Finite Group Databasebeta

← #188972

Subgroups — #188972

Order 512.

1009 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60233C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#21938C211yesmaximal, characteristic, complemented
4#17592C211yesmaximal, characteristic
5C2×#1513C211yesmaximal, characteristic
6#8629C214yesmaximal, complemented
7#8626C214yesmaximal, complemented
8#8107C211yesmaximal, characteristic, complemented
9C2×#1305C211yesmaximal, characteristic, complemented
10C2×#525C2211yescharacteristic
11#2977—22no—
12#2976—22no—
13#2974—22no—
14#2973—21no—
15#2972—21no—
16#2933C414yescomplemented
17#2929C2214yes—
18#2924C412yescomplemented
19#2880C412yescomplemented
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#1855—22no—
27#1855—22no—
28#1853—21no—
29#1853—21no—
30#1853—22no—
31#1810C2214yescomplemented
32#1807C2212yescomplemented
33#1807C2212yescomplemented
34#1520—21no—
35#1519—21no—
36#1513—22no—
37#1513C2214yes—
38#1513C2212yes—
39#1513C2212yes—
40#1504—22no—
41C2×#328—21no—
42C2×#327—21no—
43#1305C2214yescomplemented
44#1305C2212yescomplemented
45#1305C2212yescomplemented
46C22×#41C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#131C2×C414yescomplemented
49C22×M16—21no—
50C22×M16—21no—
51C22×M16C2×C412yes—
52C23×C8C2312yes—
53#525—21no—
54#525—21no—
55#525—21no—
56#525—21no—
57#525C2×C412yes—
58#525D412yes—
59#525Q812yes—
60#525C2312yes—
61C2×#123—21no—
62C2×C22⋊Q8D411yescharacteristic
63C2×C22⋊Q8Q811yescharacteristic
64C2×#121D411yescharacteristic
65C2×#121Q811yescharacteristic
66C2×#120—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×#117C2×C411yescharacteristic
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74#436—24no—
75#435—24no—
76#435—24no—
77#435—24no—
78#435—24no—
79C8×D4—24no—
80C8×D4—24no—
81C8×D4—24no—
82#433—24no—
83#432—22no—
84#431—22no—
85C2×#108—24no—
86C2×#106—22no—
87C2×#106—22no—
88#424C2×C414yescomplemented
89#408—24no—
90#408C2×C414yescomplemented
91#407—21no—
92#407—21no—
93#407—22no—
94#407—21no—
95#407—21no—
96#407—21no—
97#407—21no—
98#407—21no—
99#407—21no—
100#407C2×C412yes—
101#407C2312yes—
102#407C2312yes—
103#407C2312yes—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8C2×C412yes—
112C2×C22⋊C8C2×C412yes—
113C2×C22⋊C8D411yescharacteristic
114C2×C22⋊C8D411yescharacteristic
115C2×C22⋊C8D412yes—
116C2×C22⋊C8D412yes—
117C2×C22⋊C8D412yes—
118#405C2×C414yescomplemented
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380C2311yescharacteristic
126#380C2312yes—
127#380C2311yescharacteristic
128C2×#95—21no—
129C2×#95—21no—
130#341—24no—
131#340—22no—
132#340—22no—
133#328—24no—
134#328—22no—
135#328—22no—
136#327—24no—
137#327—22no—
138#327—22no—
139C25#4111yescharacteristic
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—82no—
143C2×#41C4⋊C412yes—
144C2×#41C22×C412yes—
145C22×Q8C4⋊C411yescharacteristic
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C4⋊C411yescharacteristic
150C23×C4—21no—
151C23×C4—41no—
152C23×C4—41no—
153C23×C4—82no—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×Q811yescharacteristic
163C23×C4C2×Q811yescharacteristic
164C23×C4#4111yescharacteristic
165C23×C4#4111yescharacteristic
166C23×C4#4111yescharacteristic
167C23×C4C2411yescharacteristic
168#131—84no—
169C2×M16—22no—
170C2×M16—24no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—41no—
174C2×M16—41no—
175C2×M16—82no—
176C2×M16—82no—
177C2×M16—82no—
178C2×M16C22×C414yes—
179C22×C8—22no—
180C22×C8—24no—
181C22×C8—41no—
182C22×C8—41no—
183C22×C8—82no—
184C22×C8C22×C414yes—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D414yes—
188#123—22no—
189#123—22no—
190#123—22no—
191#123—22no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8C4⋊C412yes—
197C22⋊Q8C4⋊C412yes—
198C22⋊Q8C2×D412yes—
199C22⋊Q8C2×Q812yes—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204#121C4⋊C412yes—
205#121C4⋊C412yes—
206#121C2×D412yes—
207#121C2×Q812yes—
208#120—22no—
209#120—22no—
210#120—22no—
211#120—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220#117—21no—
221#117—21no—
222#117C4⋊C412yes—
223#117C22×C412yes—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C4⋊C411yescharacteristic
