Finite Group Databasebeta

← C2×#9299 #199047

Subgroups — C2×#9299 #199047

Order 512.

1798 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3205C211yesmaximal, characteristic
2C2×#2328C211yesmaximal, characteristic, complemented
3C2×#1963C211yesmaximal, characteristic, complemented
4C2×#1545C211yesmaximal, characteristic, complemented
5C2×#1517C211yesmaximal, characteristic, complemented
6#9299C218yesmaximal, direct product factor
7C2×#1286C211yesmaximal, characteristic, complemented
8C2×#1274C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×#127C2211yescharacteristic
11C22×C4×D4—21no—
12#3205—22no—
13#3205—22no—
14#3205C414yescomplemented
15#3205C2214yes—
16C2×#493C412yescomplemented
17C2×#486C412yescomplemented
18C2×#459—22no—
19C2×#447—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×#104—21no—
24C22×#104—21no—
25#2328—22no—
26#2328—22no—
27#2328—22no—
28#2328C2214yescomplemented
29C2×#390—21no—
30C2×#386—22no—
31C2×#386—22no—
32C2×#382C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1963—22no—
35#1963C2214yescomplemented
36#1545—22no—
37#1545C2214yescomplemented
38C2×#331—22no—
39#1517—22no—
40#1517C2214yescomplemented
41C2×#327—22no—
42#1286C2214yescomplemented
43#1274C2214yescomplemented
44C22×#41—21no—
45C23×D4—21no—
46C23×D4—21no—
47C23×D4D412yes—
48C24×C4—22no—
49C22×D8—24no—
50C22×M16—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532C2×C412yescomplemented
58#532C2312yes—
59C2×C4⋊Q8C2×C411yescharacteristic, complemented
60C2×#127—21no—
61C2×#127—21no—
62C2×#127—21no—
63C2×#127—21no—
64C2×#127—21no—
65C2×#127—21no—
66C2×#127C2×C412yes—
67C2×#127C2×C412yes—
68C2×#127C2×C411yescharacteristic, complemented
69C2×#127D412yes—
70C2×#127D412yes—
71C2×#127D412yes—
72C2×#127C2312yes—
73C2×#124—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×C4×D4—24no—
77C2×C4×D4—24no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—44no—
81C2×C4×D4—44no—
82C22×C4⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89#493—24no—
90#493C2×C418yescomplemented
91#486—24no—
92#486C2×C418yescomplemented
93#459—28no—
94#459—28no—
95#447—28no—
96#447—28no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431C2×C412yescomplemented
103#431C2312yes—
104#431C2312yes—
105#423—21no—
106#423—21no—
107#423—21no—
108#423C2312yes—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×#104—24no—
114C2×#104—24no—
115C2×#104—44no—
116C2×#104—44no—
117C2×#102—24no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8C2×C412yescomplemented
122#390—24no—
123#390—24no—
124#386—28no—
125#386—28no—
126#386—28no—
127#386—28no—
128#382—21no—
129#382—21no—
130#382D412yes—
131#382C2312yes—
132#376—21no—
133#376—21no—
134#376D412yes—
135#376C2312yes—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95D412yes—
140#331—28no—
141#327—28no—
142C25—22no—
143C2×#41—22no—
144C2×#41—22no—
145C2×#41—41no—
146C2×#41—41no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—42no—
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—41no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—84no—
176C22×D4—84no—
177C22×D4C22⋊C414yes—
178C22×D4C22⋊C414yes—
179C22×D4C2×D414yes—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—28no—
184C23×C4—48no—
185C23×C4—41no—
186C23×C4—84no—
187C23×C4—84no—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C2×D8—216no—
192C2×D8—88no—
193C2×M16—22no—
194C2×M16—41no—
195C2×M16—42no—
196C2×M16—42no—
197C22×C8—22no—
198C22×C8—41no—
199C22×C8—41no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8C22×C412yescomplemented
204#127—21no—
205#127—22no—
206#127—22no—
207#127—22no—
208#127—22no—
209#127—21no—
210#127—21no—
211#127—22no—
212#127—22no—
213#127—22no—
214#127—22no—
215#127—22no—
216#127—22no—
217#127—22no—
218#127—22no—
219#127C22⋊C414yes—
220#127C22⋊C414yes—
221#127C22×C414yes—
222#127C22×C412yescomplemented
223#127C2×D414yes—
224#124—22no—
225#124—22no—
226#124—41no—
227#124—41no—
228#121—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—41no—
233#121—41no—
234#121—41no—
235#121—41no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—48no—
241C4×D4—416no—
242C4×D4—41no—
243C4×D4—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—48no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—84no—
260C2×C22⋊C4—84no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42D814yescomplemented
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42C2411yescharacteristic
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—41no—
287C4⋊C8—41no—
288C4⋊C8—41no—
289#104—48no—
290#104—48no—
291#102—216no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—41no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8—41no—
299C4×C8C22×C414yescomplemented
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95C2×D414yes—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—44no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24—41no—
318C24#12011yescharacteristic
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—41no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—48no—
