Finite Group Databasebeta

← C2×#11875 #482043

Subgroups — C2×#11875 #482043

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic, complemented
3C2×#1459C211yesmaximal, characteristic, complemented
4#11875C218yesmaximal, direct product factor
5C2×#1344C212yesmaximal, complemented
6C2×#1343C212yesmaximal, complemented
7C22×#121—21no—
8C22×C4×D4C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#99—21no—
11C22×C22⋊C8—21no—
12#2535—22no—
13#2535—22no—
14#2535C414yescomplemented
15#2535C2214yes—
16C2×#397C411yescharacteristic, complemented
17C2×#397C2211yescharacteristic
18C2×#392C411yescharacteristic, complemented
19C2×#392C2211yescharacteristic
20C2×#386—21no—
21C2×#386—21no—
22C2×#382C411yescharacteristic, complemented
23C2×#380—21no—
24C2×#380—21no—
25#1799—22no—
26#1799C2214yescomplemented
27C2×#342C2212yes—
28#1459—22no—
29#1459C2214yescomplemented
30C2×#324—21no—
31C2×#323C2212yes—
32#1344C2218yescomplemented
33#1343C2218yescomplemented
34C23×D4—21no—
35C23×D4—21no—
36C23×D4C2×C411yescharacteristic
37C23×D4D411yescharacteristic
38C24×C4—21no—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C23×C8—41no—
42C2×#121—24no—
43C2×#121—24no—
44C2×#121—44no—
45C2×#121—44no—
46C2×C4×D4—22no—
47C2×C4×D4—41no—
48C2×C4×D4—41no—
49C2×C4×D4—42no—
50C2×C4×D4—42no—
51C2×C4×D4—41no—
52C2×C4×D4—41no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2×C414yescomplemented
55C2×C4×D4D412yes—
56C2×C4×D4D412yes—
57C2×C4×D4C2312yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—41no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C42C2×C411yescharacteristic
71C2×#99—24no—
72C2×#99—44no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—41no—
77#397—22no—
78#397—21no—
79#397—21no—
80#397C2×C412yes—
81#397C2×C414yescomplemented
82#397C2312yes—
83#392—22no—
84#392—21no—
85#392—21no—
86#392C2×C412yes—
87#392C2×C414yescomplemented
88#392C2312yes—
89#386—24no—
90#386—24no—
91#386—24no—
92#386—24no—
93#382—22no—
94#382C2×C414yescomplemented
95#380—24no—
96#380—24no—
97#380—24no—
98#380—42no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—41no—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105#342—22no—
106#342C2314yes—
107#324—24no—
108#323—22no—
109#323C2314yes—
110C25—21no—
111C25C22⋊C411yescharacteristic
112C25C2×D411yescharacteristic
113C22×D4—22no—
114C22×D4—24no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4—42no—
120C22×D4—41no—
121C22×D4—41no—
122C22×D4—44no—
123C22×D4—44no—
124C22×D4—44no—
125C22×D4—44no—
126C22×D4—42no—
127C22×D4—42no—
128C22×D4—82no—
129C22×D4—82no—
130C22×D4C22⋊C412yes—
131C22×D4D814yes—
132C22×D4SD1614yes—
133C22×D4C22×C412yes—
134C22×D4C2×D412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C412yes—
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C22×C8—42no—
162C22×C8—44no—
163C22×C8—84no—
164#121—416no—
165#121—48no—
166#121—48no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—44no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—44no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—84no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22×C412yes—
216C2×C22⋊C4C22×C411yescharacteristic
217C2×C22⋊C4C2×D411yescharacteristic
218C2×C22⋊C4C2×D411yescharacteristic
219C2×C22⋊C4C2411yescharacteristic
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C22⋊C412yes—
224C2×C42C22×C412yes—
225#99—48no—
226C22⋊C8—24no—
227C22⋊C8—42no—
228C22⋊C8—44no—
229#95—22no—
230#95—24no—
231#95—24no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—44no—
236#95C22×C412yes—
237#95C22×C412yes—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—24no—
245C24—22no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24#10214yes—
258C24C2×C22⋊C411yescharacteristic
259C24C2×C22⋊C412yes—
260C24C2×C22⋊C411yescharacteristic
261C24C2×D812yes—
