Finite Group Databasebeta

← C2×#9766 #205585

Subgroups — C2×#9766 #205585

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2C2×#1746C211yesmaximal, characteristic, complemented
3C2×#1568C211yesmaximal, characteristic, complemented
4C2×#1531C211yesmaximal, characteristic, complemented
5C2×#1530C211yesmaximal, characteristic
6C2×#1506C211yesmaximal, characteristic, complemented
7C2×#1505C211yesmaximal, characteristic
8#9766C218yesmaximal, direct product factor
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3167—22no—
12#3167—22no—
13#3167C414yescomplemented
14#3167C2214yes—
15C2×#498C411yescharacteristic, complemented
16C2×#492C411yescharacteristic, complemented
17C2×#487C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#465—21no—
20C2×#465—21no—
21C2×C4×SD16—21no—
22C2×C4×SD16—21no—
23C2×#432C2211yescharacteristic
24C2×#432C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#1746—22no—
29#1746C2214yescomplemented
30C2×#336—21no—
31C2×#336—21no—
32#1568—22no—
33#1568C2214yescomplemented
34C2×#335—21no—
35C2×#334—21no—
36#1531—22no—
37#1531C2214yescomplemented
38#1530—22no—
39#1530C2214yes—
40C2×#330—22no—
41C2×#329—22no—
42#1506—22no—
43#1506C2214yescomplemented
44#1505—22no—
45#1505C2214yes—
46C2×#328—22no—
47C2×#327—22no—
48C22×#41—21no—
49C22×SD16—21no—
50C22×SD16—21no—
51C23×C8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533C2×C412yes—
61#533C2312yes—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67#532—21no—
68#532C2×C412yes—
69#532C2312yes—
70C2×C4⋊Q8C2×C411yescharacteristic
71C2×C4⋊Q8C2×C411yescharacteristic
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×#127C2×C411yescharacteristic
74C2×#125—21no—
75C2×#124—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×#121—21no—
79C2×#121—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×#117—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#498—22no—
88#498C2×C414yescomplemented
89#492—22no—
90#492C2×C414yescomplemented
91#487—22no—
92#487C2×C414yescomplemented
93#486—22no—
94#486C2×C414yescomplemented
95#465—24no—
96#465—24no—
97#465—24no—
98#465—24no—
99C4×SD16—24no—
100C4×SD16—24no—
101C4×SD16—24no—
102C4×SD16—24no—
103#432—21no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#432—21no—
109#432C2312yes—
110#432C2312yes—
111C2×#106—21no—
112C2×#106—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×#97—21no—
122C2×#97—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97D412yes—
127C2×#97D412yes—
128C2×#97C2312yes—
129C2×C4×C8C2×C411yescharacteristic
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C412yes—
132C2×C4×C8D412yes—
133C2×C4×C8D411yescharacteristic
134C2×C4×C8D411yescharacteristic
135C2×C4×C8D412yes—
136C2×C4×C8C2312yes—
137#376—21no—
138#376—21no—
139#376D412yes—
140#376C2312yes—
141C2×#95—21no—
142C2×#95D412yes—
143#336—24no—
144#336—24no—
145#335—24no—
146#334—24no—
147#330—28no—
148#329—28no—
149#328—28no—
150#327—28no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—41no—
154C2×#41—41no—
155C2×#41—84no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—41no—
163C23×C4—22no—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C2×SD16—24no—
169C2×SD16—24no—
170C2×SD16—44no—
171C2×SD16—44no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8—41no—
180C22×C8—41no—
181C22×C8C22⋊C411yescharacteristic
182C22×C8C22⋊C411yescharacteristic
183C22×C8C22⋊C411yescharacteristic
184C22×C8C22⋊C411yescharacteristic
185C22×C8C22⋊C412yes—
186C22×C8C22⋊C412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8C22×C412yes—
199C4⋊Q8C22×C412yes—
200C4⋊Q8C22×C412yes—
201#127—21no—
202#127—21no—
203#127—21no—
204#127C22×C412yes—
205#125—22no—
206#125—22no—
207#125—41no—
208#125—41no—
209#124—22no—
210#124—22no—
211#124—41no—
212#124—41no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—41no—
218C22⋊Q8—41no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221#121—22no—
222#121—22no—
223#121—22no—
224#121—22no—
225#121—41no—
226#121—41no—
227#121—41no—
228#121—41no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—41no—
242C4×D4—41no—
243C4×D4—41no—
244C4×D4—41no—
245#117—22no—
246#117—22no—
247#117—41no—
248#117—41no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—41no—
