Finite Group Databasebeta

← C2×#31440 #7504717

Subgroups — C2×#31440 #7504717

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic
2C2×#2442C211yesmaximal, characteristic
3#31440C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2099C211yesmaximal, characteristic, complemented
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic
8C2×#2077C211yesmaximal, characteristic
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#1992C211yesmaximal, characteristic, complemented
11C2×#1949C211yesmaximal, characteristic, complemented
12C2×#1944C211yesmaximal, characteristic, complemented
13C2×#1924C211yesmaximal, characteristic
14C2×#1918C211yesmaximal, characteristic, complemented
15C2×#1852C211yesmaximal, characteristic
16C2×#1824C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×#97C2211yescharacteristic
27#2497—24no—
28#2497C2218yes—
29#2442—24no—
30#2442C2218yes—
31C2×#400—21no—
32C2×#400—21no—
33C2×#398—21no—
34C2×#396—21no—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#388—21no—
40C2×#387—21no—
41C2×#385—21no—
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#380—21no—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2218yescomplemented
51#2099C2218yescomplemented
52#2088C2218yescomplemented
53#2077C2218yes—
54#2077C2218yes—
55#2059C2218yescomplemented
56#1992C2218yescomplemented
57#1949C2218yescomplemented
58#1944C2218yescomplemented
59#1924C2218yes—
60#1918C2218yescomplemented
61#1852C2218yes—
62#1824C2218yes—
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
77C24×C4—21no—
78C24×C4C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#125—24no—
84C2×#125—22no—
85C2×#125—42no—
86C2×#125C2314yes—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4D411yescharacteristic
112C22×C22⋊C4D411yescharacteristic
113C22×C42—21no—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C2×#107—24no—
117C2×#107—42no—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#106—24no—
121C2×#106—42no—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8—24no—
125C2×C4⋊C8C2314yes—
126C2×C22⋊C8—24no—
127C2×C22⋊C8—42no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130C2×#97—42no—
131C2×#97C2314yes—
132#400—28no—
133#400—28no—
134#398—28no—
135#396—28no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389C2314yes—
141#389C2314yes—
142#389C2314yes—
143#389C2314yes—
144#388—28no—
145#387—28no—
146#385—28no—
147#384—22no—
148#384C2314yes—
149#382—22no—
150#382C2314yes—
151#380—22no—
152#380—22no—
153#380—22no—
154#380—28no—
155#380C2314yes—
156#380C2314yes—
157#380C2314yes—
158C4×C4⋊C4—22no—
159C4×C4⋊C4C2314yes—
160C4×C22⋊C4D414yes—
161C4×C22⋊C4C2314yes—
162C2×#95—21no—
163C2×#95—21no—
164C2×#95—21no—
165C2×#95—21no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95D411yescharacteristic
173C2×#95D411yescharacteristic
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188C25C2×D411yescharacteristic
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—82no—
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8#4114yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231#125—28no—
232#125—44no—
233#125—48no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
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—28no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4#4114yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—82no—
294C2×C22⋊C4—82no—
295C2×C22⋊C4C2×D414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—82no—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2412yes—
306C2×C42C2412yes—
307#107—28no—
308#107—44no—
309#106—28no—
310#106—44no—
311C4⋊C8—28no—
312C22⋊C8—28no—
313C22⋊C8—44no—
314#97—44no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95C2×D414yes—
326#95C2×D414yes—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24#13714yes—
334C24C22×D412yes—
335C24C22×D411yescharacteristic
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—48no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
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—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—84no—
399C22×C4—816no—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C42−1+412yes—
410C22×C42−1+412yes—
411C22×C42−1+412yes—
412C22×C4C2511yescharacteristic
413C2×C8—28no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—44no—
426C2×C8—44no—
427C2×C8—44no—
428C2×C8C2×#4114yes—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4C2×#4114yes—
462C22⋊C4—28no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—24no—
468C42—24no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—84no—
474C23—28no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—28no—
479C23—28no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23#53612yes—
488C23#54612yes—
489C23#54712yes—
490C23C2×#13714yes—
491C23C2×#13712yes—
492C23#57512yes—
493C23#57612yes—
494C23#57712yes—
495C23C23×D411yescharacteristic
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—48no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
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—816no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4#53612yes—
555C2×C4#53914yes—
556C2×C4#54114yes—
557C2×C4#54612yes—
558C2×C4#54712yes—
559C2×C4#55614yes—
560C2×C4#56314yes—
561C2×C4C22×#4111yescharacteristic
562C2×C4C22×#4111yescharacteristic
563C2×C4C2×H(5,𝔽2)11yescharacteristic
564C2×C4C2×2−1+411yescharacteristic
565C2×C4C2×2−1+411yescharacteristic
566C2×C4C2×2−1+411yescharacteristic
567C8—24no—
568C8—42no—
569C8—42no—
570C8—42no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—416no—
578C22—42no—
579C22#268914yes—
580C22#323614yes—
581C22#324114yes—
582C22#331614yes—
583C22#332214yes—
584C22#334614yes—
585C22#334814yes—
586C22C2×#53611yescharacteristic
587C22C2×#54611yescharacteristic
588C22C2×#54711yescharacteristic
589C22C22×#13711yescharacteristic
590C22C2×#57511yescharacteristic
591C22C2×#57611yescharacteristic
592C22C2×#57711yescharacteristic
593C4—42no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—44no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4#268914yes—
607C4C2×#53911yescharacteristic
608C4C2×#54111yescharacteristic
609C4C2×#55611yescharacteristic
610C4C2×#56311yescharacteristic
611C2—44no—
612C2#3144018yesdirect product factor
613C2C2×#268911yescharacteristic
614C2C2×#323611yescharacteristic
615C2C2×#324111yescharacteristic
616C2C2×#331611yescharacteristic
617C2C2×#332211yescharacteristic
618C2C2×#334611yescharacteristic
619C2C2×#334811yescharacteristic
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. #31440C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2099C2semidirect
6. C2×#2088C2semidirect
9. C2×#2059C2semidirect
10. C2×#1992C2semidirect
11. C2×#1949C2semidirect
12. C2×#1944C2semidirect
14. C2×#1918C2semidirect
50. #2101C22semidirect
51. #2099C22semidirect
52. #2088C22semidirect
55. #2059C22semidirect
56. #1992C22semidirect
57. #1949C22semidirect
58. #1944C22semidirect
60. #1918C22semidirect
612. C2#31440direct (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×#24422
#314402
C2×#21012
C2×#20992
C2×#20882
C2×#20772
C2×#20592
C2×#19922
C2×#19492
C2×#19442
C2×#19242
C2×#19182
C2×#18522
C2×#18242