Finite Group Databasebeta

← #205160

Subgroups — #205160

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#21370C211yesmaximal, characteristic, complemented
3#17663C211yesmaximal, characteristic, complemented
4#17647C211yesmaximal, characteristic, complemented
5#17463C211yesmaximal, characteristic, complemented
6C2×#1484C211yesmaximal, characteristic, complemented
7#17326C211yesmaximal, characteristic, complemented
8#9726C214yesmaximal, complemented
9#9724C214yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3245—22no—
13#3226C412yes—
14#3226C412yes—
15#3205C412yes—
16#3205C412yes—
17#3186—22no—
18#3169C2214yes—
19#3100—21no—
20#3100—21no—
21#3100—22no—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#1950—22no—
28#1855—22no—
29C4×#108C2214yescomplemented
30#1536—22no—
31#1533—22no—
32#1530—22no—
33#1527—22no—
34#1525C2214yescomplemented
35#1525C2214yescomplemented
36#1484—22no—
37#1484C2212yescomplemented
38#1484C2212yescomplemented
39#1484C2212yescomplemented
40#1484C2212yescomplemented
41#1480—22no—
42#1478—22no—
43C2×D4⋊C8—22no—
44#1454—22no—
45#1452—21no—
46#1451C2212yescomplemented
47#1450—21no—
48C8×M16C2212yescomplemented
49C22×#41—21no—
50C22×#41—21no—
51C22×M16—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532C2×C412yes—
65#532C2×C412yes—
66#532C2312yes—
67#532C2312yes—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×#127C2×C411yescharacteristic
71C2×#127C2×C411yescharacteristic
72C2×#124—21no—
73C2×#124—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121—21no—
77C2×#121—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C22×C42C2311yescharacteristic
83#500C2×C414yes—
84#493C2×C414yes—
85#486C2×C414yes—
86#486C2×C414yes—
87#480—22no—
88#480—22no—
89#480—24no—
90#474—22no—
91#474—22no—
92#474—24no—
93#466—24no—
94#466—24no—
95#459—22no—
96#459—22no—
97#459—24no—
98#447—22no—
99#447—22no—
100#447—24no—
101C4×D8—24no—
102C4×D8—24no—
103#432—21no—
104#432—21no—
105#432D412yes—
106#432D412yes—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431C2×C412yes—
114#431C2×C412yes—
115#431D412yes—
116#431D411yescharacteristic
117#431D411yescharacteristic
118#431D412yes—
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#108—24no—
122#423—21no—
123#423—21no—
124C2×C4⋊C8D412yes—
125C2×C4⋊C8D412yes—
126#407—21no—
127#407—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8D412yes—
130C4×M16—21no—
131C4×M16—21no—
132C4×M16—21no—
133C4×M16C2312yes—
134C4×M16C2312yes—
135C4×M16C2312yes—
136C2×#97—21no—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139C2×C4×C8C2312yes—
140#341—24no—
141#329—28no—
142#329—28no—
143D4⋊C8—24no—
144D4⋊C8—24no—
145D4⋊C8—24no—
146D4⋊C8—24no—
147#322—22no—
148#322—22no—
149#322—22no—
150C82—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—41no—
156C2×#41—41no—
157C2×#41—41no—
158C2×#41—41no—
159C2×#41—84no—
160C2×#41—84no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—41no—
170C22×D4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4C2×D411yescharacteristic
174C23×C4#4111yescharacteristic
175C23×C4#4111yescharacteristic
176C2×D8—24no—
177C2×D8—24no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—44no—
182C2×M16—41no—
183C22×C8—22no—
184C22×C8—41no—
185C22×C8C2×D414yes—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8C22×C412yes—
193C4⋊Q8C22×C412yes—
194#127—21no—
195#127—21no—
196#127—21no—
197#127—21no—
198#127—21no—
199#127—21no—
200#127C22×C412yes—
201#127C22×C412yes—
202#124—22no—
203#124—22no—
204#124—22no—
205#124—22no—
206#124—41no—
207#124—41no—
208#124—41no—
209#124—41no—
210#121—22no—
211#121—22no—
212#121—22no—
213#121—22no—
214#121—22no—
215#121—22no—
216#121—22no—
217#121—22no—
218#121—41no—
219#121—41no—
220#121—41no—
221#121—41no—
222#121—41no—
223#121—41no—
224#121—41no—
225#121—41no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42C2411yescharacteristic
262#108—48no—
263#107—24no—
