Finite Group Databasebeta

← #206282

Subgroups — #206282

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21864C211yesmaximal, characteristic, complemented
4#21860C211yesmaximal, characteristic, complemented
5#21859C211yesmaximal, characteristic
6#21805C211yesmaximal, characteristic
7C2×#1454C211yesmaximal, characteristic, complemented
8#9841C211yesmaximal, characteristic, complemented
9#9838C211yesmaximal, characteristic, complemented
10#9834C211yesmaximal, characteristic, complemented
11#9830C211yesmaximal, characteristic, complemented
12#9830C211yesmaximal, characteristic, complemented
13#9828C211yesmaximal, characteristic, complemented
14#9823C211yesmaximal, characteristic, complemented
15#9822C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3228—24no—
19#3228C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#482—22no—
25C2×#478—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1983—22no—
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1956C2211yescharacteristic
43#1956C2211yescharacteristic
44#1956C2211yescharacteristic
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1947—22no—
49#1874—22no—
50#1873—22no—
51#1872—22no—
52#1852—22no—
53#1845C2211yescharacteristic
54#1845C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57#1805—22no—
58#1803—22no—
59#1802—22no—
60#1788—22no—
61#1454—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C22×#41—21no—
67#533—22no—
68#533—22no—
69#533—22no—
70#533C2×C414yes—
71#532—22no—
72#532—22no—
73#532—22no—
74#532C2×C414yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×#127C2311yescharacteristic
79C2×#125—21no—
80C2×#124—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—21no—
83C2×#121—21no—
84C2×#121—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×#117—21no—
90C2×#117—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C42C2311yescharacteristic
95#500—22no—
96#500C2×C414yescomplemented
97#498—22no—
98#498C2×C414yescomplemented
99#486—22no—
100#486—22no—
101#486C2×C414yescomplemented
102#486C2×C414yescomplemented
103#482—216no—
104#478—216no—
105#431—22no—
106#431—22no—
107#431—22no—
108#431—22no—
109#431C2×C414yes—
110#431D414yes—
111C2×#107—22no—
112C2×#107—22no—
113C2×#107—22no—
114C2×#106—22no—
115C2×C4⋊C8D412yes—
116C2×C4⋊C8D412yes—
117C2×C4⋊C8D412yes—
118C2×#103—22no—
119C2×#103—22no—
120C2×#103—22no—
121C2×#103—22no—
122C2×#103—22no—
123C2×#103—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×#102—22no—
127C2×#102—22no—
128C2×#102—22no—
129C2×#102—22no—
130C2×C22⋊C8D412yes—
131C2×C22⋊C8D412yes—
132C2×C22⋊C8D412yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139#384—21no—
140#384—21no—
141#384—21no—
142#384C2311yescharacteristic
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#384C2311yescharacteristic
146#382—21no—
147#380—21no—
148#380—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#322—22no—
160#322—22no—
161#322—22no—
162C82—22no—
163C2×#41—24no—
164C2×#41—42no—
165C2×#41—82no—
166C2×#41—82no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C412yes—
196C4⋊Q8C22×C412yes—
197#127—21no—
198#127—21no—
199#127—21no—
200#127C22×C412yes—
201#125—24no—
202#125—42no—
203#124—24no—
204#124—42no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—42no—
208C22⋊Q8—42no—
209#121—24no—
210#121—24no—
211#121—42no—
212#121—42no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—42no—
216C4×Q8—42no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—42no—
220C4×D4—42no—
221#117—24no—
222#117—42no—
223#117—82no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4—82no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42C22×C412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264#107—28no—
265#107—82no—
266#107—82no—
267#106—28no—
268C4⋊C8—24no—
269C4⋊C8—24no—
270C4⋊C8C22⋊C418yes—
271#103—82no—
272#103—82no—
273#103—82no—
274#103—82no—
275#103—82no—
276#103—82no—
277#102—28no—
278#102—28no—
279#102—82no—
280#102—82no—
281#102—82no—
282#102—82no—
283C22⋊C8—24no—
284C22⋊C8—24no—
285C22⋊C8C22⋊C418yes—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8C22×C412yes—
295C4×C8C22×C412yes—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298#95—41no—
299#95—41no—
300C24—41no—
301C24#12711yescharacteristic
302#41—88no—
303#41—88no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—44no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—82no—
330C2×D4—82no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4C2×C22⋊C412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12412yes—
356C22×C4#12711yescharacteristic
357C22×C4#12711yescharacteristic
358C22×C4#12711yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8C2×C22⋊C414yes—
379C2×C8C2×C22⋊C414yes—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—82no—
413C42—44no—
414C42—82no—
415C42C2×C22⋊C412yes—
416C42C4×D412yes—
417C42C4×D412yes—
418C42#12112yes—
419C42#12112yes—
420C42#12412yes—
421C42C23×C411yescharacteristic
422C42C22×D411yescharacteristic
423C42C2×#4111yescharacteristic
424C42C2×#4111yescharacteristic
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—42no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23#39012yes—
439C23C2×#12711yescharacteristic
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—82no—
444Q8—82no—
445Q8—82no—
446D4—44no—
447D4—44no—
448D4—44no—
449D4—88no—
450D4—82no—
451D4—82no—
452D4—82no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—88no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—164no—
488C2×C4#39012yes—
489C2×C4#39012yes—
490C2×C4#39012yes—
491C2×C4C22×C22⋊C411yescharacteristic
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C2×C4×D411yescharacteristic
494C2×C4C2×#12111yescharacteristic
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12411yescharacteristic
497C2×C4#57612yes—
498C2×C4#57612yes—
499C2×C4#57712yes—
500C2×C4#57712yes—
501C2×C4#57812yes—
502C2×C4#57812yes—
503C8—24no—
504C8—24no—
505C8—42no—
506C8—42no—
507C8—42no—
508C8—42no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—161no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22C2×#39011yescharacteristic
524C22#315411yescharacteristic
525C22#315411yescharacteristic
526C22#315511yescharacteristic
527C22#315511yescharacteristic
528C22#315611yescharacteristic
529C22#315611yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—161no—
537C4—161no—
538C4—161no—
539C4—161no—
540C4#289812yes—
541C4#294612yes—
542C4#295112yes—
543C4#306112yes—
544C4#306212yes—
545C4#313712yes—
546C2—42no—
547C2—162no—
548C2#1973511yescharacteristic
549C2#1973911yescharacteristic
550C2#1974011yescharacteristic
551C2#1974411yescharacteristic
552C2#2120511yescharacteristic
553C2#2123011yescharacteristic
554C2#2123511yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21982C2semidirect
3. #21864C2semidirect
4. #21860C2semidirect
7. C2×#1454C2semidirect
8. #9841C2semidirect
9. #9838C2semidirect
10. #9834C2semidirect
11. #9830C2semidirect
12. #9830C2semidirect
13. #9828C2semidirect
14. #9823C2semidirect
15. #9822C2semidirect
19. #3228C4semidirect
96. #500C2×C4semidirect
98. #498C2×C4semidirect
101. #486C2×C4semidirect
102. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#219822
#218642
#218602
#218592
#218052
C2×#14542
#98412
#98382
#98342
#98302
#98282
#98232
#98222