Finite Group Databasebeta

← #501774

Subgroups — #501774

Order 512.

838 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2233C211yesmaximal, characteristic, complemented
2C22×#357C211yesmaximal, characteristic, complemented
3#24267C211yesmaximal, characteristic
4D4×#107C211yesmaximal, characteristic, complemented
5#13527C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13381C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13131C211yesmaximal, characteristic, complemented
11#12991C211yesmaximal, characteristic, complemented
12#12976C211yesmaximal, characteristic, complemented
13#12920C211yesmaximal, characteristic, complemented
14#12835C211yesmaximal, characteristic, complemented
15#12731C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic, complemented
23#2408—21no—
24D4×C4⋊C4—21no—
25#2233—24no—
26C2×#367C2211yescharacteristic, complemented
27C2×#367C2211yescharacteristic, complemented
28C2×#367C2211yescharacteristic, complemented
29C2×#367C2211yescharacteristic, complemented
30C2×#357—24no—
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic, complemented
41C2×C22⋊C16C2211yescharacteristic, complemented
42#2054—21no—
43#1999—21no—
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1947—21no—
47#1852—21no—
48#1852—21no—
49#1845C2211yescharacteristic, complemented
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic, complemented
53#1786—21no—
54C4×#107C2211yescharacteristic, complemented
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1374C2211yescharacteristic, complemented
60#1374C2211yescharacteristic, complemented
61C23×D4—21no—
62C22×D8—22no—
63C22×D8—22no—
64C22×D8—21no—
65C22×D8—22no—
66C22×D8—22no—
67C22×D8—41no—
68C22×D8—41no—
69C22×D8C2311yescharacteristic
70C22×D8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×#121—21no—
74C2×#121—21no—
75C2×#120—41no—
76C2×C4×D4—41no—
77C2×C4×D4D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C16—22no—
81C22×C16—22no—
82C22×C16—41no—
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#466—22no—
88#466—22no—
89#466—41no—
90#466—41no—
91C8×D4—22no—
92C8×D4—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—21no—
96C2×#107—21no—
97C2×#107—21no—
98C2×#107—41no—
99C2×#107—41no—
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×C22⋊C8D411yescharacteristic
112C2×C22⋊C8D411yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#394—41no—
115#392—41no—
116#390—21no—
117#390—21no—
118#389—41no—
119#385—41no—
120#385—41no—
121#384—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#379—41no—
126C4×C4⋊C4—21no—
127#367—22no—
128#367—22no—
129#367—41no—
130#367—41no—
131#357—22no—
132#357—22no—
133#357—22no—
134#357—22no—
135#357—22no—
136#357—41no—
137#357—41no—
138#357D414yescomplemented
139C22⋊C16—22no—
140C22⋊C16—22no—
141C22⋊C16—41no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146C25—41no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4—42no—
154C22×D4—41no—
155C22×D4—42no—
156C22×D4—81no—
157C22×D4—81no—
158C22×D4D811yescharacteristic
159C23×C4—41no—
160C23×C4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×D8—28no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—44no—
175C2×D8—44no—
176C2×D8—42no—
177C2×D8—42no—
178C2×D8—44no—
179C2×D8—42no—
180C2×D8C2×D412yes—
181C2×D8#4112yes—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8#4112yes—
196C22×C8#4112yes—
197C22×C8C2411yescharacteristic
198#121—42no—
199#121—42no—
200#121—81no—
201#121—81no—
202#120—84no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—84no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4D811yescharacteristic
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4D811yescharacteristic
231C2×C22⋊C4D811yescharacteristic
232C2×C42—41no—
233C2×C42C2×D411yescharacteristic
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—44no—
246C2×C16C2×D412yes—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—42no—
259#107—42no—
260#107—81no—
261#107C2×D412yes—
262C4⋊C8—22no—
263#102—42no—
264#102—42no—
265#102—81no—
266#102—81no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C4×C8—21no—
270C4×C8—21no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—41no—
276C24—41no—
277C24—82no—
278C24—82no—
