Finite Group Databasebeta

← #393350

Subgroups — #393350

Order 512.

1853 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic
3C2×#1996C212yesmaximal, complemented
4C2×#1799C211yesmaximal, characteristic
5#10915C214yesmaximal, complemented
6#10854C212yesmaximal, complemented
7#10754C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C2×#518C2212yes—
10C22×C4×D4C2211yescharacteristic
11C22×#99C2212yes—
12#2855—21no—
13#2848—21no—
14#2627—21no—
15#2626—21no—
16#2579C414yescomplemented
17#2564—22no—
18#2487C412yescomplemented
19#2417—22no—
20#2409C412yescomplemented
21#2315C2212yes—
22#2308C2212yes—
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25#2275—24no—
26#2274—21no—
27#2273—21no—
28#2273C2212yes—
29C42×D4—24no—
30C4×#117—21no—
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4C2212yes—
33#1996—24no—
34#1996—24no—
35#1799—22no—
36#1799—22no—
37C2×#342C2218yes—
38C2×#342C2218yes—
39C2×#342C2212yes—
40C23×Q8C2×C411yescharacteristic, complemented
41C23×D4C2311yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53#518—24no—
54#518—28no—
55#518Q818yescomplemented
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—28no—
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C412yes—
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×#117—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2×C412yescomplemented
69C22×C4⋊C4C2×C411yescharacteristic, complemented
70C22×C4⋊C4D411yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2311yescharacteristic
77C22×C22⋊C4C2314yes—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42D411yescharacteristic
81C2×#99—24no—
82C2×#99—28no—
83#400—22no—
84#399—22no—
85#398—24no—
86#396—24no—
87#395—22no—
88#393—22no—
89#391—22no—
90#389—22no—
91#387—24no—
92#386—24no—
93#386C2×C414yes—
94#386C2×C414yes—
95#385—24no—
96#385—22no—
97#385—22no—
98#385C2×C414yescomplemented
99#385C2×C414yescomplemented
100#381—22no—
101#380—22no—
102#380—21no—
103#380—21no—
104#380—24no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380C2×C412yescomplemented
110#380C2×C412yescomplemented
111#379—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—28no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—28no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—28no—
123C4×C22⋊C4D414yes—
124C4×C22⋊C4D414yes—
125#376—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C43—22no—
131#342—24no—
132#342—216no—
133#342—216no—
134C25C2×D411yescharacteristic
135C25C2411yescharacteristic
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—82no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—24no—
144C22×D4C22×C412yes—
145C22×D4C2×D412yes—
146C22×D4C2×Q814yes—
147C22×D4#4114yes—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—82no—
157C23×C4—82no—
158C23×C4C22⋊C412yes—
159C23×C4C22⋊C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—42no—
185C22⋊Q8—84no—
186C4×D4—232no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—232no—
196#117—28no—
197#117—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—84no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22×C414yes—
244C2×C22⋊C4C22×C412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D418yes—
247C2×C22⋊C4C2×Q818yes—
248C2×C22⋊C4#4118yes—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—84no—
259C2×C42—84no—
260C2×C42C22⋊C412yes—
261C2×C42C22⋊C412yes—
262#99—48no—
263#95—24no—
264#95—24no—
265#95—22no—
266#95—22no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—42no—
274#95—42no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24—81no—
281C24—81no—
282C24C2×C22⋊C412yes—
283C24C22⋊Q814yes—
284C24C23×C411yescharacteristic
285C24C22×D411yescharacteristic
286C24C22×D411yescharacteristic
287C24C22×D412yes—
288C24C22×Q812yes—
289C24C2×#4112yes—
290C24C2×#4112yes—
291C24C2×#4114yes—
292C2×Q8—44no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—84no—
297C2×Q8—82no—
298C2×Q8—84no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—24no—
306C2×D4—28no—
307C2×D4C4×Q818yes—
308C2×D4#12414yes—
309C2×D4C4⋊Q814yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—216no—
317C22×C4—28no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—28no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—48no—
340C22×C4—82no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4—82no—
344C22×C4—81no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—82no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4#12011yescharacteristic
373C22×C4#12011yescharacteristic
374C22×C4#12012yes—
375C22×C4#12112yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4C22⋊Q814yes—
380C22×C4#12311yescharacteristic
381C22×C4#12311yescharacteristic
382C22×C4#12312yes—
383C4⋊C4—216no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—44no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—82no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C22⋊C4—216no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—28no—
402C22⋊C4—28no—
403C22⋊C4—28no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—28no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C22⋊C4C22⋊Q8116yescomplemented
417C42—216no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—48no—
423C42—84no—
424C42—84no—
425C42—84no—
426C23—24no—
427C23—22no—
428C23—24no—
429C23—22no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—81no—
445C23—81no—
446C23—82no—
447C23—84no—
448C23—82no—
449C23—84no—
450C23—84no—
451C23#39114yes—
452C23C22×C22⋊C411yescharacteristic
453C23C2×C4×D412yes—
454C23C2×C4×Q812yes—
455C23C4×#4112yes—
456C23C2×#12011yescharacteristic
457C23C2×#12112yes—
458C23C2×C22⋊Q812yes—
459C23C2×C22⋊Q814yes—
460C23C2×#12311yescharacteristic
461C23#52514yes—
462C23#52512yes—
463C23C2×#12411yescharacteristic
464C23#52914yes—
465C23C2×C4⋊Q811yescharacteristic
466C23#53212yes—
467C23#53314yes—
468Q8—82no—
469Q8—84no—
470Q8—82no—
471D4—28no—
472D4—28no—
473D4—216no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—216no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—88no—
519C2×C4—88no—
520C2×C4—816no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4—164no—
527C2×C4—168no—
528C2×C4#38611yescharacteristic
529C2×C4#38611yescharacteristic
530C2×C4#38611yescharacteristic
531C2×C4#38612yes—
532C2×C4#38612yes—
533C2×C4#38611yescharacteristic
534C2×C4#39114yes—
535C22—21no—
536C22—21no—
537C22—22no—
538C22—22no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—21no—
545C22—24no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—41no—
555C22—41no—
556C22—44no—
557C22—44no—
558C22—48no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—84no—
563C22C2×#38611yescharacteristic
564C22C2×#39111yescharacteristic
565C22#229711yescharacteristic
566C22C4×C22⋊Q814yes—
567C22#284812yes—
568C22#285012yes—
569C22#285512yes—
570C22#285712yes—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—22no—
576C4—24no—
577C4—416no—
578C4—42no—
579C4—42no—
580C4—84no—
581C4—82no—
582C4—82no—
583C4—164no—
584C4#191312yes—
585C2—22no—
586C2—21no—
587C2—22no—
588C2—21no—
589C2—22no—
590C2—22no—
591C2—24no—
592C2—44no—
593C2#1470211yescharacteristic
594C2C2×#191311yescharacteristic
595C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1996C2semidirect
5. #10915C2semidirect
6. #10854C2semidirect
16. #2579C4semidirect
18. #2487C4semidirect
20. #2409C4semidirect
40. C23×Q8C2×C4semidirect
55. #518Q8semidirect
68. C22×C4⋊C4C2×C4semidirect
69. C22×C4⋊C4C2×C4semidirect
98. #385C2×C4semidirect
99. #385C2×C4semidirect
109. #380C2×C4semidirect
110. #380C2×C4semidirect
416. C22⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#350102
C4×#5182
C2×#19962
C2×#17992
#109152
#108542
#107542