Finite Group Databasebeta

← #224592

Subgroups — #224592

Order 512.

1021 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic, complemented
3#13818C211yesmaximal, characteristic, complemented
4#9713C212yesmaximal, complemented
5#9712C211yesmaximal, characteristic
6#9682C212yesmaximal, complemented
7#9680C211yesmaximal, characteristic
8#8572C211yesmaximal, characteristic, complemented
9#8571C212yesmaximal, complemented
10#8218C212yesmaximal, complemented
11C4×#328C211yesmaximal, characteristic, complemented
12#2856C2211yescharacteristic
13#2850C2212yes—
14#2597—22no—
15#2571C411yescharacteristic, complemented
16#2548C411yescharacteristic, complemented
17#2513—21no—
18#2513C412yescomplemented
19#2504—21no—
20#2447C412yescomplemented
21#2444—22no—
22#2366C411yescharacteristic, complemented
23#2361—21no—
24#2357—21no—
25#2355C411yescharacteristic, complemented
26C4×C22⋊Q8C2211yescharacteristic
27C4×#121C2212yes—
28C2×#389—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1939—21no—
32#1917—22no—
33#1797C2212yes—
34#1797C2211yescharacteristic
35#1770—21no—
36#1749C2212yes—
37C4×C22⋊C8C2211yescharacteristic
38C2×#328C2211yescharacteristic
39C2×#328C2211yescharacteristic
40C2×#328C2212yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2211yescharacteristic
48C4×#95C2211yescharacteristic
49C4×#95C2212yes—
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C2×#127C2×C411yescharacteristic
54C2×#124C2×C412yes—
55C2×C22⋊Q8C2311yescharacteristic
56C2×#121—21no—
57C2×#121—21no—
58C2×#121C2312yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42—22no—
69C22×C42C2311yescharacteristic
70C22×C42C2312yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8C2311yescharacteristic
74C2×C22⋊C8C2312yes—
75C2×C22⋊C8C2311yescharacteristic
76C2×#97—22no—
77C2×C4×C8—21no—
78#400—22no—
79#397—22no—
80#397—22no—
81#397—22no—
82#397—22no—
83#394—22no—
84#391C2×C412yes—
85#391C2×C411yescharacteristic
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#390C2×C412yes—
92#390C2×C412yes—
93#390C2×C412yes—
94#389—28no—
95#389—21no—
96#389C2×C412yes—
97#389C2×C411yescharacteristic
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#387—21no—
104#387—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#385—21no—
110#385—22no—
111#384C2×C411yescharacteristic
112#382—21no—
113#382—21no—
114#382C2×C412yes—
115#381—22no—
116#380C2311yescharacteristic
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—44no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95D412yes—
135C2×#95C2312yes—
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138#336—21no—
139#336—22no—
140#336—21no—
141#336—22no—
142#328—22no—
143#328—22no—
144#328—42no—
145#327—24no—
146#327—24no—
147#327—42no—
148#327—42no—
149C22×Q8C22×C411yescharacteristic
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22×C412yes—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166#127—42no—
167#124—82no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—22no—
176#121—42no—
177#121—42no—
178#121—81no—
179#121—81no—
180C4×Q8—42no—
181C4×D4—44no—
182C4×D4—82no—
183C4×D4—82no—
184C4×D4—82no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C42—24no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42—82no—
232C2×C42—82no—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C4⋊C8—42no—
242C4⋊C8—82no—
243C22⋊C8—21no—
244C22⋊C8—22no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—22no—
248C22⋊C8#4112yes—
249#97—82no—
250C4×C8—42no—
251#95—28no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—21no—
266#95—21no—
267#95—22no—
268#95—42no—
269#95—44no—
270#95—81no—
271#95C22⋊C412yes—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C22⋊C412yes—
275#95#4112yes—
276C24—22no—
277C24C22×D411yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—81no—
297C2×D4—81no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4#13712yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C42−1+411yescharacteristic
359C2×C8—22no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—41no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—81no—
385C4⋊C4—82no—
386C4⋊C4—81no—
387C4⋊C4—88no—
388C4⋊C4#10414yes—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C42—22no—
405C42—22no—
406C42—42no—
407C42—42no—
408C42—41no—
409C42—41no—
410C42—42no—
411C42—42no—
412C42—48no—
413C42—42no—
414C42—42no—
415C42—82no—
416C42—88no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23C22×C22⋊C411yescharacteristic
429C23#54811yescharacteristic
430C23#55011yescharacteristic
431C23C2×AGL(1,ℤ/8)11yescharacteristic
432C23C2×#13711yescharacteristic
433C23#57611yescharacteristic
434C23#57711yescharacteristic
435Q8—81no—
436Q8—81no—
437D4—82no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4—82no—
443D4—84no—
444D4—162no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—164no—
517C2×C4#41012yes—
518C2×C4#41712yes—
519C2×C4#41712yes—
520C2×C4C2×#10414yes—
521C2×C4C2×#10412yes—
522C2×C4C4×#4111yescharacteristic
523C2×C4#52011yescharacteristic
524C2×C4#54811yescharacteristic
525C2×C4#55011yescharacteristic
526C8—42no—
527C8—81no—
528C8—82no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22#234111yescharacteristic
549C22C2×#41011yescharacteristic
550C22#288311yescharacteristic
551C22C2×#41711yescharacteristic
552C22#289711yescharacteristic
553C22C22×#10411yescharacteristic
554C22#290111yescharacteristic
555C22#337311yescharacteristic
556C22#337511yescharacteristic
557C22#339911yescharacteristic
558C22#340011yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—24no—
565C4—44no—
566C4—42no—
567C4—41no—
568C4—41no—
569C4—42no—
570C4—42no—
571C4—82no—
572C4—84no—
573C4—81no—
574C4—82no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4#150914yes—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—21no—
583C2—162no—
584C2C2×#150911yescharacteristic
585C2#1754711yescharacteristic
586C2#1761011yescharacteristic
587C2#1763011yescharacteristic
588C2#2253511yescharacteristic
589C2#2254611yescharacteristic
590C2#2255011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
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. #22629C2semidirect
3. #13818C2semidirect
4. #9713C2semidirect
6. #9682C2semidirect
8. #8572C2semidirect
9. #8571C2semidirect
10. #8218C2semidirect
11. C4×#328C2semidirect
15. #2571C4semidirect
16. #2548C4semidirect
18. #2513C4semidirect
20. #2447C4semidirect
22. #2366C4semidirect
25. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#365872
#226292
#138182
#97132
#97122
#96822
#96802
#85722
#85712
#82182
C4×#3282