Finite Group Databasebeta

← C2×#11783 #481610

Subgroups — C2×#11783 #481610

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2543C211yesmaximal, characteristic
2C2×C4×#100C211yesmaximal, characteristic, complemented
3C2×#1465C211yesmaximal, characteristic, complemented
4#11783C218yesmaximal, direct product factor
5C2×#1346C212yesmaximal, complemented
6C2×#1345C212yesmaximal, complemented
7C22×#121—21no—
8C22×#121C411yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C22×#100—21no—
11C2×#407—21no—
12C2×C4×M16—21no—
13#2543—22no—
14#2543—22no—
15#2543C414yes—
16#2543C2214yes—
17C2×#398C2211yescharacteristic
18C2×#397C411yescharacteristic
19C2×#394C2211yescharacteristic
20C2×#392C411yescharacteristic
21C2×#388—21no—
22C2×#386—21no—
23C2×#386—21no—
24C22×#95C411yescharacteristic
25C4×#100—22no—
26C4×#100C2214yescomplemented
27C2×#343C2212yes—
28#1465—22no—
29#1465C2214yescomplemented
30C2×#324C2212yes—
31C2×#323—21no—
32#1346C2218yescomplemented
33#1345C2218yescomplemented
34C23×D4—21no—
35C23×D4—21no—
36C23×D4C2×C411yescharacteristic
37C23×D4D411yescharacteristic
38C24×C4—21no—
39C24×C4C2×C411yescharacteristic
40C24×C4D411yescharacteristic
41C22×M16—22no—
42C22×M16—41no—
43C2×#121—24no—
44C2×#121—24no—
45C2×#121—22no—
46C2×#121—48no—
47C2×#121—42no—
48C2×#121—41no—
49C2×#121—41no—
50C2×#121C2×C414yes—
51C2×C4×D4—41no—
52C2×C4×D4—41no—
53C2×C4×D4—42no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4D412yes—
56C2×C4×D4D412yes—
57C2×C4×D4C2312yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4D411yescharacteristic
68C22×C42D411yescharacteristic
69C2×#100—28no—
70C2×#100—24no—
71#407—24no—
72#407—42no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—41no—
75C4×M16—24no—
76C4×M16—42no—
77C2×#97—41no—
78C2×C4×C8—41no—
79#398—21no—
80#398—21no—
81#398C2×C412yes—
82#398C2312yes—
83#397—22no—
84#397C2×C414yes—
85#394—21no—
86#394—21no—
87#394C2×C412yes—
88#394C2312yes—
89#392—22no—
90#392C2×C414yes—
91#388—24no—
92#388—24no—
93#386—24no—
94#386—24no—
95#386—24no—
96#386—24no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C414yes—
104#343—22no—
105#343C2314yes—
106#324—22no—
107#324C2314yes—
108#323—24no—
109C25—21no—
110C25C22⋊C411yescharacteristic
111C25C22⋊C411yescharacteristic
112C22×D4—24no—
113C22×D4—24no—
114C22×D4—24no—
115C22×D4—21no—
116C22×D4—24no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—44no—
122C22×D4—44no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4—44no—
126C22×D4—44no—
127C22×D4—82no—
128C22×D4—82no—
129C22×D4C22⋊C412yes—
130C22×D4C22×C412yes—
131C22×D4C2×D412yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C2×M16—28no—
157C2×M16—24no—
158C2×M16—42no—
159C2×M16—44no—
160C2×M16—44no—
161C2×M16—84no—
162C22×C8—42no—
163C22×C8—42no—
164#121—44no—
165#121—44no—
166#121—416no—
167#121—416no—
168#121—44no—
169#121—44no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2411yescharacteristic
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22×C412yes—
212C2×C22⋊C4C2×D412yes—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42C22⋊C412yes—
217C2×C42C2×D412yes—
218#100—216no—
219C22⋊C8—24no—
220C22⋊C8—42no—
221C22⋊C8—44no—
222#97—44no—
223C4×C8—44no—
224#95—24no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—24no—
229#95—22no—
230#95C22×C412yes—
231#95C22×C412yes—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—21no—
236C24—22no—
237C24—21no—
238C24—22no—
239C24—22no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—41no—
249C24—41no—
250C24—42no—
251C24—42no—
252C24#10414yes—
253C24C2×C22⋊C411yescharacteristic
254C24C2×C22⋊C412yes—
255C24C2×C22⋊C412yes—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—48no—
263C2×D4—48no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—48no—
267C2×D4—48no—
268C2×D4—48no—
269C2×D4—48no—
270C2×D4—48no—
271C2×D4—84no—
272C2×D4—82no—
273C2×D4—816no—
274C2×D4—84no—
275C2×D4—84no—
276C2×D4—82no—
277C2×D4—88no—
278C2×D4—88no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—48no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—48no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—816no—
327C22×C4#9911yescharacteristic
328C22×C4#9911yescharacteristic
329C22×C4#9912yes—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C412yes—
335C22×C4AGL(1,ℤ/8)14yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339M16—28no—
340M16—48no—
341M16—44no—
342M16—88no—
343C2×C8—44no—
344C2×C8—48no—
345C2×C8—44no—
346C2×C8—84no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—42no—
382C23—48no—
383C23—48no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—48no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—81no—
403C23—81no—
404C23—84no—
405C23—84no—
406C23—82no—
407C23—84no—
408C23—84no—
409C23—84no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23C2×#9911yescharacteristic
416C23C2×#9911yescharacteristic
417C23#41812yes—
418C23C2×#10412yes—
419C23C2×#10414yes—
420C23C22×C22⋊C411yescharacteristic
421D4—48no—
422D4—48no—
423D4—816no—
424D4—84no—
425D4—84no—
426D4—816no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—816no—
472C2×C4—84no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4C2×#9912yes—
476C2×C4C2×#9912yes—
477C2×C4#41812yes—
478C2×C4C2×AGL(1,ℤ/8)12yes—
479C8—48no—
480C8—84no—
481C22—24no—
482C22—24no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—24no—
488C22—44no—
489C22—44no—
490C22—48no—
491C22—44no—
492C22—48no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—82no—
499C22—84no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—84no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—164no—
509C22—164no—
510C22#150712yes—
511C22#150712yes—
512C22#212112yes—
513C22#213212yes—
514C22C22×#9911yescharacteristic
515C22C2×#41811yescharacteristic
516C22C22×#10411yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—44no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—82no—
528C4—82no—
529C2—21no—
530C2—21no—
531C2—22no—
532C2—44no—
533C2—82no—
534C2—162no—
535C2#1178314yesdirect product factor
536C2C2×#150711yescharacteristic
537C2C2×#212111yescharacteristic
538C2C2×#213211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#2543C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#100C2semidirect
3. C2×#1465C2semidirect
4. #11783C2direct (N × H)
5. C2×#1346C2semidirect
6. C2×#1345C2semidirect
26. C4×#100C22semidirect
29. #1465C22semidirect
32. #1346C22semidirect
33. #1345C22semidirect
535. C2#11783direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25432
C2×C4×#1002
C2×#14652
#117832
C2×#13462
C2×#13452