplemented
58#1852C2218yes—
59#1850C2218yes—
60#1824C2218yes—
61#1808C2218yes—
62C2×#340C2211yescharacteristic
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2218yes—
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C24×C4—21no—
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C23×C8—24no—
82C23×C8—24no—
83C23×C8—42no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2314yes—
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—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—42no—
109C22×C4⋊C4Q818yescomplemented
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4D411yescharacteristic
123C22×C22⋊C4D411yescharacteristic
124C22×C42—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C2×#107—24no—
129C2×#107—24no—
130C2×#107—42no—
131C2×#107—42no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×C4⋊C8—24no—
137C2×C4⋊C8—42no—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C22⋊C8—24no—
141C2×C22⋊C8C2314yes—
142#398—28no—
143#395—28no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#387—28no—
153#387—28no—
154#385—28no—
155#384—22no—
156#384C2314yes—
157#380—22no—
158#380—22no—
159#380—28no—
160#380C2314yes—
161#380C2314yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2314yes—
166C4×C22⋊C4—28no—
167C4×C22⋊C4D414yes—
168C4×C22⋊C4C2314yes—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95D411yescharacteristic
178C2×#95D411yescharacteristic
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193C25C2×D411yescharacteristic
194C22×Q8—24no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8—82no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—82no—
216C23×C4—82no—
217C23×C4—82no—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D412yes—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×Q814yes—
225C23×C4#4114yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—28no—
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×C8—28no—
241C22×C8—44no—
242C22×C8—416no—
243C22×C8#4114yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C4⋊Q8—28no—
249C4⋊Q8—44no—
250C4⋊Q8—48no—
251C22⋊Q8—432no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
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—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2×Q814yes—
297C2×C4⋊C4C2×Q814yes—
298C2×C4⋊C4C2×Q8116yescomplemented
299C2×C4⋊C4#4114yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4Q1618yes—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D414yes—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—82no—
323C2×C42—82no—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2412yes—
327C2×C42C2412yes—
328#107—28no—
329#107—28no—
330#107—44no—
331#107—44no—
332C4⋊C8—28no—
333C4⋊C8—44no—
334C22⋊C8—28no—
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#95C2×D414yes—
344#95C2×D414yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24C22⋊Q814yes—
350C24C2×Q1614yes—
351C24AGL(1,ℤ/8)14yes—
352C24C22×D412yes—
353C24C22×D411yescharacteristic
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—84no—
357C2×Q8—816no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—48no—
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×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—816no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—816no—
420C22×C4C22⋊Q814yes—
421C22×C4C22⋊Q818yes—
422C22×C4C2×Q1614yes—
423C22×C4C2×Q1614yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×Q814yes—
431C22×C4C22×Q812yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439C2×C8—28no—
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—416no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8—44no—
455C2×C8C2×#4114yes—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—216no—
459C4⋊C4—24no—
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—22no—
472C4⋊C4—22no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4C22×Q814yes—
487C4⋊C4C22×Q814yes—
488C4⋊C4C2×#4114yes—
489C22⋊C4—28no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4C2×Q16116yes—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—84no—
500C42—84no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—28no—
509C23—22no—
510C23—22no—
511C23#477116yes—
512C23C2×C22⋊Q814yes—
513C23C2×C22⋊Q812yes—
514C23#54612yes—
515C23#54712yes—
516C23C22×Q1614yes—
517C23C22×Q1612yes—
518C23C2×AGL(1,ℤ/8)14yes—
519C23C2×AGL(1,ℤ/8)12yes—
520C23#57612yes—
521C23#57712yes—
522C23C23×D411yescharacteristic
523Q8—816no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—216no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
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—816no—
574C2×C4—816no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4C2×C22⋊Q814yes—
581C2×C4C2×C22⋊Q812yes—
582C2×C4#53314yes—
583C2×C4#54414yes—
584C2×C4#54612yes—
585C2×C4#54712yes—
586C2×C4#55114yes—
587C2×C4#55214yes—
588C2×C4C22×Q1614yes—
589C2×C4C22×Q1614yes—
590C2×C4C23×Q811yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C22×#4111yescharacteristic
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C2×C4C2×2−1+411yescharacteristic
596C8—24no—
597C8—42no—
598C8—42no—
599C8—42no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—232no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22#243814yes—
608C22C2×#47718yes—
609C22C2×#47714yes—
610C22#308114yes—
611C22#329914yes—
612C22#331814yes—
613C22#333814yes—
614C22#334514yes—
615C22C22×C22⋊Q811yescharacteristic
616C22C2×#54611yescharacteristic
617C22C2×#54711yescharacteristic
618C22C23×Q1611yescharacteristic
619C22C22×AGL(1,ℤ/8)11yescharacteristic
620C22C2×#57611yescharacteristic
621C22C2×#57711yescharacteristic
622C4—24no—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4#243814yes—
636C4C2×#53311yescharacteristic
637C4C2×#54411yescharacteristic
638C4C2×#55111yescharacteristic
639C4C2×#55211yescharacteristic
640C2—28no—
641C2#2599418yesdirect product factor
642C2C2×#243811yescharacteristic
643C2C22×#47711yescharacteristic
644C2C2×#308111yescharacteristic
645C2C2×#329911yescharacteristic
646C2C2×#331811yescharacteristic
647C2C2×#333811yescharacteristic
648C2C2×#334511yescharacteristic
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. #25994C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2087C2semidirect
6. C2×#2087C2semidirect
7. C2×#2057C2semidirect
8. C2×#1987C2semidirect
9. C2×#1947C2semidirect
10. C2×#1921C2semidirect
11. C2×#1921C2semidirect
50. #2100C22semidirect
51. #2087C22semidirect
52. #2087C22semidirect
53. #2057C22semidirect
54. #1987C22semidirect
55. #1947C22semidirect
56. #1921C22semidirect
57. #1921C22semidirect
109. C22×C4⋊C4Q8semidirect
298. C2×C4⋊C4C2×Q8semidirect
641. C2#25994direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#23452
#259942
C2×#21002
C2×#20872
C2×#20572
C2×#19872
C2×#19472
C2×#19212
C2×#18522
C2×#18502
C2×#18242
C2×#18082
C22×#3402