Finite Group Databasebeta

← #483258

Subgroups — #483258

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62418C211yesmaximal, characteristic
2#33503C211yesmaximal, characteristic, complemented
3#17829C211yesmaximal, characteristic, complemented
4#11988C211yesmaximal, characteristic, complemented
5#11985C211yesmaximal, characteristic, complemented
6#11984C211yesmaximal, characteristic, complemented
7#11966C211yesmaximal, characteristic, complemented
8#11965C211yesmaximal, characteristic, complemented
9#11963C211yesmaximal, characteristic, complemented
10#11954C211yesmaximal, characteristic, complemented
11#11951C211yesmaximal, characteristic, complemented
12#11950C211yesmaximal, characteristic, complemented
13#11932C211yesmaximal, characteristic, complemented
14#11930C211yesmaximal, characteristic, complemented
15#11929C211yesmaximal, characteristic, complemented
16#3511—21no—
17#3393—21no—
18#3393C412yes—
19#3285—21no—
20#3285—21no—
21#3285C412yes—
22#3285C412yes—
23#3283—21no—
24#3283C412yes—
25#3210—21no—
26#3210—21no—
27#3206—21no—
28#3170C2211yescharacteristic
29#3170C2211yescharacteristic
30#3169C2211yescharacteristic
31#3147C2211yescharacteristic
32#3147C2211yescharacteristic
33C2×#494C2211yescharacteristic
34C8×#41C2211yescharacteristic
35#2925—21no—
36#2183—21no—
37#2169—21no—
38#2168—21no—
39#2168—21no—
40#2163C2211yescharacteristic
41#2163C2211yescharacteristic
42#2162C2211yescharacteristic
43#2150C2211yescharacteristic
44#2150C2211yescharacteristic
45C2×C4⋊C16C2211yescharacteristic
46#1572—21no—
47#1572—21no—
48#1570—21no—
49#1568C2211yescharacteristic
50#1568C2211yescharacteristic
51#1567C2211yescharacteristic
52#1564C2211yescharacteristic
53#1564C2211yescharacteristic
54C2×#335C2211yescharacteristic
55#1364C2211yescharacteristic
56#1364C2211yescharacteristic
57#1364C2211yescharacteristic
58#1364C2211yescharacteristic
59#1362C2211yescharacteristic
60#1362C2211yescharacteristic
61#1362C2211yescharacteristic
62#1362C2211yescharacteristic
63#1362C2211yescharacteristic
64#1362C2211yescharacteristic
65#1362C2211yescharacteristic
66#1362C2211yescharacteristic
67#1361C2211yescharacteristic
68#1361C2211yescharacteristic
69#1361C2211yescharacteristic
70#1361C2211yescharacteristic
71C2×#135—21no—
72C2×#135—21no—
73C2×#135C2×C412yes—
74C2×#135C2×C412yes—
75C22×Q16—21no—
76C22×Q16C2×C412yes—
77#556—22no—
78#555—41no—
79#554—41no—
80#553—41no—
81#553—41no—
82#548—22no—
83#548—22no—
84#547—22no—
85#542—41no—
86#542—41no—
87#537—41no—
88#533—21no—
89#533—21no—
90#532—21no—
91#532—21no—
92#532—21no—
93C2×C4⋊Q8—21no—
94#520—41no—
95C4×#41D411yescharacteristic
96C2×M32—21no—
97C2×M32—21no—
98C22×C16—21no—
99#500C2311yescharacteristic
100#495—21no—
101#495—21no—
102#495C2×C412yes—
103#495C2×C412yes—
104#494—21no—
105#494—21no—
106#494C2×C412yes—
107#494C2311yescharacteristic
108#494C2311yescharacteristic
109#493C2311yescharacteristic
110#492—21no—
111#492—21no—
112#487—21no—
113#487—21no—
114#486—21no—
115#467—21no—
116#467—21no—
117#467C2×C412yes—
118#467C2×C412yes—
119#466—21no—
120#466C2×C412yes—
121#464—22no—
122#464—22no—
123#464—22no—
124#464—22no—
125#461—22no—
126#461—22no—
127#452—22no—
128#452—22no—
129#452—22no—
130#452—22no—
131#448—22no—
132#448—22no—
133C8×Q8C2×C411yescharacteristic
134C8×Q8D411yescharacteristic
135C4×Q16—21no—
136C4×Q16—21no—
137C4×Q16C2×C412yes—
138C4×Q16C2×C412yes—
139C4×D8—21no—
140C4×D8C2×C412yes—
141C8×D4C2×C411yescharacteristic
142C8×D4C2×C411yescharacteristic
143C8×D4C2×C411yescharacteristic
144C8×D4D411yescharacteristic
145C8×D4D411yescharacteristic
146C8×D4D411yescharacteristic
147#433—41no—
148#433—41no—
149#432—21no—
150#432—21no—
151#431—21no—
152#431—21no—
153#431D411yescharacteristic
154#431D411yescharacteristic
155#431D411yescharacteristic
