Finite Group Databasebeta

← #219903

Subgroups — #219903

Order 512.

1003 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36133C211yesmaximal, characteristic
2#22603C211yesmaximal, characteristic
3C2×#1515C212yesmaximal
4#13921C211yesmaximal, characteristic
5#9711C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic
7#9153C212yesmaximal
8#9116C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8333C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12C2×#533C2212yes—
13C2×#432C2212yes—
14#2857C2211yescharacteristic
15#2855C2212yes—
16#2551C411yescharacteristic, complemented
17#2546C411yescharacteristic, complemented
18#2506—21no—
19#2462—22no—
20#2451C412yescomplemented
21#2446C412yescomplemented
22#2440—21no—
23#2438—22no—
24#2365—21no—
25#2365C411yescharacteristic, complemented
26Q8×C4⋊C4C411yescharacteristic, complemented
27#2344—21no—
28C4×C22⋊Q8C2211yescharacteristic
29C2×#383C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1939—21no—
32#1917—22no—
33#1817C2212yes—
34#1797C2211yescharacteristic
35#1770—21no—
36C4×C22⋊C8C2211yescharacteristic
37#1515—28no—
38C2×#328C2211yescharacteristic
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#328C2212yes—
42C2×#328C2212yes—
43C2×#328C2212yes—
44C2×#328C2211yescharacteristic
45#1443—22no—
46C4⋊#95—21no—
47C4⋊#95—21no—
48C4⋊#95C2211yescharacteristic
49#1434C2212yes—
50C4×#95C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2211yescharacteristic
53#1294C2212yes—
54#533—24no—
55#533—24no—
56#533—42no—
57#533—42no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×#125—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2312yes—
65C2×C22⋊Q8C2311yescharacteristic
66C2×C4×Q8—22no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C411yescharacteristic
70C2×#117—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#432—24no—
78#432—42no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C22⋊C8C2312yes—
82C2×C22⋊C8C2311yescharacteristic
83C2×C22⋊C8C2311yescharacteristic
84C2×#97—22no—
85C2×C4×C8—21no—
86#400—22no—
87#398—22no—
88#398—22no—
89#398—22no—
90#395—22no—
91#395—22no—
92#391—22no—
93#391C2×C412yes—
94#391C2×C412yes—
95#391C2×C412yes—
96#391C2×C411yescharacteristic
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—22no—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C411yescharacteristic
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—21no—
112#387—22no—
113#387—22no—
114#385—22no—
115#385—21no—
116#384—21no—
117#384C2×C411yescharacteristic
118#383—24no—
119#383—42no—
120#381—22no—
121#380—21no—
122#380—21no—
123#380C2311yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
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—22no—
151#328—24no—
152#328—24no—
153#328—42no—
154#328—42no—
155#328D414yes—
156C22×Q8—22no—
157C22×Q8C22×C412yes—
158C22×Q8C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—44no—
176#125—44no—
177#125—44no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C22⋊Q8C2×D412yes—
187C22⋊Q8#4112yes—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—44no—
193C4×Q8—44no—
194C4×Q8—42no—
195#117—44no—
196#117—44no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
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—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
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⋊C412yes—
222C2×C4⋊C4C22⋊C411yescharacteristic
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×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42C22×C412yes—
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C412yes—
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C4⋊C8—44no—
252C4⋊C8—42no—
253C22⋊C8—21no—
254C22⋊C8—22no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—22no—
258C22⋊C8C2×D412yes—
259#97—44no—
260C4×C8—42no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—21no—
266#95—21no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—44no—
283#95—82no—
284#95—81no—
285#95C22⋊C412yes—
286#95C22⋊C412yes—
287#95C22⋊C412yes—
288#95C22⋊C412yes—
289#95C2×D412yes—
290C24C22×D411yescharacteristic
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—82no—
301C2×Q8C4×D412yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4#13712yes—
356C22×C4#13712yes—
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C2×C8—22no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—41no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—81no—
398C4⋊C4—82no—
399C4⋊C4—81no—
400C4⋊C4—84no—
401C4⋊C4C4×D412yes—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—41no—
419C42—41no—
420C42—44no—
421C42—44no—
422C42—42no—
423C42—42no—
424C42—84no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23C22×C22⋊C411yescharacteristic
430C23D4211yescharacteristic
431C23#54611yescharacteristic
432C23C2×#13711yescharacteristic
433C23C2×#13711yescharacteristic
434C23#57711yescharacteristic
435C23#57811yescharacteristic
436Q8—42no—
437Q8—42no—
438Q8—82no—
439Q8—82no—
440Q8—88no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4#41012yes—
519C2×C4#41912yes—
520C2×C4#41912yes—
521C2×C4#42112yes—
522C2×C4C2×C4×D411yescharacteristic
523C2×C4#51811yescharacteristic
524C2×C4D4211yescharacteristic
525C2×C4#54611yescharacteristic
526C8—42no—
527C8—82no—
528C8—81no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22D4×C22⋊C411yescharacteristic
542C22C2×#41011yescharacteristic
543C22#288411yescharacteristic
544C22C2×#41911yescharacteristic
545C22#289811yescharacteristic
546C22C2×#42111yescharacteristic
547C22#290111yescharacteristic
548C22#327611yescharacteristic
549C22#327811yescharacteristic
550C22#332811yescharacteristic
551C22#332911yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—22no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—41no—
564C4—41no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—81no—
570C4—81no—
571C4—162no—
572C4#151514yescomplemented
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2C2×#151511yescharacteristic
578C2#1760111yescharacteristic
579C2#1761211yescharacteristic
580C2#1763211yescharacteristic
581C2#2240611yescharacteristic
582C2#2243011yescharacteristic
583C2#2243511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
16. #2551C4semidirect
17. #2546C4semidirect
20. #2451C4semidirect
21. #2446C4semidirect
25. #2365C4semidirect
26. Q8×C4⋊C4C4semidirect
572. C4#1515semidirect

Maximal subgroups

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

subgroupindex
#361332
#226032
C2×#15152
#139212
#97112
#96792
#91532
#91162
#85722
#83332
C4×#3282