348C2×D4—44no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—84no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—82no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4C2×C22⋊C418yes—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—416no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—41no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4—832no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4C4×D411yescharacteristic
413C22×C4C4×D411yescharacteristic
414C22×C4#12011yescharacteristic
415C22×C4#12011yescharacteristic
416C22×C4#12011yescharacteristic
417C22×C4#12111yescharacteristic
418C22×C4#12111yescharacteristic
419C22×C4#12111yescharacteristic
420C22×C4#12111yescharacteristic
421C22×C4#12311yescharacteristic
422C22×C4#12311yescharacteristic
423C22×C4C2×D812yes—
424C22×C4#13512yes—
425C22×C4AGL(1,ℤ/8)12yes—
426C22×C4AGL(1,ℤ/8)12yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430D8—816no—
431M16—42no—
432M16—42no—
433M16—82no—
434C2×C8—24no—
435C2×C8—44no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—82no—
439C2×C8—82no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—44no—
447C22⋊C4—416no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C42—22no—
455C42—24no—
456C42—22no—
457C42—22no—
458C42—42no—
459C42C2×C22⋊C411yescharacteristic
460C42C2×C22⋊C411yescharacteristic
461C42C4×D411yescharacteristic
462C42C4×D411yescharacteristic
463C42C4×D411yescharacteristic
464C42C4×D411yescharacteristic
465C42#12111yescharacteristic
466C42#12111yescharacteristic
467C42#12111yescharacteristic
468C42#12111yescharacteristic
469C42#12311yescharacteristic
470C42#12311yescharacteristic
471C42C2×D814yescomplemented
472C42C23×C411yescharacteristic
473C42C22×D411yescharacteristic
474C42C2×#4111yescharacteristic
475C42C2×#4111yescharacteristic
476C23—22no—
477C23—22no—
478C23—28no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—48no—
484C23—48no—
485C23—48no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—88no—
493C23—88no—
494C23—88no—
495C23—88no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23#38611yescharacteristic
500C23#38611yescharacteristic
501C23#44712yes—
502C23#44912yes—
503C23C2×#12011yescharacteristic
504Q8—42no—
505Q8—42no—
506Q8—82no—
507D4—48no—
508D4—44no—
509D4—48no—
510D4—48no—
511D4—44no—
512D4—42no—
513D4—44no—
514D4—42no—
515D4—44no—
516D4—44no—
517D4—84no—
518D4—84no—
519D4—84no—
520D4—84no—
521D4—84no—
522D4—84no—
523D4—84no—
524D4—82no—
525D4—82no—
526D4—82no—
527D4—84no—
528D4—82no—
529D4—82no—
530D4—82no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—832no—
555C2×C4—84no—
556C2×C4—88no—
557C2×C4—82no—
558C2×C4—84no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4#38611yescharacteristic
562C2×C4#38611yescharacteristic
563C2×C4#38611yescharacteristic
564C2×C4#38611yescharacteristic
565C2×C4#38611yescharacteristic
566C2×C4#38611yescharacteristic
567C2×C4C4×D812yes—
568C2×C4C4×D812yes—
569C2×C4#44212yes—
570C2×C4#44212yes—
571C2×C4#44712yes—
572C2×C4#44912yes—
573C2×C4#45912yes—
574C2×C4#45912yes—
575C2×C4#46312yes—
576C2×C4#46312yes—
577C2×C4C22×C22⋊C411yescharacteristic
578C2×C4C2×C4×D411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C2×#12111yescharacteristic
581C2×C4C2×#12111yescharacteristic
582C2×C4C2×#12311yescharacteristic
583C2×C4C22×D811yescharacteristic
584C2×C4C2×#13511yescharacteristic
585C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
586C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
587C8—44no—
588C8—82no—
589C8—82no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—24no—
595C22—24no—
596C22—48no—
597C22—48no—
598C22—48no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—816no—
604C22—816no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—164no—
609C22#187912yes—
610C22#187912yes—
611C22#190412yes—
612C22#191112yes—
613C22C2×#38611yescharacteristic
614C22C2×#44711yescharacteristic
615C22C2×#44911yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—44no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4#187912yes—
629C4#187912yes—
630C4C2×C4×D811yescharacteristic
631C4C2×#44211yescharacteristic
632C4C2×#45911yescharacteristic
633C4C2×#46311yescharacteristic
634C2—21no—
635C2—21no—
636C2—22no—
637C2—88no—
638C2—162no—
639C2#929914yesdirect product factor
640C2C2×#187911yescharacteristic
641C2C2×#190411yescharacteristic
642C2C2×#191111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3205C2
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×#2328C2semidirect
3. C2×#1963C2semidirect
4. C2×#1545C2semidirect
5. C2×#1517C2semidirect
6. #9299C2direct (N × H)
7. C2×#1286C2semidirect
8. C2×#1274C2semidirect
14. #3205C4semidirect
16. C2×#493C4semidirect
17. C2×#486C4semidirect
28. #2328C22semidirect
35. #1963C22semidirect
37. #1545C22semidirect
40. #1517C22semidirect
42. #1286C22semidirect
43. #1274C22semidirect
57. #532C2×C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
68. C2×#127C2×C4semidirect
90. #493C2×C4semidirect
92. #486C2×C4semidirect
102. #431C2×C4semidirect
121. C2×C4×C8C2×C4semidirect
203. C4⋊Q8C22×C4semidirect
222. #127C22×C4semidirect
266. C2×C42D8semidirect
299. C4×C8C22×C4semidirect
471. C42C2×D8semidirect
639. C2#9299direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32052
C2×#23282
C2×#19632
C2×#15452
C2×#15172
#92992
C2×#12862
C2×#12742