Finite Group Databasebeta

← #501379

Subgroups — #501379

Order 512.

823 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
2C22×C22⋊C16C211yesmaximal, characteristic, complemented
3#32930C211yesmaximal, characteristic
4#20658C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic, complemented
6#13493C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic, complemented
10#13059C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic, complemented
12#12921C211yesmaximal, characteristic, complemented
13#12868C211yesmaximal, characteristic, complemented
14#12835C211yesmaximal, characteristic, complemented
15#12834C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C23×C16C2211yescharacteristic
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
35C2×C22⋊C16—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—
81C22×C16—22no—
82C22×C16—22no—
83C22×C16—22no—
84C22×C16—22no—
85C22×C16—41no—
86C22×C16—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—
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—21no—
246C2×C16—44no—
247C2×C16—44no—
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—22no—
280C24—22no—
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—28no—
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—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23#46612yes—
441C23#48012yes—
442C23#50812yes—
443C23#50812yes—
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—22no—
521C22—22no—
522C22—22no—
523C22—416no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—161no—
533C22—161no—
534C22—161no—
535C22—161no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22#223414yescomplemented
540C22C2×#46611yescharacteristic
541C22C2×#48011yescharacteristic
542C22#318611yescharacteristic
543C22C2×#50811yescharacteristic
544C22C2×#50811yescharacteristic
545C22C2×#50911yescharacteristic
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—24no—
562C2—42no—
563C2—42no—
564C2—321no—
565C2#2455011yescharacteristic
566C2#3360011yescharacteristic
567C2#3360511yescharacteristic
568C2C2×#223411yescharacteristic
569C2#3364511yescharacteristic
570C2#3375411yescharacteristic
571C2#3375711yescharacteristic
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. C22×C22⋊C16C2semidirect
4. #20658C2semidirect
5. #13494C2semidirect
6. #13493C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13059C2semidirect
10. #13059C2semidirect
11. #12992C2semidirect
12. #12921C2semidirect
13. #12868C2semidirect
14. #12835C2semidirect
15. #12834C2semidirect
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
539. 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×C22⋊C162
#329302
#206582
#134942
#134932
#133812
#133802
#130592
#129922
#129212
#128682
#128352
#128342