Finite Group Databasebeta

← #197884

Subgroups — #197884

Order 512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3112C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic, complemented
5C2×#1551C211yesmaximal, characteristic
6#9230C212yesmaximal, complemented
7#9224C212yesmaximal, complemented
8#9222C212yesmaximal, complemented
9#9216C212yesmaximal, complemented
10#8040C211yesmaximal, characteristic, complemented
11#7879C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3112—24no—
15#3112C418yescomplemented
16C2×#480—21no—
17C2×#477C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×#460C2211yescharacteristic
20C2×#460C2211yescharacteristic
21C2×#448—21no—
22#2984—24no—
23#2984—24no—
24C2×#446—22no—
25C2×#446C2212yes—
26C2×C8×Q8C2212yes—
27C2×#440—21no—
28C2×#440—21no—
29C2×#432—21no—
30C2×#432C2211yescharacteristic
31C2×#432C2211yescharacteristic
32C2×#431—21no—
33C2×#431C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#103—21no—
36C2×#384C2211yescharacteristic
37#1990C2212yes—
38#1957C2212yes—
39#1874—21no—
40#1874—21no—
41#1803—21no—
42#1803—21no—
43#1803C2212yes—
44#1803C2212yes—
45#1802—22no—
46C2×#340—21no—
47#1551—24no—
48C2×#333C2212yes—
49C2×#331C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1277C2212yes—
53#1277C2212yes—
54C22×#41—21no—
55C23×Q8D411yescharacteristic
56C22×SD16—21no—
57C22×SD16—21no—
58C23×C8—21no—
59#532—22no—
60#532—22no—
61#532—22no—
62#532C2×C414yes—
63C2×C4⋊Q8C2311yescharacteristic
64C2×#127C2311yescharacteristic
65C2×#124—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C414yes—
73C2×C4×Q8D414yes—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#480—28no—
88#477—22no—
89#477—22no—
90#477C2×C414yescomplemented
91#477C2×C414yescomplemented
92#460—22no—
93#460—22no—
94#460C2×C414yescomplemented
95#460C2×C414yescomplemented
96#448—28no—
97#446—216no—
98#446—24no—
99#446—24no—
100C8×Q8—24no—
101C8×Q8—24no—
102#440—28no—
103#440—28no—
104#432—28no—
105#432—22no—
106#432—22no—
107#432C2×C414yes—
108#432D414yes—
109#431—28no—
110#431—22no—
111#431—22no—
112C2×#107—21no—
113C2×#107—21no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8D412yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2312yes—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—22no—
127C2×#103—22no—
128C2×#103—44no—
129C2×#102—21no—
130C2×#102—21no—
131C2×#102—22no—
132C2×#102—22no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8D411yescharacteristic
136C2×C22⋊C8D411yescharacteristic
137C2×#97—22no—
138C2×#97—21no—
139C2×#97—21no—
140C2×C4×C8—22no—
141C2×C4×C8C2312yes—
142#384—21no—
143#384—21no—
144#384—21no—
145#384C2312yes—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#333—24no—
151#331—24no—
152C2×#41—24no—
153C2×#41—42no—
154C2×#41—82no—
155C2×#41—82no—
156C22×Q8—21no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—21no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8C22⋊C414yes—
164C22×Q8C2×D411yescharacteristic
165C22×Q8C2×D411yescharacteristic
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—41no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4#4111yescharacteristic
177C2×SD16—24no—
178C2×SD16—24no—
179C2×SD16—42no—
180C2×SD16—42no—
181C2×SD16—42no—
182C2×SD16—42no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8C2×D412yes—
191C22×C8C2×D412yes—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8C22×C412yes—
196#127—21no—
197#127—21no—
198#127—21no—
199#127C22×C412yes—
200#124—24no—
201#124—42no—
202#121—24no—
203#121—24no—
204#121—42no—
205#121—42no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—24no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8C22⋊C414yes—
217C4×Q8C22⋊C414yes—
218C4×Q8C22×C412yes—
219C4×Q8C22×C412yes—
220C4×Q8C2×D412yes—
221C4×Q8C2×D412yes—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—42no—
225C4×D4—42no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C22⋊C414yes—
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42C22×C412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42#4111yescharacteristic
263C2×C42#4112yes—
264C2×C42C2411yescharacteristic
265#107—24no—
266#107—24no—
