Finite Group Databasebeta

← C2×#31474 #7504805

Subgroups — C2×#31474 #7504805

Order 512.

1852 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×#2552C211yesmaximal, characteristic, complemented
2C2×#2454C211yesmaximal, characteristic
3#31474C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2099C211yesmaximal, characteristic, complemented
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2071C211yesmaximal, characteristic
8C2×#2071C211yesmaximal, characteristic
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#1992C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic
12C2×#1926C211yesmaximal, characteristic
13C2×#1921C211yesmaximal, characteristic, complemented
14C2×#1917C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic
16C2×#1802C211yesmaximal, characteristic
17C22×#125—21no—
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#97C2211yescharacteristic
28#2552—24no—
29#2552C2218yescomplemented
30#2454—24no—
31#2454C2218yes—
32C2×#399—21no—
33C2×#398—21no—
34C2×#398—21no—
35C2×#396—21no—
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#389—21no—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#385—21no—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#380—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2100C2218yescomplemented
51#2099C2218yescomplemented
52#2088C2218yescomplemented
53#2071C2218yes—
54#2071C2218yes—
55#2059C2218yescomplemented
56#1992C2218yescomplemented
57#1936C2218yes—
58#1926C2218yes—
59#1921C2218yescomplemented
60#1917C2218yescomplemented
61#1862C2218yes—
62#1802C2218yes—
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4—21no—
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#125—28no—
84C2×#125—24no—
85C2×#125—22no—
86C2×#125—42no—
87C2×#125—48no—
88C2×#125C2314yes—
89C2×C4×D4—24no—
90C2×C4×D4—22no—
91C2×C4×D4—42no—
92C2×C4×D4C2314yes—
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—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×#107—24no—
124C2×#107C2314yes—
125C2×#106—42no—
126C2×#106C2314yes—
127C2×C4⋊C8—42no—
128C2×C4⋊C8C2314yes—
129C2×#102—24no—
130C2×#102—24no—
131C2×#102—24no—
132C2×#102—42no—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#97—42no—
138C2×#97C2314yes—
139#399—28no—
140#398—28no—
141#398—28no—
142#396—28no—
143#390—22no—
144#390—22no—
145#390C2314yes—
146#390C2314yes—
147#389—28no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2314yes—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#385—28no—
157#384—22no—
158#384—22no—
159#384—22no—
160#384C2314yes—
161#384C2314yes—
162#384C2314yes—
163#380—28no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4D414yes—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
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#336C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C25—21no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4#4114yes—
197C22×D4C2412yes—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—24no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—82no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2411yescharacteristic
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—22no—
227C22×C8—22no—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232#125—28no—
233#125—44no—
234#125—48no—
235#125—432no—
236C4×D4—28no—
237C4×D4—44no—
238C4×D4—48no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—21no—
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—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4#4114yes—
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—24no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—82no—
303C2×C22⋊C4—82no—
304C2×C22⋊C4—82no—
305C2×C42—22no—
306C2×C42—22no—
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×C42C2×D412yes—
316C2×C42C2412yes—
317C2×C42C2412yes—
318C2×C42C2412yes—
319#107—28no—
320#106—44no—
321C4⋊C8—44no—
322#102—28no—
323#102—28no—
324#102—28no—
325#102—44no—
326#97—44no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95C2×D414yes—
333#95C2×D414yes—
334C24—24no—
335C24—22no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24C22×D411yescharacteristic
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—28no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—42no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—44no—
355C2×D4C2×#4114yes—
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—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—22no—
379C22×C4—22no—
380C22×C4—48no—
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—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—48no—
402C22×C4—84no—
403C22×C4—816no—
404C22×C4AGL(1,ℤ/8)14yes—
405C22×C4#13714yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C42−1+412yes—
417C22×C42−1+412yes—
418C22×C42−1+412yes—
419C22×C4C2511yescharacteristic
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
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—44no—
433C2×C8—44no—
434C2×C8—44no—
435C2×C8—44no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—22no—
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—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
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—48no—
464C4⋊C4—48no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4C2×#4114yes—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C42—24no—
477C42—24no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—48no—
486C23—24no—
487C23—48no—
488C23—48no—
489C23—42no—
490C23—48no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23#53612yes—
501C23#54612yes—
502C23#54812yes—
503C23C2×AGL(1,ℤ/8)12yes—
504C23C2×#13712yes—
505C23#57612yes—
506C23#57712yes—
507C23C23×D411yescharacteristic
508D4—24no—
509D4—24no—
510D4—42no—
511D4—48no—
512D4—42no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—48no—
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#53612yes—
565C2×C4#53914yes—
566C2×C4#54114yes—
567C2×C4#54612yes—
568C2×C4#54812yes—
569C2×C4#55214yes—
570C2×C4#56314yes—
571C2×C4C2×AGL(1,ℤ/8)14yes—
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—42no—
580C8—42no—
581C8—42no—
582C8—42no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—816no—
591C22#269014yes—
592C22#323914yes—
593C22#324114yes—
594C22#330614yes—
595C22#333314yes—
596C22#336614yes—
597C22#337214yes—
598C22C2×#53611yescharacteristic
599C22C2×#54611yescharacteristic
600C22C2×#54811yescharacteristic
601C22C22×AGL(1,ℤ/8)11yescharacteristic
602C22C22×#13711yescharacteristic
603C22C2×#57611yescharacteristic
604C22C2×#57711yescharacteristic
605C4—24no—
606C4—24no—
607C4—42no—
608C4—42no—
609C4—44no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4#269014yes—
619C4C2×#53911yescharacteristic
620C4C2×#54111yescharacteristic
621C4C2×#55211yescharacteristic
622C4C2×#56311yescharacteristic
623C2—84no—
624C2#3147418yesdirect product factor
625C2C2×#269011yescharacteristic
626C2C2×#323911yescharacteristic
627C2C2×#324111yescharacteristic
628C2C2×#330611yescharacteristic
629C2C2×#333311yescharacteristic
630C2C2×#336611yescharacteristic
631C2C2×#337211yescharacteristic
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×#2552C2semidirect
3. #31474C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2099C2semidirect
6. C2×#2088C2semidirect
9. C2×#2059C2semidirect
10. C2×#1992C2semidirect
13. C2×#1921C2semidirect
14. C2×#1917C2semidirect
29. #2552C22semidirect
50. #2100C22semidirect
51. #2099C22semidirect
52. #2088C22semidirect
55. #2059C22semidirect
56. #1992C22semidirect
59. #1921C22semidirect
60. #1917C22semidirect
624. C2#31474direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24542
#314742
C2×#21002
C2×#20992
C2×#20882
C2×#20712
C2×#20592
C2×#19922
C2×#19362
C2×#19262
C2×#19212
C2×#19172
C2×#18622
C2×#18022