Finite Group Databasebeta

← #355491

Subgroups — #355491

Order 512.

1769 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#34264C211yesmaximal, characteristic
2C2×#1913C212yesmaximal
3#20392C211yesmaximal, characteristic, complemented
4#20354C211yesmaximal, characteristic, complemented
5#10938C212yesmaximal, complemented
6#10918C212yesmaximal, complemented
7#10829C212yesmaximal, complemented
8#10761C212yesmaximal, complemented
9#10745C212yesmaximal, complemented
10C2×#534C2211yescharacteristic
11C2×#518C2212yes—
12C2×#518C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#99C2212yes—
15C22×#99C2214yes—
16#2364—22no—
17#2361—22no—
18#2359—22no—
19#2358—22no—
20#2322C412yescomplemented
21#2321C412yescomplemented
22#2308C2211yescharacteristic
23C4×#123C412yescomplemented
24C4×#120C412yescomplemented
25C2×#386C2211yescharacteristic
26C2×#385C2211yescharacteristic
27#2273C2211yescharacteristic
28#2273C2212yes—
29#1913C4116yescomplemented
30C4×#99—22no—
31C4×#99—22no—
32#1740—22no—
33C2×#342—22no—
34C2×#342C2212yes—
35C2×#342C2212yes—
36C2×#342C2212yes—
37C2×#342C2214yes—
38C2×#342C2214yes—
39C22⋊#95C2214yes—
40C23×D4C2311yescharacteristic
41C23×D4C2311yescharacteristic
42C24×C4C2×C411yescharacteristic, complemented
43C24×C4C2311yescharacteristic
44#534—24no—
45#534C2×C418yes—
46C2×#123—21no—
47C2×#123C2×C412yes—
48C2×#121—22no—
49C2×#120—21no—
50C2×#120C2×C412yes—
51#518—24no—
52#518—44no—
53#518—44no—
54#518C2×C418yescomplemented
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—44no—
59C2×C4×D4—44no—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2×C412yes—
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×#117—22no—
65C2×#117D412yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2×C411yescharacteristic, complemented
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4C2×C411yescharacteristic, complemented
71C22×C22⋊C4D412yes—
72C22×C22⋊C4C2311yescharacteristic
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2311yescharacteristic
76C22×C22⋊C4C2311yescharacteristic
77C22×C22⋊C4C2311yescharacteristic
78C22×C42C2×C411yescharacteristic, complemented
79C2×#99—24no—
80C2×#99—24no—
81C2×#99—44no—
82C2×#99—44no—
83C2×#99C2×C418yescomplemented
84C2×#99C2×C4116yescomplemented
85#390—22no—
86#389—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#387—22no—
92#387—22no—
93#386—22no—
94#386—22no—
95#386—21no—
96#386—21no—
97#386—22no—
98#386—22no—
99#386C2×C412yes—
100#386C2×C412yes—
101#385—22no—
102#385—22no—
103#385—22no—
104#385C2×C412yescomplemented
105#385C2×C412yescomplemented
106#381C2×C412yescomplemented
107#379C2×C412yescomplemented
108C4×C4⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—44no—
116C4×C22⋊C4C2×C412yescomplemented
117C4×C22⋊C4C2×C412yescomplemented
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—44no—
121#342—24no—
122#342—24no—
123#342—24no—
124#342—44no—
125#342C2×C4116yescomplemented
126C25C22×C411yescharacteristic
127C25C2×D411yescharacteristic
128C25C2411yescharacteristic
129C22×D4—24no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—24no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4C22×C414yes—
140C22×D4C22×C412yes—
141C22×D4C22×C414yes—
142C22×D4C2×D412yes—
143C22×D4#4114yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4C22⋊C412yes—
157C23×C4C22⋊C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166#123—24no—
167#123—28no—
168#123C4218yescomplemented
169#121—216no—
170#120—28no—
171#120—24no—
172#120C4218yescomplemented
173C4×D4—416no—
174C4×D4—44no—
175C4×D4—44no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—48no—
179C4×D4—48no—
180C4×D4—416no—
181C4×D4—84no—
182#117—28no—
183#117—44no—
184#117—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22×C418yes—
227C2×C22⋊C4C22×C418yes—
228C2×C22⋊C4C22×C412yes—
229C2×C22⋊C4C22×C412yes—
230C2×C22⋊C4C22×C412yes—
231C2×C22⋊C4C22×C414yes—
232C2×C22⋊C4C2×D412yes—
233C2×C22⋊C4C2×D412yes—
234C2×C22⋊C4C2×D412yes—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C22⋊C412yes—
243C2×C42C22⋊C412yes—
244#99—216no—
245#99—416no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—42no—
255#95—42no—
256#95—88no—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—81no—
