Finite Group Databasebeta

← C2×#32965 #6629068

Subgroups — C2×#32965 #6629068

Order 512.

4378 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C22×#475C211yesmaximal, characteristic, complemented
3C22×#476C211yesmaximal, characteristic, complemented
4C2×#2628C211yesmaximal, characteristic, complemented
5#32965C2116yesmaximal, direct product factor
6C2×#1865C212yesmaximal, complemented
7C2×#1864C212yesmaximal, complemented
8C2×#1853C212yesmaximal, complemented
9C2×#1805C211yesmaximal, characteristic, complemented
10C2×#1804C211yesmaximal, characteristic, complemented
11C2×#1798C211yesmaximal, characteristic, complemented
12C2×#1788C212yesmaximal
13C23×#41C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×#520—28no—
16C2×#520—28no—
17C2×#520—24no—
18C2×#520C2218yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4—21no—
22C22×C4×D4—21no—
23C22×C4×D4C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C22×#117—21no—
26C22×#117C2211yescharacteristic
27C23×C4⋊C4—21no—
28C23×C4⋊C4C2211yescharacteristic
29C2×#475—28no—
30C2×#475C2218yescomplemented
31C2×#476—28no—
32C2×#476C2218yescomplemented
33C22×#106C2212yes—
34C22×#106C2212yes—
35C22×C4⋊C8C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37C22×#103C2212yes—
38C22×#103C2212yes—
39C22×#102C2212yes—
40C22×#102C2212yes—
41C22×C22⋊C8C2212yes—
42#2628—24no—
43#2628C2218yescomplemented
44C2×#400—21no—
45C2×#399—21no—
46C2×#395—21no—
47C2×#393—21no—
48C2×#389C2212yes—
49C2×#382C2211yescharacteristic
50C2×#380C2212yes—
51#1865C22116yescomplemented
52#1864C22116yescomplemented
53#1853C22116yescomplemented
54#1805C2218yescomplemented
55#1804C2218yescomplemented
56#1798C2218yescomplemented
57#1788C22116yes—
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C22×#41—24no—
62C22×#41—24no—
63C22×#41—24no—
64C22×#41—24no—
65C22×#41—22no—
66C22×#41—24no—
67C22×#41Q818yescomplemented
68C22×#41C2314yes—
69C23×Q8—21no—
70C23×Q8C2311yescharacteristic
71C23×D4—21no—
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C24×C4—21no—
75C24×C4—21no—
76C24×C4—21no—
77C24×C4C2311yescharacteristic
78C23×C8C2312yes—
79C23×C8C2312yes—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8C2314yes—
84#520—264no—
85#520—232no—
86#520—232no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8D418yes—
93C2×C4×Q8C2314yes—
94C2×C4×Q8C2314yes—
95C2×C4×D4—216no—
96C2×C4×D4—216no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—24no—
104C2×C4×D4D418yes—
105C2×C4×D4C2314yes—
106C2×C4×D4C2314yes—
107C2×#117—216no—
108C2×#117—28no—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117D414yes—
112C2×#117C2314yes—
113C22×C4⋊C4—216no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—28no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#476—232no—
140#475—232no—
141C2×#106—28no—
142C2×#106—28no—
143C2×#106C2318yes—
144C2×#106C2318yes—
145C2×C4⋊C8—24no—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149C2×#103—28no—
150C2×#103—28no—
151C2×#103C2318yes—
152C2×#103C2318yes—
153C2×#102—28no—
154C2×#102—28no—
155C2×#102C2318yes—
156C2×#102C2318yes—
157C2×C22⋊C8—28no—
158C2×C22⋊C8C2318yes—
159#400—28no—
160#399—28no—
161#395—28no—
162#393—28no—
163#389—24no—
164#389C2318yes—
165#382—22no—
166#382C2314yes—
167#380—24no—
168#380C2318yes—
169C2×#95—21no—
170C2×#95—22no—
171C2×#95—22no—
172C2×#95—21no—
173#340C2318yes—
174#340C2318yes—
175#336C2318yes—
176C25—21no—
177C25C2×D411yescharacteristic
178C2×#41—216no—
179C2×#41—216no—
180C2×#41—232no—
181C2×#41—232no—
182C2×#41—232no—
183C2×#41—28no—
184C2×#41—28no—
185C2×#41—232no—
186C2×#41—28no—
187C2×#41—28no—
188C2×#41—216no—
189C2×#41—28no—
190C2×#41C2×Q8116yescomplemented
191C22×Q8—22no—
192C22×Q8—21no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—44no—
197C22×Q8—84no—
198C22×Q8C2×D414yes—
199C22×Q8C2×Q814yes—
200C22×Q8#4114yes—
201C22×Q8C2412yes—
202C22×D4—28no—
203C22×D4—28no—
204C22×D4—28no—
205C22×D4—24no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2×D414yes—
211C22×D4C2×Q814yes—
212C22×D4#4114yes—
213C22×D4C2412yes—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—21no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—28no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—28no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—84no—
234C23×C4—84no—
235C23×C4—84no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×Q814yes—
243C23×C4#4114yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C22×C8—24no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—48no—
257C4×Q8—216no—
258C4×Q8—28no—
259C4×Q8—28no—
260C4×Q8—216no—
261C4×Q8—28no—
262C4×Q8C2×D4116yes—
263C4×D4—264no—
264C4×D4—264no—
265C4×D4—216no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—216no—
269C4×D4—28no—
270C4×D4C2×D4116yes—
271#117—264no—
272#117—216no—
273#117—28no—
