Finite Group Databasebeta

← C2×#9969 #232201

Subgroups — C2×#9969 #232201

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic, complemented
2C22×#481C211yesmaximal, characteristic
3C2×#2639C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic, complemented
5C22×#327C211yesmaximal, characteristic
6C2×#1445C211yesmaximal, characteristic, complemented
7#9969C218yesmaximal, direct product factor
8C2×#1280C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C2×#484—24no—
13C2×#484—22no—
14C2×#484—24no—
15C2×#484C2214yescomplemented
16C2×#481—24no—
17C2×#481—22no—
18C2×#481—24no—
19C2×#481C2214yes—
20C22×#107—21no—
21C22×#106—21no—
22C22×#103—21no—
23C22×#102—21no—
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26#2639—22no—
27#2639—22no—
28#2639—22no—
29#2639C2214yes—
30C2×#400—21no—
31C2×#397—21no—
32C2×#394—21no—
33C2×#393—21no—
34C2×#386—22no—
35C22×#95—22no—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C2×#328—24no—
39C2×#328C2214yescomplemented
40C2×#327—24no—
41C2×#327C2214yes—
42#1445—22no—
43#1445C2214yescomplemented
44#1280C2214yescomplemented
45C23×Q8D411yescharacteristic
46C23×D4—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4D412yes—
51C24×C4C2311yescharacteristic
52C23×C8—21no—
53C23×C8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8D412yes—
61C2×C22⋊Q8C2312yes—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121D412yes—
69C2×#121C2312yes—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4D414yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87#484—216no—
88#484—28no—
89#484—28no—
90#481—216no—
91#481—28no—
92#481—28no—
93C2×#107—28no—
94C2×#107—24no—
95C2×#106—28no—
96C2×#106—24no—
97C2×#103—28no—
98C2×#103—24no—
99C2×#102—28no—
100C2×#102—24no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8D414yes—
106C2×C22⋊C8D414yes—
107C2×C22⋊C8C2312yes—
108C2×C22⋊C8C2312yes—
109#400—24no—
110#400—24no—
111#397—24no—
112#397—24no—
113#394—24no—
114#394—24no—
115#393—24no—
116#393—24no—
117#386—28no—
118#386—28no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—28no—
129C2×#95—48no—
130C2×#95D412yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133#328—28no—
134#327—28no—
135C25—21no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—44no—
141C22×Q8C2×D412yes—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—44no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4C2×D412yes—
153C23×C4—24no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D414yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
187C22×C8—24no—
188C22×C8—24no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—44no—
192C22×C8—44no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—42no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203#121—24no—
204#121—24no—
205#121—22no—
206#121—22no—
207#121—22no—
208#121—22no—
209#121—42no—
210#121—42no—
211#121—42no—
212#121—42no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—84no—
254C2×C22⋊C4—84no—
255#107—216no—
256#106—216no—
257#103—216no—
258#102—216no—
259C22⋊C8—24no—
260C22⋊C8—24no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8C2×D414yes—
264C22⋊C8C2×D414yes—
265#95—28no—
266#95—28no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—416no—
274#95—42no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—22no—
279C24—42no—
280C24—44no—
281C24—42no—
282C24—42no—
283C24—44no—
284C24#12011yescharacteristic
285C24#12011yescharacteristic
286C24#12112yes—
287C24#12712yes—
288C24#13514yes—
289C24#13714yes—
290C24C22×D411yescharacteristic
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—44no—
294C2×Q8—48no—
295C2×Q8—42no—
296C2×Q8—82no—
297C2×Q8—84no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—44no—
301C2×D4—48no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—48no—
308C2×D4—42no—
309C2×D4—82no—
310C2×D4—84no—
311C2×D4—88no—
312C2×D4—84no—
313C2×D4—84no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—416no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—48no—
339C22×C4—416no—
340C22×C4—44no—
341C22×C4—416no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—88no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—84no—
369C22×C4—88no—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4#12011yescharacteristic
373C22×C4#12011yescharacteristic
374C22×C4#12012yes—
375C22×C4#12012yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4#12712yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—48no—
393C2×C8—48no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—42no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—88no—
437C23—88no—
438C23—84no—
439C23—84no—
440C23—84no—
441C23#39212yes—
442C23#39212yes—
443C23#45214yes—
444C23#45214yes—
445C23#45312yes—
446C23#45412yes—
447C23#45814yes—
448C23#46418yes—
449C23#49514yes—
450C23#49714yes—
451C23C2×#12011yescharacteristic
452C23C2×#12111yescharacteristic
453C23C2×#12711yescharacteristic
454C23C2×#13512yes—
455C23C2×#13712yes—
456Q8—44no—
457Q8—82no—
458Q8—84no—
459D4—44no—
460D4—82no—
461D4—84no—
462D4—82no—
463D4—82no—
464D4—84no—
465D4—88no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—82no—
500C2×C4—84no—
501C2×C4—816no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4—88no—
505C2×C4—82no—
506C2×C4—88no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—84no—
510C2×C4—88no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4#39212yes—
515C2×C4#39212yes—
516C2×C4C2×#12011yescharacteristic
517C2×C4C2×#12011yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C8—44no—
521C8—44no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—24no—
531C22—22no—
532C22—22no—
533C22—24no—
534C22—28no—
535C22—216no—
536C22—28no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—84no—
541C22—84no—
542C22—168no—
543C22#161518yes—
544C22#161818yes—
545C22#200614yes—
546C22#201514yes—
547C22#201514yes—
548C22#202214yes—
549C22C2×#39211yescharacteristic
550C22C2×#45212yes—
551C22C2×#45311yescharacteristic
552C22C2×#45411yescharacteristic
553C22C2×#45812yes—
554C22C2×#46412yes—
555C22C2×#49511yescharacteristic
556C22C2×#49711yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—22no—
562C4—22no—
563C4—84no—
564C4—82no—
565C4—84no—
566C4—82no—
567C4—82no—
568C4—84no—
569C2—21no—
570C2—21no—
571C2—22no—
572C2—24no—
573C2—162no—
574C2#996918yesdirect product factor
575C2C2×#161512yes—
576C2C2×#161812yes—
577C2C2×#200611yescharacteristic
578C2C2×#201511yescharacteristic
579C2C2×#202211yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#484C2semidirect
4. C22×#328C2semidirect
6. C2×#1445C2semidirect
7. #9969C2direct (N × H)
8. C2×#1280C2semidirect
15. C2×#484C22semidirect
39. C2×#328C22semidirect
43. #1445C22semidirect
44. #1280C22semidirect
574. C2#9969direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4842
C22×#4812
C2×#26392
C22×#3282
C22×#3272
C2×#14452
#99692
C2×#12802