Finite Group Databasebeta

← #360729

Subgroups — #360729

Order 512.

1548 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
1#34183C211yesmaximal, characteristic
2C2×#1860C212yesmaximal
3C2×#1859C212yesmaximal
4#20801C211yesmaximal, characteristic, complemented
5#20649C211yesmaximal, characteristic
6C2×#1759C211yesmaximal, characteristic
7#14354C211yesmaximal, characteristic, complemented
8#11340C212yesmaximal, complemented
9#11339C212yesmaximal, complemented
10#10857C211yesmaximal, characteristic, complemented
11#10840C211yesmaximal, characteristic, complemented
12C2×#520C2212yes—
13C2×C4×#41C2211yescharacteristic
14C2×#428C2212yes—
15C2×#428C2212yes—
16C2×#423C2211yescharacteristic
17C22×#104C2212yes—
18C2×#417C2211yescharacteristic
19C2×#417C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21Q8×C4⋊C4—22no—
22D4×C4⋊C4—22no—
23D4×C4⋊C4—24no—
24#2317—22no—
25#2317C2212yes—
26#2285—21no—
27#2285C2211yescharacteristic
28C2×#384—21no—
29C2×#384C2212yes—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2212yes—
33C2×#376C2211yescharacteristic
34#1860—28no—
35#1859—28no—
36#1847—21no—
37#1847—21no—
38#1846—22no—
39#1805—22no—
40#1804—22no—
41#1759C418yescomplemented
42C2×#343C2212yes—
43C2×#343C2212yes—
44#1736C2211yescharacteristic
45#1736C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49#1444—24no—
50C22×#41C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54#520—24no—
55#520—24no—
56#520—42no—
57#520—42no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41C2×C414yes—
61C4×#41Q814yes—
62C2×C4×Q8—22no—
63C2×C4×Q8D412yes—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—24no—
67C2×C4×D4D412yes—
68C2×#117—22no—
69C2×#117—21no—
70C2×#117C2312yes—
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4D414yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C42—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#428—24no—
83#428—24no—
84#428—42no—
85#428—42no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#106—22no—
89C2×#106—22no—
90#423—22no—
91#423—22no—
92C2×C4⋊C8—22no—
93C2×#104C2×C418yes—
94C2×#104D418yes—
95C2×#104Q818yes—
96#417—24no—
97#417—24no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#102—21no—
101C2×#102—21no—
102C4×M16—22no—
103C4×M16C2×C414yescomplemented
104C2×#97—21no—
105C2×C4×C8—21no—
106#389—28no—
107#389—24no—
108#385—28no—
109#385—24no—
110#384—24no—
111#384—22no—
112#384—21no—
113#384—22no—
114#384—21no—
115#384—24no—
116#384—42no—
117#384D411yescharacteristic
118#384D411yescharacteristic
119#382—22no—
120#382—22no—
121#382—21no—
122#382—21no—
123#382—24no—
124#382—42no—
125#382D412yes—
126#380—24no—
127#380—22no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—22no—
130#376—22no—
131#376C2×C414yescomplemented
132C2×#95—22no—
133C2×#95—24no—
134C2×#95—22no—
135#343—24no—
136#343—24no—
137#340—22no—
138#340—22no—
139#339—24no—
140#339—24no—
141#339C2×C418yescomplemented
142C2×#41—21no—
143C2×#41—21no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41C22×C412yes—
147C2×#41C2×D414yes—
148C2×#41C2×Q812yes—
149C2×#41C2×Q814yes—
150C22×Q8C2×D411yescharacteristic
151C22×D4—21no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C2×D414yes—
160C23×C4C2×D412yes—
161C23×C4C2411yescharacteristic
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16C22×C414yes—
174C2×M16C2×D414yes—
175C2×M16C2×Q814yes—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C4×Q8—24no—
180C4×Q8—44no—
181C4×Q8—44no—
182C4×Q8C4⋊C418yescomplemented
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—24no—
186C4×D4—48no—
187C4×D4—48no—
188C4×D4—44no—
189C4×D4—44no—
190C4×D4C4⋊C418yescomplemented
191#117—21no—
192#117—21no—
193#117—24no—
194#117—24no—
195#117—22no—
196#117—22no—
197#117—22no—
198#117—22no—
199#117—44no—
200#117—44no—
201#117C22×C412yes—
202#117C2×Q812yes—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—88no—
228C2×C4⋊C4—84no—
229C2×C4⋊C4—88no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42C22×C414yes—
247C2×C42C22×C412yes—
248C2×C42C2×D414yes—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×Q814yes—
