Finite Group Databasebeta

← #206301

Subgroups — #206301

Order 512.

1028 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3185C211yesmaximal, characteristic
2#21864C211yesmaximal, characteristic
3#21835C211yesmaximal, characteristic
4#21400C211yesmaximal, characteristic
5#21318C211yesmaximal, characteristic
6#20824C211yesmaximal, characteristic
7C2×C8×M16C211yesmaximal, characteristic
8#9841C211yesmaximal, characteristic, complemented
9#9840C211yesmaximal, characteristic, complemented
10#9840C211yesmaximal, characteristic, complemented
11#9839C211yesmaximal, characteristic, complemented
12#9825C211yesmaximal, characteristic, complemented
13#9821C211yesmaximal, characteristic, complemented
14#9821C211yesmaximal, characteristic, complemented
15#9819C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3185—24no—
19#3185C418yescomplemented
20C2×#498C2211yescharacteristic
21C2×#494C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#470—22no—
25C2×#441—22no—
26C2×#431C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#1991—22no—
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1990C2211yescharacteristic
42#1990C2211yescharacteristic
43#1990C2211yescharacteristic
44#1989—22no—
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1956C2211yescharacteristic
49#1892—22no—
50#1873—22no—
51#1845C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1825—22no—
56#1809—22no—
57#1805—22no—
58#1776—22no—
59C4×#103—22no—
60#1735—22no—
61C8×M16—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C22×Q16—22no—
67C22×M16D412yes—
68C23×C8D412yes—
69#533—22no—
70#533—22no—
71#533—22no—
72#533—22no—
73#533—22no—
74#533—22no—
75#533C2×C414yes—
76#533C2×C414yes—
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×#125—21no—
82C2×#125—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×#117—21no—
92C2×#117—21no—
93C2×#117—21no—
94C2×#117—21no—
95C22×C42C2311yescharacteristic
96#498—22no—
97#498C2×C414yes—
98#494—22no—
99#494C2×C414yes—
100#487—22no—
101#487—22no—
102#487C2×C414yes—
103#487C2×C414yes—
104#470—216no—
105#441—216no—
106#431—22no—
107#431—22no—
108C2×#107—22no—
109C2×#106—22no—
110C2×#106—22no—
111C2×C4⋊C8D412yes—
112C2×#103—22no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—22no—
117C2×#103—22no—
118C2×C22⋊C8D412yes—
119C4×M16C2×C414yes—
120C4×M16D414yes—
121C2×#97D412yes—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8D412yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#382—21no—
136#382—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140#380—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145#340—22no—
146#340—22no—
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—
161C82—22no—
162C82—22no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×Q16—28no—
171C2×Q16—44no—
172C2×Q16—44no—
173C2×M16—24no—
174C2×M16—24no—
175C2×M16C22⋊C418yes—
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—42no—
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⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C22×C412yes—
198C4⋊Q8C22×C412yes—
199C4⋊Q8C22×C412yes—
200C4⋊Q8C22×C412yes—
201#125—24no—
202#125—24no—
203#125—42no—
204#125—42no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8—24no—
209C22⋊Q8—42no—
210C22⋊Q8—42no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—42no—
218C4×Q8—42no—
219C4×Q8—42no—
220C4×Q8—42no—
221#117—24no—
222#117—24no—
223#117—42no—
224#117—42no—
225#117—44no—
226#117—82no—
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—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
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×C42—41no—
258C2×C42—41no—
259C2×C42C22×C412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4112yes—
265C2×C42#4112yes—
266C2×C42C2411yescharacteristic
267#107—44no—
268#106—28no—
269#106—44no—
270C4⋊C8—24no—
271#103—28no—
272#103—28no—
273#103—44no—
274#103—44no—
275#103—82no—
276#103—82no—
277C22⋊C8—24no—
278#97C22⋊C418yes—
279C4×C8—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—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4C2×C22⋊C412yes—
328C22×C4C4×D412yes—
329C22×C4C4×D412yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4#12412yes—
333C22×C4#12711yescharacteristic
334C22×C4#12711yescharacteristic
335C22×C4#12711yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339Q16—416no—
340M16—216no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—48no—
355C2×C8C2×C22⋊C414yes—
356C2×C8C2×C22⋊C414yes—
357C2×C8#13514yes—
358C2×C8#13514yes—
359C2×C8#13714yes—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C42—44no—
397C42—44no—
398C42—82no—
399C42—82no—
400C42C2×C22⋊C412yes—
401C42C4×D412yes—
402C42C4×D412yes—
403C42#12112yes—
404C42#12112yes—
405C42#12412yes—
406C42C23×C411yescharacteristic
407C42C22×D411yescharacteristic
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23#39012yes—
417C23C2×#12711yescharacteristic
418Q8—44no—
419Q8—44no—
420Q8—44no—
421Q8—44no—
422Q8—82no—
423Q8—82no—
424Q8—82no—
425Q8—82no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
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#39012yes—
463C2×C4#39012yes—
464C2×C4#39012yes—
465C2×C4C22×C22⋊C411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×#12111yescharacteristic
470C2×C4C2×#12411yescharacteristic
471C2×C4C2×#13512yes—
472C2×C4C2×#13512yes—
473C2×C4C2×#13712yes—
474C2×C4#57512yes—
475C2×C4#57712yes—
476C2×C4#57812yes—
477C8—24no—
478C8—24no—
479C8—42no—
480C8—42no—
481C8—42no—
482C8#41618yes—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22C2×#39011yescharacteristic
491C22#314711yescharacteristic
492C22#314711yescharacteristic
493C22#315211yescharacteristic
494C22#315311yescharacteristic
495C22#315411yescharacteristic
496C22#315611yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—161no—
505C4—161no—
506C4—161no—
507C4—161no—
508C4C2×#41612yes—
509C4C4×#13712yes—
510C4#294712yes—
511C4#305612yes—
512C4#306312yes—
513C4#313112yes—
514C2—42no—
515C2#1964211yescharacteristic
516C2#1964311yescharacteristic
517C2#1964811yescharacteristic
518C2#1965011yescharacteristic
519C2#2119511yescharacteristic
520C2#2123411yescharacteristic
521C2#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. #9840C2semidirect
11. #9839C2semidirect
12. #9825C2semidirect
13. #9821C2semidirect
14. #9821C2semidirect
15. #9819C2semidirect
19. #3185C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31852
#218642
#218352
#214002
#213182
#208242
C2×C8×M162
#98412
#98402
#98392
#98252
#98212
#98192