Finite Group Databasebeta

← #198764

Subgroups — #198764

Order 512.

1269 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2#34245C211yesmaximal, characteristic
3#21825C211yesmaximal, characteristic
4#17578C211yesmaximal, characteristic
5C2×#1508C211yesmaximal, characteristic
6#9276C218yesmaximal, complemented
7#8012C211yesmaximal, characteristic
8#7925C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C22×C22⋊Q8C2211yescharacteristic
11#3189—22no—
12#3189—22no—
13#3189C414yescomplemented
14#3189C2214yes—
15C2×#484—21no—
16C2×#484C412yescomplemented
17C2×#467—22no—
18C2×#451—21no—
19C2×#451C412yescomplemented
20C2×#407C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2368C2214yes—
23#2362—22no—
24#2344—21no—
25#2342—22no—
26Q8×C22⋊C4—21no—
27#2332—22no—
28#2313—21no—
29C4×C22⋊Q8—22no—
30#2280C2211yescharacteristic
31#2280C2211yescharacteristic
32#2274—21no—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1996—22no—
35#1996C2214yes—
36#1819—22no—
37#1520C2214yes—
38#1511—22no—
39#1508—21no—
40#1508—22no—
41#1508—21no—
42#1508—22no—
43C2×#328C2214yes—
44#1294C2214yes—
45#1282C2214yes—
46C23×Q8C2×C411yescharacteristic
47C23×Q8D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×Q16—22no—
50C22×M16—21no—
51C23×C8—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⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8C2×C412yes—
69C2×C22⋊Q8C2×C412yes—
70C2×C22⋊Q8C2×C412yes—
71C2×C22⋊Q8D412yes—
72C2×C22⋊Q8D412yes—
73C2×C22⋊Q8D412yes—
74C2×C22⋊Q8C2312yes—
75C2×C22⋊Q8C2312yes—
76C2×C4×Q8—22no—
77C2×#117—22no—
78C2×#117—21no—
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42—21no—
88#484—24no—
89#484—24no—
90#484—24no—
91#484C2×C418yes—
92#467—28no—
93#467—28no—
94#451—24no—
95#451—24no—
96#451—24no—
97#451C2×C418yes—
98C2×#107—22no—
99C2×#103—22no—
100C2×#103—22no—
101#407—21no—
102#407—21no—
103#407—21no—
104#407—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8C2×C412yes—
110C2×C22⋊C8C2×C412yes—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113#391—22no—
114#391—24no—
115#389—22no—
116#388—24no—
117#388—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#385—24no—
122#381—22no—
123#381—21no—
124#381—21no—
125#380—22no—
126#380—21no—
127#380—21no—
128#380D412yes—
129#380C2312yes—
130#379—21no—
131#379D412yes—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—24no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4D411yescharacteristic
143C4×C22⋊C4D412yes—
144C4×C22⋊C4D411yescharacteristic
145C4×C22⋊C4C2312yes—
146C2×#95C2312yes—
147#342—28no—
148#328—28no—
149#328—24no—
150#328—24no—
151C25C2×D411yescharacteristic
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—82no—
161C22×Q8—82no—
162C22×Q8C22⋊C412yes—
163C22×Q8C22⋊C412yes—
164C22×Q8C22×C412yes—
165C22×Q8C2×D412yes—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—82no—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C23×C4#4111yescharacteristic
183C23×C4#4111yescharacteristic
184C23×C4#4111yescharacteristic
185C23×C4C2411yescharacteristic
186C2×Q16—28no—
187C2×Q16—84no—
188C2×M16—41no—
189C2×M16—41no—
190C2×M16—82no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—82no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—41no—
217C22⋊Q8—41no—
218C22⋊Q8—41no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8—41no—
223C22⋊Q8—41no—
224C22⋊Q8C22⋊C414yes—
225C22⋊Q8C22⋊C414yes—
226C22⋊Q8C22×C414yes—
227C22⋊Q8C2×D414yes—
228C4×Q8—84no—
229#117—48no—
230#117—82no—
231#117—82no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4C22⋊C412yes—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4C2×D414yes—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—41no—
289C2×C42—82no—
290#107—28no—
291#103—28no—
292#103—28no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300C22⋊C8C22×C414yes—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—41no—
310#95C2×D412yes—
311#95C2×D412yes—
312C24—24no—
313C24—42no—
314C24—81no—
315C24C4×D411yescharacteristic
316C24C4×D411yescharacteristic
317C24#12011yescharacteristic
318C24#12011yescharacteristic
319C24#12111yescharacteristic
320C24#12111yescharacteristic
321C24C22×D411yescharacteristic
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×Q8—82no—
335C2×Q8—82no—
336C2×Q8—82no—
337C2×Q8—82no—
338C2×Q8—82no—
339C2×Q8—82no—
340C2×Q8—82no—
341C2×Q8C2×C22⋊C414yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4#12011yescharacteristic
390C22×C4#12011yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12111yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12111yescharacteristic
397C22×C4#12311yescharacteristic
398C22×C4#12311yescharacteristic
399C22×C4#12311yescharacteristic
400C22×C4#12311yescharacteristic
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407Q16—88no—
408M16—82no—
409M16—82no—
410C2×C8—24no—
411C2×C8—44no—
412C2×C8—42no—
413C2×C8—82no—
414C2×C8—82no—
415C2×C8—82no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—48no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—84no—
434C4⋊C4—82no—
435C4⋊C4C2×C22⋊C414yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—28no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—44no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C22⋊C4—82no—
453C42—44no—
454C42—48no—
455C42—82no—
456C42—81no—
457C42—81no—
458C42—82no—
459C42—84no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—88no—
481C23#38611yescharacteristic
482C23#38611yescharacteristic
483C23#38611yescharacteristic
484C23#38611yescharacteristic
485C23C2×C4×D411yescharacteristic
486C23C2×#12011yescharacteristic
487C23C2×#12111yescharacteristic
488C23#57814yes—
489Q8—44no—
490Q8—44no—
491Q8—44no—
492Q8—44no—
493Q8—82no—
494Q8—82no—
495Q8—82no—
496Q8—82no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—88no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4#38611yescharacteristic
556C2×C4#38611yescharacteristic
557C2×C4#38611yescharacteristic
558C2×C4#38611yescharacteristic
559C2×C4C22×C22⋊C411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C8—44no—
564C8—82no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22#190612yes—
590C22C2×#38611yescharacteristic
591C22#295012yes—
592C22#301112yes—
593C22#304412yes—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—44no—
603C4—44no—
604C4—84no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—162no—
609C4#191312yes—
610C2—22no—
611C2—22no—
612C2—22no—
613C2—22no—
614C2—41no—
615C2—41no—
616C2—41no—
617C2#2093511yescharacteristic
618C2C2×#190611yescharacteristic
619C2C2×#191311yescharacteristic
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. #9276C2semidirect
13. #3189C4semidirect
16. C2×#484C4semidirect
19. C2×#451C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
#342452
#218252
#175782
C2×#15082
#92762
#80122
#79252