Finite Group Databasebeta

← #6528567

Subgroups — #6528567

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35180C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#29330C214yesmaximal, complemented
4#27977C214yesmaximal, complemented
5#27750C214yesmaximal, complemented
6#27748C214yesmaximal, complemented
7#27730C214yesmaximal, complemented
8#27728C214yesmaximal, complemented
9#21860C211yesmaximal, characteristic, complemented
10#21859C211yesmaximal, characteristic
11#21386C211yesmaximal, characteristic
12#21385C211yesmaximal, characteristic, complemented
13#20533C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2850C2214yes—
19#2639—22no—
20#2592—22no—
21#2571—22no—
22#2571—22no—
23#2571C2214yes—
24#2571C2214yes—
25#2568C2212yes—
26#2563C2212yes—
27#2562—22no—
28#2548C2212yes—
29#2547C2212yes—
30C2×#400—22no—
31#2311—21no—
32#2311—21no—
33C2×#389—21no—
34C2×#389C2212yes—
35C2×#384C2211yescharacteristic
36C2×#382—21no—
37C2×#382C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#2101C2218yes—
41#2074C2218yes—
42#2073C2218yes—
43#2042C2218yes—
44#1990C2212yes—
45#1990C2212yes—
46#1988C2212yes—
47#1987C2212yes—
48#1982C2212yes—
49#1981C2212yes—
50#1958C2218yes—
51#1957C2212yes—
52#1956C2212yes—
53#1934C2218yes—
54#1933C2218yes—
55#1874C2212yes—
56#1874C2214yes—
57#1874C2212yes—
58#1872C2212yes—
59#1872C2214yes—
60#1872C2212yes—
61#1835C2218yes—
62#1803C2214yes—
63#1802C2214yes—
64C8×C4⋊C4C2214yes—
65C8×C4⋊C4C2214yes—
66C4×#103C2214yes—
67C4×#102C2214yes—
68C4×C22⋊C8C2214yes—
69C22×#41—21no—
70#533—24no—
71#533—24no—
72#532—24no—
73#532—24no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#127C2311yescharacteristic
78C2×#125—21no—
79C2×#124—21no—
80C2×#124C2314yes—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8C2312yes—
83C2×#121—21no—
84C2×#121C2312yes—
85C2×C4×Q8—21no—
86C2×C4×Q8C2312yes—
87C2×C4×D4—21no—
88C2×C4×D4C2312yes—
89C2×#117—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C42—21no—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101#431—24no—
102#431—24no—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×#103C2318yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#102C2318yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2312yes—
115C2×C4×C8C2318yes—
116#400—24no—
117#400—28no—
118#400—28no—
119#397—24no—
120#394—24no—
121#393—24no—
122#391—22no—
123#391—22no—
124#391C2314yes—
125#390—22no—
126#390—22no—
127#390C2314yes—
128#389—24no—
129#389—24no—
130#389—24no—
131#389C2318yes—
132#388—24no—
133#388—24no—
134#386—24no—
135#386—24no—
136#384—21no—
137#384—21no—
138#384C2312yes—
139#384C2312yes—
140#382—24no—
141#382—24no—
142#382—24no—
143#382C2314yes—
144#382C2318yes—
145#380—22no—
146#380—22no—
147#380—22no—
148#380—22no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4D412yes—
152C4×C4⋊C4D412yes—
153C4×C4⋊C4C2312yes—
154C4×C4⋊C4C2312yes—
155C4×C4⋊C4C2312yes—
156C4×C4⋊C4C2312yes—
157C2×#95—21no—
158C2×#95—22no—
159C2×#95—22no—
160C2×#95—22no—
161C2×#95D412yes—
162#340C2318yes—
163#340C2318yes—
164#340C2314yes—
165#340C2314yes—
166#340C2314yes—
167#340C2314yes—
168#336C2318yes—
169C2×#41—44no—
170C2×#41—44no—
171C2×#41—44no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8C2412yes—
175C22×D4—22no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C2412yes—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C22×C8C2414yes—
189C22×C8C2414yes—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196#127—22no—
197#127—22no—
198#125—44no—
199#124—48no—
200#124—44no—
201C22⋊Q8—24no—
202C22⋊Q8—24no—
