Finite Group Databasebeta

← #501791

Subgroups — #501791

Order 512.

803 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×#2232C211yesmaximal, characteristic, complemented
2C2×#2140C211yesmaximal, characteristic, complemented
3#24267C211yesmaximal, characteristic
4#22656C211yesmaximal, characteristic, complemented
5#13528C211yesmaximal, characteristic, complemented
6#13482C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13381C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13130C211yesmaximal, characteristic, complemented
11#12991C211yesmaximal, characteristic, complemented
12#12977C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14#12836C211yesmaximal, characteristic, complemented
15#12731C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C22×#107C2211yescharacteristic, complemented
23#2408—21no—
24#2352—21no—
25#2232—24no—
26C2×#367C2211yescharacteristic, complemented
27C2×#367C2211yescharacteristic, complemented
28C2×#366C2211yescharacteristic, complemented
29C2×#366C2211yescharacteristic, complemented
30#2140—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—
79C2×M32—22no—
80C2×M32—22no—
81C2×M32—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—41no—
128#367—41no—
129#366—22no—
130#366—22no—
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
231M32—44no—
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—21no—
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—41no—
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—21no—
417C23—42no—
418C23—42no—
419C23—88no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—164no—
433C23—164no—
434C23—164no—
435C23—164no—
436C23—164no—
437C23—164no—
438C23#48012yes—
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—41no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—1616no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22—161no—
533C22—161no—
534C22C2×#48011yescharacteristic
535C22D4×D811yescharacteristic
536C22#329411yescharacteristic
537C22C2×#50911yescharacteristic
538C22#341411yescharacteristic
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—42no—
556C2—42no—
557C2—162no—
558C2—162no—
559C2#2486111yescharacteristic
560C2#3364811yescharacteristic
561C2#3365011yescharacteristic
562C2#3383011yescharacteristic
563C2#3384111yescharacteristic
564C2#3387711yescharacteristic
565C2#3388911yescharacteristic
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×#2232C2semidirect
2. C2×#2140C2semidirect
4. #22656C2semidirect
5. #13528C2semidirect
6. #13482C2semidirect
7. #13381C2semidirect
8. #13381C2semidirect
9. #13131C2semidirect
10. #13130C2semidirect
11. #12991C2semidirect
12. #12977C2semidirect
13. #12921C2semidirect
14. #12836C2semidirect
15. #12731C2semidirect
22. C22×#107C22semidirect
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

Maximal subgroups

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

subgroupindex
C2×#22322
C2×#21402
#242672
#226562
#135282
#134822
#133812
#131312
#131302
#129912
#129772
#129212
#128362
#127312