Finite Group Databasebeta

← #483231

Subgroups — #483231

Order 512.

722 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60718C211yesmaximal, characteristic
2#33423C211yesmaximal, characteristic, complemented
3#17799C211yesmaximal, characteristic, complemented
4#11987C211yesmaximal, characteristic, complemented
5#11986C211yesmaximal, characteristic, complemented
6#11983C211yesmaximal, characteristic, complemented
7#11982C211yesmaximal, characteristic, complemented
8#11975C211yesmaximal, characteristic, complemented
9#11972C211yesmaximal, characteristic, complemented
10#11966C211yesmaximal, characteristic, complemented
11#11963C211yesmaximal, characteristic, complemented
12#11953C211yesmaximal, characteristic, complemented
13#11952C211yesmaximal, characteristic, complemented
14#11949C211yesmaximal, characteristic, complemented
15#11948C211yesmaximal, characteristic, complemented
16#3511—21no—
17#3182—21no—
18#3182—21no—
19#3175—21no—
20#3175—21no—
21#3175C412yes—
22#3175C412yes—
23#3170C2211yescharacteristic
24#3170C2211yescharacteristic
25#3169C2211yescharacteristic
26#3169C2211yescharacteristic
27C2×#500C2211yescharacteristic
28#3147C2211yescharacteristic
29#3141—21no—
30#3061—21no—
31#3061C412yes—
32#2989—21no—
33#2951—21no—
34#2951C412yes—
35C4×#131C2211yescharacteristic
36#2164—21no—
37#2164—21no—
38#2163C2211yescharacteristic
39#2163C2211yescharacteristic
40#2162C2211yescharacteristic
41#2162C2211yescharacteristic
42#2150C2211yescharacteristic
43C2×C4⋊C16C2211yescharacteristic
44#2113—21no—
45#2108—21no—
46#1569—21no—
47#1569—21no—
48#1568C2211yescharacteristic
49#1568C2211yescharacteristic
50#1567C2211yescharacteristic
51#1567C2211yescharacteristic
52#1565—21no—
53#1564C2211yescharacteristic
54C2×#335C2211yescharacteristic
55#1365C2211yescharacteristic
56#1365C2211yescharacteristic
57#1365C2211yescharacteristic
58#1365C2211yescharacteristic
59#1365C2211yescharacteristic
60#1365C2211yescharacteristic
61#1365C2211yescharacteristic
62#1365C2211yescharacteristic
63#1362C2211yescharacteristic
64#1362C2211yescharacteristic
65#1362C2211yescharacteristic
66#1362C2211yescharacteristic
67#1361C2211yescharacteristic
68#1361C2211yescharacteristic
69#1361C2211yescharacteristic
70#1361C2211yescharacteristic
71C2×#135—21no—
72C2×#135C2×C412yes—
73C2×#131C2×C411yescharacteristic
74C2×#131D411yescharacteristic
75#556—41no—
76#555—41no—
77#554—41no—
78#553—41no—
79#553—41no—
80#548—41no—
81#548—41no—
82#547—41no—
83#542—22no—
84#542—22no—
85#537—22no—
86#533—21no—
87#533—21no—
88#532—21no—
89#532—21no—
90#532—21no—
91C2×C4⋊Q8—21no—
92#520—22no—
93C4×#41D411yescharacteristic
94C2×M32—21no—
95C22×C16—21no—
96#500—21no—
97#500—21no—
98#500C2×C412yes—
99#500C2311yescharacteristic
100#500C2311yescharacteristic
101#495—21no—
102#495C2×C412yes—
103#494C2311yescharacteristic
104#493C2311yescharacteristic
105#492—21no—
106#489—22no—
107#489—22no—
108#487—21no—
109#487—21no—
110#486—21no—
111#486—21no—
112#468—22no—
113#468—22no—
114#468—22no—
115#468—22no—
116#467—21no—
117#467—21no—
118#467C2×C412yes—
119#467C2×C412yes—
120#466—21no—
121#466—21no—
122#466C2×C412yes—
123#466C2×C412yes—
124#446—41no—
125#440—22no—
126#440—22no—
127#440—22no—
128#440—22no—
129C4×Q16—21no—
130C4×Q16—21no—
131C4×Q16C2×C412yes—
132C4×Q16C2×C412yes—
133C4×D8—21no—
134C4×D8—21no—
135C4×D8C2×C412yes—
136C4×D8C2×C412yes—
137#436—41no—
138#435—41no—
139#435—41no—
140#432—21no—
141#432—21no—
142#431—21no—
143C2×#107—21no—
144C2×#107C2×C412yes—
145#423—21no—
146#423—21no—
147C2×C4⋊C8—21no—
148#417—22no—
149#417—22no—
150#405C2×C411yescharacteristic
151#405C2×C411yescharacteristic
152#405C2×C411yescharacteristic
153#405D411yescharacteristic
154#405D411yescharacteristic
