Finite Group Databasebeta

← #204622

Subgroups — #204622

Order 512.

1079 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35241C211yesmaximal, characteristic
2#20535C211yesmaximal, characteristic, complemented
3C2×#1518C211yesmaximal, characteristic, complemented
4C2×#1521C211yesmaximal, characteristic
5C2×#1515C211yesmaximal, characteristic, complemented
6C2×#1512C211yesmaximal, characteristic
7#14356C211yesmaximal, characteristic, complemented
8#9683C212yesmaximal, complemented
9#9682C212yesmaximal
10#9681C212yesmaximal
11#9679C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×#529C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2212yes—
17#2850C2212yes—
18#2664—21no—
19#2664—21no—
20#2568C411yescharacteristic, complemented
21#2565C412yescomplemented
22#2547C411yescharacteristic, complemented
23#2546C411yescharacteristic, complemented
24#2545C412yescomplemented
25#2542C411yescharacteristic, complemented
26#2498—21no—
27#2498—21no—
28#2417—22no—
29#2323—22no—
30C2×#383C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1771—22no—
34#1770—21no—
35C8×C4⋊C4—21no—
36#1749C2212yes—
37C4×C22⋊C8C2212yes—
38#1521—24no—
39#1518—24no—
40#1515—24no—
41#1512—24no—
42C2×#328C2212yes—
43C2×#328C2212yes—
44C2×#328C2212yes—
45C2×#328C2211yescharacteristic
46C2×#328C2211yescharacteristic
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#327C2212yes—
50C2×#327C2211yescharacteristic
51C2×#327C2211yescharacteristic
52#1444—21no—
53#1442—21no—
54#1441—22no—
55C4⋊#95C2212yes—
56#1434C2212yes—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64#529—22no—
65#529—22no—
66#529—22no—
67#529—22no—
68C2×#126—21no—
69C2×#125—21no—
70C2×#125C2×C411yescharacteristic
71C2×#124C2×C411yescharacteristic
72C2×#124C2×C412yes—
73C2×#123—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8C2312yes—
77C2×C22⋊Q8C2311yescharacteristic
78C2×#121C2311yescharacteristic
79C2×C4×Q8—21no—
80C2×C4×Q8C2×C412yes—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4C2×C411yescharacteristic
84C2×#117—21no—
85C2×#117—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4D412yes—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#432—22no—
92#432D414yes—
93#431—22no—
94#431D414yes—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C22⋊C8C2311yescharacteristic
98C2×C22⋊C8C2311yescharacteristic
99C2×C22⋊C8C2312yes—
100C2×#97—21no—
101C2×#97—22no—
102C2×C4×C8—21no—
103C2×C4×C8—22no—
104#400—21no—
105#400—21no—
106#399—21no—
107#399—21no—
108#398—22no—
109#398—22no—
110#398—21no—
111#398—21no—
112#397—21no—
113#397—21no—
114#396—22no—
115#396—22no—
116#391—22no—
117#391C2×C412yes—
118#391C2×C412yes—
119#391C2×C412yes—
120#390C2×C412yes—
121#389—22no—
122#389—22no—
123#389—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#387—22no—
129#387—22no—
130#386—22no—
131#386—22no—
132#384C2×C412yes—
133#383—22no—
134#383—22no—
135#382—22no—
136#382—22no—
137#382C2×C412yes—
138#380—22no—
139#380—22no—
140#380—22no—
141C4×C4⋊C4C2×C412yes—
142C4×C4⋊C4C2×C411yescharacteristic
143C4×C4⋊C4C2×C411yescharacteristic
144C4×C4⋊C4D412yes—
145C4×C4⋊C4D411yescharacteristic
146C4×C4⋊C4D411yescharacteristic
147C2×#95—22no—
148C2×#95—21no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95D411yescharacteristic
154C2×#95D411yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158#336—22no—
159#336—22no—
160#328—24no—
161#328—24no—
162#328—24no—
163#328—24no—
164#328—24no—
165#328—22no—
166#328—22no—
167#327—24no—
168#327—22no—
169#327—22no—
170C22×Q8—21no—
171C22×Q8C22×C411yescharacteristic
172C22×Q8C22×C412yes—
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—82no—
192#126—42no—
193#126—42no—
194#125—22no—
195#125—22no—
196#125—42no—
