Finite Group Databasebeta

← #199784

Subgroups — #199784

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2#34359C211yesmaximal, characteristic
3#21825C211yesmaximal, characteristic
4#17578C211yesmaximal, characteristic
5C2×#1511C211yesmaximal, characteristic
6#9342C218yesmaximal, complemented
7#8108C211yesmaximal, characteristic
8#7943C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C22×C22⋊Q8C2211yescharacteristic
11#3195—22no—
12#3195—22no—
13#3195C414yescomplemented
14#3195C2214yes—
15C2×#483—21no—
16C2×#483C412yescomplemented
17C2×#470—22no—
18C2×#451—21no—
19C2×#451C412yescomplemented
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22#2378C2214yes—
23#2368—22no—
24#2344—22no—
25#2342—22no—
26#2332—22no—
27#2329—22no—
28#2313—21no—
29#2310—21no—
30#2280C2211yescharacteristic
31#2280C2211yescharacteristic
32C2×#381C2211yescharacteristic
33#1996—22no—
34#1996C2214yes—
35#1819—22no—
36#1520—22no—
37#1511—22no—
38#1511—22no—
39#1511C2214yes—
40#1508C2214yes—
41C2×#328—22no—
42#1305C2214yes—
43#1284C2214yes—
44C23×Q8C2×C411yescharacteristic
45C23×Q8D411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×Q16—22no—
48C22×M16—21no—
49C22×M16—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⋊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⋊Q8C2×C412yes—
67C2×C22⋊Q8C2×C412yes—
68C2×C22⋊Q8C2×C412yes—
69C2×C22⋊Q8D412yes—
70C2×C22⋊Q8D412yes—
71C2×C22⋊Q8D412yes—
72C2×C22⋊Q8C2312yes—
73C2×C22⋊Q8C2312yes—
74C2×#117—22no—
75C2×#117—21no—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84#483—24no—
85#483—24no—
86#483—24no—
87#483C2×C418yescomplemented
88#470—28no—
89#470—28no—
90#451—24no—
91#451—24no—
92#451—24no—
93#451C2×C418yescomplemented
94C2×#106—22no—
95C2×#103—22no—
96C2×#103—22no—
97#407—21no—
98#407—21no—
99#407—21no—
100#407—21no—
101#407—21no—
102#407C2×C412yes—
103#407C2312yes—
104#407C2312yes—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8C2×C412yes—
109#391—24no—
110#391—24no—
111#389—24no—
112#388—24no—
113#388—24no—
114#387—24no—
115#387—24no—
116#387—24no—
117#387—24no—
118#385—24no—
119#381—22no—
120#381—21no—
121#381—21no—
122#381—21no—
123#381—21no—
124#381D412yes—
125#381C2312yes—
126#380—22no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380C2312yes—
131#379—21no—
132#379D412yes—
133C4×C4⋊C4—24no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4C2312yes—
140C2×#95—21no—
141C2×#95D412yes—
142#342—28no—
143#328—28no—
144#328—28no—
145C25C2×D411yescharacteristic
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—82no—
155C22×Q8—82no—
156C22×Q8C22⋊C412yes—
157C22×Q8C22⋊C412yes—
158C22×Q8C22×C412yes—
159C22×Q8C2×D412yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—82no—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4111yescharacteristic
176C23×C4#4111yescharacteristic
177C23×C4#4111yescharacteristic
178C23×C4#4111yescharacteristic
179C23×C4C2411yescharacteristic
180C2×Q16—28no—
181C2×Q16—84no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—41no—
186C2×M16—82no—
187C2×M16—82no—
188C22×C8—22no—
189C22×C8—41no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
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—41no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C22⋊Q8—41no—
217C22⋊Q8—41no—
218C22⋊Q8C22⋊C414yes—
219C22⋊Q8C22⋊C414yes—
220C22⋊Q8C22×C414yes—
221C22⋊Q8C2×D414yes—
222#117—84no—
223#117—82no—
224#117—82no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4C22⋊C412yes—
250C2×C4⋊C4C22⋊C412yes—
251C2×C4⋊C4C22×C412yes—
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—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4C2×D414yes—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278#106—28no—
279#103—28no—
280#103—28no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—41no—
287C22⋊C8—41no—
288C22⋊C8C22×C414yes—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—41no—
300#95C2×D414yes—
301C24—42no—
302C24—42no—
303C24—42no—
304C24C4×D411yescharacteristic
305C24C4×D411yescharacteristic
306C24#12011yescharacteristic
307C24#12011yescharacteristic
308C24#12111yescharacteristic
309C24#12111yescharacteristic
310C24C22×D411yescharacteristic
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×Q8—81no—
325C2×Q8—81no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8C2×C22⋊C414yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4#12011yescharacteristic
379C22×C4#12011yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4#12111yescharacteristic
386C22×C4#12311yescharacteristic
387C22×C4#12311yescharacteristic
388C22×C4#12311yescharacteristic
389C22×C4#12311yescharacteristic
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396Q16—88no—
397M16—82no—
398M16—82no—
399M16—82no—
400M16—82no—
401C2×C8—24no—
402C2×C8—44no—
403C2×C8—42no—
404C2×C8—82no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—84no—
421C4⋊C4—82no—
422C4⋊C4—84no—
423C4⋊C4—82no—
424C4⋊C4C2×C22⋊C414yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C42—84no—
442C42—82no—
443C42—84no—
444C42—82no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—88no—
467C23#38611yescharacteristic
468C23#38611yescharacteristic
469C23#38611yescharacteristic
470C23#38611yescharacteristic
471C23C2×C4×D411yescharacteristic
472C23C2×#12011yescharacteristic
473C23C2×#12111yescharacteristic
474Q8—44no—
475Q8—44no—
476Q8—82no—
477Q8—82no—
478Q8—82no—
479Q8—82no—
480Q8—82no—
481Q8—82no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—164no—
537C2×C4#38611yescharacteristic
538C2×C4#38611yescharacteristic
539C2×C4#38611yescharacteristic
540C2×C4#38611yescharacteristic
541C2×C4C22×C22⋊C411yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×#12111yescharacteristic
544C2×C4C2×#12311yescharacteristic
545C8—44no—
546C8—82no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22#190512yes—
573C22#191312yes—
574C22C2×#38611yescharacteristic
575C22#302212yes—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—44no—
585C4—44no—
586C4—84no—
587C4—84no—
588C4—82no—
589C4—82no—
590C4—162no—
591C2—22no—
592C2—22no—
593C2—22no—
594C2—41no—
595C2—41no—
596C2—41no—
597C2—41no—
598C2#2094411yescharacteristic
599C2C2×#190511yescharacteristic
600C2C2×#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. #9342C2semidirect
13. #3195C4semidirect
16. C2×#483C4semidirect
19. C2×#451C4semidirect
87. #483C2×C4semidirect
93. #451C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
#343592
#218252
#175782
C2×#15112
#93422
#81082
#79432