Finite Group Databasebeta

← #493700

Subgroups — #493700

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34200C211yesmaximal, characteristic
2C2×#2120C212yesmaximal
3#20640C211yesmaximal, characteristic, complemented
4#20634C211yesmaximal, characteristic, complemented
5#12796C214yesmaximal, complemented
6#12793C214yesmaximal, complemented
7#12575C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C2×#520C2211yescharacteristic
10C23×C4⋊C4C411yescharacteristic, complemented
11#2905—21no—
12C2×#413C2212yes—
13C2×#410C2212yes—
14#2361—22no—
15#2360—22no—
16#2352—21no—
17#2352—21no—
18D4×C4⋊C4—21no—
19D4×C4⋊C4—21no—
20#2322C412yescomplemented
21#2322C2212yes—
22#2317C411yescharacteristic, complemented
23#2316—21no—
24#2311C2211yescharacteristic
25C4×#120C412yescomplemented
26C4×#120C2212yes—
27#2280C411yescharacteristic, complemented
28#2274C411yescharacteristic, complemented
29#2120—28no—
30C2×#352C2214yes—
31C2×AΣL(1,𝔽16)C2214yes—
32#1801—22no—
33#1800C2214yes—
34#1799—22no—
35#1799C2214yes—
36C2×#343—21no—
37#1740—21no—
38#1390C2218yes—
39C22×#41C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C22×M16—22no—
43#525—22no—
44#525—22no—
45#525D414yes—
46#525Q814yes—
47C2×#123C2×C411yescharacteristic
48C2×#123C2311yescharacteristic
49C2×C22⋊Q8—21no—
50C2×#121—21no—
51C2×#120C2×C411yescharacteristic
52C2×#120C2311yescharacteristic
53#520—44no—
54C2×C4×Q8D412yes—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4C2312yes—
62C2×#117—22no—
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77#424—44no—
78#423—24no—
79#413—24no—
80#413—24no—
81C2×#101—22no—
82C2×#100C2314yes—
83#410—24no—
84#410—24no—
85C2×#99—22no—
86C2×#99C2314yes—
87#391—21no—
88#390—21no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—44no—
93#388—22no—
94#388—22no—
95#387—22no—
96#387—22no—
97#386—22no—
98#386—22no—
99#386C2×C412yes—
100#386C2312yes—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385C2×C412yes—
106#385C2×C412yes—
107#384—21no—
108#382—21no—
109#381C2×C412yes—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—24no—
115#379C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4C2×C412yes—
124#352—28no—
125AΣL(1,𝔽16)—28no—
126#343—24no—
127#342—24no—
128C25C22×C411yescharacteristic
129C2×#41—42no—
130C2×#41—42no—
131C2×#41C2×D412yes—
132C2×#41C2×Q812yes—
133C2×#41#4112yes—
134C2×#41#4112yes—
135C22×Q8C2×D411yescharacteristic
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C22×C411yescharacteristic
139C22×D4C2×D411yescharacteristic
140C22×D4C2411yescharacteristic
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—44no—
149C23×C4—84no—
150C23×C4C22⋊C412yes—
151C23×C4C4⋊C414yescomplemented
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C22×C412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C2×M16—24no—
160C2×M16—42no—
161C2×M16—44no—
162C2×M16—44no—
163C22×C8—42no—
164#123—22no—
165#123—21no—
166#123—21no—
167#123C4⋊C414yescomplemented
168#123C2×D412yes—
169#123C2×Q812yes—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172#121—24no—
173#121—24no—
174#120—22no—
175#120—21no—
176#120—21no—
177#120C4⋊C414yescomplemented
178#120C2×D412yes—
179#120C2×Q812yes—
180C4×Q8—44no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—44no—
188C4×D4—84no—
189C4×D4—84no—
190#117—22no—
191#117—24no—
192#117—42no—
193#117—42no—
194#117—42no—
195#117—44no—
196#117—44no—
197#117C4⋊C414yescomplemented
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C2×D412yes—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4C22⋊C412yes—
235C2×C22⋊C4C22×C411yescharacteristic
236C2×C22⋊C4C22×C411yescharacteristic
237C2×C22⋊C4C22×C412yes—
238C2×C22⋊C4C22×C412yes—
