Finite Group Databasebeta

← #204729

Subgroups — #204729

Order 512.

983 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35139C211yesmaximal, characteristic
2#21356C211yesmaximal, characteristic, complemented
3#20467C211yesmaximal, characteristic, complemented
4C2×#1535C211yesmaximal, characteristic
5C2×#1534C211yesmaximal, characteristic, complemented
6C2×#1529C211yesmaximal, characteristic
7C2×#1528C211yesmaximal, characteristic, complemented
8#9690C212yesmaximal, complemented
9#9688C212yesmaximal, complemented
10#9687C212yesmaximal, complemented
11#9686C212yesmaximal, complemented
12C2×#529C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#432C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2861C2211yescharacteristic
19#2861C2211yescharacteristic
20#2859C2211yescharacteristic
21#2859C2211yescharacteristic
22#2664—22no—
23#2611—22no—
24#2566C411yescharacteristic
25#2566C411yescharacteristic
26#2565C411yescharacteristic
27#2565C411yescharacteristic
28#2504C411yescharacteristic
29#2503C411yescharacteristic
30#2502C411yescharacteristic
31#2501C411yescharacteristic
32#2413—22no—
33#2322—22no—
34C4×#117C2211yescharacteristic
35#1939—22no—
36#1851—22no—
37#1768—22no—
38#1767C2212yes—
39#1767C2212yes—
40#1750—22no—
41#1745C2212yes—
42C4×#97C2212yes—
43#1535—24no—
44#1534—24no—
45#1529—24no—
46#1528—24no—
47C2×#330C2212yes—
48C2×#330C2212yes—
49C2×#330C2212yes—
50C2×#330C2212yes—
51C2×#330C2212yes—
52C2×#330C2212yes—
53C2×#329C2212yes—
54C2×#329C2212yes—
55C22×M16—21no—
56#529—22no—
57#529—22no—
58#529—22no—
59#529—22no—
60C2×#126—21no—
61C2×#125—21no—
62C2×#125—21no—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125—22no—
66C2×#125—22no—
67C2×#125C2×C411yescharacteristic
68C2×#125C2×C411yescharacteristic
69C2×#125C2311yescharacteristic
70C2×#125C2311yescharacteristic
71C2×#125C2311yescharacteristic
72C2×#124C2311yescharacteristic
73C2×#123—21no—
74C2×#123—22no—
75C2×C22⋊Q8C2×C411yescharacteristic
76C2×#121C2×C411yescharacteristic
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4C2×C411yescharacteristic
80C2×#117—21no—
81C2×#117D412yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C42C2311yescharacteristic
87#432—22no—
88#432—22no—
89#432—22no—
90#432—22no—
91#432D414yes—
92#432D414yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C4×M16—22no—
102C4×M16—22no—
103C2×#97—22no—
104C2×#97C2312yes—
105C2×#97C2312yes—
106C2×#97C2312yes—
107C2×#97C2312yes—
108C2×C4×C8—21no—
109C2×C4×C8—22no—
110#400—22no—
111#399—22no—
112#398—22no—
113#398—22no—
114#398—22no—
115#397—22no—
116#396—22no—
117#396—22no—
118#391C2×C411yescharacteristic
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#390C2×C411yescharacteristic
124#390C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#388—22no—
131#388—22no—
132#388—22no—
133#388—22no—
134#387—22no—
135#387—22no—
136#386—22no—
137#385—22no—
138#385—22no—
139#385—22no—
140#384C2×C411yescharacteristic
141#384C2×C411yescharacteristic
142#384C2×C411yescharacteristic
143#384C2×C411yescharacteristic
144#382C2×C411yescharacteristic
145#382C2×C411yescharacteristic
146#382C2×C411yescharacteristic
147#382C2×C411yescharacteristic
148#381—22no—
149C4×C4⋊C4C2×C411yescharacteristic
150C4×C4⋊C4C2×C411yescharacteristic
151C4×C4⋊C4D412yes—
152C4×C22⋊C4C2×C411yescharacteristic
153C4×C22⋊C4C2×C411yescharacteristic
154C4×C22⋊C4D412yes—
155#376D412yes—
156C43C2311yescharacteristic
157C43C2311yescharacteristic
158#336—22no—
159#336—22no—
160#336—24no—
161#330—24no—
162#330—24no—
163#330—24no—
164#330—24no—
165#330—24no—
166#330—24no—
167#329—24no—
168#329—24no—
169C22×Q8C22×C411yescharacteristic
170C22×D4—21no—
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C2×M16—44no—
179C2×M16—42no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184#126—42no—
