Finite Group Databasebeta

← #205688

Subgroups — #205688

Order 512.

1187 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
1C2×#3205C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3#17750C211yesmaximal, characteristic, complemented
4#17741C211yesmaximal, characteristic, complemented
5#17717C211yesmaximal, characteristic, complemented
6C2×#1545C211yesmaximal, characteristic, complemented
7#17325C211yesmaximal, characteristic, complemented
8#9780C212yesmaximal, complemented
9#9774C212yesmaximal, complemented
10#9773C212yesmaximal, complemented
11#9769C212yesmaximal, complemented
12C2×#532C2211yescharacteristic
13C22×#127C2211yescharacteristic
14#3205—21no—
15#3205—21no—
16#3205—22no—
17#3205C414yes—
18#3205C414yes—
19C2×#493C2212yes—
20C2×#486C2212yes—
21C2×#459—22no—
22C2×#447—22no—
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28#1950—22no—
29#1855—22no—
30#1848C2212yescomplemented
31#1837C2212yescomplemented
32#1560—22no—
33#1555—22no—
34#1553—22no—
35#1551—22no—
36#1549C2211yescharacteristic, complemented
37#1549C2211yescharacteristic, complemented
38#1548C2212yescomplemented
39#1545—22no—
40#1545—22no—
41#1545C2212yes—
42#1545C2212yes—
43#1545C2212yes—
44#1545C2212yes—
45#1545C2211yescharacteristic, complemented
46#1545C2212yescomplemented
47#1545C2211yescharacteristic, complemented
48#1540—22no—
49C2×#331—22no—
50#1453—22no—
51#1452—21no—
52#1452C2212yescomplemented
53#1450—21no—
54#1450C2212yescomplemented
55C22×#41—21no—
56C23×D4—21no—
57C23×D4—21no—
58C23×D4—21no—
59C22×D8—24no—
60C22×M16—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67#532C2×C412yes—
68#532C2×C412yes—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127—21no—
71C2×#127—21no—
72C2×#127—21no—
73C2×#127—21no—
74C2×#127—21no—
75C2×#127—21no—
76C2×#127—21no—
77C2×#127—21no—
78C2×#127—21no—
79C2×#127C2×C412yes—
80C2×#127C2×C412yes—
81C2×#127C2312yes—
82C2×#127C2311yescharacteristic
83C2×#124—21no—
84C2×#121—21no—
85C2×#121—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C22×C42C2311yescharacteristic
89#493—22no—
90#493—22no—
91#493C2×C418yes—
92#486—22no—
93#486—22no—
94#486C2×C418yes—
95#459—24no—
96#459—24no—
97#459—28no—
98#447—24no—
99#447—24no—
100#447—28no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432D412yes—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431C2×C412yes—
111#431C2×C412yes—
112#431D412yes—
113#431D411yescharacteristic
114#431D411yescharacteristic
115#431D412yes—
116#431C2312yes—
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119C2×#108—24no—
120#423—21no—
121#423—21no—
122#423—21no—
123#423D412yes—
124#423C2312yes—
125#423C2311yescharacteristic
126#423C2311yescharacteristic
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8D412yes—
130C2×C4⋊C8D412yes—
131C2×#102—24no—
132#407—21no—
133#407—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8D412yes—
136C4×M16—21no—
137C4×M16—21no—
138C2×C4×C8—21no—
139C2×C4×C8C2312yes—
140#341—24no—
141#333—24no—
142#332—22no—
143#332—22no—
144#331—24no—
145#331—24no—
146#331—22no—
147#331—22no—
148#331—24no—
149#331—24no—
150#331—24no—
151#322—22no—
152#322—22no—
153#322—22no—
154C82—22no—
155C25—41no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—41no—
159C2×#41—41no—
160C2×#41—84no—
161C22×Q8—21no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—82no—
185C22×D4—82no—
186C22×D4—84no—
187C22×D4—84no—
188C23×C4—41no—
189C23×C4C2×D411yescharacteristic
190C23×C4#4111yescharacteristic
191C23×C4#4111yescharacteristic
192C2×D8—28no—
193C2×D8—28no—
194C2×D8—84no—
195C2×D8—84no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—41no—
201C2×M16—82no—
202C22×C8—22no—
203C22×C8—41no—
204C22×C8C2×D414yes—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C22×C412yes—
209#127—22no—
210#127—22no—
211#127—22no—
212#127—22no—
213#127—22no—
214#127—22no—
215#127—22no—
216#127—22no—
217#127—22no—
218#127—21no—
219#127—21no—
220#127—21no—
221#127—41no—
222#127—41no—
223#127—41no—
224#127—41no—
225#127—41no—
226#127—41no—
227#127C22×C414yes—
228#127C22×C412yes—
229#124—22no—
230#124—22no—
231#124—41no—
232#124—41no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—41no—
238#121—41no—
239#121—41no—
240#121—41no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—41no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42C2411yescharacteristic
265#108—84no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—22no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8C22⋊C412yes—