262C24C2×SD1612yes—
263C24C22×D411yescharacteristic
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—48no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—48no—
281C2×D4—44no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—44no—
285C2×D4—48no—
286C2×D4—82no—
287C2×D4—84no—
288C2×D4—82no—
289C2×D4—816no—
290C2×D4—88no—
291C2×D4—88no—
292C2×D4#10218yes—
293C2×D4C2×D814yes—
294C2×D4C2×SD1614yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—48no—
334C22×C4—44no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—88no—
338C22×C4—88no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4#9911yescharacteristic
349C22×C4#9911yescharacteristic
350C22×C4#9912yes—
351C22×C4#10214yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C412yes—
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C2×C8—44no—
360C2×C8—88no—
361C2×C8—84no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—48no—
367C4⋊C4—88no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—48no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—84no—
384C22⋊C4—82no—
385C22⋊C4—84no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C23—22no—
391C23—22no—
392C23—21no—
393C23—22no—
394C23—21no—
395C23—22no—
396C23—24no—
397C23—22no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—24no—
403C23—22no—
404C23—24no—
405C23—41no—
406C23—42no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—48no—
412C23—44no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—42no—
418C23—48no—
419C23—48no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—84no—
433C23—84no—
434C23—82no—
435C23—84no—
436C23—84no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23C2×#9911yescharacteristic
441C23C2×#9911yescharacteristic
442C23C2×#10212yes—
443C23C2×#10212yes—
444C23C2×#10214yes—
445C23#42112yes—
446C23C22×C22⋊C411yescharacteristic
447C23C22×D811yescharacteristic
448C23C22×SD1611yescharacteristic
449D4—48no—
450D4—44no—
451D4—44no—
452D4—44no—
453D4—44no—
454D4—48no—
455D4—84no—
456D4—84no—
457D4—816no—
458D4—88no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—816no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—168no—
509C2×C4C2×#9912yes—
510C2×C4C2×#9912yes—
511C2×C4C2×#10214yes—
512C8—84no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—24no—
518C22—24no—
519C22—21no—
520C22—21no—
521C22—22no—
522C22—24no—
523C22—41no—
524C22—41no—
525C22—44no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—41no—
533C22—42no—
534C22—41no—
535C22—44no—
536C22—48no—
537C22—44no—
538C22—44no—
539C22—81no—
540C22—81no—
541C22—82no—
542C22—82no—
543C22—84no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—88no—
548C22—82no—
549C22—88no—
550C22—82no—
551C22#151912yes—
552C22#151912yes—
553C22#212012yes—
554C22#212712yes—
555C22C22×#9911yescharacteristic
556C22C22×#10211yescharacteristic
557C22C2×#42111yescharacteristic
558C4—22no—
559C4—22no—
560C4—24no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—164no—
570C2—21no—
571C2—21no—
572C2—22no—
573C2—41no—
574C2—41no—
575C2—42no—
576C2—44no—
577C2—84no—
578C2#1187514yesdirect product factor
579C2C2×#151911yescharacteristic
580C2C2×#212011yescharacteristic
581C2C2×#212711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1799C2semidirect
3. C2×#1459C2semidirect
4. #11875C2direct (N × H)
5. C2×#1344C2semidirect
6. C2×#1343C2semidirect
8. C22×C4×D4C4semidirect
14. #2535C4semidirect
16. C2×#397C4semidirect
18. C2×#392C4semidirect
22. C2×#382C4semidirect
26. #1799C22semidirect
29. #1459C22semidirect
32. #1344C22semidirect
33. #1343C22semidirect
54. C2×C4×D4C2×C4semidirect
81. #397C2×C4semidirect
87. #392C2×C4semidirect
94. #382C2×C4semidirect
578. C2#11875direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25352
C2×#17992
C2×#14592
#118752
C2×#13442
C2×#13432