Finite Group Databasebeta

← #230418

Subgroups — #230418

Order 512.

635 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23016C211yesmaximal, characteristic, complemented
2#23015C211yesmaximal, characteristic, complemented
3#19125C211yesmaximal, characteristic
4#18073C211yesmaximal, characteristic, complemented
5#17826C211yesmaximal, characteristic, complemented
6#9852C211yesmaximal, characteristic, complemented
7#9625C211yesmaximal, characteristic, complemented
8#9582C211yesmaximal, characteristic, complemented
9#9578C211yesmaximal, characteristic, complemented
10#8995C211yesmaximal, characteristic, complemented
11#8995C211yesmaximal, characteristic, complemented
12#8932C211yesmaximal, characteristic, complemented
13#8932C211yesmaximal, characteristic, complemented
14#8279C211yesmaximal, characteristic, complemented
15#8193C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3035C2211yescharacteristic
18#3035C2211yescharacteristic
19C8×#41C2211yescharacteristic
20C2×#444—21no—
21C2×#443—21no—
22#2935C2211yescharacteristic
23#2935C2211yescharacteristic
24#2906C2211yescharacteristic
25#2906C2211yescharacteristic
26#1962C2211yescharacteristic
27#1962C2211yescharacteristic
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1938—21no—
31#1937—21no—
32#1876—21no—
33#1876—21no—
34#1866—21no—
35#1866—21no—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×#104C2211yescharacteristic
39#1744C2211yescharacteristic
40#1682C411yescharacteristic, complemented
41#1682C411yescharacteristic, complemented
42#1681C411yescharacteristic, complemented
43#1681C411yescharacteristic, complemented
44#1674C411yescharacteristic, complemented
45#1673C411yescharacteristic, complemented
46#1668C411yescharacteristic, complemented
47#1667C411yescharacteristic, complemented
48#1594—21no—
49#1586—21no—
50#1586—21no—
51#1580—21no—
52#1570—21no—
53#1570—21no—
54#1570C2211yescharacteristic
55#1570C2211yescharacteristic
56#1566—21no—
57#1566C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×#334—21no—
60C2×#334C2211yescharacteristic
61#1540C2211yescharacteristic
62#1540C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65#1480C2211yescharacteristic
66#1480C2211yescharacteristic
67#1476C2211yescharacteristic
68#1451C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1278C2211yescharacteristic
72#1277C2211yescharacteristic
73#1277C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135—21no—
76C2×#135—21no—
77C2×#131—21no—
78C2×#131—21no—
79C2×#131—21no—
80C2×#131—21no—
81#532C2311yescharacteristic
82#532C2311yescharacteristic
83C4×#41C2311yescharacteristic
84#500C2×C411yescharacteristic
85#493C2×C411yescharacteristic
86#486C2×C411yescharacteristic
87#486C2×C411yescharacteristic
88#466—21no—
89#466—21no—
90#464—21no—
91#464—21no—
92#463—21no—
93#463—21no—
94#462C2×C411yescharacteristic
95#462C2×C411yescharacteristic
96#461C2×C411yescharacteristic
97#461C2×C411yescharacteristic
98#460C2×C411yescharacteristic
99#460C2×C411yescharacteristic
100#459C2×C411yescharacteristic
101#459C2×C411yescharacteristic
102C8×Q8C2×C411yescharacteristic
103C8×Q8C2×C411yescharacteristic
104#444—44no—
105#443—44no—
106C4×D8—21no—
107C4×D8—21no—
108#436—21no—
109#436—21no—
110#435—41no—
111#435—41no—
112C8×D4—21no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4—21no—
116C8×D4C2×C411yescharacteristic
117C8×D4C2×C411yescharacteristic
118#431—21no—
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#108—21no—
122C2×#108—21no—
123#424—21no—
124#424—21no—
125#424—41no—
126#424—41no—
127#423—21no—
128#423—21no—
129#423C2311yescharacteristic
130#423C2311yescharacteristic
131C2×C4⋊C8—21no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×#104—21no—
138C2×#104—21no—
139C2×#104—21no—
140C2×#104—21no—
141C2×#104D411yescharacteristic
142C2×#104D411yescharacteristic
143#416—21no—
144#416—21no—
145#408—21no—
146#408—21no—
147C4×M16C2311yescharacteristic
148C2×#97—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C43C2311yescharacteristic
154#341—21no—
155#341—21no—
156#339D411yescharacteristic
157#339D411yescharacteristic
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#335C2×C411yescharacteristic
163#335C2×C411yescharacteristic
164#335C2×C411yescharacteristic
165#335C2×C411yescharacteristic
166#334—21no—
167#334—21no—
168#334—21no—
169#334—21no—
170#334—21no—
171#334—21no—
172#334D411yescharacteristic
173#334D411yescharacteristic
174#334D411yescharacteristic
175#334D411yescharacteristic
176#332C2×C411yescharacteristic
177#332C2×C411yescharacteristic
178#332C2×C411yescharacteristic
179#332C2×C411yescharacteristic
180#331C2×C411yescharacteristic
181#331C2×C411yescharacteristic
182#331C2×C411yescharacteristic
183#331C2×C411yescharacteristic
184Q8⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188D4⋊C8—21no—
189D4⋊C8—21no—
190D4⋊C8—41no—
191D4⋊C8—41no—
192#322—21no—
193#322—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41C2×D411yescharacteristic
197#135—42no—
198#135—42no—
199#135—42no—
200#135—42no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×SD16—21no—
204C2×SD16—21no—
205C2×SD16—21no—
206C2×SD16—21no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—21no—
211#131—42no—
212#131—42no—
213#131—42no—
214#131—42no—
215#131—81no—
216#131—81no—
217#131—81no—
218#131—81no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—41no—