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42#4111yescharacteristic
276C2×C42C2411yescharacteristic
277#106—24no—
278#106—24no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—41no—
282C4⋊C8—41no—
283#103—24no—
284#103—24no—
285#102—24no—
286#102—24no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291#97—22no—
292#97—22no—
293#97—22no—
294#97—22no—
295#97—22no—
296#97—41no—
297#97—41no—
298#97C2×D414yes—
299C4×C8C22⋊C412yes—
300C4×C8C22⋊C412yes—
301C4×C8C22⋊C412yes—
302C4×C8C22⋊C412yes—
303C4×C8C22×C412yes—
304C4×C8C22×C412yes—
305C4×C8C2×D412yes—
306C4×C8C2×D412yes—
307#95—22no—
308#95—22no—
309#95—22no—
310#95C2×D414yes—
311C24—41no—
312C24#12111yescharacteristic
313#41—84no—
314#41—84no—
315#41—84no—
316#41—84no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—82no—
326C2×Q8—82no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4—84no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—84no—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12411yescharacteristic
366C22×C4#12411yescharacteristic
367C22×C4#12711yescharacteristic
368C22×C4#12711yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372SD16—48no—
373SD16—48no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—44no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—44no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—82no—
390C2×C8—82no—
391C2×C8C2×C22⋊C412yes—
392C2×C8C2×C22⋊C412yes—
393C2×C8C2×C22⋊C412yes—
394C2×C8C2×C22⋊C412yes—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C42—42no—
418C42—42no—
419C42—44no—
420C42—42no—
421C42—42no—
422C42C2×C22⋊C411yescharacteristic
423C42C2×C22⋊C411yescharacteristic
424C42C4×D411yescharacteristic
425C42C4×D411yescharacteristic
426C42C4×D411yescharacteristic
427C42C4×D411yescharacteristic
428C42#12111yescharacteristic
429C42#12111yescharacteristic
430C42#12411yescharacteristic
431C42#12411yescharacteristic
432C42#12711yescharacteristic
433C42#12711yescharacteristic
434C42C23×C411yescharacteristic
435C42C22×D411yescharacteristic
436C42C22×D411yescharacteristic
437C42C2×#4111yescharacteristic
438C23—22no—
439C23—22no—
440C23—42no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23#39011yescharacteristic
445C23#39011yescharacteristic
446C23#47114yes—
447C23C2×#12111yescharacteristic
448Q8—42no—
449Q8—42no—
450Q8—82no—
451Q8—82no—
452Q8—82no—
453Q8—82no—
454Q8—82no—
455D4—42no—
456D4—42no—
457D4—82no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—84no—
462D4—84no—
463D4—82no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—88no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4#39011yescharacteristic
496C2×C4#39011yescharacteristic
497C2×C4#39011yescharacteristic
498C2×C4#39011yescharacteristic
499C2×C4#39011yescharacteristic
500C2×C4#39011yescharacteristic
501C2×C4#42112yes—
502C2×C4#42112yes—
503C2×C4C22×C22⋊C411yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×#12111yescharacteristic
507C2×C4C2×#12411yescharacteristic
508C2×C4C2×#12711yescharacteristic
509C8—22no—
510C8—22no—
511C8—22no—
512C8—22no—
513C8—44no—
514C8—82no—
515C8—82no—
516C22—28no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—164no—
525C22#196312yes—
526C22#197612yes—
527C22#197712yes—
528C22#197712yes—
529C22C2×#39011yescharacteristic
530C22C2×#47112yes—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—84no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4#196312yes—
546C4C2×#42111yescharacteristic
547C4C2×#42111yescharacteristic
548C2—22no—
549C2—22no—
550C2—162no—
551C2#976614yesdirect product factor
552C2C2×#196311yescharacteristic
553C2C2×#197611yescharacteristic
554C2C2×#197711yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1746C2semidirect
3. C2×#1568C2semidirect
4. C2×#1531C2semidirect
6. C2×#1506C2semidirect
8. #9766C2direct (N × H)
13. #3167C4semidirect
15. C2×#498C4semidirect
16. C2×#492C4semidirect
17. C2×#487C4semidirect
18. C2×#486C4semidirect
29. #1746C22semidirect
33. #1568C22semidirect
37. #1531C22semidirect
43. #1506C22semidirect
88. #498C2×C4semidirect
90. #492C2×C4semidirect
92. #487C2×C4semidirect
94. #486C2×C4semidirect
551. C2#9766direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31672
C2×#17462
C2×#15682
C2×#15312
C2×#15302
C2×#15062
C2×#15052
#97662