Finite Group Databasebeta

← #196012

Subgroups — #196012

Order 512.

1421 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic, complemented
2C2×#2636C211yesmaximal, characteristic
3#21786C211yesmaximal, characteristic, complemented
4C2×#1473C212yesmaximal, complemented
5#9085C212yesmaximal, complemented
6#9083C212yesmaximal, complemented
7#9081C212yesmaximal, complemented
8#9079C212yesmaximal, complemented
9#7914C212yesmaximal, complemented
10C22×#126C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#2973—28no—
13C2×#435—24no—
14C2×#432C2212yes—
15C2×#431C2212yes—
16C2×#407C2212yes—
17C22×C22⋊C8C2212yes—
18#2636—24no—
19#2636—24no—
20C2×#400C411yescharacteristic
21C2×#399C411yescharacteristic
22C2×#396C411yescharacteristic
23C2×#396C411yescharacteristic
24C2×#394C411yescharacteristic
25C2×#394C411yescharacteristic
26C2×#394C411yescharacteristic
27C2×#392C411yescharacteristic
28C2×#386C2211yescharacteristic
29C2×#386C2211yescharacteristic
30C2×#385C2211yescharacteristic
31C2×#385C2211yescharacteristic
32C2×#382C2211yescharacteristic
33#1954—24no—
34#1779C2212yes—
35#1779C2212yes—
36#1473—28no—
37C2×#324C2214yes—
38C2×#323C2214yes—
39#1280C2214yes—
40#1280C2214yes—
41C23×D4C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C23×D4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C23×C8—22no—
48C2×#126—22no—
49C2×#126—22no—
50C2×#126—24no—
51C2×#126—24no—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4D414yes—
55C2×C4×D4D414yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#435—232no—
79#432—24no—
80#432D418yescomplemented
81#431—24no—
82#431—24no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85#407—28no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—28no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2314yes—
94C2×C22⋊C8C2314yes—
95C2×#97—22no—
96C2×C4×C8—22no—
97#400—24no—
98#399—24no—
99#396—24no—
100#396—24no—
101#394—24no—
102#394—24no—
103#394—24no—
104#392—24no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#382—22no—
114#382—22no—
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2311yescharacteristic
122C2×#95C2311yescharacteristic
123#324—28no—
124#323—28no—
125C25C22×C411yescharacteristic
126C25C22×C411yescharacteristic
127C25C22×C411yescharacteristic
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—81no—
136C22×D4—81no—
137C22×D4—81no—
138C22×D4—81no—
139C22×D4C22⋊C414yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—81no—
145C23×C4—81no—
146C23×C4—81no—
147C23×C4—81no—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2411yescharacteristic
164C2×M16—28no—
165C2×M16—82no—
166C2×M16—82no—
167C22×C8—24no—
168C22×C8—28no—
169C22×C8—24no—
170C22×C8—82no—
171C22×C8—82no—
172#126—44no—
173#126—44no—
174#126—44no—
175#126—44no—
176C4×D4—28no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4C22⋊C414yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—81no—
219C2×C22⋊C4—81no—
220C2×C22⋊C4—81no—
221C2×C22⋊C4—81no—
222C2×C22⋊C4—81no—
223C2×C22⋊C4—81no—
224C2×C22⋊C4C22⋊C414yes—
225C2×C22⋊C4C22⋊C414yes—
226C2×C42C2×D414yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42#4114yes—
230C4⋊C8—28no—
231C4⋊C8—44no—
232C22⋊C8—28no—
233C22⋊C8—28no—
234C22⋊C8—24no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—82no—
238C22⋊C8—82no—
239C22⋊C8—82no—
240C22⋊C8—82no—
241C22⋊C8C2×D418yes—
242#97—28no—
243C4×C8—44no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—21no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24C2×C22⋊C412yes—
276C24C2×C22⋊C412yes—
277C24#13114yes—
278C24C23×C411yescharacteristic
279C2×D4—28no—
280C2×D4—28no—
281C2×D4—44no—
282C2×D4—416no—
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—88no—
290C2×D4—88no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
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—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C4×D414yes—
354C22×C4C4×D414yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4C2×#4112yes—
359M16—88no—
360C2×C8—28no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—88no—
367C4⋊C4—28no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C22⋊C4—28no—
378C22⋊C4—28no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—41no—
382C22⋊C4—42no—
383C22⋊C4—41no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C42#12014yes—
393C42#12114yes—
394C42#12114yes—
395C42#12314yes—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—28no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—44no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23#40814yes—
442C23C22×C22⋊C411yescharacteristic
443C23C2×C4×D412yes—
444C23C2×#13112yes—
445C23#56812yes—
446D4—416no—
447D4—88no—
448D4—88no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—28no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
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—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4#38614yes—
504C2×C4#38614yes—
505C2×C4C2×#12011yescharacteristic
506C2×C4C2×#12112yes—
507C2×C4C2×#12311yescharacteristic
508C8—48no—
509C8—84no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—88no—
531C22—88no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—164no—
540C22C2×#38611yescharacteristic
541C22C2×#40811yescharacteristic
542C22#287911yescharacteristic
543C22#288311yescharacteristic
544C22#288311yescharacteristic
545C22#288811yescharacteristic
546C22#288911yescharacteristic
547C22#293314yes—
548C4—24no—
549C4—24no—
550C4—21no—
551C4—22no—
552C4—21no—
553C4—42no—
554C4—42no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C2—24no—
563C2—44no—
564C2—82no—
565C2—161no—
566C2#1738611yescharacteristic
567C2#1738711yescharacteristic
568C2#1740511yescharacteristic
569C2#1740611yescharacteristic
570C2#2089711yescharacteristic
571C2#2098511yescharacteristic
572C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#392C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2973C2semidirect
3. #21786C2semidirect
4. C2×#1473C2semidirect
5. #9085C2semidirect
6. #9083C2semidirect
7. #9081C2semidirect
8. #9079C2semidirect
9. #7914C2semidirect
80. #432D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
C2×#26362
#217862
C2×#14732
#90852
#90832
#90812
#90792
#79142