156#423—21no—
157C2×C4⋊C8—21no—
158#408—41no—
159#405—41no—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C4⋊C16—21no—
164C4⋊C16—21no—
165C4⋊C16—21no—
166C4⋊C16C2311yescharacteristic
167C4⋊C16C2311yescharacteristic
168C4⋊C16C2311yescharacteristic
169C4⋊C16C2311yescharacteristic
170C22⋊C16—21no—
171C22⋊C16—21no—
172C22⋊C16—21no—
173#346—21no—
174#346—21no—
175C4×C16—21no—
176#335—21no—
177#335—21no—
178#335C2311yescharacteristic
179#335C2311yescharacteristic
180#335C2311yescharacteristic
181#335C2311yescharacteristic
182#334—21no—
183#334—21no—
184#334—21no—
185#334—21no—
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—41no—
189C2×#41D811yescharacteristic
190C22×Q8—22no—
191#135—81no—
192#135—81no—
193#135—81no—
194#135—81no—
195#135—82no—
196#135—82no—
197C2×Q16—22no—
198C2×Q16—22no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—81no—
202C2×Q16—81no—
203C2×Q16—82no—
204C2×Q16C22×C412yes—
205C2×Q16C22×C412yes—
206C2×SD16—22no—
207C2×SD16—22no—
208C2×SD16—22no—
209C2×SD16—22no—
210C2×D8—21no—
211C2×D8C22×C412yes—
212C2×M16—41no—
213C22×C8—41no—
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C4⋊Q8—22no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—41no—
225C4⋊Q8—41no—
226#127—21no—
227#126—42no—
228#126—42no—
229#125—41no—
230#125—41no—
231#124—22no—
232#124—22no—
233#124—42no—
234#124—42no—
235#124—41no—
236#123—42no—
237#123—42no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—42no—
241C22⋊Q8—41no—
242C22⋊Q8—41no—
243C22⋊Q8—42no—
244C22⋊Q8—42no—
245#121—22no—
246#121—42no—
247#121—42no—
248#121—42no—
249#121—41no—
250#121—41no—
251#121—41no—
252#121—41no—
253#121—41no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—41no—
257C4×Q8—41no—
258C4×Q8C22⋊C411yescharacteristic
259C4×Q8SD1611yescharacteristic
260C4×D4—22no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—41no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4C22⋊C411yescharacteristic
269C4×D4C22⋊C411yescharacteristic
270C4×D4C22⋊C411yescharacteristic
271C4×D4SD1611yescharacteristic
272C4×D4SD1611yescharacteristic
273C4×D4SD1611yescharacteristic
274#117—41no—
275#117—41no—
276#117—42no—
277#117D811yescharacteristic
278#117D811yescharacteristic
279#117D811yescharacteristic
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—42no—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286M32—81no—
287M32—81no—
288C2×C16—21no—
289C2×C16—21no—
290C2×C16—21no—
291C2×C16—21no—
292C2×C16—81no—
293#107—21no—
294#107C22×C412yes—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—41no—
300C4⋊C8—41no—
301C4⋊C8C22⋊C411yescharacteristic
302C4⋊C8C22⋊C411yescharacteristic
303C4⋊C8C22⋊C411yescharacteristic
304C4⋊C8C22⋊C411yescharacteristic
305C4⋊C8C22⋊C411yescharacteristic
306C4⋊C8C22⋊C411yescharacteristic
307#103—22no—
308#103—22no—
309#103—22no—
310#103—22no—
311#102—22no—
312#102—22no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8—41no—
316C22⋊C8—41no—
317C22⋊C8C22⋊C411yescharacteristic
318C22⋊C8C22⋊C411yescharacteristic
319C22⋊C8C22⋊C411yescharacteristic
320C22⋊C8C22⋊C411yescharacteristic
321C22⋊C8C22⋊C411yescharacteristic
322C22⋊C8C22⋊C411yescharacteristic
323#97—41no—
324#97—41no—
325C4×C8—41no—
326C4×C8—41no—
327C4×C8C22×C411yescharacteristic
328C4×C8C22×C411yescharacteristic
329C4×C8C22×C411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8C2×D411yescharacteristic
336C4×C8C2411yescharacteristic
337#41—41no—
338#41—41no—
339#41—82no—
340#41—82no—
341#41—82no—
342#41—82no—
343#41—82no—
344#41—82no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—42no—
349C2×Q8—82no—
350C2×Q8—82no—
351C2×Q8#10211yescharacteristic
352C2×D4—22no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4#10211yescharacteristic
359C2×D4#10211yescharacteristic
360C2×D4#10211yescharacteristic