277C4⋊C8C22⋊C412yes—
278C4⋊C8C22⋊C414yes—
279C4⋊C8C2×D412yes—
280C4⋊C8C2×D412yes—
281C4⋊C8C2×D414yes—
282#102—28no—
283#102—28no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—22no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291C22⋊C8C22⋊C412yes—
292C22⋊C8C22⋊C412yes—
293C22⋊C8C22⋊C414yes—
294#97—22no—
295#97—41no—
296C4×C8—22no—
297C4×C8—21no—
298C4×C8—22no—
299C4×C8—21no—
300C4×C8—41no—
301C4×C8C22×C414yes—
302C4×C8C2×D412yes—
303C4×C8C2×D412yes—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—84no—
309C24—84no—
310C24—84no—
311C24#12411yescharacteristic
312#41—84no—
313#41—84no—
314#41—84no—
315#41—84no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—82no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—82no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—84no—
359C2×D4—82no—
360C2×D4—84no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—84no—
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4C4×D411yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12411yescharacteristic
384C22×C4#12411yescharacteristic
385C22×C4#12411yescharacteristic
386C22×C4#12711yescharacteristic
387C22×C4#12711yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391D8—88no—
392D8—88no—
393M16—42no—
394M16—44no—
395M16—42no—
396M16—82no—
397M16—82no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—44no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—82no—
408C2×C8C2×C22⋊C414yes—
409C2×C8C2×C22⋊C414yes—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C42C2×C22⋊C411yescharacteristic
419C42C2×C22⋊C411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42C4×D411yescharacteristic
423C42C4×D411yescharacteristic
424C42#12111yescharacteristic
425C42#12111yescharacteristic
426C42#12111yescharacteristic
427C42#12111yescharacteristic
428C42#12711yescharacteristic
429C42#12711yescharacteristic
430C42C23×C411yescharacteristic
431C42C22×D411yescharacteristic
432C42C22×D411yescharacteristic
433C42C22×D411yescharacteristic
434C23—22no—
435C23—22no—
436C23—41no—
437C23—42no—
438C23—44no—
439C23—82no—
440C23—84no—
441C23—84no—
442C23—84no—
443C23—84no—
444C23—84no—
445C23—84no—
446C23—84no—
447C23—82no—
448C23—84no—
449C23—82no—
450C23—84no—
451C23—1616no—
452C23#39011yescharacteristic
453C23#39011yescharacteristic
454C23C2×#12411yescharacteristic
455Q8—82no—
456Q8—82no—
457D4—44no—
458D4—48no—
459D4—82no—
460D4—84no—
461D4—84no—
462D4—82no—
463D4—84no—
464D4—84no—
465D4—84no—
466D4—84no—
467D4—84no—
468D4—84no—
469D4—84no—
470D4—81no—
471D4—82no—
472D4—81no—
473D4—82no—
474D4—84no—
475D4—84no—
476D4—82no—
477D4—82no—
478D4—84no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—84no—
499C2×C4—82no—
500C2×C4—84no—
501C2×C4#39011yescharacteristic
502C2×C4#39011yescharacteristic
503C2×C4#39011yescharacteristic
504C2×C4#39011yescharacteristic
505C2×C4#39011yescharacteristic
506C2×C4#39011yescharacteristic
507C2×C4#47214yes—
508C2×C4C22×C22⋊C411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×#12111yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12711yescharacteristic
514C2×C4#57612yes—
515C2×C4#57712yes—
516C8—24no—
517C8—24no—
518C8—42no—
519C8—42no—
520C8—44no—
521C8—81no—
522C8—82no—
523C8—81no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—44no—
532C22—84no—
533C22—82no—
534C22—84no—
535C22—82no—
536C22—82no—
537C22—84no—
538C22—164no—
539C22—168no—
540C22—168no—
541C22—168no—
542C22#197812yes—
543C22C2×#39011yescharacteristic
544C22#313612yes—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—82no—
553C4#197812yes—
554C4#289712yes—
555C4C2×#47212yes—
556C4#306812yes—
557C4#315411yescharacteristic
558C4#315511yescharacteristic
559C2—22no—
560C2—22no—
561C2—41no—
562C2—162no—
563C2—164no—
564C2#2120611yescharacteristic
565C2C2×#197811yescharacteristic
566C2#2124911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3205C2
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. #21993C2semidirect
3. #17750C2semidirect
4. #17741C2semidirect
5. #17717C2semidirect
6. C2×#1545C2semidirect
7. #17325C2semidirect
8. #9780C2semidirect
9. #9774C2semidirect
10. #9773C2semidirect
11. #9769C2semidirect
30. #1848C22semidirect
31. #1837C22semidirect
36. #1549C22semidirect
37. #1549C22semidirect
38. #1548C22semidirect
45. #1545C22semidirect
46. #1545C22semidirect
47. #1545C22semidirect
52. #1452C22semidirect
54. #1450C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32052
#219932
#177502
#177412
#177172
C2×#15452
#173252
#97802
#97742
#97732
#97692