Finite Group Databasebeta

← #205614

Subgroups — #205614

Order 512.

933 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60701C211yesmaximal, characteristic
2#20367C211yesmaximal, characteristic, complemented
3#17797C211yesmaximal, characteristic, complemented
4#17646C211yesmaximal, characteristic
5C2×#1531C211yesmaximal, characteristic, complemented
6#17545C211yesmaximal, characteristic, complemented
7#17543C211yesmaximal, characteristic
8#9766C214yesmaximal, complemented
9#9765C214yesmaximal, complemented
10#3444C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3185C412yescomplemented
13#3182C412yescomplemented
14#3181—22no—
15#3172—22no—
16#3170C2212yes—
17#3167C2212yes—
18#3165C411yescharacteristic, complemented
19#3152C411yescharacteristic, complemented
20#3141C411yescharacteristic, complemented
21#3139C411yescharacteristic, complemented
22#3056—22no—
23C4×#137—22no—
24#2920C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1746C2214yes—
30#1745—22no—
31#1729—22no—
32#1569—22no—
33#1568C2214yes—
34#1565—22no—
35#1532—22no—
36#1531—22no—
37#1531C2212yes—
38#1531C2214yes—
39#1530C2212yes—
40#1524—22no—
41C2×#330—22no—
42#1509—22no—
43#1508—22no—
44#1506C2212yes—
45#1506C2214yes—
46#1505C2212yes—
47#1503—22no—
48#1501—22no—
49C2×#137—22no—
50C22×M16D412yes—
51C23×C8—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533C2×C411yescharacteristic
59#533C2×C412yes—
60#533C2×C411yescharacteristic
61#533C2312yes—
62#533C2311yescharacteristic
63#532C2×C411yescharacteristic
64#532C2×C411yescharacteristic
65#532C2311yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68#529—21no—
69#529—21no—
70#529—21no—
71#529—21no—
72#529—21no—
73#529—21no—
74C2×#125—21no—
75#525—21no—
76#525—21no—
77#525—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C4×#41—21no—
81C4×#41—21no—
82C4×#41—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8—21no—
85C2×#117—21no—
86C2×#117—21no—
87C22×C42C2311yescharacteristic
88#501—22no—
89#500C2×C412yes—
90#498C2×C412yes—
91#497—22no—
92#494C2×C412yes—
93#492C2×C412yes—
94#491—22no—
95#489—22no—
96#487C2×C412yes—
97#487C2×C412yes—
98#487C2×C412yes—
99#486C2×C412yes—
100#470—24no—
101#468—24no—
102#467—24no—
103#465—24no—
104#441—24no—
105#440—24no—
106C4×Q16—24no—
107C4×SD16—24no—
108#432—21no—
109#432—21no—
110#432—21no—
111#432—21no—
112#432C2312yes—
113#432C2312yes—
114#431—21no—
115#431—21no—
116#428—22no—
117#423—21no—
118C2×C4⋊C8—21no—
119#419—22no—
120#417—22no—
121#407—21no—
122C2×C22⋊C8—21no—
123C4×M16—21no—
124C4×M16C2×C411yescharacteristic
125C4×M16C2×C411yescharacteristic
126C4×M16C2×C412yes—
127C4×M16D412yes—
128C4×M16D412yes—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97—21no—
133C2×#97D412yes—
134C2×#97D412yes—
135C2×#97D412yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4D412yes—
140C4×C22⋊C4—21no—
141C4×C22⋊C4D412yes—
142#376C2312yes—
143C43—21no—
144#336—28no—
145#335—24no—
146#334—24no—
147#330—28no—
148#330—24no—
149#329—24no—
150#328—24no—
151#328—28no—
152#327—24no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161#137—48no—
162C2×Q16—24no—
163C2×SD16—24no—
164C2×M16—44no—
165C2×M16—41no—
166C2×M16C22⋊C412yes—
167C2×M16C22⋊C412yes—
168C2×M16C22⋊C414yes—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—44no—
173C22×C8—41no—
174C22×C8C2×D414yes—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C22×C411yescharacteristic
184C4⋊Q8C22×C411yescharacteristic
185C4⋊Q8C22×C411yescharacteristic
186#127C22×C411yescharacteristic
187#126—41no—
188#126—41no—
189#125—21no—
190#125—22no—
191#125—21no—
192#125—22no—
193#125—21no—
194#125—41no—
195#125—41no—
196#124—21no—
197#124—21no—
198#124—21no—
199#123—41no—
200#123—41no—
201#123—41no—
202C22⋊Q8—22no—
203C22⋊Q8—21no—
204C22⋊Q8—22no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216#121—21no—
217#121—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221#121—21no—
222#120—41no—
223C4×Q8—22no—
224C4×Q8—21no—
225C4×Q8—22no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—22no—
230C4×Q8—21no—
