Finite Group Databasebeta

← #6489035

Subgroups — #6489035

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34629C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic
3#32967C212yesmaximal
4#32915C212yesmaximal
5#30140C214yesmaximal, complemented
6#29135C212yesmaximal, complemented
7#28391C212yesmaximal, complemented
8#28287C212yesmaximal, complemented
9#28255C212yesmaximal, complemented
10#27682C212yesmaximal
11#24514C214yesmaximal
12#24212C212yesmaximal
13#22154C212yesmaximal, complemented
14#22148C212yesmaximal, complemented
15#20812C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#519C2212yes—
18C2×#479C2214yes—
19C2×#462C2212yes—
20C2×#462C2212yes—
21C2×#462C2214yes—
22C2×#451C2214yes—
23C2×#433C2212yes—
24#2655—22no—
25#2652—22no—
26#2647C2211yescharacteristic
27#2645C2212yes—
28#2642C2211yescharacteristic
29#2629C2211yescharacteristic
30#2625C2212yes—
31#2621C2211yescharacteristic
32#2402—22no—
33#2401—22no—
34#2365—22no—
35#2365—22no—
36#2344—22no—
37#2344—22no—
38#2327C2212yes—
39#2327C2212yes—
40#2310C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×#125C2211yescharacteristic
45C4×C22⋊Q8C2212yes—
46#2295C2212yes—
47#2285C2211yescharacteristic
48C4×#117C2211yescharacteristic
49#2101C2214yes—
50#2097C2214yes—
51#2068C2214yes—
52#2066C2214yes—
53#2056C2214yes—
54#2056C2214yes—
55#2043C2214yes—
56#2016C2214yes—
57#1981C2212yes—
58#1981C2212yes—
59#1971C2214yes—
60#1969C2214yes—
61#1948C2212yes—
62#1947C2212yes—
63#1935C2214yes—
64#1933C2214yes—
65#1878C2214yes—
66#1878C2214yes—
67#1873C2214yes—
68#1865C2214yes—
69#1863C2214yes—
70#1854C2214yes—
71#1842C2212yes—
72#1832C2212yes—
73#1805C2214yes—
74#1803C2212yes—
75#1803C2212yes—
76#1803C2214yes—
77#1798C2212yes—
78#1797C2212yes—
79#1787C2214yes—
80#1768C2212yes—
81#1767C2211yescharacteristic
82C4×C4⋊C8C2211yescharacteristic
83C23×Q8C2312yes—
84C22×Q16C2314yes—
85C22×Q16C2314yes—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#125C2311yescharacteristic
93C2×C22⋊Q8C2312yes—
94#519—48no—
95C2×C4×Q8—24no—
96C2×C4×Q8C2314yes—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2314yes—
99C2×#117—24no—
100C2×#117C2312yes—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C42C2311yescharacteristic
106#479—48no—
107#462—28no—
108#462—28no—
109#462—48no—
110#451—48no—
111#433—44no—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×C22⋊C8C2314yes—
127C2×#97C2312yes—
128C2×C4×C8C2312yes—
129#401—22no—
130#400—22no—
131#400—22no—
132#400—22no—
133#400—22no—
134#400—22no—
135#400—22no—
136#400—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#397—22no—
142#397—22no—
143#397—22no—
144#396—22no—
145#391—24no—
146#391C2314yes—
147#391C2314yes—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—24no—
166#387—24no—
167#387—24no—
168#387—24no—
169#385—22no—
170#384C2312yes—
171#384C2311yescharacteristic
172#382C2311yescharacteristic
173#382C2312yes—
174#381—24no—
175#380—21no—
176#380—21no—
177#380C2312yes—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4D412yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4D412yes—
193C4×C22⋊C4D412yes—
194#376—22no—
195#376C2311yescharacteristic
196#376C2311yescharacteristic
197C43C2311yescharacteristic
198C43C2311yescharacteristic
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2312yes—
205#336C2312yes—
206C22×Q8—28no—
207C22×Q8—22no—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—44no—
211C22×Q8C2412yes—
212C22×Q8C2414yes—
213C23×C4—22no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C2×Q16—28no—
218C2×Q16—28no—
219C2×Q16—44no—
220C2×Q16—44no—
221C2×Q16—44no—
222C2×Q16—44no—
223C22×C8C2414yes—
224C22×C8C2414yes—
225C4⋊Q8—28no—
226C4⋊Q8—28no—
227C4⋊Q8—28no—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230C4⋊Q8#4118yes—
231#125—24no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—28no—
237C4×Q8—28no—
238C4×Q8—44no—
239C4×Q8—44no—
240C4×Q8—416no—
241#117—22no—
242#117—22no—
243#117—24no—
244#117—416no—
245#117—44no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#107—44no—
295#106—44no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—28no—
299C4⋊C8—44no—
300#103—28no—
301#103—28no—
302#103—44no—
303#103—44no—
304#103—44no—
305#103—44no—
306#103—44no—
307#103—44no—
308C22⋊C8—28no—
309#97—42no—
310C4×C8—42no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335#95C2×D412yes—
336C24C22×D411yescharacteristic
337C2×Q8—28no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8C2×#4118yes—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4H(5,𝔽2)12yes—
379C22×C42−1+412yes—
380C22×C4C2511yescharacteristic
381Q16—88no—
382Q16—88no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—28no—
392C4⋊C4—28no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C4⋊C4—44no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—82no—
421C42—22no—
422C42—22no—
423C42—24no—
424C42—24no—
425C42—22no—
426C42—21no—
427C42—21no—
428C42—22no—
429C42—21no—
430C42—21no—
431C42—42no—
432C42—42no—
433C42—44no—
434C42—48no—
435C42—44no—
436C42—48no—
437C42—44no—
438C42C2×#4114yes—
439C42H(5,𝔽2)14yes—
440C23—22no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23#53712yes—
445C23C23×D411yescharacteristic
446Q8—28no—
447Q8—28no—
448Q8—416no—
449Q8—44no—
450Q8—44no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—42no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—44no—
477C2×C4—416no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4#52512yes—
497C2×C4#52512yes—
498C2×C4#53514yes—
499C2×C4#53712yes—
500C2×C4#57814yes—
501C2×C4C22×#4112yes—
502C2×C4C2×H(5,𝔽2)11yescharacteristic
503C2×C4C2×H(5,𝔽2)12yes—
504C2×C4C2×2−1+411yescharacteristic
505C8—44no—
506C8—44no—
507C22—22no—
508C22—24no—
509C22—41no—
510C22—41no—
511C22—42no—
512C22C2×#52511yescharacteristic
513C22C2×#53511yescharacteristic
514C22C2×#53711yescharacteristic
515C22C2×#57812yes—
516C22#358612yes—
517C4—22no—
518C4—22no—
519C4—24no—
520C4—41no—
521C4—41no—
522C4—48no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—84no—
530C4—82no—
531C4#312714yes—
532C4#351912yes—
533C4#352012yes—
534C2—42no—
535C2#3523611yescharacteristic
536C2C2×#312711yescharacteristic
537C2#6056611yescharacteristic
538C2#6083712yes—
539C2#6092512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #30140C2semidirect
6. #29135C2semidirect
7. #28391C2semidirect
8. #28287C2semidirect
9. #28255C2semidirect
13. #22154C2semidirect
14. #22148C2semidirect

Maximal subgroups

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

subgroupindex
#346292
#342092
#329672
#329152
#301402
#291352
#283912
#282872
#282552
#276822
#245142
#242122
#221542
#221482
#208122