Finite Group Databasebeta

← #205426

Subgroups — #205426

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#21892C211yesmaximal, characteristic, complemented
3#21807C211yesmaximal, characteristic
4#20545C211yesmaximal, characteristic, complemented
5C2×#1574C211yesmaximal, characteristic, complemented
6C2×#1547C211yesmaximal, characteristic
7C2×#1542C211yesmaximal, characteristic, complemented
8#9747C211yesmaximal, characteristic, complemented
9#9746C211yesmaximal, characteristic, complemented
10#9741C211yesmaximal, characteristic, complemented
11#9740C211yesmaximal, characteristic, complemented
12#9738C211yesmaximal, characteristic, complemented
13#9735C211yesmaximal, characteristic, complemented
14#9731C211yesmaximal, characteristic, complemented
15#9729C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3228—24no—
19#3228C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#482—21no—
25C2×#482—21no—
26C2×#478—21no—
27C2×#478—21no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32#2283C2211yescharacteristic
33#1987C2211yescharacteristic
34#1987C2211yescharacteristic
35#1982C2211yescharacteristic
36#1982C2211yescharacteristic
37#1981C2211yescharacteristic
38#1981C2211yescharacteristic
39#1981C2211yescharacteristic
40#1981C2211yescharacteristic
41#1958—22no—
42#1894—22no—
43#1805—22no—
44#1788—22no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1768—21no—
50#1768—21no—
51#1752—21no—
52C8×C22⋊C4—21no—
53#1574—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1547—24no—
59#1542—24no—
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×M16—21no—
70C23×C8—21no—
71#533—22no—
72#533—22no—
73#533—22no—
74#533C2×C414yes—
75#532—22no—
76#532—22no—
77#532—22no—
78#532C2×C414yes—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#127C2311yescharacteristic
83C2×#125—21no—
84C2×#124—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×#121—21no—
88C2×#121—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×#117—21no—
94C22×C42C2311yescharacteristic
95#500—22no—
96#500C2×C414yes—
97#498—22no—
98#498C2×C414yes—
99#486—22no—
100#486—22no—
101#486C2×C414yes—
102#486C2×C414yes—
103#482—28no—
104#482—28no—
105#478—28no—
106#478—28no—
107#431—22no—
108#431—22no—
109#431C2×C414yes—
110#431D414yes—
111C2×#107—21no—
112C2×#107—21no—
113C2×#106—21no—
114C2×#106—21no—
115#423—22no—
116#423—22no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8D414yes—
121C2×C4⋊C8D411yescharacteristic
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×#103—22no—
130C2×#103—22no—
131C2×#103—22no—
132C2×#103—22no—
133C2×#102—21no—
134C2×#102—21no—
135C2×#102—21no—
136C2×#102—21no—
137C2×#102—22no—
138C2×#102—22no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×#97—21no—
143C2×#97—21no—
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8—21no—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#383—21no—
150#382D412yes—
151#381—21no—
152#380D412yes—
153C4×C4⋊C4C2311yescharacteristic
154C4×C4⋊C4C2311yescharacteristic
155C4×C22⋊C4—21no—
156#376—21no—
157#340—22no—
158#340—22no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#335—22no—
164#335—22no—
165#334—22no—
166#334—22no—
167#333—22no—
168#333—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#331—22no—
174#331—22no—
175C2×#41—24no—
176C2×#41—42no—
177C2×#41—82no—
178C2×#41—82no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—41no—
186C23×C4—41no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4111yescharacteristic
190C2×M16—42no—
191C2×M16—82no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—24no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—81no—
200C22×C8—81no—
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8C22×C412yes—
215C4⋊Q8C22×C412yes—
216C4⋊Q8C22×C412yes—
217#127—21no—
218#127—21no—
219#127—21no—
220#127C22×C412yes—
221#125—24no—
222#125—42no—
223#124—24no—
224#124—42no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—42no—
228C22⋊Q8—42no—
229#121—24no—
230#121—24no—
231#121—42no—
232#121—42no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—42no—
