Finite Group Databasebeta

← C2×#31690 #7505980

Subgroups — C2×#31690 #7505980

Order 512.

1890 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3#31690C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic
8C2×#2077C211yesmaximal, characteristic
9C2×#2058C211yesmaximal, characteristic, complemented
10C2×#1988C211yesmaximal, characteristic, complemented
11C2×#1941C211yesmaximal, characteristic, complemented
12C2×#1923C211yesmaximal, characteristic
13C2×#1919C211yesmaximal, characteristic, complemented
14C2×#1853C211yesmaximal, characteristic
15C2×#1834C211yesmaximal, characteristic, complemented
16C2×#1823C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C23×C4⋊C4C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2497—24no—
29#2497C2218yes—
30#2440—24no—
31#2440C2218yes—
32C2×#400—21no—
33C2×#398—21no—
34C2×#396—21no—
35C2×#395—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#388—21no—
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#2100C2218yescomplemented
51#2088C2218yescomplemented
52#2084C2218yescomplemented
53#2077C2218yes—
54#2077C2218yes—
55#2058C2218yescomplemented
56#1988C2218yescomplemented
57#1941C2218yescomplemented
58#1923C2218yes—
59#1919C2218yescomplemented
60#1853C2218yes—
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×C22⋊Q8—28no—
89C2×C22⋊Q8—48no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—42no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C2×#106—24no—
121C2×#106—24no—
122C2×#106—42no—
123C2×#106—42no—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8—42no—
129C2×C4⋊C8C2314yes—
130C2×C22⋊C8—24no—
131C2×C22⋊C8—42no—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C4×C8—24no—
135C2×C4×C8C2314yes—
136#400—28no—
137#398—28no—
138#396—28no—
139#395—28no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389C2314yes—
145#389C2314yes—
146#389C2314yes—
147#389C2314yes—
148#388—28no—
149#387—28no—
150#385—28no—
151#384—22no—
152#384—22no—
153#384C2314yes—
154#384C2314yes—
155#380—22no—
156#380—22no—
157#380—22no—
158#380C2314yes—
159#380C2314yes—
160#380C2314yes—
161C4×C4⋊C4—22no—
162C4×C4⋊C4C2314yes—
163C4×C22⋊C4D414yes—
164C4×C22⋊C4C2314yes—
165C2×#95—21no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95D411yescharacteristic
174C2×#95D411yescharacteristic
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189C25C2×D411yescharacteristic
190C22×Q8—24no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8—82no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
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—24no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—82no—
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C22×C8—22no—
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#4114yes—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C4⋊Q8—28no—
237C4⋊Q8—44no—
238C4⋊Q8—48no—
239C22⋊Q8—432no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
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—28no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—82no—
301C2×C22⋊C4—82no—
302C2×C22⋊C4SD1618yes—
303C2×C22⋊C4C2×D414yes—
304C2×C22⋊C4C2×D414yes—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—82no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2412yes—
314C2×C42C2412yes—
315#106—28no—
316#106—28no—
317#106—44no—
318#106—44no—
319C4⋊C8—44no—
320C22⋊C8—28no—
321C22⋊C8—44no—
322C4×C8—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95C2×D414yes—
332#95C2×D414yes—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24C2×SD1614yes—
340C24C22×D412yes—
341C24C22×D411yescharacteristic
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—84no—
345C2×Q8—816no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
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—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
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—48no—
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—84no—
406C22×C4—816no—
407C22×C4C2×SD1614yes—
408C22×C4C2×SD1614yes—
409C22×C4#13714yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C42−1+412yes—
420C22×C42−1+412yes—
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—44no—
436C2×C8—44no—
437C2×C8—44no—
438C2×C8C2×#4114yes—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
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—48no—
456C4⋊C4—48no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4C2×#4114yes—
473C22⋊C4—28no—
474C22⋊C4—48no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4C2×SD16116yes—
478C42—24no—
479C42—24no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—84no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—28no—
490C23—28no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23#53612yes—
499C23#54612yes—
500C23#54712yes—
501C23C22×SD1614yes—
502C23C22×SD1612yes—
503C23C2×#13712yes—
504C23#57612yes—
505C23#57712yes—
506C23C23×D411yescharacteristic
507Q8—816no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—42no—
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—816no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4#53612yes—
564C2×C4#54114yes—
565C2×C4#54214yes—
566C2×C4#54612yes—
567C2×C4#54712yes—
568C2×C4#55214yes—
569C2×C4#56314yes—
570C2×C4C22×SD1614yes—
571C2×C4C22×SD1614yes—
572C2×C4C2×#13714yes—
573C2×C4C22×#4111yescharacteristic
574C2×C4C22×#4111yescharacteristic
575C2×C4C2×H(5,𝔽2)11yescharacteristic
576C2×C4C2×2−1+411yescharacteristic
577C2×C4C2×2−1+411yescharacteristic
578C2×C4C2×2−1+411yescharacteristic
579C8—24no—
580C8—42no—
581C8—42no—
582C8—42no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—416no—
590C22—42no—
591C22#269814yes—
592C22#322914yes—
593C22#324114yes—
594C22#329514yes—
595C22#332914yes—
596C22#333514yes—
597C22#334614yes—
598C22C2×#53611yescharacteristic
599C22C2×#54611yescharacteristic
600C22C2×#54711yescharacteristic
601C22C23×SD1611yescharacteristic
602C22C22×#13711yescharacteristic
603C22C2×#57611yescharacteristic
604C22C2×#57711yescharacteristic
605C4—24no—
606C4—24no—
607C4—24no—
608C4—42no—
609C4—44no—
610C4—42no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4#269814yes—
619C4C2×#54111yescharacteristic
620C4C2×#54211yescharacteristic
621C4C2×#55211yescharacteristic
622C4C2×#56311yescharacteristic
623C2—44no—
624C2#3169018yesdirect product factor
625C2C2×#269811yescharacteristic
626C2C2×#322911yescharacteristic
627C2C2×#324111yescharacteristic
628C2C2×#329511yescharacteristic
629C2C2×#332911yescharacteristic
630C2C2×#333511yescharacteristic
631C2C2×#334611yescharacteristic
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. #31690C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2088C2semidirect
6. C2×#2084C2semidirect
9. C2×#2058C2semidirect
10. C2×#1988C2semidirect
11. C2×#1941C2semidirect
13. C2×#1919C2semidirect
15. C2×#1834C2semidirect
50. #2100C22semidirect
51. #2088C22semidirect
52. #2084C22semidirect
55. #2058C22semidirect
56. #1988C22semidirect
57. #1941C22semidirect
59. #1919C22semidirect
61. #1834C22semidirect
624. C2#31690direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24972
C2×#24402
#316902
C2×#21002
C2×#20882
C2×#20842
C2×#20772
C2×#20582
C2×#19882
C2×#19412
C2×#19232
C2×#19192
C2×#18532
C2×#18342
C2×#18232