Finite Group Databasebeta

← #231535

Subgroups — #231535

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23235C211yesmaximal, characteristic, complemented
2#23234C211yesmaximal, characteristic, complemented
3#19613C211yesmaximal, characteristic
4#18067C211yesmaximal, characteristic, complemented
5#18067C211yesmaximal, characteristic, complemented
6#9754C211yesmaximal, characteristic, complemented
7#9751C211yesmaximal, characteristic, complemented
8#9717C211yesmaximal, characteristic, complemented
9#9717C211yesmaximal, characteristic, complemented
10#9056C211yesmaximal, characteristic, complemented
11#9056C211yesmaximal, characteristic, complemented
12#9053C211yesmaximal, characteristic
13#9053C211yesmaximal, characteristic
14#8294C211yesmaximal, characteristic, complemented
15#8290C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3169C2211yescharacteristic
18#3167C2211yescharacteristic
19#3167C2211yescharacteristic
20#3131C2211yescharacteristic
21#3131C2211yescharacteristic
22#2962C2211yescharacteristic
23#2962C2211yescharacteristic
24#2935C2211yescharacteristic
25#2935C2211yescharacteristic
26#1986C2211yescharacteristic
27#1986C2211yescharacteristic
28#1980C2211yescharacteristic
29#1980C2211yescharacteristic
30#1938—21no—
31#1937—21no—
32#1929—21no—
33#1929—21no—
34#1866—21no—
35#1866—21no—
36C4×#108C2211yescharacteristic
37C4×#106C2211yescharacteristic
38#1770C2211yescharacteristic
39C8×C4⋊C4C2211yescharacteristic
40#1765—21no—
41#1764—21no—
42#1720C411yescharacteristic, complemented
43#1720C411yescharacteristic, complemented
44#1719C411yescharacteristic, complemented
45#1719C411yescharacteristic, complemented
46#1718C411yescharacteristic, complemented
47#1718C411yescharacteristic, complemented
48#1717C411yescharacteristic, complemented
49#1717C411yescharacteristic, complemented
50#1590—21no—
51#1590—21no—
52#1583—21no—
53#1583—21no—
54#1582—21no—
55#1582—21no—
56C8×SD16—21no—
57C8×SD16—21no—
58#1564C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1503C2211yescharacteristic
61#1503C2211yescharacteristic
62#1502C2211yescharacteristic
63#1502C2211yescharacteristic
64#1481C2211yescharacteristic
65#1481C2211yescharacteristic
66#1480C2211yescharacteristic
67#1480C2211yescharacteristic
68C8×M16C2211yescharacteristic
69C8×M16C2211yescharacteristic
70C2×#322C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#131—21no—
76C2×#131—21no—
77#533C2311yescharacteristic
78#533C2311yescharacteristic
79#532C2311yescharacteristic
80#532C2311yescharacteristic
81#500C2×C411yescharacteristic
82#500C2×C411yescharacteristic
83#498C2×C411yescharacteristic
84#498C2×C411yescharacteristic
85#494C2×C411yescharacteristic
86#493C2×C411yescharacteristic
87#492C2×C411yescharacteristic
88#492C2×C411yescharacteristic
89#488—21no—
90#488—21no—
91#488—21no—
92#488—21no—
93#487C2×C411yescharacteristic
94#487C2×C411yescharacteristic
95#487C2×C411yescharacteristic
96#487C2×C411yescharacteristic
97#487C2×C411yescharacteristic
98#487C2×C411yescharacteristic
99#487C2×C411yescharacteristic
100#487C2×C411yescharacteristic
101#486C2×C411yescharacteristic
102#486C2×C411yescharacteristic
103#486C2×C411yescharacteristic
104#486C2×C411yescharacteristic
105#486C2×C411yescharacteristic
106#486C2×C411yescharacteristic
107#486C2×C411yescharacteristic
108#486C2×C411yescharacteristic
109#467—41no—
110#466—41no—
111#465—21no—
112#465—21no—
113#465—21no—
114#465—21no—
115#446—21no—
116#446—21no—
117#446—41no—