253C2×C42C2×Q812yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256#107—44no—
257#107—44no—
258#106—44no—
259#106—44no—
260C4⋊C8—44no—
261#104C4⋊C4132yescomplemented
262#103—42no—
263#103—42no—
264#102—42no—
265#102—42no—
266#97—24no—
267C4×C8—24no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—48no—
272#95—42no—
273#95—42no—
274C24—21no—
275C24C22×D411yescharacteristic
276#41—24no—
277#41—24no—
278#41—24no—
279#41—24no—
280#41—24no—
281#41—24no—
282#41C2×C4⋊C418yes—
283#41C4⋊Q818yes—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8C2×C4⋊C412yes—
288C2×Q8#12712yes—
289C2×Q8C4⋊Q812yes—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4C2×C4⋊C412yes—
298C2×D4#12712yes—
299C2×D4C4⋊Q812yes—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—28no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—88no—
330C22×C4C4×D414yes—
331C22×C4#12012yes—
332C22×C4#12012yes—
333C22×C4C22⋊Q814yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q811yescharacteristic
339C22×C4C22×Q812yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4H(5,𝔽2)12yes—
343C22×C42−1+412yes—
344M16—24no—
345M16—24no—
346M16—24no—
347M16—42no—
348M16—42no—
349M16—42no—
350M16—42no—
351M16C2×C4⋊C418yescomplemented
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C4⋊C4—22no—
362C4⋊C4—28no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—84no—
382C4⋊C4—88no—
383C4⋊C4—816no—
384C4⋊C4C2×C4⋊C414yes—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—28no—
388C22⋊C4—48no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C42—22no—
392C42—24no—
393C42—22no—
394C42—24no—
395C42—24no—
396C42—28no—
397C42—48no—
398C42—42no—
399C42—42no—
400C42C2×C4⋊C414yes—
401C42C2×C4⋊C418yescomplemented
402C42C4×D412yes—
403C42C4×D412yes—
404C42C22⋊Q812yes—
405C42C22⋊Q812yes—
406C42#12314yes—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—42no—
413C23—42no—
414C23C2×C4×D411yescharacteristic
415C23C2×#12011yescharacteristic
416C23C2×C22⋊Q811yescharacteristic
417C23D4212yes—
418C23D4×Q812yes—
419Q8—22no—
420Q8—22no—
421Q8—22no—
422Q8#38414yes—
423D4—28no—
424D4—28no—
425D4—22no—
426D4—22no—
427D4—22no—
428D4#38414yes—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—416no—
443C2×C4—44no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—168no—
464C2×C4#38514yes—
465C2×C4#38514yes—
466C2×C4C22×C4⋊C411yescharacteristic
467C2×C4C22×C4⋊C412yes—
468C2×C4C2×C4×D411yescharacteristic
469C2×C4#52012yes—
470C2×C4C2×C22⋊Q811yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×#12711yescharacteristic
473C2×C4C2×C4⋊Q812yes—
474C2×C4C2×C4⋊Q811yescharacteristic
475C2×C4#53511yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#54412yes—
478C2×C4D4212yes—
479C2×C4#54712yes—
480C2×C4#54712yes—
481C2×C4D4×Q812yes—
482C2×C4#55412yes—
483C2×C4#55412yes—
484C8—28no—
485C8—81no—
486C8—81no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—24no—
492C22—24no—
493C22—44no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22C2×#38511yescharacteristic
498C22D4×C4⋊C412yes—
499C22#243311yescharacteristic
500C22#243411yescharacteristic
501C22#244012yes—
502C22#301912yes—
503C22#302012yes—
504C4—24no—
505C4—24no—
506C4—24no—
507C4—48no—
508C4—44no—
509C4—84no—
510C4—82no—
511C4—82no—
512C4—161no—
513C4—161no—
514C4C2×#38411yescharacteristic
515C4#231711yescharacteristic
516C4D4×C4⋊C412yes—
517C4#244012yes—
518C4#250111yescharacteristic
519C4#250211yescharacteristic
520C2—22no—
521C2—22no—
522C2—44no—
523C2#1388011yescharacteristic
524C2#2086412yes—
525C2#2574911yescharacteristic
526C2#2575011yescharacteristic
527C2#2583212yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20801C2semidirect
7. #14354C2semidirect
8. #11340C2semidirect
9. #11339C2semidirect
10. #10857C2semidirect
11. #10840C2semidirect
41. #1759C4semidirect
103. C4×M16C2×C4semidirect
131. #376C2×C4semidirect
141. #339C2×C4semidirect
182. C4×Q8C4⋊C4semidirect
190. C4×D4C4⋊C4semidirect
261. #104C4⋊C4semidirect
351. M16C2×C4⋊C4semidirect
401. C42C2×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341832
C2×#18602
C2×#18592
#208012
#206492
C2×#17592
#143542
#113402
#113392
#108572
#108402