Finite Group Databasebeta

← C2×#9754 #205476

Subgroups — C2×#9754 #205476

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3131C211yesmaximal, characteristic
2C2×#1770C211yesmaximal, characteristic, complemented
3C2×#1564C211yesmaximal, characteristic, complemented
4C2×#1503C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic
6C2×#1481C211yesmaximal, characteristic, complemented
7C2×#1480C211yesmaximal, characteristic
8#9754C218yesmaximal, direct product factor
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3131—22no—
12#3131—22no—
13#3131C414yescomplemented
14#3131C2214yes—
15C2×#488—21no—
16C2×#488—21no—
17C2×#487C411yescharacteristic, complemented
18C2×#487C411yescharacteristic, complemented
19C2×#486C411yescharacteristic, complemented
20C2×#486C411yescharacteristic, complemented
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#416—21no—
25C2×#416—21no—
26C22×#97—21no—
27C22×C4×C8C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1770—22no—
30#1770C2214yescomplemented
31C2×#336—21no—
32#1564—22no—
33#1564C2214yescomplemented
34C2×#335—21no—
35C2×#334—21no—
36#1503—22no—
37#1503C2214yescomplemented
38#1502—22no—
39#1502C2214yes—
40C2×#328—22no—
41C2×#327—22no—
42#1481—22no—
43#1481C2214yescomplemented
44#1480—22no—
45#1480C2214yes—
46C2×Q8⋊C8—22no—
47C2×D4⋊C8—22no—
48C22×#41—21no—
49C23×C8—21no—
50C23×C8D411yescharacteristic
51C23×C8D411yescharacteristic
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533C2×C412yes—
59#533C2312yes—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532C2×C412yes—
67#532C2312yes—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4⋊Q8C2×C411yescharacteristic
71C2×#127C2×C411yescharacteristic
72C2×#125—21no—
73C2×#124—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×#121—21no—
77C2×#121—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#488—24no—
87#488—24no—
88#488—24no—
89#488—24no—
90#487—22no—
91#487—22no—
92#487C2×C414yescomplemented
93#487C2×C414yescomplemented
94#486—22no—
95#486—22no—
96#486C2×C414yescomplemented
97#486C2×C414yescomplemented
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431C2312yes—
105#431C2312yes—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8D412yes—
112C2×C4⋊C8D412yes—
113C2×C4⋊C8C2312yes—
114#416—24no—
115#416—24no—
116#416—24no—
117#416—24no—
118C2×#103—21no—
119C2×#103—21no—
120C2×#103—21no—
121C2×#103—21no—
122C2×#102—21no—
123C2×#102—21no—
124C2×#102—21no—
125C2×#102—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×#97—24no—
129C2×#97—44no—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132C2×C4×C8C2×C411yescharacteristic
133C2×C4×C8C2×C411yescharacteristic
134C2×C4×C8C2×C412yes—
135C2×C4×C8D412yes—
136C2×C4×C8D411yescharacteristic
137C2×C4×C8D411yescharacteristic
138C2×C4×C8D412yes—
139C2×C4×C8C2312yes—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4D412yes—
143C4×C4⋊C4C2312yes—
144C2×#95D412yes—
145#336—24no—
146#335—24no—
147#334—24no—
148#328—28no—
149#327—28no—
150Q8⋊C8—28no—
151D4⋊C8—28no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—42no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—42no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—41no—
167C23×C4—22no—
168C23×C4—41no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4#4111yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8—41no—
182C22×C8—41no—
183C22×C8C22⋊C411yescharacteristic
184C22×C8C22⋊C411yescharacteristic
185C22×C8C22⋊C411yescharacteristic
186C22×C8C22⋊C411yescharacteristic
187C22×C8C22⋊C412yes—
188C22×C8C22⋊C412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8C22×C412yes—
201C4⋊Q8C22×C412yes—
202C4⋊Q8C22×C412yes—
203#127—21no—
204#127—21no—
205#127—21no—
206#127C22×C412yes—
207#125—22no—
208#125—22no—
209#125—41no—
210#125—41no—
211#124—22no—
212#124—22no—
213#124—41no—
214#124—41no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8—41no—
223#121—22no—
224#121—22no—
225#121—22no—
226#121—22no—
227#121—41no—
228#121—41no—
229#121—41no—
230#121—41no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8—41no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—41no—
247#117—22no—
248#117—22no—
249#117—41no—
250#117—41no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—41no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42C2411yescharacteristic
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—41no—
288C4⋊C8—41no—
289C4⋊C8C2×D414yes—
290#103—24no—
291#103—24no—
292#103—24no—
293#103—24no—