118#446—41no—
119C8×Q8—21no—
120C8×Q8—21no—
121C4×Q16—41no—
122C4×SD16—21no—
123C4×SD16—21no—
124C4×SD16—21no—
125C4×SD16—21no—
126C4×D8—41no—
127#436—21no—
128#436—21no—
129#435—41no—
130#435—41no—
131C8×D4—21no—
132C8×D4—21no—
133#431C2311yescharacteristic
134#431C2311yescharacteristic
135C2×#108D411yescharacteristic
136C2×#108D411yescharacteristic
137C2×#106—21no—
138#424—21no—
139#424—21no—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×#104—21no—
143C2×#104—21no—
144C2×#104—21no—
145C2×#104—21no—
146#416—21no—
147#416—21no—
148#416—21no—
149#416—21no—
150#408—21no—
151#408—21no—
152#405—21no—
153#405—21no—
154C4×M16C2311yescharacteristic
155C4×M16C2311yescharacteristic
156C2×#97—21no—
157C2×C4×C8—21no—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C4×C4⋊C4C2311yescharacteristic
162C4×C4⋊C4C2311yescharacteristic
163#341D411yescharacteristic
164#341D411yescharacteristic
165#340—21no—
166#339—21no—
167#339—21no—
168#339—21no—
169#339—21no—
170#336—21no—
171#336—21no—
172#335—21no—
173#334—21no—
174#334D411yescharacteristic
175#334D411yescharacteristic
176#334D411yescharacteristic
177#334D411yescharacteristic
178#328—41no—
179#328—41no—
180#327—41no—
181#327—41no—
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184Q8⋊C8—21no—
185Q8⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188D4⋊C8—21no—
189D4⋊C8—21no—
190#322—21no—
191#322—21no—
192#322C2×C411yescharacteristic
193#322C2×C411yescharacteristic
194#322C2×C411yescharacteristic
195#322C2×C411yescharacteristic
196C82—21no—
197C82—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×Q16—41no—
201C2×SD16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×SD16—21no—
205C2×D8—41no—
206#131—42no—
207#131—42no—
208#131—81no—
209#131—81no—
210C2×M16—41no—
211C2×M16—41no—
212C2×M16C2×D411yescharacteristic
213C2×M16C2×D411yescharacteristic
214C2×M16C2×D411yescharacteristic
215C2×M16C2×D411yescharacteristic
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—41no—
220C22×C8—41no—
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8#4111yescharacteristic
224C22×C8#4111yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226C4⋊Q8C22×C411yescharacteristic
227C4⋊Q8C22×C411yescharacteristic
228C4⋊Q8C22×C411yescharacteristic
229C4⋊Q8C22×C411yescharacteristic
230C4⋊Q8C22×C411yescharacteristic
231#127C22×C411yescharacteristic
232#127C22×C411yescharacteristic
233#125—21no—
234#125—21no—
235#124—21no—
236#124—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241#121—21no—
242#121—21no—
243#121—21no—
244#121—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253#117—21no—
254#117—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42C2411yescharacteristic
260#108—22no—
261#108—22no—
262#107—41no—
263#107—41no—
264#106—21no—
265#106—21no—
266#106—21no—
267#106—21no—
268#106—42no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279C4⋊C8C2×D411yescharacteristic
280C4⋊C8C2×D411yescharacteristic
281#104—42no—
282#104—42no—
283#104—81no—
284#104—81no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#103—21no—
294#103—21no—
295#103—21no—
296#103—21no—
297#103—41no—
298#103—41no—
299#102—21no—
300#102—21no—
301#102—21no—
302#102—21no—
303#102—21no—
304#102—21no—
305#102—21no—
306#102—21no—
307#102—21no—
308#102—21no—
309#102—21no—
310#102—21no—
311#102—41no—
312#102—41no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C22⋊C8—41no—
316C22⋊C8—41no—
317C22⋊C8—41no—
318C22⋊C8—41no—
319#97—21no—
320#97—21no—