155#405D411yescharacteristic
156C4×M16D411yescharacteristic
157C4×M16D411yescharacteristic
158C4×M16D411yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C4⋊C16—21no—
163C4⋊C16—21no—
164C4⋊C16C2311yescharacteristic
165C4⋊C16C2311yescharacteristic
166C4⋊C16C2311yescharacteristic
167C4⋊C16C2311yescharacteristic
168C22⋊C16—21no—
169C22⋊C16—21no—
170C22⋊C16—21no—
171C22⋊C16—21no—
172#346—21no—
173#346—21no—
174C4×C16—21no—
175C4×C16—21no—
176#335—21no—
177#335—21no—
178#335—21no—
179#335C2311yescharacteristic
180#335C2311yescharacteristic
181#335C2311yescharacteristic
182#335C2311yescharacteristic
183#334—21no—
184#334—21no—
185#334—21no—
186C2×#41—22no—
187C2×#41—41no—
188C2×#41—41no—
189C2×#41C22⋊C411yescharacteristic
190C22×Q8—41no—
191#135—42no—
192#135—42no—
193#135—44no—
194C2×Q16—21no—
195C2×Q16C22×C412yes—
196C2×SD16—22no—
197C2×SD16—22no—
198C2×D8—21no—
199C2×D8C22×C412yes—
200#131—41no—
201#131—41no—
202#131—42no—
203C2×M16—41no—
204C2×M16—41no—
205C2×M16C22⋊C411yescharacteristic
206C2×M16C22⋊C411yescharacteristic
207C2×M16C22⋊C411yescharacteristic
208C2×M16C22⋊C411yescharacteristic
209C2×M16C22⋊C411yescharacteristic
210C2×M16C22⋊C411yescharacteristic
211C22×C8—41no—
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C4⋊Q8—22no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—41no—
223C4⋊Q8—41no—
224#127—21no—
225#126—42no—
226#126—42no—
227#125—41no—
228#125—41no—
229#124—22no—
230#124—42no—
231#124—42no—
232#124—41no—
233#124—41no—
234#123—42no—
235#123—42no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C22⋊Q8—42no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C22⋊Q8—42no—
242C22⋊Q8—42no—
243#121—22no—
244#121—22no—
245#121—42no—
246#121—42no—
247#121—42no—
248#121—41no—
249#121—41no—
250#121—41no—
251#121—41no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—41no—
255C4×Q8—41no—
256C4×Q8D811yescharacteristic
257C4×Q8SD1611yescharacteristic
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—41no—
263C4×D4—41no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4D811yescharacteristic
267C4×D4D811yescharacteristic
268C4×D4D811yescharacteristic
269C4×D4SD1611yescharacteristic
270C4×D4SD1611yescharacteristic
271C4×D4SD1611yescharacteristic
272#117—41no—
273#117—41no—
274#117—42no—
275#117C22⋊C411yescharacteristic
276#117C22⋊C411yescharacteristic
277#117C22⋊C411yescharacteristic
278C2×C4⋊C4—22no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—42no—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284M32—42no—
285C2×C16—21no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—42no—
290#107—22no—
291#107—22no—
292#107—21no—
293#107—21no—
294#107C22×C412yes—
295#107C22×C412yes—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—41no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303#103—22no—
304#103—22no—
305#103—22no—
306#103—22no—
307#102—22no—
308#102—22no—
309#102—22no—
310#102—22no—
311C22⋊C8—41no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314#97—41no—
315#97—41no—
316#97C22⋊C411yescharacteristic
317#97C22⋊C411yescharacteristic
318#97C22⋊C411yescharacteristic
319#97C22⋊C411yescharacteristic
320#97C22⋊C411yescharacteristic
321#97C22⋊C411yescharacteristic
322C4×C8—41no—
323C4×C8C22×C411yescharacteristic
324C4×C8C22×C411yescharacteristic
325C4×C8C22×C411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2411yescharacteristic
333#41—41no—
334#41—41no—
335#41—44no—
336#41—44no—
337#41—162no—
338#41—162no—
339C2×Q8—22no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—162no—
344C2×Q8#10211yescharacteristic
345C2×D4—22no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4#10211yescharacteristic
352C2×D4#10211yescharacteristic
353C2×D4#10211yescharacteristic
