Finite Group Databasebeta

← #196002

Subgroups — #196002

Order 512.

1439 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic, complemented
2C2×#2630C211yesmaximal, characteristic
3#21738C211yesmaximal, characteristic, complemented
4C2×#1468C212yesmaximal, complemented
5#9086C212yesmaximal, complemented
6#9084C212yesmaximal, complemented
7#9081C212yesmaximal, complemented
8#9079C212yesmaximal, complemented
9#7915C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#2973—28no—
13C2×#435—24no—
14C2×#432C2212yes—
15C2×#407C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18#2630—24no—
19#2630—24no—
20C2×#398C411yescharacteristic
21C2×#397C411yescharacteristic
22C2×#396C411yescharacteristic
23C2×#396C411yescharacteristic
24C2×#394C411yescharacteristic
25C2×#394C411yescharacteristic
26C2×#394C411yescharacteristic
27C2×#392C411yescharacteristic
28C2×#388C2211yescharacteristic
29C2×#387C2211yescharacteristic
30C2×#386C2211yescharacteristic
31C2×#385C2211yescharacteristic
32C2×#380C2211yescharacteristic
33#1869—24no—
34#1817C2212yes—
35#1779C2212yes—
36#1468—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—
48C23×C8—22no—
49C2×#123—22no—
50C2×#123—22no—
51C2×#123—24no—
52C2×#123—24no—
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4D414yes—
56C2×C4×D4D414yes—
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⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
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⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#435—232no—
80#432—28no—
81C2×C4⋊C8—22no—
82#407—24no—
83#407D418yescomplemented
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—28no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2314yes—
93C2×C22⋊C8C2314yes—
94C2×#97—22no—
95#398—24no—
96#397—24no—
97#396—24no—
98#396—24no—
99#394—24no—
100#394—24no—
101#394—24no—
102#392—24no—
103#388—22no—
104#388—22no—
105#387—22no—
106#387—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#385—22no—
111#380—22no—
112#380—22no—
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119C2×#95C2311yescharacteristic
120C2×#95C2311yescharacteristic
121#336—24no—
122#324—28no—
123#323—28no—
124C25C22×C411yescharacteristic
125C25C22×C411yescharacteristic
126C25C22×C411yescharacteristic
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—42no—
134C22×D4—81no—
135C22×D4—81no—
136C22×D4—81no—
137C22×D4—81no—
138C22×D4C22⋊C414yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—81no—
142C23×C4—81no—
143C23×C4—81no—
144C23×C4—81no—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
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×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D414yes—
161C23×C4#4114yes—
162C23×C4C2411yescharacteristic
163C2×M16—28no—
164C2×M16—82no—
165C2×M16—82no—
166C22×C8—24no—
167C22×C8—28no—
168C22×C8—24no—
169C22×C8—44no—
170C22×C8—82no—
171C22×C8—82no—
172C22×C8—82no—
173C22×C8—82no—
174#123—44no—
175#123—44no—
176#123—44no—
177#123—44no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—28no—
181C4×D4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4C22⋊C414yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—81no—
221C2×C22⋊C4—81no—
222C2×C22⋊C4—81no—
223C2×C22⋊C4—81no—
224C2×C22⋊C4—81no—
225C2×C22⋊C4—81no—
226C2×C22⋊C4C22⋊C414yes—
227C2×C22⋊C4C22⋊C414yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C4⋊C8—28no—
233C22⋊C8—28no—
234C22⋊C8—28no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—24no—
238C22⋊C8—44no—
239C22⋊C8—82no—
240C22⋊C8—82no—
241C22⋊C8C2×D418yes—
242#97—28no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24C2×C22⋊C412yes—
274C24C2×C22⋊C412yes—
275C24C4×D414yes—
276C24#13114yes—
277C24C23×C411yescharacteristic
278C2×D4—28no—
279C2×D4—28no—
280C2×D4—44no—
281C2×D4—416no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—88no—
289C2×D4—88no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
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—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
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—41no—
341C22×C4—41no—
342C22×C4—84no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C4×D414yes—
351C22×C4#12014yes—
352C22×C4#12114yes—
353C22×C4#12114yes—
354C22×C4#12314yes—
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—
367C2×C8—88no—
368C4⋊C4—28no—
369C4⋊C4—44no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—44no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—41no—
384C22⋊C4—42no—
385C22⋊C4—41no—
386C22⋊C4—44no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C42—22no—
394C42—24no—
395C42—22no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—28no—
413C23—28no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—44no—
426C23—82no—
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#38614yes—
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—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
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—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4#38614yes—
506C2×C4C2×#12011yescharacteristic
507C2×C4C2×#12112yes—
508C2×C4C2×#12311yescharacteristic
509C8—48no—
510C8—84no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—24no—
518C22—24no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—88no—
532C22—88no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—164no—
541C22C2×#38611yescharacteristic
542C22C2×#40811yescharacteristic
543C22#287911yescharacteristic
544C22#288311yescharacteristic
545C22#288311yescharacteristic
546C22#288811yescharacteristic
547C22#288911yescharacteristic
548C22#293314yes—
549C4—21no—
550C4—22no—
551C4—21no—
552C4—24no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C2—24no—
564C2—44no—
565C2—44no—
566C2—161no—
567C2#1738611yescharacteristic
568C2#1738711yescharacteristic
569C2#1740711yescharacteristic
570C2#1740811yescharacteristic
571C2#2089711yescharacteristic
572C2#2098411yescharacteristic
573C2#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. #21738C2semidirect
4. C2×#1468C2semidirect
5. #9086C2semidirect
6. #9084C2semidirect
7. #9081C2semidirect
8. #9079C2semidirect
9. #7915C2semidirect
83. #407D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
C2×#26302
#217382
C2×#14682
#90862
#90842
#90812
#90792
#79152