279C24—82no—
280C24—82no—
281C24—82no—
282C24—161no—
283C24—161no—
284C24C2×D811yescharacteristic
285C24C2×D811yescharacteristic
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—48no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—84no—
302C2×D4—84no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—84no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4—161no—
312C2×D4—161no—
313C2×D4SD3214yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—161no—
335C22×C4—161no—
336C22×C4#12312yes—
337C22×C4C2×D811yescharacteristic
338C22×C4C2×D811yescharacteristic
339C22×C4C2×D811yescharacteristic
340C22×C4C2×D811yescharacteristic
341C22×C4#13712yes—
342C22×C4C22×D411yescharacteristic
343D8—22no—
344D8—22no—
345D8—22no—
346D8—22no—
347D8—48no—
348D8—41no—
349D8—41no—
350D8—41no—
351D8—41no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8#12312yes—
369C2×C8C22×D411yescharacteristic
370C2×C8C2×#4111yescharacteristic
371C2×C8C2×#4111yescharacteristic
372C2×C8C2×#4111yescharacteristic
373C2×C8H(5,𝔽2)11yescharacteristic
374C2×C8H(5,𝔽2)11yescharacteristic
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—161no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—161no—
402C22⋊C4—161no—
403C42—21no—
404C42—21no—
405C42—82no—
406C16—22no—
407C16—41no—
408C16—41no—
409C16—41no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—42no—
421C23—42no—
422C23—88no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—164no—
436C23—164no—
437C23—164no—
438C23—164no—
439C23—164no—
440C23—164no—
441C23#48012yes—
442C23C2×SD3212yes—
443C23C2×SD3212yes—
444C23#50912yes—
445C23C22×D811yescharacteristic
446D4—28no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—88no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—81no—
456D4—164no—
457D4—164no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—164no—
495C2×C4—164no—
496C2×C4#48012yes—
497C2×C4C2×SD3212yes—
498C2×C4C2×#12311yescharacteristic
499C2×C4D4211yescharacteristic
500C2×C4#54611yescharacteristic
501C2×C4C2×#13711yescharacteristic
502C2×C4#57611yescharacteristic
503C2×C4#57711yescharacteristic
504C8—22no—
505C8—41no—
506C8—41no—
507C8#52511yescharacteristic
508C8#53811yescharacteristic
509C8#54611yescharacteristic
510C8#55111yescharacteristic
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—1616no—
531C22—161no—
532C22—161no—
533C22—161no—
534C22—161no—
535C22—161no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22#223314yescomplemented
540C22C2×#48011yescharacteristic
541C22D4×D811yescharacteristic
542C22#329411yescharacteristic
543C22C22×SD3211yescharacteristic
544C22C2×#50911yescharacteristic
545C22#341511yescharacteristic
546C22#341611yescharacteristic
547C4—22no—
548C4—22no—
549C4—41no—
550C4—41no—
551C4—82no—
552C4—161no—
553C4—161no—
554C4—161no—
555C4—161no—
556C4—161no—
557C4#241811yescharacteristic
558C4#309611yescharacteristic
559C4#326611yescharacteristic
560C4#331411yescharacteristic
561C2—24no—
562C2—42no—
563C2—162no—
564C2—162no—
565C2#2486111yescharacteristic
566C2C2×#223311yescharacteristic
567C2#3365011yescharacteristic
568C2#3383011yescharacteristic
569C2D4×SD3211yescharacteristic
570C2#3386211yescharacteristic
571C2#3387311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#24267C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2233C2semidirect
2. C22×#357C2semidirect
4. D4×#107C2semidirect
5. #13527C2semidirect
6. #13480C2semidirect
7. #13381C2semidirect
8. #13381C2semidirect
9. #13131C2semidirect
10. #13131C2semidirect
11. #12991C2semidirect
12. #12976C2semidirect
13. #12920C2semidirect
14. #12835C2semidirect
15. #12731C2semidirect
22. C22×#107C22semidirect
26. C2×#367C22semidirect
27. C2×#367C22semidirect
28. C2×#367C22semidirect
29. C2×#367C22semidirect
40. C2×C22⋊C16C22semidirect
41. C2×C22⋊C16C22semidirect
49. #1845C22semidirect
52. #1808C22semidirect
54. C4×#107C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1374C22semidirect
60. #1374C22semidirect
138. #357D4semidirect
539. C22#2233semidirect

Maximal subgroups

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

subgroupindex
C2×#22332
C22×#3572
#242672
D4×#1072
#135272
#134802
#133812
#131312
#129912
#129762
#129202
#128352
#127312