294#102—24no—
295#102—24no—
296#102—24no—
297#102—24no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—41no—
301C22⋊C8—41no—
302#97—48no—
303C4×C8—22no—
304C4×C8—22no—
305C4×C8—41no—
306C4×C8—41no—
307C4×C8C22⋊C412yes—
308C4×C8C22⋊C412yes—
309C4×C8C22⋊C412yes—
310C4×C8C22⋊C412yes—
311C4×C8C22×C412yes—
312C4×C8C22×C412yes—
313C4×C8C2×D412yes—
314C4×C8C2×D412yes—
315#95—22no—
316#95C2×D414yes—
317C24—41no—
318C24#12111yescharacteristic
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323#41—44no—
324#41—44no—
325#41—82no—
326#41—82no—
327#41—82no—
328#41—82no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—44no—
338C2×Q8—82no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—44no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12411yescharacteristic
378C22×C4#12411yescharacteristic
379C22×C4#12711yescharacteristic
380C22×C4#12711yescharacteristic
381C22×C4#13512yes—
382C22×C4#13512yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4#13712yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—82no—
406C2×C8—82no—
407C2×C8C2×C22⋊C412yes—
408C2×C8C2×C22⋊C412yes—
409C2×C8C2×C22⋊C412yes—
410C2×C8C2×C22⋊C412yes—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—44no—
440C42C2×C22⋊C411yescharacteristic
441C42C2×C22⋊C411yescharacteristic
442C42C4×D411yescharacteristic
443C42C4×D411yescharacteristic
444C42C4×D411yescharacteristic
445C42C4×D411yescharacteristic
446C42#12111yescharacteristic
447C42#12111yescharacteristic
448C42#12411yescharacteristic
449C42#12411yescharacteristic
450C42#12711yescharacteristic
451C42#12711yescharacteristic
452C42C23×C411yescharacteristic
453C42C22×D411yescharacteristic
454C42C22×D411yescharacteristic
455C42C2×#4111yescharacteristic
456C23—22no—
457C23—22no—
458C23—42no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23#39011yescharacteristic
463C23#39011yescharacteristic
464C23#46312yes—
465C23#46412yes—
466C23C2×#12111yescharacteristic
467Q8—42no—
468Q8—42no—
469Q8—44no—
470Q8—82no—
471Q8—82no—
472Q8—82no—
473Q8—82no—
474D4—42no—
475D4—42no—
476D4—44no—
477D4—82no—
478D4—82no—
479D4—84no—
480D4—82no—
481D4—82no—
482D4—82no—
483D4—82no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—88no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#39011yescharacteristic
509C2×C4#39011yescharacteristic
510C2×C4#39011yescharacteristic
511C2×C4#39011yescharacteristic
512C2×C4#39011yescharacteristic
513C2×C4#39011yescharacteristic
514C2×C4#41612yes—
515C2×C4#41612yes—
516C2×C4#41712yes—
517C2×C4#41712yes—
518C2×C4#44414yes—
519C2×C4#46312yes—
520C2×C4#46312yes—
521C2×C4#46312yes—
522C2×C4#46412yes—
523C2×C4#46412yes—
524C2×C4#46412yes—
525C2×C4C22×C22⋊C411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×#12111yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4C2×#12711yescharacteristic
531C2×C4C2×#13511yescharacteristic
532C2×C4C2×#13511yescharacteristic
533C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
534C2×C4C2×#13711yescharacteristic
535C8—22no—
536C8—22no—
537C8—22no—
538C8—22no—
539C8—44no—
540C8—82no—
541C8—82no—
542C22—24no—
543C22—24no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—164no—
552C22#197512yes—
553C22#197612yes—
554C22#198312yes—
555C22#198312yes—
556C22C2×#39011yescharacteristic
557C22C2×#46311yescharacteristic
558C22C2×#46411yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—84no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4#197512yes—
570C4#198312yes—
571C4#198312yes—
572C4C2×#41611yescharacteristic
573C4C2×#41711yescharacteristic
574C4C2×#44412yes—
575C4C2×#46311yescharacteristic
576C4C2×#46411yescharacteristic
577C2—21no—
578C2—22no—
579C2—21no—
580C2—162no—
581C2#975414yesdirect product factor
582C2C2×#197511yescharacteristic
583C2C2×#197611yescharacteristic
584C2C2×#198311yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#390
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1770C2semidirect
3. C2×#1564C2semidirect
4. C2×#1503C2semidirect
6. C2×#1481C2semidirect
8. #9754C2direct (N × H)
13. #3131C4semidirect
17. C2×#487C4semidirect
18. C2×#487C4semidirect
19. C2×#486C4semidirect
20. C2×#486C4semidirect
30. #1770C22semidirect
33. #1564C22semidirect
37. #1503C22semidirect
43. #1481C22semidirect
92. #487C2×C4semidirect
93. #487C2×C4semidirect
96. #486C2×C4semidirect
97. #486C2×C4semidirect
581. C2#9754direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31312
C2×#17702
C2×#15642
C2×#15032
C2×#15022
C2×#14812
C2×#14802
#97542