Finite Group Databasebeta

← C2×#22092 #6375622

Subgroups — C2×#22092 #6375622

Order 512.

2252 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C2×#2323C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic
4#22092C2116yesmaximal, direct product factor
5C2×#1943C211yesmaximal, characteristic, complemented
6C2×#1941C211yesmaximal, characteristic, complemented
7C2×#1935C211yesmaximal, characteristic, complemented
8C2×#1933C211yesmaximal, characteristic, complemented
9C2×#1921C211yesmaximal, characteristic, complemented
10C2×#1920C211yesmaximal, characteristic, complemented
11C2×#1872C211yesmaximal, characteristic
12C2×#1865C211yesmaximal, characteristic, complemented
13C2×#1865C211yesmaximal, characteristic, complemented
14C2×#1853C211yesmaximal, characteristic
15C2×#1825C211yesmaximal, characteristic
16C2×#1750C211yesmaximal, characteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#485C4116yescomplemented
23C2×#485C2218yes—
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2323—24no—
37#2323C2218yes—
38#2317—24no—
39#2317C2218yes—
40C2×#391C2211yescharacteristic
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#388—21no—
48C2×#388—21no—
49C2×#387—21no—
50C2×#385—21no—
51C2×#385—21no—
52C2×#384—21no—
53C2×#382—21no—
54C2×#380C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#1943C2218yescomplemented
57#1941C2218yescomplemented
58#1935C2218yescomplemented
59#1933C2218yescomplemented
60#1921C2218yescomplemented
61#1920C2218yescomplemented
62#1872C2218yes—
63#1865C2218yescomplemented
64#1865C2218yescomplemented
65#1853C2218yes—
66#1825C2218yes—
67#1750C2218yes—
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8C2×C418yes—
88C2×C22⋊Q8C2314yes—
89C2×C4×Q8—22no—
90C2×C4×Q8—42no—
91C2×C4×Q8—42no—
92C2×C4×Q8C2314yes—
93C2×#117—22no—
94C2×#117—24no—
95C2×#117—42no—
96C2×#117—42no—
97C2×#117C2×C418yes—
98C2×#117D414yes—
99C2×#117C2314yes—
100C2×#117C2314yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—42no—
111C22×C4⋊C4—42no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4D411yescharacteristic
126C22×C22⋊C4D411yescharacteristic
127C22×C42—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#485C2×C4164yescomplemented
133C2×#107C2×C418yescomplemented
134C2×#107C2314yes—
135C2×#106—42no—
136C2×#106—42no—
137C2×#106C2×C418yescomplemented
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103C2×C418yescomplemented
144C2×#103C2×C418yescomplemented
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8—24no—
150C2×C22⋊C8C2×C418yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97—42no—
154C2×#97C2314yes—
155C2×C4×C8—42no—
156C2×C4×C8C2314yes—
157#391—22no—
158#391C2314yes—
159#389—28no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#389C2314yes—
169#389C2314yes—
170#388—28no—
171#388—28no—
172#387—28no—
173#385—28no—
174#385—28no—
175#384—28no—
176#382—28no—
177#380—22no—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2314yes—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25C2×D411yescharacteristic
201C22×Q8—21no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8C22×C414yes—
208C22×Q8C2412yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—24no—
222C23×C4—44no—
223C23×C4—82no—
224C23×C4—82no—
225C23×C4C22×C414yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4#4114yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8C22×C414yes—
246C22×C8C22×C414yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22⋊Q8—216no—
252C22⋊Q8—28no—
253C22⋊Q8C22×C4116yes—
254C4×Q8—44no—
255C4×Q8—44no—
256C4×Q8—48no—
257#117—216no—
258#117—28no—
259#117—48no—
260#117—44no—
261#117—44no—
262#117C22×C4116yes—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C22×C414yes—
303C2×C4⋊C4C22×C414yes—
304C2×C4⋊C4C22×C414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—82no—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D414yes—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42—82no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338#107C22×C4116yescomplemented