264#107—24no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270C4⋊C8C22⋊C412yes—
271C4⋊C8C22⋊C412yes—
272C4⋊C8C22⋊C414yes—
273C4⋊C8C2×D412yes—
274C4⋊C8C2×D412yes—
275#102—24no—
276#102—24no—
277#102—24no—
278#102—24no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286C22⋊C8C22⋊C412yes—
287C22⋊C8C22⋊C412yes—
288C22⋊C8C22⋊C414yes—
289#97—22no—
290#97—22no—
291#97—41no—
292#97C2×D414yes—
293C4×C8—22no—
294C4×C8—21no—
295C4×C8—22no—
296C4×C8—21no—
297C4×C8—22no—
298C4×C8—22no—
299C4×C8—41no—
300C4×C8C22×C414yes—
301C4×C8C2×D412yes—
302C4×C8C2×D412yes—
303C24—41no—
304C24—41no—
305C24#12411yescharacteristic
306#41—84no—
307#41—84no—
308#41—84no—
309#41—84no—
310#41—84no—
311#41—84no—
312#41—84no—
313#41—84no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—82no—
319C2×Q8—82no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—84no—
343C2×D4—84no—
344C22×C4—24no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12411yescharacteristic
366C22×C4#12411yescharacteristic
367C22×C4#12411yescharacteristic
368C22×C4#12711yescharacteristic
369C22×C4#12711yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373D8—84no—
374D8—84no—
375M16—24no—
376M16—44no—
377M16—42no—
378M16—44no—
379M16—44no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—42no—
386C2×C8—44no—
387C2×C8—42no—
388C2×C8C2×C22⋊C414yes—
389C2×C8C2×C22⋊C414yes—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C42C2×C22⋊C411yescharacteristic
407C42C2×C22⋊C411yescharacteristic
408C42C4×D411yescharacteristic
409C42C4×D411yescharacteristic
410C42C4×D411yescharacteristic
411C42C4×D411yescharacteristic
412C42#12111yescharacteristic
413C42#12111yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12711yescharacteristic
417C42#12711yescharacteristic
418C42C23×C411yescharacteristic
419C42C22×D411yescharacteristic
420C42C22×D411yescharacteristic
421C42C22×D411yescharacteristic
422C23—22no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23#39011yescharacteristic
434C23#39011yescharacteristic
435C23C2×#12411yescharacteristic
436Q8—82no—
437Q8—82no—
438Q8—82no—
439Q8—82no—
440D4—42no—
441D4—42no—
442D4—44no—
443D4—44no—
444D4—44no—
445D4—82no—
446D4—82no—
447D4—82no—
448D4—82no—
449D4—82no—
450D4—82no—
451D4—82no—
452D4—84no—
453D4—84no—
454D4—84no—
455D4—84no—
456D4—82no—
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—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—84no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4#39011yescharacteristic
483C2×C4#39011yescharacteristic
484C2×C4#39011yescharacteristic
485C2×C4#39011yescharacteristic
486C2×C4#39011yescharacteristic
487C2×C4#39011yescharacteristic
488C2×C4#47214yes—
489C2×C4C22×C22⋊C411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×#12111yescharacteristic
494C2×C4C2×#12711yescharacteristic
495C8—24no—
496C8—24no—
497C8—44no—
498C8—44no—
499C8—42no—
500C8—42no—
501C8—82no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—164no—
518C22—164no—
519C22#197812yes—
520C22C2×#39011yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—82no—
530C4—82no—
531C4#197812yes—
532C4#290112yes—
533C4#295512yes—
534C4C2×#47212yes—
535C2—22no—
536C2—41no—
537C2—41no—
538C2—162no—
539C2—162no—
540C2#2124211yescharacteristic
541C2C2×#197811yescharacteristic
542C2#2125311yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60940C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21370C2semidirect
3. #17663C2semidirect
4. #17647C2semidirect
5. #17463C2semidirect
6. C2×#1484C2semidirect
7. #17326C2semidirect
8. #9726C2semidirect
9. #9724C2semidirect
29. C4×#108C22semidirect
34. #1525C22semidirect
35. #1525C22semidirect
37. #1484C22semidirect
38. #1484C22semidirect
39. #1484C22semidirect
40. #1484C22semidirect
46. #1451C22semidirect
48. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
#609402
#213702
#176632
#176472
#174632
C2×#14842
#173262
#97262
#97242