231C4×Q8—22no—
232C4×Q8—21no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—41no—
249#117—21no—
250#117—22no—
251#117—21no—
252#117—22no—
253#117—21no—
254#117—44no—
255#117—41no—
256#117—41no—
257#117—41no—
258#117—41no—
259#117—41no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—41no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42C2411yescharacteristic
286#107—24no—
287#106—24no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290C4⋊C8—41no—
291C4⋊C8—41no—
292#103—24no—
293#103—24no—
294#103—24no—
295#102—24no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300#97—22no—
301#97—22no—
302#97—22no—
303#97—22no—
304#97—22no—
305#97—41no—
306#97C22⋊C414yes—
307#97C22⋊C412yes—
308#97C22⋊C412yes—
309#97C2×D414yes—
310C4×C8—41no—
311C4×C8C22×C412yes—
312C4×C8C22×C412yes—
313C4×C8C2×D414yes—
314#95—22no—
315#95C2×D414yes—
316C24#12111yescharacteristic
317#41—44no—
318#41—82no—
319#41—82no—
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—44no—
328C2×Q8—82no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—81no—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4C4×D411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12411yescharacteristic
356C22×C4#12411yescharacteristic
357C22×C4#12711yescharacteristic
358C22×C4#12711yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362Q16—48no—
363SD16—48no—
364M16—24no—
365M16—24no—
366M16—44no—
367M16—82no—
368C2×C8—24no—
369C2×C8—44no—
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—44no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—82no—
376C2×C8C2×C22⋊C414yes—
377C2×C8C2×C22⋊C414yes—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—41no—
406C22⋊C4—42no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C42—44no—
416C42—41no—
417C42—42no—
418C42—41no—
419C42—41no—
420C42—42no—
421C42—82no—
422C42—82no—
423C42C2×C22⋊C411yescharacteristic
424C42C2×C22⋊C411yescharacteristic
425C42C4×D411yescharacteristic
426C42C4×D411yescharacteristic
427C42C4×D411yescharacteristic
428C42C4×D411yescharacteristic
429C42#12111yescharacteristic
430C42#12111yescharacteristic
431C42#12411yescharacteristic
432C42#12411yescharacteristic
433C42#12711yescharacteristic
434C42#12711yescharacteristic
435C42C23×C411yescharacteristic
436C42C22×D411yescharacteristic
437C42C22×D411yescharacteristic
438C42C2×#4111yescharacteristic
439C23—22no—
440C23—22no—
441C23—41no—
442C23—41no—
443C23#39011yescharacteristic
444C23#39011yescharacteristic
445C23C2×#12111yescharacteristic
446Q8—44no—
447Q8—42no—
448Q8—82no—
449Q8—81no—
450Q8—81no—
451Q8—82no—
452Q8—82no—
453D4—42no—
454D4—81no—
455D4—81no—
456D4—82no—
457D4—82no—
458D4—82no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4#39011yescharacteristic
490C2×C4#39011yescharacteristic
491C2×C4#39011yescharacteristic
492C2×C4#39011yescharacteristic
493C2×C4#39011yescharacteristic
494C2×C4#39011yescharacteristic
495C2×C4#42114yes—
496C2×C4C22×C22⋊C411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12411yescharacteristic
501C2×C4C2×#12711yescharacteristic
502C8—24no—
503C8—24no—
504C8—44no—
505C8—82no—
506C8—82no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—44no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22#196312yes—
519C22C2×#39011yescharacteristic
520C22#306912yes—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—84no—
532C4—81no—
533C4—81no—
534C4—82no—
535C4—81no—
536C4—82no—
537C4#196312yes—
538C4C2×#42112yes—
539C2—22no—
540C2—22no—
541C2—41no—
542C2—161no—
543C2C2×#196311yescharacteristic
544C2#2124411yescharacteristic
545C2#2125011yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20367C2semidirect
3. #17797C2semidirect
5. C2×#1531C2semidirect
6. #17545C2semidirect
8. #9766C2semidirect
9. #9765C2semidirect
12. #3185C4semidirect
13. #3182C4semidirect
18. #3165C4semidirect
19. #3152C4semidirect
20. #3141C4semidirect
21. #3139C4semidirect

Maximal subgroups

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

subgroupindex
#607012
#203672
#177972
#176462
C2×#15312
#175452
#175432
#97662
#97652