Finite Group Databasebeta

← C2×#11766 #481696

Subgroups — C2×#11766 #481696

Order 512.

1356 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C22×#343C211yesmaximal, characteristic, complemented
3C2×#1463C211yesmaximal, characteristic, complemented
4#11766C218yesmaximal, direct product factor
5C2×#1353C212yesmaximal, complemented
6C2×#1345C212yesmaximal, complemented
7C23×M16—21no—
8C22×#121C411yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407—21no—
11#2591—22no—
12#2591—22no—
13#2591C414yes—
14#2591C2214yes—
15C2×#400C411yescharacteristic
16C2×#400C2211yescharacteristic
17C2×#394C411yescharacteristic
18C2×#394C2211yescharacteristic
19C2×#388—21no—
20C2×#388—21no—
21C2×#386—21no—
22C2×#386—21no—
23C2×C4×C22⋊C4C411yescharacteristic
24C2×#343—22no—
25C2×#343—21no—
26C2×#343C2212yes—
27C2×#343C2214yescomplemented
28#1463—22no—
29#1463C2214yescomplemented
30C2×#324—21no—
31C2×#324C2212yes—
32#1353C2218yescomplemented
33#1345C2218yescomplemented
34C23×D4—21no—
35C23×D4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4D411yescharacteristic
39C22×M16—28no—
40C22×M16—22no—
41C22×M16—24no—
42C22×M16—42no—
43C22×M16—41no—
44C22×M16—44no—
45C22×M16—41no—
46C23×C8—41no—
47C2×#121—22no—
48C2×#121—42no—
49C2×#121—41no—
50C2×#121—41no—
51C2×#121C2×C414yes—
52C22×C4⋊C4—41no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4—42no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—21no—
71#407—24no—
72#407—42no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—41no—
75#400—22no—
76#400—21no—
77#400—21no—
78#400C2×C412yes—
79#400C2×C414yes—
80#400C2312yes—
81#394—22no—
82#394—21no—
83#394—21no—
84#394C2×C412yes—
85#394C2×C414yes—
86#394C2312yes—
87#388—24no—
88#388—24no—
89#388—24no—
90#388—24no—
91#386—24no—
92#386—24no—
93#386—24no—
94#386—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2×C414yes—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103#343—22no—
104#343—24no—
105#343C2314yes—
106#324—24no—
107#324—22no—
108#324C2314yes—
109C25—21no—
110C25C22⋊C411yescharacteristic
111C22×D4—24no—
112C22×D4—21no—
113C22×D4—41no—
114C22×D4—41no—
115C22×D4—44no—
116C22×D4—44no—
117C22×D4—42no—
118C22×D4—41no—
119C22×D4—41no—
120C22×D4—82no—
121C22×D4C22×C412yes—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C2×M16—28no—
152C2×M16—216no—
153C2×M16—24no—
154C2×M16—416no—
155C2×M16—48no—
156C2×M16—42no—
157C2×M16—44no—
158C2×M16—48no—
159C2×M16—44no—
160C2×M16—44no—
161C2×M16—84no—
162C22×C8—48no—
163C22×C8—44no—
164C22×C8—42no—
165C22×C8—42no—
166#121—44no—
167#121—44no—
168#121—44no—
169#121—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2411yescharacteristic
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C42—24no—
217C2×C42—82no—
218C22⋊C8—24no—
219C22⋊C8—42no—
220C22⋊C8—44no—
221#95—22no—
222#95—24no—
223#95—24no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95C22×C412yes—
230#95C22×C412yes—
231C24—24no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—41no—
241C24—41no—
242C24—42no—
243C24—42no—
244C24#9911yescharacteristic
245C24#9912yes—
246C24#9911yescharacteristic
247C24C2×C22⋊C412yes—
248C24C2×C22⋊C411yescharacteristic
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—48no—
252C2×D4—48no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—82no—
256C2×D4—84no—
257C2×D4—84no—
258C2×D4—82no—
259C2×D4—88no—
260C2×D4—84no—
261C2×D4—84no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4#10414yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4AGL(1,ℤ/8)12yes—
334C22×C4#13712yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338M16—28no—
339M16—416no—
340M16—48no—
341M16—44no—
342M16—88no—
343C2×C8—416no—
344C2×C8—44no—
345C2×C8—48no—
346C2×C8—44no—
347C2×C8—84no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C22⋊C4—48no—
361C22⋊C4—48no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—44no—
365C22⋊C4—48no—
366C22⋊C4—84no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—84no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—24no—
377C23—24no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—22no—
382C23—42no—
383C23—42no—
384C23—48no—
385C23—48no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—82no—
394C23—82no—
395C23—84no—
396C23—84no—
397C23—84no—
398C23—84no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23#35514yes—
404C23C2×#9912yes—
405C23C2×#9912yes—
406C23C2×#9911yescharacteristic
407C23C2×#9911yescharacteristic
408C23#41712yes—
409C23C2×#10412yes—
410C23C22×C22⋊C411yescharacteristic
411D4—84no—
412D4—84no—
413D4—88no—
414D4—88no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4#41712yes—
475C2×C4C2×#10414yes—
476C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C8—48no—
479C8—84no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—28no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—44no—
493C22—44no—
494C22—48no—
495C22—44no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—164no—
504C22—164no—
505C22#150412yes—
506C22#150412yes—
507C22#212112yes—
508C22C2×#35514yes—
509C22C2×#35512yes—
510C22C22×#9911yescharacteristic
511C22C2×#41711yescharacteristic
512C22C22×#10411yescharacteristic
513C4—21no—
514C4—21no—
515C4—22no—
516C4—24no—
517C4—41no—
518C4—42no—
519C4—41no—
520C4—42no—
521C4—41no—
522C4—41no—
523C4—82no—
524C4—82no—
525C4—82no—
526C2—21no—
527C2—21no—
528C2—22no—
529C2—24no—
530C2—42no—
531C2—162no—
532C2#1176614yesdirect product factor
533C2C2×#150411yescharacteristic
534C2C2×#212111yescharacteristic
535C2C22×#35511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#343C2semidirect
3. C2×#1463C2semidirect
4. #11766C2direct (N × H)
5. C2×#1353C2semidirect
6. C2×#1345C2semidirect
27. C2×#343C22semidirect
29. #1463C22semidirect
32. #1353C22semidirect
33. #1345C22semidirect
532. C2#11766direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25912
C22×#3432
C2×#14632
#117662
C2×#13532
C2×#13452