267C4⋊C8—28no—
268C4⋊C8—28no—
269C4⋊C8—24no—
270C4⋊C8—22no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—42no—
276C4⋊C8C22×C412yes—
277C4⋊C8C22×C412yes—
278C4⋊C8C2×D414yes—
279#103—24no—
280#103—24no—
281#103—44no—
282#103—44no—
283#102—24no—
284#102—24no—
285#102—82no—
286#102—82no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—22no—
290C22⋊C8—22no—
291#97—28no—
292#97—24no—
293#97—24no—
294C4×C8—28no—
295C4×C8—22no—
296C4×C8—22no—
297C24—41no—
298C24#12111yescharacteristic
299#41—88no—
300#41—88no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—24no—
306C2×Q8—28no—
307C2×Q8—22no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—41no—
318C2×Q8—81no—
319C2×Q8—81no—
320C2×Q8C2×C22⋊C412yes—
321C2×Q8C2×C22⋊C412yes—
322C2×Q8AGL(1,ℤ/8)12yes—
323C2×Q8#13712yes—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—44no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—82no—
340C2×D4—82no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4C2×C22⋊C412yes—
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4#12012yes—
371C22×C4#12112yes—
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12312yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379SD16—48no—
380SD16—48no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—84no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4C2×C22⋊C412yes—
422C4⋊C4C2×C22⋊C412yes—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—24no—
430C42—24no—
431C42C2×C22⋊C412yes—
432C42C4×D412yes—
433C42C4×D412yes—
434C42#12012yes—
435C42#12112yes—
436C42#12312yes—
437C42C23×C411yescharacteristic
438C42C22×D411yescharacteristic
439C42C22×D411yescharacteristic
440C42C2×#4111yescharacteristic
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—42no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23#38612yes—
453C23#47112yes—
454C23#47212yes—
455C23C2×#12111yescharacteristic
456Q8—22no—
457Q8—22no—
458Q8—22no—
459Q8—22no—
460Q8—22no—
461Q8—48no—
462Q8—41no—
463Q8—41no—
464Q8—82no—
465Q8—82no—
466Q8#41714yes—
467D4—42no—
468D4—42no—
469D4—44no—
470D4—82no—
471D4—82no—
472D4—88no—
473D4—82no—
474D4—81no—
475D4—81no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—88no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—164no—
514C2×C4#38612yes—
515C2×C4#38612yes—
516C2×C4#38612yes—
517C2×C4C22×C22⋊C411yescharacteristic
518C2×C4C2×C4×D411yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×#12011yescharacteristic
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×#12311yescharacteristic
523C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
524C2×C4C2×#13711yescharacteristic
525C2×C4#57511yescharacteristic
526C2×C4#57511yescharacteristic
527C2×C4#57712yes—
528C2×C4#57812yes—
529C8—42no—
530C8—42no—
531C8—42no—
532C8—42no—
533C8—44no—
534C8—82no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—161no—
545C22—161no—
546C22—161no—
547C22—161no—
548C22#190714yes—
549C22C2×#38611yescharacteristic
550C22#304311yescharacteristic
551C22#304411yescharacteristic
552C22#305511yescharacteristic
553C22#305611yescharacteristic
554C22C2×#47111yescharacteristic
555C22C2×#47211yescharacteristic
556C4—22no—
557C4—21no—
558C4—21no—
559C4—22no—
560C4—22no—
561C4—24no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—82no—
566C4—162no—
567C4C2×#41711yescharacteristic
568C4#289611yescharacteristic
569C4#294712yes—
570C4#301312yes—
571C4#305511yescharacteristic
572C4#305611yescharacteristic
573C2—24no—
574C2—162no—
575C2#1946911yescharacteristic
576C2#1947011yescharacteristic
577C2#1947411yescharacteristic
578C2#1947711yescharacteristic
579C2#2093011yescharacteristic
580C2#2097111yescharacteristic
581C2C2×#190711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2984C2semidirect
4. C2×#1803C2semidirect
6. #9230C2semidirect
7. #9224C2semidirect
8. #9222C2semidirect
9. #9216C2semidirect
10. #8040C2semidirect
11. #7879C2semidirect
15. #3112C4semidirect
90. #477C2×C4semidirect
91. #477C2×C4semidirect
94. #460C2×C4semidirect
95. #460C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31122
C2×#29842
#218592
C2×#18032
C2×#15512
#92302
#92242
#92222
#92162
#80402
#78792