203C22⋊Q8—44no—
204#121—24no—
205#121—24no—
206#121—44no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—44no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—44no—
213#117—44no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—84no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2414yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42C2×D414yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4114yes—
261C2×C42#4114yes—
262C2×C42C2414yes—
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265C4⋊C8—24no—
266C4⋊C8—24no—
267#103—48no—
268#103—44no—
269#103—44no—
270#102—44no—
271#102—44no—
272#102—48no—
273C22⋊C8—24no—
274C22⋊C8—24no—
275C4×C8—24no—
276C4×C8—24no—
277C4×C8—48no—
278#95—22no—
279#95—22no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95C2×D414yes—
288#95C2×D414yes—
289C24—22no—
290C24C22×D411yescharacteristic
291#41—816no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—81no—
301C2×Q8—81no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—81no—
314C2×D4—81no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4#12414yes—
344C22×C4#12414yes—
345C22×C4#13514yes—
346C22×C4#13514yes—
347C22×C4C22×D414yes—
348C22×C4C22×D414yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4C2×#4112yes—
356C22×C4C2511yescharacteristic
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—28no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—88no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C22⋊C4—84no—
397C42—28no—
398C42—44no—
399C42—44no—
400C42—44no—
401C42—44no—
402C42#12414yes—
403C42#12414yes—
404C42C2×#4112yes—
405C42C2×#4112yes—
406C42C2×#4112yes—
407C42C2×#4112yes—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23C2×#12412yes—
420C23C2×#13512yes—
421C23C2×#13512yes—
422C23C23×D411yescharacteristic
423Q8—44no—
424Q8—44no—
425Q8—84no—
426Q8—84no—
427D4—44no—
428D4—44no—
429D4—84no—
430D4—88no—
431D4—84no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—88no—
470C2×C4—88no—
471C2×C4#488116yes—
472C2×C4C2×#12412yes—
473C2×C4C2×#12412yes—
474C2×C4C2×#12412yes—
475C2×C4C2×#12412yes—
476C2×C4C2×#12412yes—
477C2×C4C2×#13512yes—
478C2×C4C2×#13512yes—
479C2×C4C2×#13512yes—
480C2×C4C2×#13512yes—
481C2×C4C22×#4111yescharacteristic
482C2×C4C22×#4111yescharacteristic
483C2×C4#58512yes—
484C2×C4#58512yes—
485C8—44no—
486C8—44no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22C2×#48814yes—
501C22C22×#12411yescharacteristic
502C22#348511yescharacteristic
503C22#348711yescharacteristic
504C22C22×#13511yescharacteristic
505C22C22×#13511yescharacteristic
506C22#358511yescharacteristic
507C22#358611yescharacteristic
508C4—22no—
509C4—22no—
510C4—22no—
511C4—44no—
512C4—44no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—82no—
517C4—84no—
518C4—82no—
519C4C2×#48814yes—
520C4C2×#48814yes—
521C4#351212yes—
522C4#351212yes—
523C2—24no—
524C2—162no—
525C2#3564911yescharacteristic
526C2C22×#48811yescharacteristic
527C2#6062111yescharacteristic
528C2#6170011yescharacteristic
529C2#6170111yescharacteristic
530C2#6188211yescharacteristic
531C2#6188511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2571C2semidirect
3. #29330C2semidirect
4. #27977C2semidirect
5. #27750C2semidirect
6. #27748C2semidirect
7. #27730C2semidirect
8. #27728C2semidirect
9. #21860C2semidirect
12. #21385C2semidirect
13. #20533C2semidirect

Maximal subgroups

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

subgroupindex
#351802
C2×#25712
#293302
#279772
#277502
#277482
#277302
#277282
#218602
#218592
#213862
#213852
#205332