Finite Group Databasebeta

← #6490407

Subgroups — #6490407

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35314C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#33105C214yesmaximal
4#30096C212yesmaximal, complemented
5#30060C212yesmaximal, complemented
6#29520C212yesmaximal, complemented
7#29512C212yesmaximal, complemented
8#29344C212yesmaximal
9#29311C212yesmaximal
10#29284C212yesmaximal, complemented
11#28216C212yesmaximal, complemented
12#27680C212yesmaximal
13#24306C212yesmaximal
14#22128C214yesmaximal, complemented
15#20785C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#479C2214yes—
18#2864C2211yescharacteristic
19#2863C2212yes—
20#2862C2211yescharacteristic
21#2628C2211yescharacteristic
22#2625C2212yes—
23#2621C2211yescharacteristic
24#2524—22no—
25#2523—22no—
26#2520—22no—
27#2517—22no—
28#2360—22no—
29#2360—22no—
30#2355—22no—
31Q8×C4⋊C4—22no—
32#2331C2211yescharacteristic
33#2330C2212yes—
34#2330C2212yes—
35#2327C2211yescharacteristic
36#2320C2212yes—
37#2317C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C2×#384C2211yescharacteristic
42#2274C2212yes—
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2099C2212yes—
46#2099C2212yes—
47#2096C2212yes—
48#2096C2212yes—
49#2085C2214yes—
50#2058C2212yes—
51#2058C2214yes—
52#2057C2212yes—
53#2043C2214yes—
54#2024C2212yes—
55#2024C2212yes—
56#1989C2214yes—
57#1948C2214yes—
58#1943C2214yes—
59#1941C2214yes—
60#1935C2214yes—
61#1933C2214yes—
62#1928C2214yes—
63#1927C2214yes—
64#1927C2214yes—
65#1927C2214yes—
66#1921C2214yes—
67#1920C2214yes—
68#1919C2214yes—
69#1919C2214yes—
70#1873C2214yes—
71#1872C2214yes—
72#1854C2214yes—
73#1853C2214yes—
74#1839C2212yes—
75#1832C2212yes—
76#1818C2212yes—
77#1803C2214yes—
78#1797C2212yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83#533—28no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#125C2312yes—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8C2312yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8C2312yes—
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100#479—216no—
101C2×#107C2314yes—
102C2×#106C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2314yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×C22⋊C8C2314yes—
111C2×#97C2312yes—
112C2×C4×C8C2312yes—
113#400—22no—
114#400—21no—
115#399—21no—
116#398—21no—
117#398—21no—
118#398—22no—
119#397—21no—
120#397—21no—
121#397—22no—
122#397—22no—
123#395—21no—
124#393—21no—
125#391—22no—
126#391—22no—
127#391—22no—
128#391—22no—
129#391C2312yes—
130#391C2312yes—
131#391C2312yes—
132#391C2312yes—
133#389—22no—
134#389—22no—
135#389—24no—
136#389—24no—
137#389C2314yes—
138#389C2314yes—
139#389C2312yes—
140#389C2312yes—
141#389C2312yes—
142#389C2312yes—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—24no—
154#387—24no—
155#387—22no—
156#385—22no—
157#385—22no—
158#385—22no—
159#384—22no—
160#384—22no—
161#384C2312yes—
162#384C2312yes—
163#383—22no—
164#383—22no—
165#382—22no—
166#382—22no—
167#382—22no—
168#382C2311yescharacteristic
169#382C2311yescharacteristic
170#380C2312yes—
171C4×C4⋊C4—24no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2312yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—24no—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4D412yes—
184#376C2311yescharacteristic
185#376C2311yescharacteristic
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202C22×Q8—22no—
203C22×Q8—21no—
204C22×Q8C2412yes—
205C23×C4—22no—
206C23×C4C2×D412yes—
207C23×C4C2411yescharacteristic
208C22×C8C2414yes—
209C22×C8C2414yes—
210C4⋊Q8—28no—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213#125—24no—
214#125#4118yes—
215C22⋊Q8—216no—
216C22⋊Q8—28no—
217C4×Q8—216no—
218C4×Q8—28no—
219C4×Q8—84no—
220C4×Q8—84no—
221#117—24no—
222#117—28no—
223#117—42no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—84no—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2414yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4C2×D412yes—
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2×D411yescharacteristic
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275#107—28no—
276#106—28no—
277#106—44no—
278#106—44no—
279C4⋊C8—28no—
280#103—28no—
281#103—28no—
282#103—44no—
283#103—44no—
284C22⋊C8—28no—
285#97—42no—
286C4×C8—42no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312#95C2×D412yes—
313C24C22×D411yescharacteristic
314C2×Q8—28no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—44no—
319C2×Q8—42no—
320C2×Q8—44no—
321C2×Q8—82no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—84no—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D412yes—
344C22×C42−1+414yes—
345C22×C4C2511yescharacteristic
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C4⋊C4—28no—
353C4⋊C4—216no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4C2×#4118yes—
379C22⋊C4—28no—
380C22⋊C4—24no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—44no—
384C22⋊C4—82no—
385C42—28no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—42no—
395C42—42no—
396C42—44no—
397C42—82no—
398C42—84no—
399C42—84no—
400C42C2×#4114yes—
401C23—22no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23#53612yes—
406C23C23×D411yescharacteristic
407Q8—416no—
408Q8—48no—
409Q8—162no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—164no—
451C2×C4#52514yes—
452C2×C4#53612yes—
453C2×C4#57714yes—
454C2×C4C22×#4112yes—
455C2×C4C2×2−1+412yes—
456C2×C4#58512yes—
457C8—44no—
458C8—44no—
459C22—22no—
460C22—24no—
461C22—41no—
462C22—41no—
463C22—42no—
464C22C2×#52511yescharacteristic
465C22C2×#53611yescharacteristic
466C22#344911yescharacteristic
467C22C2×#57712yes—
468C22#358612yes—
469C4—24no—
470C4—24no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—48no—
477C4—42no—
478C4—42no—
479C4—84no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4#351614yes—
485C2—42no—
486C2#3524511yescharacteristic
487C2#6058512yes—
488C2#6088912yes—
489C2#6101211yescharacteristic
490C2#6101811yescharacteristic
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
4. #30096C2semidirect
5. #30060C2semidirect
6. #29520C2semidirect
7. #29512C2semidirect
10. #29284C2semidirect
11. #28216C2semidirect
14. #22128C2semidirect

Maximal subgroups

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

subgroupindex
#353142
#342142
#331052
#300962
#300602
#295202
#295122
#293442
#293112
#292842
#282162
#276802
#243062
#221282
#207852