Finite Group Databasebeta

← #239746

Subgroups — #239746

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1476C211yesmaximal, characteristic
7#10423C214yesmaximal, complemented
8#10420C214yesmaximal, complemented
9#7897C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#417—21no—
15C22×C4×C8C2211yescharacteristic
16#2857—21no—
17#2857—21no—
18#2628—22no—
19#2568C2212yes—
20#2567—21no—
21#2563—21no—
22#2542C2212yes—
23#2487—22no—
24#2409—22no—
25#2317—21no—
26#2311—21no—
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#1990—22no—
30#1990C2214yes—
31#1948C2214yes—
32#1853—22no—
33#1844—21no—
34#1834—21no—
35#1803—21no—
36#1803C2212yes—
37#1802—21no—
38#1802C2212yes—
39#1736—21no—
40C2×#340—22no—
41#1549—24no—
42C2×#332C2214yes—
43#1476—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1279C2214yes—
47C22×#41D411yescharacteristic
48C23×Q8—21no—
49C22×M16—21no—
50C23×C8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8C2312yes—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C4×#41—22no—
62C4×#41—22no—
63C4×#41—22no—
64C4×#41—22no—
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431D414yes—
83C2×#106—24no—
84#423—22no—
85#423—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89#417—44no—
90C2×#103—22no—
91C2×#102—22no—
92C2×C22⋊C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98#400—24no—
99#399—24no—
100#395—24no—
101#393—24no—
102#391—22no—
103#391—22no—
104#391—22no—
105#390—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#386—24no—
111#385—24no—
112#384—22no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384D411yescharacteristic
117#384D411yescharacteristic
118#384C2312yes—
119#382—22no—
120#382—21no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125#380—21no—
126#380D412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2312yes—
129#340—22no—
130#340—22no—
131#340—24no—
132#340—28no—
133#332—28no—
134Q8⋊C8—24no—
135D4⋊C8—24no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41SD1614yes—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—82no—
147C22×Q8—82no—
148C22×Q8C2×D411yescharacteristic
149C22×D4—21no—
150C22×D4C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—41no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—42no—
162C22×C8—24no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—42no—
167C22×C8—42no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C22⋊Q8—84no—
179C22⋊Q8—84no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196#117—22no—
197#117—22no—
198#117—42no—
199#117—42no—
200#117—42no—
201#117—42no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4112yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42C2411yescharacteristic
252#106—84no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—42no—
257C4⋊C8C2×D414yes—
258#103—44no—
259#102—44no—
260C22⋊C8—24no—
261C4×C8—24no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—42no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—22no—
271C24—21no—
272C24#12011yescharacteristic
273#41—28no—
274#41—28no—
275#41—44no—
276#41—44no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—82no—
288C2×Q8—82no—
289C2×Q8—88no—
290C2×Q8—88no—
291C2×Q8C2×SD1612yes—
292C2×D4—24no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4C2×SD1612yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4#12011yescharacteristic
329C22×C4#12011yescharacteristic
330C22×C4#12011yescharacteristic
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12312yes—
335C22×C4#12312yes—
336C22×C4#12412yes—
337C22×C4C2×SD1612yes—
338C22×C4#13512yes—
339C22×C4#13712yes—
340C22×C4#13712yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344M16—84no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—42no—
348C2×C8—48no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—24no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—24no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—82no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—42no—
388C22⋊C4—84no—
389C22⋊C4—84no—
390C42—41no—
391C42—41no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42#12112yes—
397C42#12112yes—
398C42#12112yes—
399C42#12312yes—
400C42#12312yes—
401C42#12412yes—
402C42C22×D411yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23#39412yes—
418C23#44812yes—
419C23#45212yes—
420C23C2×#12011yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—41no—
424Q8—41no—
425Q8—88no—
426Q8—84no—
427Q8—84no—
428D4—22no—
429D4—22no—
430D4—48no—
431D4—41no—
432D4—41no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—48no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—88no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—84no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—164no—
468C2×C4#39412yes—
469C2×C4#39412yes—
470C2×C4#39412yes—
471C2×C4#44812yes—
472C2×C4#45212yes—
473C2×C4#45412yes—
474C2×C4#45612yes—
475C2×C4#46514yes—
476C2×C4#47014yes—
477C2×C4#48314yes—
478C2×C4#48514yes—
479C2×C4C2×#12111yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4C2×#12311yescharacteristic
484C2×C4C2×#12411yescharacteristic
485C2×C4C22×SD1611yescharacteristic
486C2×C4C2×#13511yescharacteristic
487C2×C4C2×#13711yescharacteristic
488C2×C4C2×#13711yescharacteristic
489C2×C4#57612yes—
490C2×C4#57712yes—
491C8—44no—
492C8—44no—
493C8—82no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22#203514yes—
508C22C2×#39411yescharacteristic
509C22C2×#44811yescharacteristic
510C22C2×#45211yescharacteristic
511C22#300811yescharacteristic
512C22#300911yescharacteristic
513C22C2×#45411yescharacteristic
514C22C2×#45611yescharacteristic
515C4—21no—
516C4—21no—
517C4—22no—
518C4—42no—
519C4—81no—
520C4—81no—
521C4—84no—
522C4—82no—
523C4—82no—
524C4—162no—
525C4#203514yes—
526C4#203814yes—
527C4#304512yes—
528C4C2×#46511yescharacteristic
529C4C2×#47011yescharacteristic
530C4C2×#48311yescharacteristic
531C4C2×#48511yescharacteristic
532C4#313612yes—
533C2—24no—
534C2—82no—
535C2#1864211yescharacteristic
536C2#1864711yescharacteristic
537C2#1865211yescharacteristic
538C2#1865711yescharacteristic
539C2C2×#203511yescharacteristic
540C2C2×#203811yescharacteristic
541C2#2350911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C2×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. #21981C2semidirect
3. C2×#1990C2semidirect
5. C2×#1549C2semidirect
7. #10423C2semidirect
8. #10420C2semidirect
9. #7897C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#219812
C2×#19902
#206482
C2×#15492
C2×#14762
#104232
#104202
#78972