239C2×C22⋊C4C2×D412yes—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42C22⋊C412yes—
246C2×C42C22⋊C412yes—
247C2×C42C2×D412yes—
248C4⋊C8—48no—
249#101—44no—
250#100—24no—
251#100—24no—
252#99—24no—
253#99—24no—
254#99—44no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—44no—
264C24—21no—
265C24—21no—
266C24—41no—
267C24—41no—
268C24—42no—
269C24—42no—
270C24C2×C22⋊C411yescharacteristic
271C24C2×C4⋊C412yes—
272C24C23×C411yescharacteristic
273C24C22×D411yescharacteristic
274#41—44no—
275#41—44no—
276#41—44no—
277#41—44no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8#12112yes—
281C2×Q8C22⋊Q812yes—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4C2×C4⋊C412yes—
296C2×D4#11712yes—
297C2×D4#12112yes—
298C2×D4C22⋊Q812yes—
299C2×D4C22×D411yescharacteristic
300C2×D4C22×Q811yescharacteristic
301C2×D4C2×#4111yescharacteristic
302C2×D4C2×#4111yescharacteristic
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C4⋊C412yes—
350C22×C4C2×C4⋊C412yes—
351C22×C4#11712yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4C22⋊Q812yes—
355C22×C4C22⋊Q812yes—
356C22×C4C22×D411yescharacteristic
357M16—44no—
358M16—44no—
359M16—84no—
360C2×C8—44no—
361C2×C8—84no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—84no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—164no—
381C4⋊C4#9914yes—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—42no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4C2×C4⋊C412yes—
401C22⋊C4C2×C4⋊C412yes—
402C42—24no—
403C42—24no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—84no—
408C42—82no—
409C42—82no—
410C23—22no—
411C23—22no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—22no—
417C23—21no—
418C23—21no—
419C23—42no—
420C23—41no—
421C23—44no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—44no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—48no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23#38012yes—
439C23C22×C22⋊C411yescharacteristic
440C23C22×C4⋊C411yescharacteristic
441C23C2×#11711yescharacteristic
442C23C2×#12111yescharacteristic
443C23C2×C22⋊Q811yescharacteristic
444Q8—42no—
445Q8—42no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—41no—
454D4—44no—
455D4—44no—
456D4—48no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—48no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—162no—
504C2×C4#38012yes—
505C2×C4#38012yes—
506C2×C4#38012yes—
507C2×C4C2×#9914yes—
508C2×C4C2×#9912yes—
509C2×C4#41012yes—
510C2×C4C2×#12111yescharacteristic
511C2×C4C2×C22⋊Q811yescharacteristic
512C8—84no—
513C22—21no—
514C22—21no—
515C22—24no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—41no—
525C22—41no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—41no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22C2×#38011yescharacteristic
546C22C22×#9911yescharacteristic
547C22C2×#41011yescharacteristic
548C4—24no—
549C4—21no—
550C4—22no—
551C4—21no—
552C4—22no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—82no—
561C4—164no—
562C4#179914yes—
563C2—21no—
564C2—21no—
565C2—41no—
566C2—44no—
567C2—41no—
568C2—41no—
569C2—41no—
570C2—42no—
571C2C2×#179911yescharacteristic
572C2#3334811yescharacteristic
573C2#3334911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#525C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20640C2semidirect
4. #20634C2semidirect
5. #12796C2semidirect
6. #12793C2semidirect
7. #12575C2semidirect
10. C23×C4⋊C4C4semidirect
20. #2322C4semidirect
22. #2317C4semidirect
25. C4×#120C4semidirect
27. #2280C4semidirect
28. #2274C4semidirect
151. C23×C4C4⋊C4semidirect
167. #123C4⋊C4semidirect
177. #120C4⋊C4semidirect
197. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342002
C2×#21202
#206402
#206342
#127962
#127932
#125752