Finite Group Databasebeta

← #219916

Subgroups — #219916

Order 512.

1045 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36116C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic, complemented
3C2×#1521C212yesmaximal
4#13921C211yesmaximal, characteristic, complemented
5#9711C211yesmaximal, characteristic, complemented
6#9679C211yesmaximal, characteristic, complemented
7#9153C212yesmaximal, complemented
8#9112C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic
10#8330C212yesmaximal
11C4×#327C211yesmaximal, characteristic
12C2×#529C2212yes—
13C2×#431C2212yes—
14#2857C2211yescharacteristic
15#2855C2212yes—
16#2552C411yescharacteristic, complemented
17#2547C411yescharacteristic, complemented
18#2496—21no—
19#2486—21no—
20#2478—22no—
21#2453C412yescomplemented
22#2447C412yescomplemented
23#2419—22no—
24#2365C411yescharacteristic, complemented
25D4×C4⋊C4—21no—
26D4×C4⋊C4C411yescharacteristic, complemented
27#2344—21no—
28C4×#121C2211yescharacteristic
29C2×#383C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1939—21no—
32#1916—22no—
33#1817C2212yes—
34#1797C2211yescharacteristic
35C8×C4⋊C4—21no—
36C4×C22⋊C8C2211yescharacteristic
37#1521—28no—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#328C2212yes—
42C2×#327C2211yescharacteristic
43C2×#327C2212yes—
44C2×#327C2211yescharacteristic
45#1443—22no—
46C4⋊#95—21no—
47C4⋊#95—21no—
48C4⋊#95C2211yescharacteristic
49#1434C2212yes—
50C4×#95C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2211yescharacteristic
53#1294C2212yes—
54C2×C4⋊Q8C2×C411yescharacteristic
55#529—24no—
56#529—24no—
57#529—42no—
58#529—42no—
59C2×#126—22no—
60C2×#125C2×C412yes—
61C2×#124C2×C412yes—
62C2×#123—22no—
63C2×C22⋊Q8C2312yes—
64C2×#121C2311yescharacteristic
65C2×C4×Q8C2×C412yes—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C411yescharacteristic
70C2×#117—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#431—24no—
79#431—42no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8C2312yes—
83C2×C22⋊C8C2311yescharacteristic
84C2×C22⋊C8C2311yescharacteristic
85C2×C4×C8—22no—
86C2×C4×C8—21no—
87#400—22no—
88#398—22no—
89#398—22no—
90#397—22no—
91#397—22no—
92#397—22no—
93#391C2×C412yes—
94#391C2×C412yes—
95#391C2×C412yes—
96#390C2×C411yescharacteristic
97#389—21no—
98#389—22no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2×C411yescharacteristic
105#388—21no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#386—21no—
111#385—22no—
112#385—22no—
113#385—22no—
114#384—21no—
115#384C2×C411yescharacteristic
116#383—24no—
117#383—42no—
118#381—22no—
119#381—22no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380C2311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D412yes—
133C4×C4⋊C4D411yescharacteristic
134C4×C22⋊C4—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95D412yes—
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145#336—21no—
146#336—22no—
147#336—22no—
148#336—21no—
149#328—24no—
150#328—24no—
151#328—42no—
152#328—42no—
153#327—24no—
154#327—22no—
155#327D414yes—
156C22×Q8C22×C412yes—
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C4⋊Q8—42no—
173#126—44no—
174#126—44no—
175#125—24no—
176#125—42no—
177#124—24no—
178#124—42no—
179#123—44no—
180#123—44no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185#121—21no—
186#121—21no—
187#121C2×D412yes—
188#121#4112yes—
189C4×Q8—24no—
190C4×Q8—42no—
191C4×D4—24no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—42no—
197C4×D4—42no—
198#117—44no—
199#117—44no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C22×C412yes—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C412yes—
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C4⋊C8—44no—
256C4⋊C8—42no—
257C22⋊C8—21no—
258C22⋊C8—22no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—22no—
262C22⋊C8C2×D412yes—
263C4×C8—44no—
264C4×C8—42no—
265#95—21no—
266#95—22no—
267#95—21no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95—21no—
279#95—21no—
280#95—22no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95—44no—
285#95—82no—
286#95—81no—
287#95C22⋊C412yes—
288#95C22⋊C412yes—
289#95C22⋊C412yes—
290#95C22⋊C412yes—
291#95C2×D412yes—
292C24—21no—
293C24C22×D411yescharacteristic
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4C4×D412yes—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—82no—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4#13512yes—
367C22×C4#13512yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C2×C8—22no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—41no—
383C4⋊C4—21no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—82no—
405C4⋊C4—81no—
406C4⋊C4—84no—
407C4⋊C4C4×D412yes—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—42no—
419C22⋊C4—81no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—41no—
431C42—41no—
432C42—44no—
433C42—42no—
434C42—84no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23C22×C22⋊C411yescharacteristic
447C23D4211yescharacteristic
448C23#54611yescharacteristic
449C23C2×#13511yescharacteristic
450C23C2×#13511yescharacteristic
451C23#57711yescharacteristic
452C23#57811yescharacteristic
453Q8—82no—
454Q8—82no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
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—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—88no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—84no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4#41012yes—
538C2×C4#41612yes—
539C2×C4#41612yes—
540C2×C4#42112yes—
541C2×C4C2×C4×D411yescharacteristic
542C2×C4#51811yescharacteristic
543C2×C4D4211yescharacteristic
544C2×C4#54611yescharacteristic
545C8—42no—
546C8—82no—
547C8—81no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22D4×C22⋊C411yescharacteristic
568C22C2×#41011yescharacteristic
569C22#288411yescharacteristic
570C22C2×#41611yescharacteristic
571C22#289811yescharacteristic
572C22C2×#42111yescharacteristic
573C22#290111yescharacteristic
574C22#328311yescharacteristic
575C22#328411yescharacteristic
576C22#330211yescharacteristic
577C22#330311yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—22no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—81no—
594C4—82no—
595C4—81no—
596C4—162no—
597C4#152114yescomplemented
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—82no—
603C2C2×#152111yescharacteristic
604C2#1760111yescharacteristic
605C2#1761211yescharacteristic
606C2#1761611yescharacteristic
607C2#2240611yescharacteristic
608C2#2241711yescharacteristic
609C2#2243511yescharacteristic
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. #22599C2semidirect
4. #13921C2semidirect
5. #9711C2semidirect
6. #9679C2semidirect
7. #9153C2semidirect
8. #9112C2semidirect
16. #2552C4semidirect
17. #2547C4semidirect
21. #2453C4semidirect
22. #2447C4semidirect
24. #2365C4semidirect
26. D4×C4⋊C4C4semidirect
597. C4#1521semidirect

Maximal subgroups

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

subgroupindex
#361162
#225992
C2×#15212
#139212
#97112
#96792
#91532
#91122
#85712
#83302
C4×#3272