339#106—44no—
340#106—44no—
341#106C22×C4116yescomplemented
342#103—44no—
343#103—44no—
344#103C22×C4116yescomplemented
345#103C22×C4116yescomplemented
346C22⋊C8—28no—
347C22⋊C8C22×C4116yes—
348#97—44no—
349C4×C8—44no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24C4×D414yes—
366C24C22×D412yes—
367C24C22×D411yescharacteristic
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8C23×C414yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—816no—
441C22×C4C4×D414yes—
442C22×C4C4×D418yes—
443C22×C4C23×C414yes—
444C22×C4C23×C412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4114yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C42−1+412yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458C2×C8—28no—
459C2×C8—28no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—44no—
471C2×C8—44no—
472C2×C8C23×C414yes—
473C2×C8C23×C414yes—
474C4⋊C4—216no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4C23×C414yes—
503C4⋊C4C23×C414yes—
504C4⋊C4C23×C414yes—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—84no—
510C42—216no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—48no—
518C42—84no—
519C23—28no—
520C23—22no—
521C23—22no—
522C23—28no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23C2×C4×D414yes—
533C23C2×C4×D412yes—
534C23#53412yes—
535C23#53612yes—
536C23#57712yes—
537C23#57712yes—
538C23#57712yes—
539C23#57812yes—
540C23C23×D411yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—48no—
544Q8—42no—
545Q8—42no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—216no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—816no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4C2×C4×D414yes—
604C2×C4C2×C4×D412yes—
605C2×C4#52014yes—
606C2×C4#52014yes—
607C2×C4#53412yes—
608C2×C4#53612yes—
609C2×C4#53914yes—
610C2×C4#53914yes—
611C2×C4C24×C411yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C8—24no—
618C8—24no—
619C8—42no—
620C8—42no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22#232214yes—
630C22#294714yes—
631C22#295214yes—
632C22#320014yes—
633C22#320114yes—
634C22#324214yes—
635C22#324414yes—
636C22C22×C4×D411yescharacteristic
637C22C2×#53411yescharacteristic
638C22C2×#53611yescharacteristic
639C22C2×#57711yescharacteristic
640C22C2×#57711yescharacteristic
641C22C2×#57711yescharacteristic
642C22C2×#57811yescharacteristic
643C4—24no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#232214yes—
657C4C2×#52011yescharacteristic
658C4C2×#52011yescharacteristic
659C4C2×#53911yescharacteristic
660C4C2×#53911yescharacteristic
661C2—44no—
662C2#2209218yesdirect product factor
663C2C2×#232211yescharacteristic
664C2C2×#294711yescharacteristic
665C2C2×#295211yescharacteristic
666C2C2×#320011yescharacteristic
667C2C2×#320111yescharacteristic
668C2C2×#324211yescharacteristic
669C2C2×#324411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #22092C2direct (N × H)
5. C2×#1943C2semidirect
6. C2×#1941C2semidirect
7. C2×#1935C2semidirect
8. C2×#1933C2semidirect
9. C2×#1921C2semidirect
10. C2×#1920C2semidirect
12. C2×#1865C2semidirect
13. C2×#1865C2semidirect
22. C2×#485C4semidirect
56. #1943C22semidirect
57. #1941C22semidirect
58. #1935C22semidirect
59. #1933C22semidirect
60. #1921C22semidirect
61. #1920C22semidirect
63. #1865C22semidirect
64. #1865C22semidirect
132. #485C2×C4semidirect
133. C2×#107C2×C4semidirect
137. C2×#106C2×C4semidirect
143. C2×#103C2×C4semidirect
144. C2×#103C2×C4semidirect
338. #107C22×C4semidirect
341. #106C22×C4semidirect
344. #103C22×C4semidirect
345. #103C22×C4semidirect
662. C2#22092direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4852
C2×#23232
C2×#23172
#220922
C2×#19432
C2×#19412
C2×#19352
C2×#19332
C2×#19212
C2×#19202
C2×#18722
C2×#18652
C2×#18532
C2×#18252
C2×#17502