Finite Group Databasebeta

← #198461

Subgroups — #198461

Order 512.

1320 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3#21825C211yesmaximal, characteristic
4#20700C211yesmaximal, characteristic
5#17578C211yesmaximal, characteristic
6#9256C218yesmaximal, complemented
7#7982C211yesmaximal, characteristic
8#7927C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#483—21no—
12C2×#483—22no—
13C2×#483—22no—
14C2×#483—22no—
15C2×#483—21no—
16C2×#483C414yescomplemented
17C2×#483C2214yes—
18C22×#106C412yescomplemented
19C22×#103C412yescomplemented
20C2×#407C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2345—22no—
23#2345—22no—
24#2344—22no—
25#2344C2214yes—
26#2316—21no—
27#2313—22no—
28#2280—22no—
29#2280C2211yescharacteristic
30#2280C2211yescharacteristic
31#2280C2211yescharacteristic
32#2274—21no—
33#1996—22no—
34#1996C2214yes—
35#1824—22no—
36#1823C2214yes—
37#1819—22no—
38#1736—22no—
39#1520C2214yes—
40#1511—22no—
41#1508—22no—
42#1289C2214yes—
43#1282C2214yes—
44C23×Q8C2×C411yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C22×M16—21no—
48C23×C8C2×C412yes—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2312yes—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—24no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4C2×C412yes—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D412yes—
78C22×C4⋊C4D412yes—
79C22×C4⋊C4D412yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83#483—24no—
84#483—24no—
85#483—28no—
86#483—28no—
87#483—24no—
88#483—24no—
89C2×#106—24no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106C2×C418yes—
93C2×#103—24no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103C2×C418yes—
97#407—21no—
98#407—21no—
99#407—21no—
100#407—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8C2×C412yes—
107C2×C22⋊C8C2312yes—
108C2×C22⋊C8C2312yes—
109#389—24no—
110#389—24no—
111#387—28no—
112#387—28no—
113#381—21no—
114#381—24no—
115#381—21no—
116#381D412yes—
117#380—24no—
118#380—24no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380D412yes—
123#380C2312yes—
124#380C2312yes—
125#379—21no—
126#379—21no—
127#379D412yes—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—24no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4D412yes—
137C4×C22⋊C4C2312yes—
138#342—28no—
139#340—28no—
140#328—28no—
141C25C2×D411yescharacteristic
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—44no—
148C22×Q8C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—44no—
160C23×C4—84no—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4111yescharacteristic
171C23×C4#4111yescharacteristic
172C23×C4#4111yescharacteristic
173C23×C4#4111yescharacteristic
174C23×C4C2411yescharacteristic
175C2×M16—44no—
176C2×M16—41no—
177C2×M16—41no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—44no—
182C22×C8C22×C414yes—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199#117—44no—
200#117—44no—
201#117—44no—
202#117—88no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4C22⋊C414yes—
243C2×C4⋊C4C22⋊C412yes—
244C2×C4⋊C4C22⋊C412yes—
245C2×C4⋊C4C22⋊C412yes—
246C2×C4⋊C4C22⋊C412yes—
247C2×C4⋊C4C22⋊C414yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4C2×D414yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—28no—
271C2×C42—41no—
272#106—28no—
273#106—28no—
274#103—28no—
275#103—28no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C22⋊C8—41no—
283C22⋊C8—41no—
284#95—28no—
285#95—28no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—41no—
291#95C2×D414yes—
292C24—24no—
293C24—42no—
294C24—42no—
295C24C4×D411yescharacteristic
296C24C4×D411yescharacteristic
297C24#12011yescharacteristic
298C24#12011yescharacteristic
299C24#12111yescharacteristic
300C24#12111yescharacteristic
301C24C22×D411yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—44no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8—82no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—48no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—82no—
349C22×C4—84no—
350C22×C4—82no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12311yescharacteristic
374C22×C4#12311yescharacteristic
375C22×C4#12311yescharacteristic
376C22×C4#12311yescharacteristic
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383M16—82no—
384M16—82no—
385C2×C8—24no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—44no—
389C2×C8—44no—
390C2×C8—42no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—88no—
413C4⋊C4C2×C22⋊C414yes—
414C4⋊C4C2×C22⋊C414yes—
415C22⋊C4—28no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C42—44no—
429C42—44no—
430C42—82no—
431C42—88no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—416no—
453C23#38611yescharacteristic
454C23#38611yescharacteristic
455C23#38611yescharacteristic
456C23#38611yescharacteristic
457C23C2×C4×D411yescharacteristic
458C23C2×#12011yescharacteristic
459C23C2×#12111yescharacteristic
460C23#57714yes—
461Q8—44no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—48no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—82no—
507C2×C4—84no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—84no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#38611yescharacteristic
522C2×C4#38611yescharacteristic
523C2×C4#38611yescharacteristic
524C2×C4#38611yescharacteristic
525C2×C4#41014yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×#12111yescharacteristic
529C2×C4C2×#12311yescharacteristic
530C8—44no—
531C8—82no—
532C22—28no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22#189912yes—
556C22#190512yes—
557C22C2×#38611yescharacteristic
558C22#294912yes—
559C22#300912yes—
560C22#304212yes—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—44no—
568C4—44no—
569C4—84no—
570C4—84no—
571C4—82no—
572C4—82no—
573C4—162no—
574C4#189912yes—
575C4C2×#41012yes—
576C2—22no—
577C2—22no—
578C2—22no—
579C2—22no—
580C2—22no—
581C2—41no—
582C2—41no—
583C2#2093311yescharacteristic
584C2C2×#189911yescharacteristic
585C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9256C2semidirect
16. C2×#483C4semidirect
18. C22×#106C4semidirect
19. C22×#103C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
#341782
#218252
#207002
#175782
#92562
#79822
#79272