236C4×Q8—42no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—42no—
240C4×D4—42no—
241#117—24no—
242#117—42no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—41no—
260C2×C42C22×C412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42#4111yescharacteristic
266C2×C42#4112yes—
267C2×C42C2411yescharacteristic
268#107—24no—
269#107—24no—
270#106—24no—
271#106—24no—
272C4⋊C8—24no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—42no—
284C4⋊C8C22⋊C414yes—
285C4⋊C8C22⋊C414yes—
286C4⋊C8C2×D412yes—
287C4⋊C8C2×D412yes—
288#103—82no—
289#103—82no—
290#103—82no—
291#103—82no—
292#102—24no—
293#102—24no—
294#102—24no—
295#102—24no—
296#102—82no—
297#102—82no—
298C22⋊C8—42no—
299C22⋊C8C22⋊C414yes—
300C22⋊C8C22⋊C414yes—
301#97—81no—
302#97—81no—
303C4×C8—42no—
304C4×C8—81no—
305C4×C8—81no—
306C4×C8C22×C412yes—
307C4×C8C22×C412yes—
308C4×C8C2×D412yes—
309C4×C8C2×D412yes—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—21no—
314#95—21no—
315C24—41no—
316C24#12111yescharacteristic
317#41—88no—
318#41—84no—
319#41—84no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—82no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—44no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—82no—
345C2×D4—84no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—44no—
359C22×C4—84no—
360C22×C4C2×C22⋊C412yes—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#12112yes—
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12412yes—
368C22×C4#12712yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372M16—84no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—84no—
396C2×C8C2×C22⋊C412yes—
397C2×C8C2×C22⋊C412yes—
398C2×C8C2×C22⋊C412yes—
399C2×C8C2×C22⋊C412yes—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—44no—
427C42C2×C22⋊C412yes—
428C42C4×D412yes—
429C42C4×D412yes—
430C42#12112yes—
431C42#12412yes—
432C42#12712yes—
433C42C23×C411yescharacteristic
434C42C22×D411yescharacteristic
435C42C22×D411yescharacteristic
436C42C2×#4111yescharacteristic
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—42no—
442C23—81no—
443C23—81no—
444C23—82no—
445C23—82no—
446C23#39012yes—
447C23C2×#12111yescharacteristic
448Q8—44no—
449Q8—44no—
450Q8—82no—
451Q8—82no—
452Q8—81no—
453Q8—81no—
454Q8—81no—
455Q8—81no—
456D4—42no—
457D4—42no—
458D4—44no—
459D4—82no—
460D4—82no—
461D4—88no—
462D4—81no—
463D4—81no—
464D4—81no—
465D4—81no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—88no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#39012yes—
494C2×C4#39012yes—
495C2×C4#39012yes—
496C2×C4C22×C22⋊C411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12411yescharacteristic
501C2×C4C2×#12711yescharacteristic
502C2×C4#57612yes—
503C2×C4#57711yescharacteristic
504C2×C4#57711yescharacteristic
505C2×C4#57712yes—
506C2×C4#57811yescharacteristic
507C2×C4#57811yescharacteristic
508C8—22no—
509C8—22no—
510C8—22no—
511C8—22no—
512C8—42no—
513C8—42no—
514C8—81no—
515C8—81no—
516C8—81no—
517C8—81no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—162no—
527C22—162no—
528C22C2×#39011yescharacteristic
529C22#304611yescharacteristic
530C22#304611yescharacteristic
531C22#306212yes—
532C22#306712yes—
533C4—21no—
534C4—21no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4#289811yescharacteristic
544C4#289811yescharacteristic
545C4#294612yes—
546C4#304211yescharacteristic
547C4#304411yescharacteristic
548C4#315411yescharacteristic
549C4#315511yescharacteristic
550C2—42no—
551C2—162no—
552C2#1958412yes—
553C2#1958612yes—
554C2#2123511yescharacteristic
555C2#2124811yescharacteristic
556C2#2126211yescharacteristic
subgroupG/H
center, socleC23C2×#121
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. #21892C2semidirect
4. #20545C2semidirect
5. C2×#1574C2semidirect
7. C2×#1542C2semidirect
8. #9747C2semidirect
9. #9746C2semidirect
10. #9741C2semidirect
11. #9740C2semidirect
12. #9738C2semidirect
13. #9735C2semidirect
14. #9731C2semidirect
15. #9729C2semidirect
19. #3228C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#218922
#218072
#205452
C2×#15742
C2×#15472
C2×#15422
#97472
#97462
#97412
#97402
#97382
#97352
#97312
#97292