Finite Group Databasebeta

← #224831

Subgroups — #224831

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36596C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic, complemented
3#13936C211yesmaximal, characteristic, complemented
4#9715C212yesmaximal, complemented
5#9712C211yesmaximal, characteristic, complemented
6#9684C212yesmaximal, complemented
7#9680C211yesmaximal, characteristic, complemented
8#8573C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic, complemented
10#8216C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic, complemented
12C2×#553—21no—
13#2856C2211yescharacteristic
14#2856C2212yes—
15#2554—22no—
16#2552C411yescharacteristic, complemented
17#2543—22no—
18#2541C411yescharacteristic, complemented
19#2512C412yescomplemented
20#2490—21no—
21#2445C412yescomplemented
22#2416—21no—
23#2366C411yescharacteristic, complemented
24#2363—21no—
25#2354C411yescharacteristic, complemented
26C4×#121C2211yescharacteristic
27C4×#121C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—21no—
30#1917—22no—
31#1797C2211yescharacteristic
32#1797C2212yes—
33#1770—21no—
34#1749C2212yes—
35C4×C22⋊C8C2211yescharacteristic
36C2×#327C2211yescharacteristic
37C2×#327C2211yescharacteristic
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2212yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43#1443—22no—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2211yescharacteristic
47#1434C2212yes—
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53#553—44no—
54#553—44no—
55C2×#127C2×C411yescharacteristic
56C2×#127C2×C412yes—
57C2×#126—22no—
58C2×#125—21no—
59C2×#123—22no—
60C2×#121—21no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121C2312yes—
67C2×#121C2311yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C412yes—
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2312yes—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8C2311yescharacteristic
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2311yescharacteristic
85C2×#97—22no—
86C2×C4×C8—21no—
87#398—22no—
88#397—22no—
89#396—22no—
90#394—22no—
91#394—22no—
92#392—22no—
93#390—21no—
94#390—21no—
95#390—22no—
96#390—22no—
97#390—21no—
98#390C2×C412yes—
99#390C2×C412yes—
100#390C2×C412yes—
101#390C2×C411yescharacteristic
102#390C2×C412yes—
103#389—21no—
104#389C2×C412yes—
105#389C2×C411yescharacteristic
106#388—22no—
107#388—21no—
108#388—22no—
109#388—22no—
110#386—22no—
111#386—21no—
112#386—22no—
113#386—22no—
114#384C2×C411yescharacteristic
115#384C2×C412yes—
116#380—21no—
117#380C2311yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95D412yes—
135C2×#95C2311yescharacteristic
136C2×#95C2312yes—
137C2×#95C2311yescharacteristic
138#336—21no—
139#336—22no—
140#336—21no—
141#336—22no—
142#327—22no—
143#327—24no—
144#327—22no—
145#327—24no—
146#327—42no—
147#327—42no—
148#327—42no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4C22×C411yescharacteristic
156C22×D4C22×C412yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170#127—42no—
171#127—82no—
172#126—48no—
173#125—44no—
174#123—48no—
175#121—22no—
176#121—21no—
177#121—22no—
178#121—21no—
179#121—22no—
180#121—21no—
181#121—22no—
182#121—21no—
183#121—44no—
184#121—44no—
185#121—44no—
186#121—44no—
187#121—42no—
188#121—42no—
189#121—82no—
190#121—81no—
191#121—82no—
192#121—81no—
193C4×D4—42no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—82no—
200#117—44no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—82no—
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C4⋊C8—42no—
254C4⋊C8—82no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—22no—
258C22⋊C8—21no—
259C22⋊C8—22no—
260C22⋊C8#4112yes—
261#97—82no—
262C4×C8—42no—
263#95—22no—
264#95—21no—
265#95—22no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—22no—
271#95—21no—
272#95—22no—
273#95—42no—
274#95—44no—
275#95—82no—
276#95—81no—
277#95—82no—
278#95—82no—
279#95C22⋊C412yes—
280#95C22⋊C412yes—
281#95C22⋊C412yes—
282#95C22⋊C412yes—
283#95#4112yes—
284C24—21no—
285C24—22no—
286C24C22×D411yescharacteristic
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—41no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—82no—
314C2×D4—81no—
315C2×D4—82no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—81no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C42−1+411yescharacteristic
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—81no—
408C4⋊C4—82no—
409C4⋊C4—84no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—42no—
419C22⋊C4—41no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—81no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C42—22no—
429C42—22no—
430C42—42no—
431C42—42no—
432C42—41no—
433C42—41no—
434C42—42no—
435C42—42no—
436C42—44no—
437C42—42no—
438C42—84no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—41no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23C22×C22⋊C411yescharacteristic
458C23#54811yescharacteristic
459C23#55011yescharacteristic
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23C2×AGL(1,ℤ/8)11yescharacteristic
462C23#57611yescharacteristic
463C23#57711yescharacteristic
464D4—81no—
465D4—82no—
466D4—81no—
467D4—82no—
468D4—82no—
469D4—81no—
470D4—82no—
471D4—81no—
472D4—82no—
473D4—81no—
474D4—82no—
475D4—81no—
476D4—84no—
477D4—164no—
478D4—162no—
479D4—164no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—84no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—164no—
551C2×C4#41012yes—
552C2×C4#41812yes—
553C2×C4#41812yes—
554C2×C4#42112yes—
555C2×C4C4×#4111yescharacteristic
556C2×C4#52011yescharacteristic
557C2×C4#54811yescharacteristic
558C2×C4#55011yescharacteristic
559C8—42no—
560C8—81no—
561C8—82no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—81no—
575C22—82no—
576C22—81no—
577C22—82no—
578C22—82no—
579C22—81no—
580C22—81no—
581C22—82no—
582C22—81no—
583C22—82no—
584C22—82no—
585C22—81no—
586C22—82no—
587C22—81no—
588C22#234111yescharacteristic
589C22C2×#41011yescharacteristic
590C22#288311yescharacteristic
591C22C2×#41811yescharacteristic
592C22#289711yescharacteristic
593C22C2×#42111yescharacteristic
594C22#290111yescharacteristic
595C22#336611yescharacteristic
596C22#336811yescharacteristic
597C22#339711yescharacteristic
598C22#339911yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—42no—
609C4—42no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—162no—
617C4#151414yes—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—161no—
623C2—162no—
624C2C2×#151411yescharacteristic
625C2#1759911yescharacteristic
626C2#1761011yescharacteristic
627C2#1762611yescharacteristic
628C2#2253511yescharacteristic
629C2#2254511yescharacteristic
630C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36596C2
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. #13936C2semidirect
4. #9715C2semidirect
5. #9712C2semidirect
6. #9684C2semidirect
7. #9680C2semidirect
8. #8573C2semidirect
9. #8571C2semidirect
10. #8216C2semidirect
11. C4×#327C2semidirect
16. #2552C4semidirect
18. #2541C4semidirect
19. #2512C4semidirect
21. #2445C4semidirect
23. #2366C4semidirect
25. #2354C4semidirect

Maximal subgroups

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

subgroupindex
#365962
#226292
#139362
#97152
#97122
#96842
#96802
#85732
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C4×#3272