Finite Group Databasebeta

← C2×#9386 #200395

Subgroups — C2×#9386 #200395

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic, complemented
2C2×#2543C211yesmaximal, characteristic
3C2×#1799C211yesmaximal, characteristic, complemented
4C2×#1519C212yesmaximal, complemented
5#9386C218yesmaximal, direct product factor
6C2×#1294C212yesmaximal, complemented
7C22×#121—21no—
8C22×#121C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10#2973—24no—
11#2973—22no—
12#2973C2214yescomplemented
13C2×#435—24no—
14C2×#432—21no—
15C22×#99—21no—
16C2×#407—21no—
17C22×C22⋊C8C2212yes—
18#2543—22no—
19#2543—22no—
20#2543C414yescomplemented
21#2543C2214yes—
22C2×#398C411yescharacteristic, complemented
23C2×#397C411yescharacteristic, complemented
24C2×#394C411yescharacteristic, complemented
25C2×#392C411yescharacteristic, complemented
26C2×#388—21no—
27C2×#386—21no—
28C2×#386—21no—
29C2×#380C2212yes—
30C22×#95C2211yescharacteristic
31#1799—22no—
32#1799C2214yescomplemented
33C2×#342—21no—
34#1519—24no—
35#1519C2218yescomplemented
36C2×#327—24no—
37#1294C2218yescomplemented
38C23×D4—21no—
39C23×D4—21no—
40C23×D4C2×C411yescharacteristic
41C23×D4D411yescharacteristic
42C24×C4—21no—
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C23×C8—22no—
47C2×#121—24no—
48C2×#121—24no—
49C2×#121—21no—
50C2×#121—21no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—48no—
56C2×#121C2×C412yes—
57C2×#121C2312yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×C4×D4C2312yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C42D411yescharacteristic
79#435—216no—
80#435—216no—
81#432—24no—
82#432—24no—
83C2×C4⋊C8—22no—
84C2×#99—24no—
85C2×#99—44no—
86#407—24no—
87#407—24no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8D414yescomplemented
93C2×C22⋊C8D414yescomplemented
94C2×C22⋊C8C2314yes—
95C2×#97—22no—
96#398—22no—
97#398C2×C414yescomplemented
98#397—22no—
99#397C2×C414yescomplemented
100#394—22no—
101#394C2×C414yescomplemented
102#392—22no—
103#392C2×C414yescomplemented
104#388—24no—
105#388—24no—
106#386—24no—
107#386—24no—
108#386—24no—
109#386—24no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380C2314yes—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C412yes—
123C2×#95C2312yes—
124#342—24no—
125#327—216no—
126C25—21no—
127C25C22⋊C411yescharacteristic
128C25C2×D411yescharacteristic
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—44no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—82no—
148C22×D4—82no—
149C22×D4C22⋊C412yes—
150C22×D4C22×C412yes—
151C22×D4C2×D412yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C412yes—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182C2×M16—28no—
183C2×M16—84no—
184C22×C8—24no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—84no—
188#121—24no—
189#121—24no—
190#121—24no—
191#121—24no—
192#121—416no—
193#121—416no—
194#121—42no—
195#121—42no—
196#121—42no—
197#121—42no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—84no—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22×C412yes—
244C2×C22⋊C4C2×D412yes—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C22⋊C412yes—
250C2×C42C2×D412yes—
251C4⋊C8—28no—
252#99—48no—
253C22⋊C8—28no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256C22⋊C8—24no—
257C22⋊C8C2×D418yescomplemented
258#97—28no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—24no—
266#95—42no—
267#95C22×C412yes—
268#95C22×C412yes—
269C24—21no—
270C24—22no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24—42no—
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C412yes—
288C24C2×C22⋊C411yescharacteristic
289C24#12112yes—
290C24#12112yes—
291C24AGL(1,ℤ/8)12yes—
292C24AGL(1,ℤ/8)14yes—
293C24#13712yes—
294C24C22×D411yescharacteristic
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—42no—
317C2×D4—84no—
318C2×D4—82no—
319C2×D4—84no—
320C2×D4—816no—
321C2×D4—84no—
322C2×D4—82no—
323C2×D4—84no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—84no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—82no—
368C22×C4—81no—
369C22×C4—82no—
370C22×C4—81no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—816no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4C4×D412yes—
390C22×C4#12011yescharacteristic
391C22×C4#12011yescharacteristic
392C22×C4#12012yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4#12311yescharacteristic
396C22×C4#12311yescharacteristic
397C22×C4#12312yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4112yes—
402M16—88no—
403C2×C8—28no—
404C2×C8—48no—
405C2×C8—84no—
406C2×C8—84no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—48no—
446C23—42no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—48no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—81no—
464C23—81no—
465C23—84no—
466C23—82no—
467C23—84no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23#38612yes—
477C23#38611yescharacteristic
478C23#38611yescharacteristic
479C23#41712yes—
480C23#41712yes—
481C23#41812yes—
482C23#41812yes—
483C23#46814yes—
484C23#46814yes—
485C23#46914yes—
486C23#46914yes—
487C23C22×C22⋊C411yescharacteristic
488C23C2×#12112yes—
489C23C2×AGL(1,ℤ/8)12yes—
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23C2×#13711yescharacteristic
492D4—48no—
493D4—44no—
494D4—44no—
495D4—84no—
496D4—84no—
497D4—816no—
498D4—88no—
499D4—84no—
500D4—84no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—88no—
547C2×C4—88no—
548C2×C4—84no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—168no—
555C2×C4#38611yescharacteristic
556C2×C4#38611yescharacteristic
557C2×C4#38612yes—
558C2×C4C2×C4×D412yes—
559C2×C4C2×#12011yescharacteristic
560C2×C4C2×#12311yescharacteristic
561C8—48no—
562C22—21no—
563C22—22no—
564C22—21no—
565C22—22no—
566C22—24no—
567C22—28no—
568C22—24no—
569C22—28no—
570C22—21no—
571C22—22no—
572C22—21no—
573C22—22no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—82no—
586C22—84no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—84no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—164no—
596C22—164no—
597C22#188412yes—
598C22#189512yes—
599C22#189514yes—
600C22#189512yes—
601C22#191312yes—
602C22C2×#38611yescharacteristic
603C22C2×#41811yescharacteristic
604C22C2×#41711yescharacteristic
605C22C2×#46812yes—
606C22C2×#46912yes—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—22no—
612C4—22no—
613C4—44no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—82no—
618C4—82no—
619C4—164no—
620C2—21no—
621C2—22no—
622C2—21no—
623C2—22no—
624C2—24no—
625C2—22no—
626C2—82no—
627C2—162no—
628C2#938614yesdirect product factor
629C2C2×#188411yescharacteristic
630C2C2×#189511yescharacteristic
631C2C2×#191311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2543C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2973C2semidirect
3. C2×#1799C2semidirect
4. C2×#1519C2semidirect
5. #9386C2direct (N × H)
6. C2×#1294C2semidirect
12. #2973C22semidirect
20. #2543C4semidirect
22. C2×#398C4semidirect
23. C2×#397C4semidirect
24. C2×#394C4semidirect
25. C2×#392C4semidirect
32. #1799C22semidirect
35. #1519C22semidirect
37. #1294C22semidirect
92. C2×C22⋊C8D4semidirect
93. C2×C22⋊C8D4semidirect
97. #398C2×C4semidirect
99. #397C2×C4semidirect
101. #394C2×C4semidirect
103. #392C2×C4semidirect
257. C22⋊C8C2×D4semidirect
628. C2#9386direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29732
C2×#25432
C2×#17992
C2×#15192
#93862
C2×#12942