Finite Group Databasebeta

← C2×#31686 #7505978

Subgroups — C2×#31686 #7505978

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3#31686C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2086C211yesmaximal, characteristic, complemented
6C2×#2083C211yesmaximal, characteristic, complemented
7C2×#2075C211yesmaximal, characteristic
8C2×#2075C211yesmaximal, characteristic
9C2×#2057C211yesmaximal, characteristic, complemented
10C2×#1988C211yesmaximal, characteristic, complemented
11C2×#1942C211yesmaximal, characteristic, complemented
12C2×#1923C211yesmaximal, characteristic
13C2×#1918C211yesmaximal, characteristic, complemented
14C2×#1852C211yesmaximal, characteristic
15C2×#1834C211yesmaximal, characteristic, complemented
16C2×#1823C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#126—21no—
19C22×C22⋊Q8—21no—
20C23×C4⋊C4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2496—24no—
30#2496C2218yes—
31#2440—24no—
32#2440C2218yes—
33C2×#400—21no—
34C2×#398—21no—
35C2×#397—21no—
36C2×#395—21no—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#387—21no—
42C2×#385—21no—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2099C2218yescomplemented
51#2086C2218yescomplemented
52#2083C2218yescomplemented
53#2075C2218yes—
54#2075C2218yes—
55#2057C2218yescomplemented
56#1988C2218yescomplemented
57#1942C2218yescomplemented
58#1923C2218yes—
59#1918C2218yescomplemented
60#1852C2218yes—
61#1834C2218yescomplemented
62#1823C2218yes—
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×C4—21no—
79C24×C4C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8C2314yes—
88C2×#126—28no—
89C2×#126—48no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—42no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C2×#107—24no—
123C2×#107—24no—
124C2×#107—42no—
125C2×#107—42no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×C4⋊C8—42no—
131C2×C4⋊C8C2314yes—
132C2×C22⋊C8—24no—
133C2×C22⋊C8—42no—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×C4×C8—24no—
137C2×C4×C8C2314yes—
138#400—28no—
139#398—28no—
140#397—28no—
141#395—28no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389C2314yes—
147#389C2314yes—
148#389C2314yes—
149#389C2314yes—
150#387—28no—
151#385—28no—
152#384—22no—
153#384—22no—
154#384C2314yes—
155#384C2314yes—
156#380—22no—
157#380—22no—
158#380—22no—
159#380C2314yes—
160#380C2314yes—
161#380C2314yes—
162C4×C4⋊C4—22no—
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×#95D411yescharacteristic
175C2×#95D411yescharacteristic
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190C25C2×D411yescharacteristic
191C22×Q8—24no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—82no—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—24no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—82no—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C22×C8—22no—
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—22no—
232C22×C8#4114yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C4⋊Q8—28no—
238C4⋊Q8—44no—
239C4⋊Q8—48no—
240#126—432no—
241C22⋊Q8—432no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
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—28no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
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—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4D818yes—
305C2×C22⋊C4C2×D414yes—
306C2×C22⋊C4C2×D414yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—44no—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2412yes—
316C2×C42C2412yes—
317#107—28no—
318#107—28no—
319#107—44no—
320#107—44no—
321C4⋊C8—44no—
322C22⋊C8—28no—
323C22⋊C8—44no—
324C4×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#95C2×D414yes—
334#95C2×D414yes—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24C2×D814yes—
342C24C22×D412yes—
343C24C22×D411yescharacteristic
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—84no—
347C2×Q8—816no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
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—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
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—48no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—84no—
408C22×C4—816no—
409C22×C4C2×D814yes—
410C22×C4C2×D814yes—
411C22×C4#13714yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+412yes—
422C22×C42−1+412yes—
423C22×C42−1+412yes—
424C22×C4C2511yescharacteristic
425C2×C8—28no—
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—22no—
436C2×C8—22no—
437C2×C8—44no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8C2×#4114yes—
441C4⋊C4—24no—
442C4⋊C4—22no—
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—22no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—84no—
474C4⋊C4C2×#4114yes—
475C22⋊C4—28no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4C2×D8116yes—
480C42—24no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—48no—
487C23—28no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—28no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23#53612yes—
501C23#54612yes—
502C23#54712yes—
503C23C22×D814yes—
504C23C22×D812yes—
505C23C2×#13712yes—
506C23#57712yes—
507C23#57812yes—
508C23C23×D411yescharacteristic
509Q8—816no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
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—816no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4#53612yes—
566C2×C4#54114yes—
567C2×C4#54214yes—
568C2×C4#54612yes—
569C2×C4#54712yes—
570C2×C4#55214yes—
571C2×C4#56314yes—
572C2×C4C22×D814yes—
573C2×C4C22×D814yes—
574C2×C4C2×#13714yes—
575C2×C4C22×#4111yescharacteristic
576C2×C4C22×#4111yescharacteristic
577C2×C4C2×H(5,𝔽2)11yescharacteristic
578C2×C4C2×2−1+411yescharacteristic
579C2×C4C2×2−1+411yescharacteristic
580C2×C4C2×2−1+411yescharacteristic
581C8—24no—
582C8—42no—
583C8—42no—
584C8—42no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—416no—
592C22—42no—
593C22#269814yes—
594C22#322614yes—
595C22#324214yes—
596C22#329414yes—
597C22#332814yes—
598C22#333414yes—
599C22#334414yes—
600C22C2×#53611yescharacteristic
601C22C2×#54611yescharacteristic
602C22C2×#54711yescharacteristic
603C22C23×D811yescharacteristic
604C22C22×#13711yescharacteristic
605C22C2×#57711yescharacteristic
606C22C2×#57811yescharacteristic
607C4—24no—
608C4—24no—
609C4—24no—
610C4—42no—
611C4—44no—
612C4—42no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4#269814yes—
621C4C2×#54111yescharacteristic
622C4C2×#54211yescharacteristic
623C4C2×#55211yescharacteristic
624C4C2×#56311yescharacteristic
625C2—44no—
626C2#3168618yesdirect product factor
627C2C2×#269811yescharacteristic
628C2C2×#322611yescharacteristic
629C2C2×#324211yescharacteristic
630C2C2×#329411yescharacteristic
631C2C2×#332811yescharacteristic
632C2C2×#333411yescharacteristic
633C2C2×#334411yescharacteristic
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. #31686C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2086C2semidirect
6. C2×#2083C2semidirect
9. C2×#2057C2semidirect
10. C2×#1988C2semidirect
11. C2×#1942C2semidirect
13. C2×#1918C2semidirect
15. C2×#1834C2semidirect
50. #2099C22semidirect
51. #2086C22semidirect
52. #2083C22semidirect
55. #2057C22semidirect
56. #1988C22semidirect
57. #1942C22semidirect
59. #1918C22semidirect
61. #1834C22semidirect
626. C2#31686direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#24402
#316862
C2×#20992
C2×#20862
C2×#20832
C2×#20752
C2×#20572
C2×#19882
C2×#19422
C2×#19232
C2×#19182
C2×#18522
C2×#18342
C2×#18232