Finite Group Databasebeta

← #501384

Subgroups — #501384

Order 512.

807 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×#2234C211yesmaximal, characteristic, complemented
2C2×#2112C211yesmaximal, characteristic, complemented
3#32930C211yesmaximal, characteristic
4#20658C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic, complemented
10#13061C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic, complemented
12#12921C211yesmaximal, characteristic, complemented
13#12868C211yesmaximal, characteristic, complemented
14#12836C211yesmaximal, characteristic, complemented
15#12836C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C22×M32C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×#466C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#427C2211yescharacteristic
24#2847—21no—
25#2280—21no—
26#2234—24no—
27C2×#367C2211yescharacteristic, complemented
28C2×#367C2211yescharacteristic, complemented
29C2×#366C2211yescharacteristic, complemented
30C2×#366C2211yescharacteristic, complemented
31C2×#357C2211yescharacteristic, complemented
32C2×#357C2211yescharacteristic, complemented
33C2×#357C2211yescharacteristic, complemented
34C2×#357C2211yescharacteristic, complemented
35#2112—24no—
36C2×C22⋊C16C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40#1893—21no—
41#1893—21no—
42#1893—21no—
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1807—21no—
51#1807—21no—
52#1735—21no—
53#1380C2211yescharacteristic, complemented
54#1380C2211yescharacteristic, complemented
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1380C2211yescharacteristic, complemented
60#1380C2211yescharacteristic, complemented
61C24×C4D411yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8—22no—
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—22no—
67C23×C8—41no—
68C23×C8—41no—
69C23×C8D411yescharacteristic
70C23×C8D411yescharacteristic
71C23×C8D411yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×#121—21no—
76C2×#121—21no—
77C2×#121—21no—
78C2×#120—41no—
79C2×#117—21no—
80C2×#117—21no—
81C2×M32—22no—
82C2×M32—22no—
83C2×M32—22no—
84C2×M32—22no—
85C2×M32—41no—
86C2×M32—41no—
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91#466—22no—
92#466—22no—
93#466—22no—
94#466—22no—
95#466—41no—
96#466—41no—
97#427—22no—
98#427—22no—
99#427—41no—
100#427—41no—
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#106—21no—
109C2×#106—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113#386—41no—
114#386—41no—
115#386—41no—
116#381—41no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#379—41no—
125C4×C22⋊C4—41no—
126C4×C22⋊C4—41no—
127#367—22no—
128#367—41no—
129#366—22no—
130#366—41no—
131#357—22no—
132#357—22no—
133#357—41no—
134#357—41no—
135C22⋊C16—22no—
136C22⋊C16—22no—
137C22⋊C16—22no—
138C22⋊C16D414yescomplemented
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146C25D811yescharacteristic
147C22×D4—21no—
148C22×D4—41no—
149C22×D4—41no—
150C22×D4—41no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4D811yescharacteristic
158C23×C4D811yescharacteristic
159C23×C4D811yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—41no—
190C22×C8—41no—
191C22×C8—41no—
192C22×C8—44no—
193C22×C8—44no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D412yes—
199C22×C8C2×D411yescharacteristic
200C22×C8#4112yes—
201C22×C8#4112yes—
202C22×C8#4112yes—
203C22×C8C2411yescharacteristic
204#121—42no—
205#121—42no—
206#121—42no—
207#121—42no—
208#121—42no—
209#121—42no—
210#120—84no—
211#117—42no—
212#117—42no—
213#117—42no—
214#117—42no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
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⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C42—41no—
235C2×C42—41no—
236M32—44no—
237M32—44no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16C2×D412yes—
249C2×C16C2×D412yes—
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—21no—
259#107—21no—
260#107—21no—
261#107—21no—
262#107—21no—
263#107—21no—
264#106—42no—
265#106—42no—
266#102—42no—
267#102—42no—
268#102—42no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24C2×D811yescharacteristic
285C24C2×D811yescharacteristic
286C24C2×D811yescharacteristic
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4—82no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4#12112yes—
337C22×C4#12312yes—
338C22×C4C2×D811yescharacteristic
339C22×C4C2×D811yescharacteristic
340C22×C4C2×D811yescharacteristic
341C22×C4#13512yes—
342C22×C4#13712yes—
343C22×C4C22×D411yescharacteristic
344D8—41no—
345D8—41no—
346D8—41no—
347D8—41no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8#12112yes—
371C2×C8#12312yes—
372C2×C8C22×D411yescharacteristic
373C2×C8C2×#4111yescharacteristic
374C2×C8C2×#4111yescharacteristic
375C2×C8C2×#4111yescharacteristic
376C2×C8H(5,𝔽2)11yescharacteristic
377C2×C8H(5,𝔽2)11yescharacteristic
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—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C42—82no—
405C42—82no—
406C16—22no—
407C16—22no—
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—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23#46612yes—
443C23#48012yes—
444C23#50912yes—
445C23C22×D811yescharacteristic
446D4—81no—
447D4—81no—
448D4—81no—
449D4—81no—
450D4—164no—
451D4—164no—
452D4—164no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
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—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4#46612yes—
497C2×C4#48012yes—
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12311yescharacteristic
500C2×C4#53411yescharacteristic
501C2×C4C2×#13511yescharacteristic
502C2×C4C2×#13711yescharacteristic
503C2×C4#57511yescharacteristic
504C8—22no—
505C8—22no—
506C8—41no—
507C8#54611yescharacteristic
508C8#54611yescharacteristic
509C8#55111yescharacteristic
510C8#55111yescharacteristic
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—416no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—161no—
534C22—161no—
535C22—161no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22—161no—
540C22C2×#46611yescharacteristic
541C22C2×#48011yescharacteristic
542C22#318611yescharacteristic
543C22C2×#50911yescharacteristic
544C22#341411yescharacteristic
545C22#341411yescharacteristic
546C22#341411yescharacteristic
547C4—22no—
548C4—22no—
549C4—41no—
550C4—161no—
551C4—161no—
552C4—161no—
553C4—161no—
554C4—161no—
555C4—161no—
556C4—161no—
557C4#241211yescharacteristic
558C4#305611yescharacteristic
559C4#309111yescharacteristic
560C4#319011yescharacteristic
561C2—42no—
562C2—42no—
563C2—42no—
564C2—321no—
565C2#2455011yescharacteristic
566C2#3360411yescharacteristic
567C2#3360511yescharacteristic
568C2#3364711yescharacteristic
569C2#3364811yescharacteristic
570C2#3376011yescharacteristic
571C2#3376311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#32930C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ 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. C2×#2112C2semidirect
4. #20658C2semidirect
5. #13494C2semidirect
6. #13492C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13061C2semidirect
10. #13061C2semidirect
11. #12992C2semidirect
12. #12921C2semidirect
13. #12868C2semidirect
14. #12836C2semidirect
15. #12836C2semidirect
27. C2×#367C22semidirect
28. C2×#367C22semidirect
29. C2×#366C22semidirect
30. C2×#366C22semidirect
31. C2×#357C22semidirect
32. C2×#357C22semidirect
33. C2×#357C22semidirect
34. C2×#357C22semidirect
53. #1380C22semidirect
54. #1380C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1380C22semidirect
60. #1380C22semidirect
138. C22⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
C2×#21122
#329302
#206582
#134942
#134922
#133812
#133802
#130612
#129922
#129212
#128682
#128362