185#126—42no—
186#125—22no—
187#125—22no—
188#125—22no—
189#125—22no—
190#125—21no—
191#125—21no—
192#125—21no—
193#125—21no—
194#125—21no—
195#125—21no—
196#125—21no—
197#125—21no—
198#125—21no—
199#125—21no—
200#125—21no—
201#125—21no—
202#125—42no—
203#125—42no—
204#125—42no—
205#125—42no—
206#124—21no—
207#124—21no—
208#124—21no—
209#124—21no—
210#123—48no—
211#123—42no—
212#123—42no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215#121—22no—
216#121—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—42no—
222C4×D4—42no—
223#117—44no—
224#117—42no—
225#117—42no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4C22⋊C412yes—
253C2×C4⋊C4C22⋊C412yes—
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4C22⋊C412yes—
270C2×C22⋊C4C22⋊C412yes—
271C2×C42—21no—
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4112yes—
286C2×C42#4112yes—
287C2×C42C2411yescharacteristic
288C4⋊C8—24no—
289C4⋊C8—24no—
290C4⋊C8—44no—
291C4⋊C8—44no—
292C4⋊C8—42no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295C22⋊C8—42no—
296#97—22no—
297#97—22no—
298#97—22no—
299#97—22no—
300#97—22no—
301#97—22no—
302#97—44no—
303#97C2×D414yes—
304#97C2×D414yes—
305C4×C8—44no—
306C4×C8—42no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95C22⋊C412yes—
316#95C22⋊C412yes—
317#95C22⋊C412yes—
318#95C22⋊C412yes—
319C24—21no—
320C24C2×C22⋊C411yescharacteristic
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—44no—
330C2×D4—42no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C412yes—
355C22×C4C4×D412yes—
356C22×C4C4×D412yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4#12412yes—
360C22×C4#12712yes—
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364M16—48no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—24no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C22⋊C4—24no—
416C22⋊C4—42no—
417C22⋊C4—44no—
418C22⋊C4—42no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C42—24no—
429C42—42no—
430C42—42no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—42no—
436C42—42no—
437C42C4×D412yes—
438C42C4×D412yes—
439C42#12112yes—
440C42#12112yes—
441C42#12412yes—
442C42#12712yes—
443C42C22×D411yescharacteristic
444C42C22×D411yescharacteristic
445C42C2×#4111yescharacteristic
446C42C2×#4111yescharacteristic
447C23—21no—
448C23—21no—
449C23—22no—
450C23—22no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23#39012yes—
457C23C22×C22⋊C411yescharacteristic
458Q8—81no—
459Q8—81no—
460D4—81no—
461D4—81no—
462D4—84no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4#39012yes—
507C2×C4#39012yes—
508C2×C4#39012yes—
509C2×C4#42114yes—
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12111yescharacteristic
514C2×C4C2×#12411yescharacteristic
515C2×C4C2×#12711yescharacteristic
516C8—44no—
517C8—44no—
518C8—44no—
519C8—82no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—42no—
524C22—42no—
525C22—81no—
526C22—82no—
527C22—82no—
528C22—81no—
529C22—81no—
530C22C2×#39011yescharacteristic
531C22#288911yescharacteristic
532C22#288911yescharacteristic
533C22C2×#42112yes—
534C22#290112yes—
535C4—21no—
536C4—22no—
537C4—21no—
538C4—24no—
539C4—21no—
540C4—21no—
541C4—22no—
542C4—21no—
543C4—21no—
544C4—22no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—81no—
555C2—42no—
556C2—161no—
557C2#1768012yes—
558C2#1768212yes—
559C2#2120911yescharacteristic
560C2#2122011yescharacteristic
561C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21356C2semidirect
3. #20467C2semidirect
5. C2×#1534C2semidirect
7. C2×#1528C2semidirect
8. #9690C2semidirect
9. #9688C2semidirect
10. #9687C2semidirect
11. #9686C2semidirect

Maximal subgroups

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

subgroupindex
#351392
#213562
#204672
C2×#15352
C2×#15342
C2×#15292
C2×#15282
#96902
#96882
#96872
#96862