265C24—81no—
266C24—81no—
267C24—81no—
268C24—81no—
269C24—81no—
270C24C2×C4212yes—
271C24C2×C22⋊C412yes—
272C24C4×D414yes—
273C24C23×C411yescharacteristic
274C24C23×C412yes—
275C24C22×D411yescharacteristic
276C24C22×D411yescharacteristic
277C24C2×#4112yes—
278C24H(5,𝔽2)14yes—
279C24H(5,𝔽2)12yes—
280C242−1+412yes—
281C2×D4—24no—
282C2×D4—28no—
283C2×D4—24no—
284C2×D4—28no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—28no—
288C2×D4—28no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—28no—
292C2×D4—28no—
293C2×D4—28no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4C2×C4214yes—
304C2×D4#11714yes—
305C2×D4C4×D418yes—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—48no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—84no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—82no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—84no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—162no—
365C22×C4—162no—
366C22×C4C2×C4212yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C4×D414yes—
373C4⋊C4—28no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—82no—
380C22⋊C4—22no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—28no—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—24no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—48no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4C2×C4214yes—
407C42—42no—
408C42—42no—
409C42—44no—
410C42—48no—
411C42—82no—
412C42—82no—
413C42—162no—
414C42—162no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—24no—
421C23—41no—
422C23—44no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—44no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—84no—
446C23—82no—
447C23—84no—
448C23—161no—
449C23—161no—
450C23C4×C22⋊C414yes—
451C23C22×C4211yescharacteristic
452C23C22×C22⋊C411yescharacteristic
453C23C2×#11711yescharacteristic
454C23C2×C4×D412yes—
455C23C2×C4×D412yes—
456C23#51812yes—
457C23#51814yes—
458C23#52012yes—
459C23#52014yes—
460C23#53412yes—
461C23#53412yes—
462C23#53511yescharacteristic
463C23#53611yescharacteristic
464C23#53712yes—
465C23#53814yes—
466C23#53914yes—
467D4—28no—
468D4—216no—
469D4—28no—
470D4—44no—
471D4—44no—
472D4—48no—
473D4—88no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4—88no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—88no—
522C2×C4—162no—
523C2×C4—162no—
524C2×C4—161no—
525C2×C4—161no—
526C2×C4—164no—
527C2×C4—164no—
528C2×C4—168no—
529C2×C4C4×C22⋊C414yes—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—24no—
539C22—42no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—42no—
547C22—42no—
548C22—41no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—84no—
557C22—82no—
558C22—84no—
559C22—84no—
560C22—82no—
561C22—84no—
562C22—81no—
563C22—81no—
564C22C2×C4×C22⋊C411yescharacteristic
565C22#227512yes—
566C22#230811yescharacteristic
567C22#231111yescharacteristic
568C22#231812yes—
569C22#232112yes—
570C22#232212yes—
571C22#232514yes—
572C4—24no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—82no—
581C4—88no—
582C4—88no—
583C4—81no—
584C2—21no—
585C2—22no—
586C2—21no—
587C2—24no—
588C2—22no—
589C2—22no—
590C2—22no—
591C2—44no—
592C2—82no—
593C2#1378411yescharacteristic
594C2#2177611yescharacteristic
595C2#2177711yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC2×#534C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20392C2semidirect
4. #20354C2semidirect
5. #10938C2semidirect
6. #10918C2semidirect
7. #10829C2semidirect
8. #10761C2semidirect
9. #10745C2semidirect
20. #2322C4semidirect
21. #2321C4semidirect
23. C4×#123C4semidirect
24. C4×#120C4semidirect
29. #1913C4semidirect
42. C24×C4C2×C4semidirect
54. #518C2×C4semidirect
67. C22×C4⋊C4C2×C4semidirect
70. C22×C22⋊C4C2×C4semidirect
78. C22×C42C2×C4semidirect
83. C2×#99C2×C4semidirect
84. C2×#99C2×C4semidirect
104. #385C2×C4semidirect
105. #385C2×C4semidirect
106. #381C2×C4semidirect
107. #379C2×C4semidirect
116. C4×C22⋊C4C2×C4semidirect
117. C4×C22⋊C4C2×C4semidirect
125. #342C2×C4semidirect
168. #123C42semidirect
172. #120C42semidirect

Maximal subgroups

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

subgroupindex
#342642
C2×#19132
#203922
#203542
#109382
#109182
#108292
#107612
#107452