222C2×M16—41no—
223C2×M16—41no—
224C2×M16—41no—
225C2×M16C2×D411yescharacteristic
226C2×M16C2×D411yescharacteristic
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—41no—
235C22×C8—41no—
236C22×C8—41no—
237C22×C8—41no—
238C22×C8#4111yescharacteristic
239C22×C8#4111yescharacteristic
240C4⋊Q8C22×C411yescharacteristic
241C4⋊Q8C22×C411yescharacteristic
242#127C22×C411yescharacteristic
243#127C22×C411yescharacteristic
244#124—21no—
245#124—21no—
246#121—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250C4×Q8C22×C411yescharacteristic
251C4×Q8#4111yescharacteristic
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C22×C411yescharacteristic
259C4×D4#4111yescharacteristic
260#117—21no—
261#117C2×D411yescharacteristic
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2411yescharacteristic
268#108—42no—
269#108—42no—
270#107—21no—
271#107—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—41no—
284C4⋊C8—41no—
285C4⋊C8—41no—
286C4⋊C8—41no—
287C4⋊C8C22×C411yescharacteristic
288C4⋊C8C22×C411yescharacteristic
289C4⋊C8C22×C411yescharacteristic
290C4⋊C8C22×C411yescharacteristic
291C4⋊C8C22×C411yescharacteristic
292C4⋊C8C22×C411yescharacteristic
293C4⋊C8C2×D411yescharacteristic
294C4⋊C8C2×D411yescharacteristic
295C4⋊C8C2×D411yescharacteristic
296C4⋊C8C2×D411yescharacteristic
297#104—22no—
298#104—22no—
299#104—42no—
300#104—42no—
301#104—81no—
302#104—81no—
303#103—21no—
304#103—21no—
305#103—21no—
306#103—21no—
307#102—21no—
308#102—21no—
309#102—21no—
310#102—21no—
311#102—21no—
312#102—21no—
313#102—21no—
314#102—21no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—41no—
320C22⋊C8—41no—
321C22⋊C8—41no—
322C22⋊C8—41no—
323#97—21no—
324#97—42no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—42no—
333C4×C8C22×C411yescharacteristic
334C4×C8C22×C411yescharacteristic
335C4×C8C2×D411yescharacteristic
336C4×C8C2×D411yescharacteristic
337#41—21no—
338#41—21no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346#41—41no—
347#41AGL(1,ℤ/8)12yes—
348#41AGL(1,ℤ/8)12yes—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8C4×D411yescharacteristic
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4C4×D411yescharacteristic
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C42−1+411yescharacteristic
372Q16—42no—
373Q16—42no—
374SD16—42no—
375SD16—42no—
376SD16—42no—
377SD16—42no—
378D8—42no—
379D8—42no—
380D8—42no—
381D8—81no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—21no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—82no—
391M16—82no—
392M16—82no—
393M16—82no—
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—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—81no—
424C2×C8—82no—
425C2×C8—82no—
426C2×C8—82no—
427C2×C8—82no—
428C2×C8C4×D411yescharacteristic
429C2×C8C4×D411yescharacteristic
430C2×C8C4×D411yescharacteristic
431C2×C8C4×D411yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4C4×D411yescharacteristic
439C4⋊C4C4×D411yescharacteristic
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—41no—
455C42—41no—
456C42—42no—
457C42C23×C411yescharacteristic
458C42C22×D411yescharacteristic
459C42C2×#4111yescharacteristic
460C42H(5,𝔽2)11yescharacteristic
461C23—21no—
462C23—41no—
463C23—41no—
464C23#54711yescharacteristic
465Q8—21no—
466Q8—41no—
467Q8—41no—
468Q8—41no—
469Q8—41no—
470Q8#44212yes—
471D4—22no—
472D4—22no—
473D4—21no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—81no—
483D4—81no—
484D4—81no—
485D4—81no—
486D4—81no—
487D4—81no—
488D4#44212yes—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4C2×C4×D411yescharacteristic
513C2×C4C2×C4×D411yescharacteristic
514C2×C4#52011yescharacteristic
515C2×C4D4211yescharacteristic
516C2×C4#54811yescharacteristic
517C2×C4#55311yescharacteristic
518C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
519C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
520C2×C4#57711yescharacteristic
521C2×C4#57711yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—81no—
539C8—81no—
540C8—81no—
541C8—81no—
542C22—21no—
543C22—21no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22#236111yescharacteristic
554C22#334511yescharacteristic
555C22#334511yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—22no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4C2×#44211yescharacteristic
567C4#295211yescharacteristic
568C4#295511yescharacteristic
569C4#295711yescharacteristic
570C4#326511yescharacteristic
571C4#326511yescharacteristic
572C2—21no—
573C2—21no—
574C2—42no—
575C2—161no—
576C2—161no—
577C2#2301711yescharacteristic
578C2#2302911yescharacteristic
579C2#2304211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#19125C2
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. #23016C2semidirect
2. #23015C2semidirect
4. #18073C2semidirect
5. #17826C2semidirect
6. #9852C2semidirect
7. #9625C2semidirect
8. #9582C2semidirect
9. #9578C2semidirect
10. #8995C2semidirect
11. #8995C2semidirect
12. #8932C2semidirect
13. #8932C2semidirect
14. #8279C2semidirect
15. #8193C2semidirect
40. #1682C4semidirect
41. #1682C4semidirect
42. #1681C4semidirect
43. #1681C4semidirect
44. #1674C4semidirect
45. #1673C4semidirect
46. #1668C4semidirect
47. #1667C4semidirect

Maximal subgroups

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

subgroupindex
#230162
#230152
#191252
#180732
#178262
#98522
#96252
#95822
#95782
#89952
#89322
#82792
#81932