274#117—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—264no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—84no—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—84no—
317C2×C22⋊C4C2×D414yes—
318C2×C22⋊C4C2×D414yes—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331#106—216no—
332#106—216no—
333C4⋊C8—28no—
334C4⋊C8—28no—
335#103—216no—
336#103—216no—
337#102—216no—
338#102—216no—
339C22⋊C8—216no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—28no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—44no—
353C24—44no—
354C24C22⋊Q814yes—
355C24C22×D412yes—
356C24C22×D411yescharacteristic
357#41—2128no—
358#41—232no—
359#41—232no—
360#41—232no—
361#41—232no—
362C2×Q8—24no—
363C2×Q8—28no—
364C2×Q8—24no—
365C2×Q8—28no—
366C2×Q8—28no—
367C2×Q8—24no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—48no—
372C2×Q8—88no—
373C2×Q8—816no—
374C2×Q8C22⋊Q8116yes—
375C2×Q8C22×D414yes—
376C2×Q8C22×Q814yes—
377C2×Q8C2×#4114yes—
378C2×D4—216no—
379C2×D4—264no—
380C2×D4—216no—
381C2×D4—216no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—24no—
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4C22⋊Q8116yes—
392C2×D4C22×D414yes—
393C2×D4C22×Q814yes—
394C2×D4C2×#4114yes—
395C22×C4—216no—
396C22×C4—264no—
397C22×C4—216no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—216no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—216no—
426C22×C4—216no—
427C22×C4—216no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—816no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—816no—
443C22×C4—816no—
444C22×C4#12014yes—
445C22×C4#12014yes—
446C22×C4C22⋊Q814yes—
447C22×C4C22⋊Q818yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×Q814yes—
457C22×C4C22×Q812yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2511yescharacteristic
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—48no—
470C2×C8—48no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—88no—
488C4⋊C4C22×D414yes—
489C4⋊C4C22×D414yes—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—88no—
495C42—216no—
496C42—216no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42#12018yes—
501C42C22×D414yes—
502C42C22×D414yes—
503C23—216no—
504C23—28no—
505C23—216no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—416no—
514C23—48no—
515C23—416no—
516C23—44no—
517C23—48no—
518C23—44no—
519C23—48no—
520C23—44no—
521C23—44no—
522C23C2×#12012yes—
523C23C2×C22⋊Q814yes—
524C23C2×C22⋊Q812yes—
525C23#52512yes—
526C23#57714yes—
527C23#57714yes—
528C23C23×D411yescharacteristic
529Q8—216no—
530Q8—24no—
531Q8—24no—
532Q8—24no—
533Q8—816no—
534Q8C2×C22⋊Q818yes—
535D4—264no—
536D4—216no—
537D4—24no—
538D4—24no—
539D4—24no—
540D4C2×C22⋊Q818yes—
541C2×C4—264no—
542C2×C4—28no—
543C2×C4—216no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—216no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—816no—
577C2×C4—816no—
578C2×C4—816no—
579C2×C4—88no—
580C2×C4—88no—
581C2×C4C2×#12014yes—
582C2×C4C2×#12014yes—
583C2×C4C2×#12012yes—
584C2×C4C2×C22⋊Q814yes—
585C2×C4C2×C22⋊Q812yes—
586C2×C4C2×C22⋊Q814yes—
587C2×C4C2×C22⋊Q814yes—
588C2×C4#52512yes—
589C2×C4#52514yes—
590C2×C4#52514yes—
591C2×C4C23×D411yescharacteristic
592C2×C4C23×D411yescharacteristic
593C2×C4C23×Q811yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C8—44no—
598C8—44no—
599C22—24no—
600C22—28no—
601C22—28no—
602C22—48no—
603C22—44no—
604C22—48no—
605C22—416no—
606C22—432no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—44no—
611C22#284814yes—
612C22#300914yes—
613C22#301014yes—
614C22#308418yes—
615C22#312518yes—
616C22C22×#12011yescharacteristic
617C22C22×C22⋊Q811yescharacteristic
618C22C2×#52511yescharacteristic
619C22C2×#57712yes—
620C22C2×#57712yes—
621C4—24no—
622C4—28no—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—84no—
630C4—84no—
631C4#284814yes—
632C4C22×C22⋊Q811yescharacteristic
633C4C22×C22⋊Q811yescharacteristic
634C4C2×#52511yescharacteristic
635C4C2×#52511yescharacteristic
636C2—44no—
637C2—48no—
638C2#3296518yesdirect product factor
639C2C2×#284811yescharacteristic
640C2C2×#300911yescharacteristic
641C2C2×#301011yescharacteristic
642C2C2×#308412yes—
643C2C2×#312512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#475C2semidirect
3. C22×#476C2semidirect
4. C2×#2628C2semidirect
5. #32965C2direct (N × H)
6. C2×#1865C2semidirect
7. C2×#1864C2semidirect
8. C2×#1853C2semidirect
9. C2×#1805C2semidirect
10. C2×#1804C2semidirect
11. C2×#1798C2semidirect
30. C2×#475C22semidirect
32. C2×#476C22semidirect
43. #2628C22semidirect
51. #1865C22semidirect
52. #1864C22semidirect
53. #1853C22semidirect
54. #1805C22semidirect
55. #1804C22semidirect
56. #1798C22semidirect
67. C22×#41Q8semidirect
190. C2×#41C2×Q8semidirect
638. C2#32965direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5202
C22×#4752
C22×#4762
C2×#26282
#329652
C2×#18652
C2×#18642
C2×#18532
C2×#18052
C2×#18042
C2×#17982
C2×#17882