Finite Group Databasebeta

← #206433

Subgroups — #206433

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60947C211yesmaximal, characteristic
2#21856C211yesmaximal, characteristic
3#21844C211yesmaximal, characteristic, complemented
4#20473C211yesmaximal, characteristic, complemented
5#17817C211yesmaximal, characteristic, complemented
6#17473C211yesmaximal, characteristic
7C2×#1493C211yesmaximal, characteristic, complemented
8#9851C212yesmaximal, complemented
9#9850C212yesmaximal, complemented
10#9848C212yesmaximal, complemented
11#9846C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#533C2211yescharacteristic
14#3248—22no—
15#3231C412yescomplemented
16#3229C411yescharacteristic, complemented
17#3226C411yescharacteristic, complemented
18#3210C412yescomplemented
19#3209C411yescharacteristic, complemented
20#3206C411yescharacteristic, complemented
21#3190—22no—
22#3170C2212yes—
23#3167C2212yes—
24#3108—22no—
25#3107—22no—
26#2920C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1994C2212yes—
32#1994C2212yes—
33#1994C2212yes—
34#1986—22no—
35#1962C2212yes—
36#1871—22no—
37#1826—22no—
38#1806—22no—
39#1770—22no—
40#1752—22no—
41#1744C2212yes—
42C4×#97C2212yes—
43#1574—22no—
44#1573—22no—
45#1567C2212yes—
46#1566C2212yes—
47#1494—22no—
48#1493—22no—
49#1493C2212yes—
50#1493C2212yes—
51#1493C2211yescharacteristic
52#1493C2211yescharacteristic
53#1492C2211yescharacteristic
54#1492C2211yescharacteristic
55#1476—22no—
56C2×Q8⋊C8—22no—
57C22×M16—21no—
58C23×C8—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533C2×C412yes—
66#533C2×C411yescharacteristic
67#533C2×C411yescharacteristic
68#533C2312yes—
69#533C2311yescharacteristic
70#532C2×C411yescharacteristic
71#532C2×C411yescharacteristic
72#532C2311yescharacteristic
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4⋊Q8C2×C411yescharacteristic
75#529—21no—
76#529—21no—
77#529—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81C2×#125—21no—
82#525—21no—
83#525—21no—
84#525—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C4×#41—21no—
88C4×#41—21no—
89C4×#41—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8—21no—
92C2×#117—21no—
93C2×#117—21no—
94C22×C42C2311yescharacteristic
95#500C2×C412yes—
96#498C2×C412yes—
97#494C2×C412yes—
98#492C2×C412yes—
99#487C2×C412yes—
100#487C2×C412yes—
101#487C2×C412yes—
102#486C2×C412yes—
103#485—24no—
104#483—22no—
105#480—22no—
106#479—24no—
107#475—22no—
108#474—22no—
109#467—24no—
110#465—24no—
111#464—24no—
112#462—22no—
113#461—22no—
114#452—24no—
115#451—22no—
116#448—22no—
117C4×Q16—24no—
118C4×SD16—24no—
119#432—21no—
120#432—21no—
121#431—21no—
122#431—21no—
123#431C2×C411yescharacteristic
124#431C2×C411yescharacteristic
125#431C2×C412yes—
126#431D412yes—
127#431D412yes—
128#423—21no—
129#423—21no—
130#423—21no—
131#423C2311yescharacteristic
132#423C2312yes—
133#423C2311yescharacteristic
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×C4⋊C8D412yes—
137C2×#104—24no—
138#407—21no—
139C2×C22⋊C8D412yes—
140C4×M16—21no—
141C2×#97—24no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8D412yes—
146C2×C4×C8D412yes—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2312yes—
150C4×C4⋊C4—21no—
151C4×C4⋊C4D412yes—
152C4×C22⋊C4—21no—
153C4×C22⋊C4D412yes—
154#376—21no—
155C43C2312yes—
156#339—24no—
157#339—24no—
158#339—24no—
159#336—24no—
160#335—24no—
161#334—24no—
162Q8⋊C8—24no—
163Q8⋊C8—24no—
164Q8⋊C8—22no—
165Q8⋊C8—22no—
166D4⋊C8—22no—
167D4⋊C8—22no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4111yescharacteristic
176C2×Q16—24no—
177C2×SD16—24no—
178C2×M16—22no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—41no—
182C2×M16—82no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—44no—
188C22×C8—41no—
189C22×C8C2×D414yes—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8C22×C412yes—
197C4⋊Q8C22×C412yes—
198C4⋊Q8C22×C411yescharacteristic
199C4⋊Q8C22×C411yescharacteristic
200C4⋊Q8C22×C411yescharacteristic
201#127C22×C411yescharacteristic
202#126—41no—
203#126—41no—
204#125—21no—
205#125—22no—
206#125—21no—
207#125—22no—
208#125—21no—
209#125—41no—
210#125—41no—
211#124—21no—
212#124—21no—
213#124—21no—
214#123—41no—
215#123—41no—
216#123—41no—
217C22⋊Q8—22no—
218C22⋊Q8—21no—
219C22⋊Q8—22no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—22no—
224C22⋊Q8—21no—
225C22⋊Q8—22no—
226C22⋊Q8—21no—
227C22⋊Q8—41no—
228C22⋊Q8—41no—
229C22⋊Q8—41no—
230C22⋊Q8—41no—
231#121—21no—
232#121—21no—
233#121—21no—
234#121—21no—
235#121—21no—
236#121—21no—
237#120—41no—
238C4×Q8—22no—
239C4×Q8—21no—
240C4×Q8—22no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—41no—