234C2×C4⋊C4C4⋊C411yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4—82no—
247C2×C42—21no—
248#108—84no—
249#106—82no—
250#106—82no—
251C4⋊C8—24no—
252C4⋊C8—24no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—24no—
258C22⋊C8—24no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267C22⋊C8C22⋊C412yes—
268C22⋊C8C22⋊C412yes—
269C22⋊C8C22⋊C414yes—
270C22⋊C8C22⋊C412yes—
271C22⋊C8C22⋊C412yes—
272C22⋊C8C22⋊C414yes—
273#97—24no—
274C4×C8—24no—
275#95—22no—
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—81no—
284C24—81no—
285C24#11711yescharacteristic
286C24#11711yescharacteristic
287C24#12111yescharacteristic
288C24#12111yescharacteristic
289C24C22⋊Q811yescharacteristic
290C24C22⋊Q811yescharacteristic
291C24C2×#4111yescharacteristic
292#41—44no—
293#41—82no—
294#41—82no—
295#41—82no—
296C2×Q8—21no—
297C2×Q8—81no—
298C2×Q8C2×C4⋊C412yes—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—81no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4C2×C4⋊C412yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—82no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C4⋊C411yescharacteristic
346C22×C4C2×C4⋊C411yescharacteristic
347C22×C4C2×C4⋊C411yescharacteristic
348C22×C4C2×C4⋊C411yescharacteristic
349C22×C4#11711yescharacteristic
350C22×C4#11711yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4C22⋊Q811yescharacteristic
358C22×C4C22⋊Q811yescharacteristic
359C22×C4C22⋊Q811yescharacteristic
360C22×C4C22⋊Q811yescharacteristic
361C22×C4C22⋊Q811yescharacteristic
362C22×C4C22⋊Q811yescharacteristic
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C2×#4111yescharacteristic
369M16—44no—
370M16—44no—
371M16—44no—
372M16—84no—
373M16—82no—
374M16—82no—
375M16—82no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—44no—
380C2×C8—44no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—84no—
384C2×C8—82no—
385C2×C8—82no—
386C2×C8—82no—
387C2×C8C2×C22⋊C414yes—
388C2×C8C2×C22⋊C414yes—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4C2×C4⋊C412yes—
402C4⋊C4C2×C4⋊C412yes—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—81no—
417C22⋊C4—81no—
418C42—22no—
419C42—22no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—81no—
445C23—81no—
446C23—84no—
447C23—84no—
448C23#38011yescharacteristic
449C23#38011yescharacteristic
450C23#38011yescharacteristic
451C23#38011yescharacteristic
452C23#47112yes—
453C23#47112yes—
454C23C2×#11711yescharacteristic
455C23C2×#12111yescharacteristic
456C23C2×C22⋊Q811yescharacteristic
457Q8—42no—
458Q8—81no—
459D4—42no—
460D4—44no—
461D4—44no—
462D4—44no—
463D4—42no—
464D4—42no—
465D4—81no—
466D4—82no—
467D4—82no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
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—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—164no—
514C2×C4—164no—
515C2×C4#38011yescharacteristic
516C2×C4#38011yescharacteristic
517C2×C4#38011yescharacteristic
518C2×C4#38011yescharacteristic
519C2×C4C22×C22⋊C411yescharacteristic
520C2×C4C22×C4⋊C411yescharacteristic
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×C22⋊Q811yescharacteristic
523C8—24no—
524C8—24no—
525C8—44no—
526C8—82no—
527C8—82no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—162no—
560C22#181512yes—
561C22#181512yes—
562C22C2×#38011yescharacteristic
563C22C2×#47111yescharacteristic
564C22C2×#47111yescharacteristic
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—162no—
576C4—161no—
577C4—161no—
578C4#288312yes—
579C2—21no—
580C2—21no—
581C2—22no—
582C2—22no—
583C2—41no—
584C2—41no—
585C2—41no—
586C2—41no—
587C2—82no—
588C2#2063311yescharacteristic
589C2#2067511yescharacteristic
590C2C2×#181511yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #21938C2semidirect
6. #8629C2semidirect
7. #8626C2semidirect
8. #8107C2semidirect
9. C2×#1305C2semidirect
16. #2933C4semidirect
18. #2924C4semidirect
19. #2880C4semidirect
31. #1810C22semidirect
32. #1807C22semidirect
33. #1807C22semidirect
43. #1305C22semidirect
44. #1305C22semidirect
45. #1305C22semidirect
48. C2×#131C2×C4semidirect
88. #424C2×C4semidirect
90. #408C2×C4semidirect
118. #405C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602332
#219482
#219382
#175922
C2×#15132
#86292
#86262
#81072
C2×#13052