Finite Group Databasebeta

← #219923

Subgroups — #219923

Order 512.

1055 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36133C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic
3C2×#1518C212yesmaximal
4#13918C211yesmaximal, characteristic
5#9711C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic
7#9152C212yesmaximal
8#9112C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8333C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12C2×#533C2212yes—
13C2×#431C2212yes—
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×#389—21no—
30C2×#382C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1939—21no—
33#1916—22no—
34#1817C2212yes—
35#1797C2211yescharacteristic
36C8×C4⋊C4—21no—
37C4×C22⋊C8C2211yescharacteristic
38#1518—28no—
39C2×#328C2211yescharacteristic
40C2×#328C2212yes—
41C2×#328C2212yes—
42C2×#328C2212yes—
43C2×#328C2212yes—
44C2×#328C2212yes—
45C2×#328C2211yescharacteristic
46#1442—22no—
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⋊C4—22no—
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#398—22no—
91#395—22no—
92#395—22no—
93#391—22no—
94#391C2×C412yes—
95#391C2×C412yes—
96#391C2×C412yes—
97#391C2×C411yescharacteristic
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—22no—
105#389—44no—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C411yescharacteristic
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—21no—
114#387—22no—
115#387—22no—
116#385—22no—
117#385—21no—
118#384—21no—
119#384C2×C411yescharacteristic
120#382—24no—
121#382—42no—
122#381—22no—
123#380—21no—
124#380—21no—
125#380C2311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D412yes—
135C4×C4⋊C4D411yescharacteristic
136C4×C22⋊C4—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—21no—
142C2×#95D412yes—
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146#336—21no—
147#336—22no—
148#336—22no—
149#336—21no—
150#328—24no—
151#328—22no—
152#328—24no—
153#328—24no—
154#328—42no—
155#328—42no—
156#328D414yes—
157C22×Q8—22no—
158C22×Q8C22×C412yes—
159C22×Q8C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—44no—
177#125—44no—
178#125—44no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8C2×D412yes—
188C22⋊Q8#4112yes—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8—44no—
194C4×Q8—44no—
195C4×Q8—42no—
196#117—44no—
197#117—44no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
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—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4SD1614yes—
227C2×C4⋊C4Q1614yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
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—21no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—21no—
278#95—21no—
279#95—22no—
280#95—21no—
281#95—21no—
282#95—22no—
283#95—44no—
284#95—42no—
285#95—44no—
286#95—44no—
287#95—81no—
288#95C22⋊C412yes—
289#95C22⋊C412yes—
290#95C22⋊C412yes—
291#95C22⋊C412yes—
292#95C2×D412yes—
293C24C22×D411yescharacteristic
294C2×Q8—21no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—82no—
304C2×Q8C4×D412yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
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—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×SD1614yes—
357C22×C4C2×SD1612yes—
358C22×C4C2×Q1614yes—
359C22×C4C2×Q1612yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—41no—
375C4⋊C4—21no—
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—88no—
401C4⋊C4—84no—
402C4⋊C4#10318yes—
403C4⋊C4#10314yes—
404C4⋊C4#10314yes—
405C4⋊C4C4×D412yes—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—41no—
423C42—41no—
424C42—44no—
425C42—44no—
426C42—42no—
427C42—42no—
428C42—84no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23C22×C22⋊C411yescharacteristic
434C23D4211yescharacteristic
435C23#54611yescharacteristic
436C23C22×SD1611yescharacteristic
437C23C22×Q1611yescharacteristic
438C23#57711yescharacteristic
439C23#57811yescharacteristic
440Q8—42no—
441Q8—42no—
442Q8—82no—
443Q8—82no—
444Q8—88no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—82no—
502C2×C4—82no—
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—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#41012yes—
521C2×C4C2×#10314yes—
522C2×C4C2×#10314yes—
523C2×C4C2×#10312yes—
524C2×C4C2×#10312yes—
525C2×C4#42112yes—
526C2×C4C2×C4×D411yescharacteristic
527C2×C4#51811yescharacteristic
528C2×C4D4211yescharacteristic
529C2×C4#54611yescharacteristic
530C8—42no—
531C8—82no—
532C8—81no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22D4×C22⋊C411yescharacteristic
546C22C2×#41011yescharacteristic
547C22#288411yescharacteristic
548C22C22×#10311yescharacteristic
549C22#289811yescharacteristic
550C22C2×#42111yescharacteristic
551C22#290111yescharacteristic
552C22D4×SD1611yescharacteristic
553C22D4×Q1611yescharacteristic
554C22#329811yescharacteristic
555C22#329911yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—24no—
561C4—24no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—41no—
568C4—41no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—81no—
574C4—81no—
575C4—162no—
576C4#151814yescomplemented
577C2—21no—
578C2—21no—
579C2—21no—
580C2—21no—
581C2C2×#151811yescharacteristic
582C2#1760111yescharacteristic
583C2#1760511yescharacteristic
584C2#1761911yescharacteristic
585C2#2240611yescharacteristic
586C2D4×#10311yescharacteristic
587C2#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
576. C4#1518semidirect

Maximal subgroups

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

subgroupindex
#361332
#225992
C2×#15182
#139182
#97112
#96792
#91522
#91122
#85722
#83332
C4×#3282