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×D811yescharacteristic
362C22×C4C2×D811yescharacteristic
363C22×C4C2×D811yescharacteristic
364C22×C4C2×SD1611yescharacteristic
365C22×C4C2×SD1611yescharacteristic
366C22×C4C2×SD1611yescharacteristic
367Q16—42no—
368Q16—41no—
369Q16—41no—
370SD16—44no—
371SD16—44no—
372D8—42no—
373D8—41no—
374D8—41no—
375M16—22no—
376M16—22no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—161no—
382M16—161no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—161no—
397C2×C8C2×C22⋊C411yescharacteristic
398C2×C8C2×C22⋊C411yescharacteristic
399C2×C8C2×C22⋊C411yescharacteristic
400C2×C8C2×C22⋊C411yescharacteristic
401C2×C8C2×C22⋊C411yescharacteristic
402C2×C8C2×C22⋊C411yescharacteristic
403C2×C8C23×C411yescharacteristic
404C2×C8C22×D411yescharacteristic
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4#10211yescharacteristic
418C4⋊C4#10211yescharacteristic
419C4⋊C4#10211yescharacteristic
420C4⋊C4#10211yescharacteristic
421C4⋊C4#10211yescharacteristic
422C4⋊C4#10211yescharacteristic
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4#10211yescharacteristic
432C22⋊C4#10211yescharacteristic
433C22⋊C4#10211yescharacteristic
434C22⋊C4#10211yescharacteristic
435C22⋊C4#10211yescharacteristic
436C22⋊C4#10211yescharacteristic
437C42—42no—
438C42—42no—
439C42C2×C22⋊C411yescharacteristic
440C42C2×C22⋊C411yescharacteristic
441C42C2×C22⋊C411yescharacteristic
442C42C2×D811yescharacteristic
443C42C2×D811yescharacteristic
444C42C2×D811yescharacteristic
445C42C2×SD1611yescharacteristic
446C42C2×SD1611yescharacteristic
447C42C2×SD1611yescharacteristic
448C42C22×D411yescharacteristic
449C16—41no—
450C16—41no—
451C16—41no—
452C16—41no—
453C23—42no—
454C23C2×#10211yescharacteristic
455C23C2×#10211yescharacteristic
456C23C2×#10211yescharacteristic
457Q8—41no—
458Q8—42no—
459Q8—42no—
460Q8—82no—
461Q8—82no—
462Q8—162no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—42no—
467D4—42no—
468D4—82no—
469D4—84no—
470D4—81no—
471D4—81no—
472D4—162no—
473D4—162no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—84no—
493C2×C4—162no—
494C2×C4—162no—
495C2×C4—162no—
496C2×C4C2×#10211yescharacteristic
497C2×C4C2×#10211yescharacteristic
498C2×C4C2×#10211yescharacteristic
499C2×C4C2×#10211yescharacteristic
500C2×C4C2×#10211yescharacteristic
501C2×C4C2×#10211yescharacteristic
502C2×C4C2×#10211yescharacteristic
503C2×C4C2×#10211yescharacteristic
504C2×C4C2×#10211yescharacteristic
505C2×C4C22×C22⋊C411yescharacteristic
506C2×C4C22×D811yescharacteristic
507C2×C4C22×SD1611yescharacteristic
508C8—21no—
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—21no—
514C8—81no—
515C8—81no—
516C8—81no—
517C8—81no—
518C8#41312yes—
519C8#57511yescharacteristic
520C8#57511yescharacteristic
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22C22×#10211yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4C2×#41311yescharacteristic
548C4#289611yescharacteristic
549C4#341611yescharacteristic
550C4#341711yescharacteristic
551C2—41no—
552C2—41no—
553C2—41no—
554C2—162no—
555C2#1770411yescharacteristic
556C2#3338211yescharacteristic
557C2#3339211yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#60718C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33423C2semidirect
3. #17799C2semidirect
4. #11987C2semidirect
5. #11986C2semidirect
6. #11983C2semidirect
7. #11982C2semidirect
8. #11975C2semidirect
9. #11972C2semidirect
10. #11966C2semidirect
11. #11963C2semidirect
12. #11953C2semidirect
13. #11952C2semidirect
14. #11949C2semidirect
15. #11948C2semidirect

Maximal subgroups

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

subgroupindex
#607182
#334232
#177992
#119872
#119862
#119832
#119822
#119752
#119722
#119662
#119632
#119532
#119522
#119492
#119482