Finite Group Databasebeta

← #206464

Subgroups — #206464

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60782C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3#17481C211yesmaximal, characteristic, complemented
4#17478C211yesmaximal, characteristic, complemented
5#17473C211yesmaximal, characteristic, complemented
6C2×#1492C211yesmaximal, characteristic, complemented
7C2×#1452C211yesmaximal, characteristic, complemented
8#9863C212yesmaximal, complemented
9#9862C212yesmaximal, complemented
10#9860C212yesmaximal, complemented
11#9858C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3212—22no—
15#3211—22no—
16#3210C412yes—
17#3209C411yescharacteristic
18#3208C411yescharacteristic
19#3207C412yes—
20#3206C411yescharacteristic
21#3205C411yescharacteristic
22#3147C2212yes—
23#3131C2212yes—
24#3035—22no—
25#3002—22no—
26#2920C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#1950—22no—
32#1855—22no—
33#1848C2212yescomplemented
34#1837C2212yescomplemented
35#1496—22no—
36#1495—22no—
37#1494—22no—
38#1493C2211yescharacteristic, complemented
39#1493C2211yescharacteristic, complemented
40#1493C2212yescomplemented
41#1492—22no—
42#1492C2212yes—
43#1492C2212yes—
44#1492C2212yes—
45#1492C2212yes—
46#1492C2211yescharacteristic, complemented
47#1492C2211yescharacteristic, complemented
48#1492C2212yescomplemented
49#1478—22no—
50#1477—22no—
51#1476—22no—
52C2×D4⋊C8—22no—
53#1452—22no—
54#1452—22no—
55C2×#322C2212yescomplemented
56C2×#322C2212yescomplemented
57C22×#41—21no—
58C2×#135—24no—
59C22×M16—21no—
60C23×C8—21no—
61#533C2×C411yescharacteristic
62#533C2×C411yescharacteristic
63#533C2311yescharacteristic
64#532—21no—
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532—21no—
70#532C2×C412yes—
71#532C2×C411yescharacteristic
72#532C2×C411yescharacteristic
73#532C2312yes—
74#532C2311yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#127C2×C411yescharacteristic
77#529—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529—21no—
83C2×#124—21no—
84#525—21no—
85#525—21no—
86#525—21no—
87C2×#121—21no—
88C2×#121—21no—
89C4×#41—21no—
90C4×#41—21no—
91C4×#41—21no—
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C22×C42C2311yescharacteristic
95#495—24no—
96#494C2×C412yes—
97#493C2×C412yes—
98#492C2×C412yes—
99#492C2×C412yes—
100#488—24no—
101#487C2×C412yes—
102#487C2×C412yes—
103#486C2×C412yes—
104#486C2×C412yes—
105#464—24no—
106#463—24no—
107#462—22no—
108#461—22no—
109#460—22no—
110#459—22no—
111#452—24no—
112#451—22no—
113#450—22no—
114#449—24no—
115#448—22no—
116#447—22no—
117#432—21no—
118#432—21no—
119#431—21no—
120#431—21no—
121#431—21no—
122#431—21no—
123#431C2×C411yescharacteristic
124#431C2×C411yescharacteristic
125#431C2×C412yes—
126#431D412yes—
127#431D412yes—
128#431D412yes—
129#431D412yes—
130C2×#108—24no—
131#423—21no—
132#423—21no—
133#423—21no—
134#423C2311yescharacteristic
135#423C2312yes—
136#423C2311yescharacteristic
137C2×C4⋊C8—21no—
138C2×C4⋊C8—21no—
139C2×C4⋊C8—21no—
140C2×C4⋊C8D412yes—
141#416—24no—
142#407—21no—
143C2×C22⋊C8—21no—
144C2×C22⋊C8D412yes—
145C4×M16—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8D412yes—
150C2×C4×C8D412yes—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155#341—24no—
156Q8⋊C8—22no—
157Q8⋊C8—22no—
158Q8⋊C8—24no—
159D4⋊C8—24no—
160D4⋊C8—24no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—24no—
164D4⋊C8—24no—
165D4⋊C8—24no—
166#322—24no—
167#322—24no—
168C2×#41—22no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—22no—
173C2×#41—41no—
174C2×#41—41no—
175C2×#41—82no—
176C2×#41—82no—
177C22×Q8—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—41no—
182C23×C4—41no—
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185C23×C4#4111yescharacteristic
186#135—84no—
187#135—84no—
188C2×Q16—24no—
189C2×SD16—24no—
190C2×SD16—24no—
191C2×D8—24no—
192C2×M16—22no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—41no—
196C2×M16—82no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—44no—
203C22×C8—41no—
204C22×C8C2×D414yes—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C22×C412yes—
209C4⋊Q8C22×C411yescharacteristic
210C4⋊Q8C22×C411yescharacteristic
211C4⋊Q8C22×C411yescharacteristic
212#127—21no—
213#127—21no—
214#127—21no—
215#127C22×C412yes—
216#127C22×C411yescharacteristic
217#126—41no—
218#126—41no—
219#125—21no—
220#125—21no—
221#125—21no—
222#124—22no—
223#124—21no—
224#124—21no—
225#124—22no—
226#124—21no—
227#124—41no—
228#124—41no—
229#123—41no—
230#123—41no—
231#123—41no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238#121—21no—
239#121—21no—
240#121—22no—
241#121—22no—
242#121—21no—
243#121—21no—
244#121—22no—
245#121—22no—
246#121—21no—
247#121—21no—
248#121—41no—
249#121—41no—
250#121—41no—
251#121—41no—