321#97—41no—
322#97—41no—
323#97—81no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—41no—
330C4×C8—41no—
331C4×C8—81no—
332C4×C8C22×C411yescharacteristic
333C4×C8C22×C411yescharacteristic
334C4×C8C22×C411yescharacteristic
335C4×C8C22×C411yescharacteristic
336C4×C8C22×C411yescharacteristic
337C4×C8C22×C411yescharacteristic
338C4×C8C2×D411yescharacteristic
339C4×C8C2×D411yescharacteristic
340C4×C8#4111yescharacteristic
341C4×C8#4111yescharacteristic
342#95—21no—
343#95—21no—
344#41—41no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—41no—
365C2×D4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373Q16—82no—
374SD16—42no—
375SD16—42no—
376SD16—81no—
377SD16—81no—
378D8—82no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—82no—
386M16—82no—
387M16#13512yes—
388M16#13512yes—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—82no—
412C2×C8—82no—
413C2×C8—81no—
414C2×C8C4×D411yescharacteristic
415C2×C8C4×D411yescharacteristic
416C2×C8C4×D411yescharacteristic
417C2×C8C4×D411yescharacteristic
418C2×C8C4×D411yescharacteristic
419C2×C8C4×D411yescharacteristic
420C2×C8C4×D411yescharacteristic
421C2×C8C4×D411yescharacteristic
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42C23×C411yescharacteristic
455C42C22×D411yescharacteristic
456C42C2×#4111yescharacteristic
457C42H(5,𝔽2)11yescharacteristic
458C23—41no—
459C23—41no—
460C23#55011yescharacteristic
461Q8—41no—
462Q8—41no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—41no—
467Q8—41no—
468Q8—41no—
469Q8—81no—
470Q8—81no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—81no—
480D4—81no—
481D4—81no—
482D4—81no—
483D4—81no—
484D4—81no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4#51811yescharacteristic
507C2×C4D4211yescharacteristic
508C2×C4#55011yescharacteristic
509C2×C4#55911yescharacteristic
510C2×C4C2×#13511yescharacteristic
511C2×C4C2×#13511yescharacteristic
512C2×C4#57711yescharacteristic
513C2×C4#57711yescharacteristic
514C8—21no—
515C8—21no—
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—21no—
521C8—21no—
522C8—22no—
523C8—22no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—81no—
533C8—81no—
534C22—21no—
535C22—21no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22#236611yescharacteristic
543C22#339411yescharacteristic
544C22#339411yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—82no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—161no—
556C4C4×#13511yescharacteristic
557C4#295211yescharacteristic
558C4#295511yescharacteristic
559C4#295611yescharacteristic
560C4#328211yescharacteristic
561C4#328211yescharacteristic
562C2—21no—
563C2—21no—
564C2—161no—
565C2—161no—
566C2#2318111yescharacteristic
567C2#2318311yescharacteristic
568C2#2319711yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2366
prime radical#3169C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23235C2semidirect
2. #23234C2semidirect
4. #18067C2semidirect
5. #18067C2semidirect
6. #9754C2semidirect
7. #9751C2semidirect
8. #9717C2semidirect
9. #9717C2semidirect
10. #9056C2semidirect
11. #9056C2semidirect
14. #8294C2semidirect
15. #8290C2semidirect
42. #1720C4semidirect
43. #1720C4semidirect
44. #1719C4semidirect
45. #1719C4semidirect
46. #1718C4semidirect
47. #1718C4semidirect
48. #1717C4semidirect
49. #1717C4semidirect

Maximal subgroups

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

subgroupindex
#232352
#232342
#196132
#180672
#97542
#97512
#97172
#90562
#90532
#82942
#82902