Finite Group Databasebeta

← #501778

Subgroups — #501778

Order 512.

809 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2234C211yesmaximal, characteristic, complemented
2C22×#357C211yesmaximal, characteristic, complemented
3#24267C211yesmaximal, characteristic
4#22656C211yesmaximal, characteristic, complemented
5#13528C211yesmaximal, characteristic, complemented
6#13482C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13130C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic, complemented
12#12976C211yesmaximal, characteristic, complemented
13#12920C211yesmaximal, characteristic, complemented
14#12834C211yesmaximal, characteristic, complemented
15#12731C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#427C2211yescharacteristic, complemented
22C22×#107C2211yescharacteristic
23#2408—21no—
24#2352—21no—
25#2234—24no—
26C2×#367C2211yescharacteristic, complemented
27C2×#367C2211yescharacteristic, complemented
28C2×#366C2211yescharacteristic, complemented
29C2×#366C2211yescharacteristic, 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#1949—21no—
47#1854—21no—
48#1853—21no—
49#1846C2211yescharacteristic, 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×D4D411yescharacteristic
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C16—22no—
80C22×C16—22no—
81C22×C16—41no—
82C22×C16C2311yescharacteristic
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86#466—22no—
87#466—22no—
88#466—41no—
89#466—41no—
90C8×D4—22no—
91C8×D4—22no—
92#427—22no—
93#427—41no—
94C2×#107—22no—
95C2×#107—21no—
96C2×#107—41no—
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#106—21no—
104C2×#106—21no—
105C2×C4⋊C8D411yescharacteristic
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×C22⋊C8D411yescharacteristic
111C2×C22⋊C8D411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#394—41no—
114#392—41no—
115#390—21no—
116#390—21no—
117#388—41no—
118#387—41no—
119#386—41no—
120#382—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#379—41no—
125C4×C4⋊C4—21no—
126C4×C22⋊C4—41no—
127#367—22no—
128#367—41no—
129#366—22no—
130#366—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×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×D8—28no—
163C2×D8—21no—
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—44no—
174C2×D8—44no—
175C2×D8—42no—
176C2×D8—42no—
177C2×D8—44no—
178C2×D8—42no—
179C2×D8C2×D412yes—
180C2×D8#4112yes—
181C22×C8—21no—
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×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8#4112yes—
195C22×C8#4112yes—
196C22×C8C2411yescharacteristic
197#121—42no—
198#121—42no—
199#121—81no—
200#121—81no—
201#120—84no—
202C4×D4—22no—
203C4×D4—22no—
204#117—42no—
205#117—81no—
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—41no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4D811yescharacteristic
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4D811yescharacteristic
227C2×C22⋊C4D811yescharacteristic
228C2×C42—41no—
229C2×C42—41no—
230C2×C42C2×D411yescharacteristic
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
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—44no—
243C2×C16C2×D412yes—
244#107—21no—
245#107—21no—
246#107—21no—
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—42no—
256#107C2×D412yes—
257#106—42no—
258#106—81no—
259C4⋊C8—22no—
260#102—42no—
261#102—42no—
262#102—81no—
263#102—81no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C4×C8—21no—
267C4×C8—21no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24—41no—
275C24—41no—
276C24—82no—
277C24—82no—
278C24—82no—
279C24—82no—
280C24—82no—
281C24—161no—
282C24—161no—
283C24C2×D811yescharacteristic
284C24C2×D811yescharacteristic
285C2×D4—22no—
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—
313C22×C4—21no—
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—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—161no—
332C22×C4#12312yes—
333C22×C4C2×D811yescharacteristic
334C22×C4C2×D811yescharacteristic
335C22×C4C2×D811yescharacteristic
336C22×C4C2×D811yescharacteristic
337C22×C4#13712yes—
338C22×C4C22×D411yescharacteristic
339D8—22no—
340D8—22no—
341D8—22no—
342D8—22no—
343D8—48no—
344D8—41no—
345D8—41no—
346D8—41no—
347D8—41no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8#12312yes—
365C2×C8C22×D411yescharacteristic
366C2×C8C2×#4111yescharacteristic
367C2×C8C2×#4111yescharacteristic
368C2×C8C2×#4111yescharacteristic
369C2×C8H(5,𝔽2)11yescharacteristic
370C2×C8H(5,𝔽2)11yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—82no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—161no—
395C22⋊C4—161no—
396C22⋊C4—161no—
397C42—21no—
398C42—21no—
399C42—82no—
400C42—161no—
401C16—22no—
402C16—41no—
403C16—41no—
404C16—41no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—42no—
417C23—42no—
418C23—88no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—164no—
432C23—164no—
433C23—164no—
434C23—164no—
435C23—164no—
436C23—164no—
437C23#48012yes—
438C23#50812yes—
439C23#50912yes—
440C23C22×D811yescharacteristic
441D4—44no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—88no—
447D4—81no—
448D4—81no—
449D4—81no—
450D4—81no—
451D4—164no—
452D4—164no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
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—164no—
491C2×C4#48012yes—
492C2×C4C2×#12311yescharacteristic
493C2×C4D4211yescharacteristic
494C2×C4#54611yescharacteristic
495C2×C4C2×#13711yescharacteristic
496C2×C4#57611yescharacteristic
497C2×C4#57711yescharacteristic
498C8—22no—
499C8—41no—
500C8—41no—
501C8#52511yescharacteristic
502C8#53811yescharacteristic
503C8#54611yescharacteristic
504C8#55111yescharacteristic
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—1616no—
525C22—161no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22—161no—
533C22#223414yescomplemented
534C22C2×#48011yescharacteristic
535C22D4×D811yescharacteristic
536C22#329411yescharacteristic
537C22C2×#50811yescharacteristic
538C22C2×#50911yescharacteristic
539C22#341511yescharacteristic
540C22#341611yescharacteristic
541C4—22no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—82no—
546C4—161no—
547C4—161no—
548C4—161no—
549C4—161no—
550C4—161no—
551C4#241811yescharacteristic
552C4#309611yescharacteristic
553C4#326611yescharacteristic
554C4#331411yescharacteristic
555C2—24no—
556C2—81no—
557C2—162no—
558C2—162no—
559C2#2486111yescharacteristic
560C2C2×#223411yescharacteristic
561C2#3365011yescharacteristic
562C2#3383111yescharacteristic
563C2#3384611yescharacteristic
564C2#3386311yescharacteristic
565C2#3388511yescharacteristic
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×#2234C2semidirect
2. C22×#357C2semidirect
4. #22656C2semidirect
5. #13528C2semidirect
6. #13482C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13131C2semidirect
10. #13130C2semidirect
11. #12992C2semidirect
12. #12976C2semidirect
13. #12920C2semidirect
14. #12834C2semidirect
15. #12731C2semidirect
21. C2×#427C22semidirect
26. C2×#367C22semidirect
27. C2×#367C22semidirect
28. C2×#366C22semidirect
29. C2×#366C22semidirect
40. C2×C22⋊C16C22semidirect
41. C2×C22⋊C16C22semidirect
49. #1846C22semidirect
52. #1808C22semidirect
54. C4×#107C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1374C22semidirect
60. #1374C22semidirect
138. #357D4semidirect
533. C22#2234semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
C22×#3572
#242672
#226562
#135282
#134822
#133812
#133802
#131312
#131302
#129922
#129762
#129202
#128342
#127312