197#125—42no—
198#124—22no—
199#124—22no—
200#124—82no—
201#123—42no—
202#123—42no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—48no—
212C22⋊Q8—42no—
213C22⋊Q8—42no—
214#121—21no—
215#121—21no—
216#121—21no—
217#121—21no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—42no—
227C4×D4—42no—
228#117—42no—
229#117—42no—
230#117—42no—
231#117—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4C22⋊C412yes—
252C2×C4⋊C4C22⋊C412yes—
253C2×C4⋊C4C22⋊C412yes—
254C2×C4⋊C4C22⋊C412yes—
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C412yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—82no—
271C2×C42C22×C412yes—
272C2×C42C22×C412yes—
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42#4112yes—
282C2×C42#4112yes—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—22no—
288C22⋊C8C2×D412yes—
289C22⋊C8C2×D412yes—
290C22⋊C8C2×D414yes—
291#97—24no—
292#97—82no—
293C4×C8—24no—
294C4×C8—82no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—21no—
305#95—21no—
306#95—22no—
307#95—21no—
308#95—21no—
309#95—44no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—82no—
314#95—82no—
315#95C22⋊C412yes—
316#95C22⋊C412yes—
317#95C22⋊C411yescharacteristic
318#95C22⋊C411yescharacteristic
319#95C22⋊C411yescharacteristic
320#95C22⋊C411yescharacteristic
321C24—21no—
322C24C22×D411yescharacteristic
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—44no—
328C2×Q8—42no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—42no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C412yes—
386C22×C4#12412yes—
387C22×C4#12412yes—
388C22×C4#12712yes—
389C22×C4#12712yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—88no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—41no—
435C22⋊C4—42no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—44no—
451C42C4×D412yes—
452C42C4×D412yes—
453C42C4×D412yes—
454C42C4×D412yes—
455C42#12112yes—
456C42#12112yes—
457C42#12112yes—
458C42#12112yes—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23C22×C22⋊C411yescharacteristic
468C23C2×#12411yescharacteristic
469C23C2×#12711yescharacteristic
470C23#57712yes—
471C23#57812yes—
472Q8—81no—
473Q8—81no—
474Q8—82no—
475Q8—82no—
476Q8—84no—
477D4—81no—
478D4—81no—
479D4—82no—
480D4—81no—
481D4—81no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4#39012yes—
546C2×C4#39012yes—
547C2×C4#39012yes—
548C2×C4#39012yes—
549C2×C4#42114yes—
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×#12111yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C8—42no—
555C8—42no—
556C8—44no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—21no—
564C22—21no—
565C22—81no—
566C22—82no—
567C22—82no—
568C22—81no—
569C22—81no—
570C22C2×#39011yescharacteristic
571C22#288311yescharacteristic
572C22#288411yescharacteristic
573C22#289812yes—
574C22C2×#42112yes—
575C22#313712yes—
576C22#315411yescharacteristic
577C22#315611yescharacteristic
578C4—22no—
579C4—22no—
580C4—24no—
581C4—22no—
582C4—22no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—82no—
592C4—81no—
593C4—82no—
594C4—81no—
595C4—161no—
596C4—162no—
597C4—161no—
598C2—21no—
599C2—21no—
600C2—22no—
601C2—161no—
602C2#1759812yes—
603C2#1760112yes—
604C2#2120511yescharacteristic
605C2#2121411yescharacteristic
606C2#2125211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20535C2semidirect
3. C2×#1518C2semidirect
5. C2×#1515C2semidirect
7. #14356C2semidirect
8. #9683C2semidirect
11. #9679C2semidirect
20. #2568C4semidirect
21. #2565C4semidirect
22. #2547C4semidirect
23. #2546C4semidirect
24. #2545C4semidirect
25. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#352412
#205352
C2×#15182
C2×#15212
C2×#15152
C2×#15122
#143562
#96832
#96822
#96812
#96792