249C4×Q8—41no—
250C4×Q8—41no—
251C4×Q8—41no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—41no—
263C4×D4—41no—
264#117—21no—
265#117—22no—
266#117—21no—
267#117—22no—
268#117—21no—
269#117—44no—
270#117—41no—
271#117—41no—
272#117—41no—
273#117—41no—
274#117—41no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—41no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—41no—
287C2×C42—41no—
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C2×D414yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42#4111yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42C2411yescharacteristic
301#107—24no—
302#106—24no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—22no—
306C4⋊C8—21no—
307C4⋊C8—22no—
308C4⋊C8—21no—
309C4⋊C8—41no—
310C4⋊C8—41no—
311C4⋊C8C22⋊C412yes—
312C4⋊C8C22⋊C412yes—
313C4⋊C8C22⋊C414yes—
314#104—84no—
315#103—24no—
316#103—24no—
317#103—24no—
318#102—24no—
319C22⋊C8—41no—
320C22⋊C8—41no—
321C22⋊C8C22⋊C412yes—
322C22⋊C8C22⋊C412yes—
323C22⋊C8C22⋊C414yes—
324#97—48no—
325#97—41no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—41no—
331C4×C8C22×C412yes—
332C4×C8C22×C412yes—
333C4×C8C2×D412yes—
334C4×C8C2×D412yes—
335C4×C8C2×D414yes—
336#95—22no—
337#95—22no—
338C24#12111yescharacteristic
339#41—42no—
340#41—42no—
341#41—44no—
342#41—82no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—44no—
351C2×Q8—82no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12411yescharacteristic
380C22×C4#12411yescharacteristic
381C22×C4#12711yescharacteristic
382C22×C4#12711yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4111yescharacteristic
386Q16—84no—
387SD16—84no—
388M16—44no—
389M16—42no—
390M16—42no—
391M16—82no—
392C2×C8—24no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8—44no—
398C2×C8—42no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—82no—
403C2×C8C2×C22⋊C414yes—
404C2×C8C2×C22⋊C414yes—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C22⋊C4—44no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—42no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C42—24no—
443C42—24no—
444C42—44no—
445C42—44no—
446C42—41no—
447C42—42no—
448C42—41no—
449C42—41no—
450C42—42no—
451C42C2×C22⋊C411yescharacteristic
452C42C2×C22⋊C411yescharacteristic
453C42C4×D411yescharacteristic
454C42C4×D411yescharacteristic
455C42C4×D411yescharacteristic
456C42C4×D411yescharacteristic
457C42#12111yescharacteristic
458C42#12111yescharacteristic
459C42#12411yescharacteristic
460C42#12411yescharacteristic
461C42#12711yescharacteristic
462C42#12711yescharacteristic
463C42C23×C411yescharacteristic
464C42C22×D411yescharacteristic
465C42C22×D411yescharacteristic
466C42C2×#4111yescharacteristic
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23#39011yescharacteristic
472C23#39011yescharacteristic
473C23C2×#12111yescharacteristic
474Q8—44no—
475Q8—42no—
476Q8—42no—
477Q8—44no—
478Q8—42no—
479Q8—82no—
480Q8—82no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—82no—
485D4—82no—
486D4—82no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4#39011yescharacteristic
520C2×C4#39011yescharacteristic
521C2×C4#39011yescharacteristic
522C2×C4#39011yescharacteristic
523C2×C4#39011yescharacteristic
524C2×C4#39011yescharacteristic
525C2×C4#44414yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12411yescharacteristic
531C2×C4C2×#12711yescharacteristic
532C2×C4#57712yes—
533C2×C4#57812yes—
534C8—24no—
535C8—24no—
536C8—24no—
537C8—42no—
538C8—82no—
539C8—81no—
540C8—81no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—44no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22#197512yes—
553C22C2×#39011yescharacteristic
554C22#306312yes—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—84no—
563C4—81no—
564C4—82no—
565C4—81no—
566C4—81no—
567C4—82no—
568C4#197512yes—
569C4#290112yes—
570C4#294712yes—
571C4C2×#44412yes—
572C4#315411yescharacteristic
573C4#315611yescharacteristic
574C2—22no—
575C2—22no—
576C2—41no—
577C2—161no—
578C2#2121011yescharacteristic
579C2#2123411yescharacteristic
580C2C2×#197511yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21844C2semidirect
4. #20473C2semidirect
5. #17817C2semidirect
7. C2×#1493C2semidirect
8. #9851C2semidirect
9. #9850C2semidirect
10. #9848C2semidirect
11. #9846C2semidirect
15. #3231C4semidirect
16. #3229C4semidirect
17. #3226C4semidirect
18. #3210C4semidirect
19. #3209C4semidirect
20. #3206C4semidirect

Maximal subgroups

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

subgroupindex
#609472
#218562
#218442
#204732
#178172
#174732
C2×#14932
#98512
#98502
#98482
#98462