252#120—41no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4—41no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—41no—
283#117—41no—
284#117—41no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C42—41no—
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42#4111yescharacteristic
301C2×C42C2411yescharacteristic
302#108—84no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—22no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—41no—
312C4⋊C8—41no—
313C4⋊C8C22⋊C412yes—
314C4⋊C8C22⋊C412yes—
315C4⋊C8C22⋊C414yes—
316#103—24no—
317#103—24no—
318#102—24no—
319#102—24no—
320C22⋊C8—22no—
321C22⋊C8—22no—
322C22⋊C8—41no—
323C22⋊C8—41no—
324C22⋊C8C22⋊C412yes—
325C22⋊C8C22⋊C412yes—
326C22⋊C8C22⋊C414yes—
327#97—41no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—22no—
333C4×C8—41no—
334C4×C8C22×C412yes—
335C4×C8C22×C412yes—
336C4×C8C2×D412yes—
337C4×C8C2×D412yes—
338C4×C8C2×D414yes—
339C4×C8C2×D414yes—
340C24—41no—
341C24#12411yescharacteristic
342#41—44no—
343#41—42no—
344#41—44no—
345#41—42no—
346#41—44no—
347#41—44no—
348#41—84no—
349#41—84no—
350#41—82no—
351#41—82no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—41no—
357C2×Q8—82no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—44no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—44no—
374C2×D4—82no—
375C2×D4—84no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4#12111yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12111yescharacteristic
397C22×C4#12111yescharacteristic
398C22×C4#12411yescharacteristic
399C22×C4#12411yescharacteristic
400C22×C4#12411yescharacteristic
401C22×C4#12711yescharacteristic
402C22×C4#12711yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406Q16—84no—
407SD16—84no—
408SD16—84no—
409D8—84no—
410M16—42no—
411M16—44no—
412M16—42no—
413M16—82no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—44no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—42no—
427C2×C8—82no—
428C2×C8C2×C22⋊C414yes—
429C2×C8C2×C22⋊C414yes—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42C2×C22⋊C411yescharacteristic
463C42C2×C22⋊C411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42C4×D411yescharacteristic
467C42C4×D411yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42#12111yescharacteristic
471C42#12111yescharacteristic
472C42#12711yescharacteristic
473C42#12711yescharacteristic
474C42C23×C411yescharacteristic
475C42C22×D411yescharacteristic
476C42C22×D411yescharacteristic
477C42C22×D411yescharacteristic
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23#39011yescharacteristic
487C23#39011yescharacteristic
488C23C2×#12411yescharacteristic
489Q8—42no—
490Q8—44no—
491Q8—42no—
492Q8—42no—
493Q8—82no—
494D4—42no—
495D4—44no—
496D4—44no—
497D4—42no—
498D4—44no—
499D4—42no—
500D4—82no—
501D4—82no—
502D4—84no—
503D4—82no—
504D4—84no—
505D4—82no—
506D4—82no—
507D4—82no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—84no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4#39011yescharacteristic
539C2×C4#39011yescharacteristic
540C2×C4#39011yescharacteristic
541C2×C4#39011yescharacteristic
542C2×C4#39011yescharacteristic
543C2×C4#39011yescharacteristic
544C2×C4#44414yes—
545C2×C4C22×C22⋊C411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12711yescharacteristic
551C2×C4#57612yes—
552C2×C4#57712yes—
553C8—24no—
554C8—24no—
555C8—44no—
556C8—42no—
557C8—42no—
558C8—81no—
559C8—82no—
560C8—81no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—44no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22—82no—
573C22—81no—
574C22—82no—
575C22—164no—
576C22#197512yes—
577C22C2×#39011yescharacteristic
578C22#313612yes—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—82no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4#197512yes—
591C4#289712yes—
592C4C2×#44412yes—
593C4#295512yes—
594C4#315411yescharacteristic
595C4#315511yescharacteristic
596C2—22no—
597C2—22no—
598C2—41no—
599C2—161no—
600C2—162no—
601C2#2120611yescharacteristic
602C2C2×#197511yescharacteristic
603C2#2124211yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60782C2
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. #21993C2semidirect
3. #17481C2semidirect
4. #17478C2semidirect
5. #17473C2semidirect
6. C2×#1492C2semidirect
7. C2×#1452C2semidirect
8. #9863C2semidirect
9. #9862C2semidirect
10. #9860C2semidirect
11. #9858C2semidirect
33. #1848C22semidirect
34. #1837C22semidirect
38. #1493C22semidirect
39. #1493C22semidirect
40. #1493C22semidirect
46. #1492C22semidirect
47. #1492C22semidirect
48. #1492C22semidirect
55. C2×#322C22semidirect
56. C2×#322C22semidirect

Maximal subgroups

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

subgroupindex
#607822
#219932
#174812
#174782
#174732
C2×#14922
C2×#14522
#98632
#98622
#98602
#98582