Finite Group Databasebeta

← #206267

Subgroups — #206267

Order 512.

1016 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21835C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic
5#21409C211yesmaximal, characteristic
6#21324C211yesmaximal, characteristic
7C2×#1453C211yesmaximal, characteristic
8#9841C211yesmaximal, characteristic, complemented
9#9840C211yesmaximal, characteristic, complemented
10#9833C211yesmaximal, characteristic, complemented
11#9828C211yesmaximal, characteristic, complemented
12#9828C211yesmaximal, characteristic, complemented
13#9826C211yesmaximal, characteristic, complemented
14#9821C211yesmaximal, characteristic, complemented
15#9819C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18#3229—24no—
19#3229C418yescomplemented
20C2×#498C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#483—22no—
25C2×#475—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#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1990C2211yescharacteristic
42#1989—22no—
43#1981—22no—
44#1956C2211yescharacteristic
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1956C2211yescharacteristic
49#1956C2211yescharacteristic
50#1949—22no—
51#1894—22no—
52#1854—22no—
53#1845C2211yescharacteristic
54#1844C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57#1825—22no—
58#1787—22no—
59#1763—22no—
60#1736—22no—
61#1453—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C23×Q8—21no—
67C22×M16D412yes—
68#533—22no—
69#533—22no—
70#533—22no—
71#533C2×C414yes—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8C2×C414yes—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×#125—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×#117—21no—
89C2×#117—21no—
90C2×#117—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C42C2311yescharacteristic
94#498—22no—
95#498—22no—
96#498C2×C414yes—
97#498C2×C414yes—
98#487—22no—
99#487—22no—
100#487C2×C414yes—
101#487C2×C414yes—
102#483—216no—
103#475—216no—
104#431—22no—
105#431—22no—
106#431C2×C414yes—
107#431D414yes—
108C2×#107—22no—
109C2×#106—22no—
110C2×#106—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×#103—22no—
121C2×#103—22no—
122C2×C22⋊C8D412yes—
123C2×C22⋊C8D412yes—
124C4×M16—22no—
125C4×M16—22no—
126C2×#97D412yes—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#382—21no—
138#382—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
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—
163C22×Q8—24no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—42no—
169C22×Q8—41no—
170C22×Q8—82no—
171C22×Q8—82no—
172C23×C4—41no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×M16—24no—
177C2×M16—24no—
178C2×M16—42no—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—42no—
201C4⋊Q8—42no—
202C4⋊Q8—42no—
203C4⋊Q8C22×C412yes—
204C4⋊Q8C22×C412yes—
205C4⋊Q8C22×C412yes—
206C4⋊Q8C22×C412yes—
207#125—24no—
208#125—42no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—42no—
216C4×Q8—42no—
217#117—24no—
218#117—42no—
219#117—44no—
220#117—82no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
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—21no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42C22×C412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4112yes—
261C2×C42#4112yes—
262C2×C42C2411yescharacteristic
263#107—82no—
264#106—28no—
265#106—28no—
266#106—82no—
267C4⋊C8—24no—
268C4⋊C8C22⋊C418yes—
269#103—28no—
270#103—28no—
271#103—44no—
272#103—44no—
273#103—82no—
274#103—82no—
275#103—82no—
276#103—82no—
277C22⋊C8—24no—
278C22⋊C8C22⋊C418yes—
279#97—24no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8C22×C412yes—
289C4×C8C22×C412yes—
290C4×C8C2×D412yes—
291C4×C8C2×D412yes—
292#95—41no—
293#95—41no—
294#95—41no—
295#95—41no—
296C24#12711yescharacteristic
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—44no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—88no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8—88no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—44no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4C2×C22⋊C412yes—
336C22×C4C4×D412yes—
337C22×C4C4×D412yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12412yes—
341C22×C4#12711yescharacteristic
342C22×C4#12711yescharacteristic
343C22×C4#12711yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347M16—48no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8C2×C22⋊C414yes—
364C2×C8C2×C22⋊C414yes—
365C2×C8#13714yes—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C22⋊C4—44no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C42—44no—
402C42—82no—
403C42—82no—
404C42C2×C22⋊C412yes—
405C42C4×D412yes—
406C42C4×D412yes—
407C42#12112yes—
408C42#12112yes—
409C42#12412yes—
410C42C23×C411yescharacteristic
411C42C22×D411yescharacteristic
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23#39012yes—
421C23C2×#12711yescharacteristic
422Q8—44no—
423Q8—44no—
424Q8—44no—
425Q8—44no—
426Q8—88no—
427Q8—82no—
428Q8—82no—
429Q8—82no—
430Q8—82no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—88no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#39012yes—
469C2×C4#39012yes—
470C2×C4#39012yes—
471C2×C4C22×C22⋊C411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×#12111yescharacteristic
476C2×C4C2×#12411yescharacteristic
477C2×C4C2×#13712yes—
478C2×C4#57512yes—
479C2×C4#57712yes—
480C2×C4#57712yes—
481C2×C4#57712yes—
482C2×C4#57812yes—
483C8—24no—
484C8—24no—
485C8—24no—
486C8—42no—
487C8—42no—
488C8—42no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22C2×#39011yescharacteristic
497C22#315211yescharacteristic
498C22#315311yescharacteristic
499C22#315411yescharacteristic
500C22#315411yescharacteristic
501C22#315411yescharacteristic
502C22#315611yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—161no—
511C4—161no—
512C4—161no—
513C4—161no—
514C4#289812yes—
515C4C4×#13712yes—
516C4#294912yes—
517C4#305612yes—
518C4#305712yes—
519C4#313712yes—
520C2—42no—
521C2#1971411yescharacteristic
522C2#1971611yescharacteristic
523C2#1971811yescharacteristic
524C2#1972011yescharacteristic
525C2#2120511yescharacteristic
526C2#2123111yescharacteristic
527C2#2123711yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#127
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
8. #9841C2semidirect
9. #9840C2semidirect
10. #9833C2semidirect
11. #9828C2semidirect
12. #9828C2semidirect
13. #9826C2semidirect
14. #9821C2semidirect
15. #9819C2semidirect
19. #3229C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#219842
#218352
#218042
#214092
#213242
C2×#14532
#98412
#98402
#98332
#98282
#98262
#98212
#98192