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×D811yescharacteristic
369C22×C4C2×D811yescharacteristic
370C22×C4C2×D811yescharacteristic
371C22×C4C2×SD1611yescharacteristic
372C22×C4C2×SD1611yescharacteristic
373C22×C4C2×SD1611yescharacteristic
374Q16—42no—
375Q16—42no—
376Q16—41no—
377Q16—41no—
378Q16—41no—
379Q16—41no—
380Q16—82no—
381Q16—82no—
382SD16—82no—
383SD16—82no—
384SD16—82no—
385SD16—82no—
386D8—42no—
387D8—41no—
388D8—41no—
389M16—82no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—82no—
404C2×C8C2×C22⋊C411yescharacteristic
405C2×C8C2×C22⋊C411yescharacteristic
406C2×C8C2×C22⋊C411yescharacteristic
407C2×C8C2×C22⋊C411yescharacteristic
408C2×C8C2×C22⋊C411yescharacteristic
409C2×C8C2×C22⋊C411yescharacteristic
410C2×C8C23×C411yescharacteristic
411C2×C8C22×D411yescharacteristic
412C4⋊C4—22no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4#10211yescharacteristic
425C4⋊C4#10211yescharacteristic
426C4⋊C4#10211yescharacteristic
427C4⋊C4#10211yescharacteristic
428C4⋊C4#10211yescharacteristic
429C4⋊C4#10211yescharacteristic
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4#10211yescharacteristic
439C22⋊C4#10211yescharacteristic
440C22⋊C4#10211yescharacteristic
441C22⋊C4#10211yescharacteristic
442C22⋊C4#10211yescharacteristic
443C22⋊C4#10211yescharacteristic
444C42—42no—
445C42—42no—
446C42C2×C22⋊C411yescharacteristic
447C42C2×C22⋊C411yescharacteristic
448C42C2×C22⋊C411yescharacteristic
449C42C2×D811yescharacteristic
450C42C2×D811yescharacteristic
451C42C2×D811yescharacteristic
452C42C2×SD1611yescharacteristic
453C42C2×SD1611yescharacteristic
454C42C2×SD1611yescharacteristic
455C42C22×D411yescharacteristic
456C16—41no—
457C16—41no—
458C16—41no—
459C16—41no—
460C23—42no—
461C23C2×#10211yescharacteristic
462C23C2×#10211yescharacteristic
463C23C2×#10211yescharacteristic
464Q8—41no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—81no—
470Q8—81no—
471Q8—162no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—42no—
476D4—42no—
477D4—82no—
478D4—84no—
479D4—81no—
480D4—81no—
481D4—162no—
482D4—162no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—162no—
503C2×C4—162no—
504C2×C4—162no—
505C2×C4C2×#10211yescharacteristic
506C2×C4C2×#10211yescharacteristic
507C2×C4C2×#10211yescharacteristic
508C2×C4C2×#10211yescharacteristic
509C2×C4C2×#10211yescharacteristic
510C2×C4C2×#10211yescharacteristic
511C2×C4C2×#10211yescharacteristic
512C2×C4C2×#10211yescharacteristic
513C2×C4C2×#10211yescharacteristic
514C2×C4C22×C22⋊C411yescharacteristic
515C2×C4C22×D811yescharacteristic
516C2×C4C22×SD1611yescharacteristic
517C8—21no—
518C8—21no—
519C8—21no—
520C8—21no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—81no—
525C8—81no—
526C8—81no—
527C8—81no—
528C8#57611yescharacteristic
529C8#57711yescharacteristic
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22C22×#10211yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4#288811yescharacteristic
557C4#289711yescharacteristic
558C4#341411yescharacteristic
559C4#341411yescharacteristic
560C2—41no—
561C2—41no—
562C2—41no—
563C2—162no—
564C2#1772011yescharacteristic
565C2#3338011yescharacteristic
566C2#3339311yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#62418C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33503C2semidirect
3. #17829C2semidirect
4. #11988C2semidirect
5. #11985C2semidirect
6. #11984C2semidirect
7. #11966C2semidirect
8. #11965C2semidirect
9. #11963C2semidirect
10. #11954C2semidirect
11. #11951C2semidirect
12. #11950C2semidirect
13. #11932C2semidirect
14. #11930C2semidirect
15. #11929C2semidirect

Maximal subgroups

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

subgroupindex
#624182
#335032
#178292
#119882
#119852
#119842
#119662
#119652
#119632
#119542
#119512
#119502
#119322
#119302
#119292