Finite Group Databasebeta

← #206271

Subgroups — #206271

Order 512.

1102 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21805C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic, complemented
5#21410C211yesmaximal, characteristic
6#21324C211yesmaximal, characteristic, complemented
7C2×#1453C211yesmaximal, characteristic, complemented
8#9838C211yesmaximal, characteristic, complemented
9#9836C211yesmaximal, characteristic, complemented
10#9834C211yesmaximal, characteristic, complemented
11#9831C211yesmaximal, characteristic, complemented
12#9830C211yesmaximal, characteristic, complemented
13#9827C211yesmaximal, characteristic, complemented
14#9823C211yesmaximal, characteristic, complemented
15#9822C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18#3226—24no—
19#3226C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#480—22no—
25C2×#474—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#1993—22no—
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1983—22no—
39#1981—22no—
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1957C2211yescharacteristic
43#1957C2211yescharacteristic
44#1956C2211yescharacteristic
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1956C2211yescharacteristic
49#1956C2211yescharacteristic
50#1949—22no—
51#1895—22no—
52#1854—22no—
53#1845C2211yescharacteristic
54#1844C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57#1788—22no—
58#1787—22no—
59#1762—22no—
60#1736—22no—
61#1453—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C22×#41—21no—
67C23×Q8—21no—
68C22×M16D412yes—
69#532—22no—
70#532—22no—
71#532—22no—
72#532C2×C414yes—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8C2×C414yes—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×#127C2311yescharacteristic
84C2×#124—21no—
85C2×#121—21no—
86C2×#121—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×#117—21no—
90C2×#117—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C42C2311yescharacteristic
94#500—22no—
95#500—22no—
96#500C2×C414yes—
97#500C2×C414yes—
98#486—22no—
99#486—22no—
100#486C2×C414yes—
101#486C2×C414yes—
102#480—216no—
103#474—216no—
104#431—22no—
105#431—22no—
106#431C2×C414yes—
107#431D414yes—
108C2×#107—22no—
109C2×#107—22no—
110C2×#107—22no—
111C2×#106—22no—
112C2×C4⋊C8D412yes—
113C2×C4⋊C8D412yes—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—22no—
117C2×#103—22no—
118C2×#103—22no—
119C2×#103—22no—
120C2×#102—22no—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×C22⋊C8D412yes—
127C2×C22⋊C8D412yes—
128C4×M16—22no—
129C4×M16—22no—
130C2×#97D412yes—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#382—21no—
142#382—21no—
143#380—21no—
144#380—21no—
145#380—21no—
146#380—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—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—24no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—42no—
172C22×Q8—41no—
173C22×Q8—82no—
174C22×Q8—82no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—41no—
179C23×C4—41no—
180C23×C4—41no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C2×M16—24no—
185C2×M16—24no—
186C2×M16—42no—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8C2×D412yes—
192C22×C8C2×D412yes—
193C4⋊Q8—24no—
194C4⋊Q8—24no—
195C4⋊Q8—24no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—42no—
206C4⋊Q8—42no—
207C4⋊Q8—42no—
208C4⋊Q8C22×C412yes—
209C4⋊Q8C22×C412yes—
210C4⋊Q8C22×C412yes—
211#127—21no—
212#127—21no—
213#127—21no—
214#127C22×C412yes—
215#124—24no—
216#124—42no—
217#121—24no—
218#121—24no—
219#121—42no—
220#121—42no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—42no—
224C4×D4—42no—
225#117—44no—
226#117—82no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
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—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42C22×C412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42#4112yes—
263C2×C42#4112yes—
264C2×C42C2411yescharacteristic
265#107—28no—
266#107—28no—
267#107—82no—
268#106—82no—
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—44no—
280#102—44no—
281#102—82no—
282#102—82no—
283C22⋊C8—24no—
284C22⋊C8C22⋊C418yes—
285#97—24no—
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—
300#95—41no—
301#95—41no—
302C24—41no—
303C24#12711yescharacteristic
304#41—88no—
305#41—88no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—44no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—88no—
320C2×Q8—82no—
321C2×Q8—82no—
322C2×Q8—88no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—44no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—82no—
337C2×D4—82no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4C2×C22⋊C412yes—
358C22×C4C4×D412yes—
359C22×C4C4×D412yes—
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4#12412yes—
363C22×C4#12711yescharacteristic
364C22×C4#12711yescharacteristic
365C22×C4#12711yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369M16—48no—
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×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8C2×C22⋊C414yes—
386C2×C8C2×C22⋊C414yes—
387C2×C8#13714yes—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C42—82no—
419C42—82no—
420C42C2×C22⋊C412yes—
421C42C4×D412yes—
422C42C4×D412yes—
423C42#12112yes—
424C42#12112yes—
425C42#12412yes—
426C42C23×C411yescharacteristic
427C42C22×D411yescharacteristic
428C42C2×#4111yescharacteristic
429C42C2×#4111yescharacteristic
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—42no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23#39012yes—
444C23C2×#12711yescharacteristic
445Q8—44no—
446Q8—44no—
447Q8—44no—
448Q8—88no—
449Q8—82no—
450Q8—82no—
451Q8—82no—
452D4—44no—
453D4—44no—
454D4—44no—
455D4—88no—
456D4—82no—
457D4—82no—
458D4—82no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—88no—
475C2×C4—88no—
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—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4#39012yes—
495C2×C4#39012yes—
496C2×C4#39012yes—
497C2×C4C22×C22⋊C411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4C2×#12111yescharacteristic
502C2×C4C2×#12411yescharacteristic
503C2×C4C2×#13712yes—
504C2×C4#57512yes—
505C2×C4#57612yes—
506C2×C4#57612yes—
507C2×C4#57712yes—
508C2×C4#57812yes—
509C8—24no—
510C8—24no—
511C8—24no—
512C8—42no—
513C8—42no—
514C8—42no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—161no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22C2×#39011yescharacteristic
530C22#315211yescharacteristic
531C22#315311yescharacteristic
532C22#315411yescharacteristic
533C22#315511yescharacteristic
534C22#315511yescharacteristic
535C22#315611yescharacteristic
536C4—22no—
537C4—22no—
538C4—22no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—161no—
543C4—161no—
544C4—161no—
545C4—161no—
546C4#289812yes—
547C4C4×#13712yes—
548C4#294812yes—
549C4#305612yes—
550C4#305812yes—
551C4#313712yes—
552C2—42no—
553C2—162no—
554C2#1972211yescharacteristic
555C2#1972611yescharacteristic
556C2#1972811yescharacteristic
557C2#1973011yescharacteristic
558C2#2120511yescharacteristic
559C2#2123311yescharacteristic
560C2#2123711yescharacteristic
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. #21984C2semidirect
4. #21804C2semidirect
6. #21324C2semidirect
7. C2×#1453C2semidirect
8. #9838C2semidirect
9. #9836C2semidirect
10. #9834C2semidirect
11. #9831C2semidirect
12. #9830C2semidirect
13. #9827C2semidirect
14. #9823C2semidirect
15. #9822C2semidirect
19. #3226C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#219842
#218052
#218042
#214102
#213242
C2×#14532
#98382
#98362
#98342
#98312
#98302
#98272
#98232
#98222