Finite Group Databasebeta

← #6499312

Subgroups — #6499312

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic
2#34281C211yesmaximal, characteristic
3#33075C214yesmaximal, complemented
4#29985C212yesmaximal, complemented
5#29983C212yesmaximal, complemented
6#29284C212yesmaximal, complemented
7#29280C212yesmaximal, complemented
8#29264C212yesmaximal, complemented
9#29255C212yesmaximal, complemented
10#29170C214yesmaximal
11#29165C212yesmaximal
12#29159C212yesmaximal
13#21967C212yesmaximal
14#21898C212yesmaximal
15#20754C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C2×#424C2212yes—
18#2868C2211yescharacteristic
19#2862C2211yescharacteristic
20#2857C2212yes—
21#2647C2212yes—
22#2628C2212yes—
23#2511—22no—
24#2492—22no—
25#2451—22no—
26#2423—22no—
27#2373—22no—
28#2373—22no—
29#2371—22no—
30#2370—22no—
31#2331C2212yes—
32#2330C2211yescharacteristic
33#2327C2211yescharacteristic
34#2323C2212yes—
35#2314C2212yes—
36#2314C2211yescharacteristic
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2211yescharacteristic
39C4×#125C2211yescharacteristic
40C4×C22⋊Q8C2212yes—
41#2295C2212yes—
42#2276C2212yes—
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2099C2212yes—
46#2099C2212yes—
47#2097C2214yes—
48#2096C2212yes—
49#2096C2212yes—
50#2058C2214yes—
51#2058C2212yes—
52#2057C2212yes—
53#2024C2214yes—
54#2024C2212yes—
55#2024C2212yes—
56#1989C2214yes—
57#1981C2214yes—
58#1949C2214yes—
59#1949C2214yes—
60#1948C2214yes—
61#1948C2214yes—
62#1921C2214yes—
63#1920C2214yes—
64#1920C2214yes—
65#1919C2214yes—
66#1894C2214yes—
67#1892C2214yes—
68#1865C2214yes—
69#1865C2214yes—
70#1865C2214yes—
71#1863C2214yes—
72#1832C2212yes—
73#1832C2212yes—
74#1831C2212yes—
75#1831C2212yes—
76#1809C2214yes—
77#1807C2214yes—
78#1805C2214yes—
79#1803C2214yes—
80#1729C2212yes—
81#1728C2211yescharacteristic
82C2×#336C2211yescharacteristic
83C22×M16C2312yes—
84C23×C8C2312yes—
85#541—24no—
86#541Q818yescomplemented
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#126—22no—
92C2×#125—22no—
93C2×#125C2311yescharacteristic
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8C2312yes—
96C2×C4×Q8—22no—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C2×#107C2314yes—
104C2×#106C2314yes—
105C2×#106C2314yes—
106C2×#106C2314yes—
107#424—28no—
108C2×C4⋊C8C2314yes—
109C2×C4⋊C8C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114#400—22no—
115#400—24no—
116#399—22no—
117#398—24no—
118#395—22no—
119#393—22no—
120#391—22no—
121#391—22no—
122#391—22no—
123#391—22no—
124#391—24no—
125#391C2312yes—
126#391C2312yes—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389C2312yes—
133#389C2314yes—
134#389C2312yes—
135#389C2314yes—
136#388—22no—
137#388—24no—
138#388—22no—
139#388—22no—
140#387—22no—
141#387—24no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—24no—
151#385—22no—
152#384—22no—
153#384C2311yescharacteristic
154#384C2312yes—
155#384C2312yes—
156#382C2311yescharacteristic
157#382C2312yes—
158#382C2312yes—
159#380—22no—
160#380C2312yes—
161#380C2314yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—22no—
177C4×C22⋊C4D412yes—
178C4×C22⋊C4D412yes—
179#376—22no—
180#376—22no—
181#376C2311yescharacteristic
182#376C2311yescharacteristic
183C43C2311yescharacteristic
184C43C2311yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2312yes—
194#336C2312yes—
195#336C2314yes—
196#336C2314yes—
197#336C2312yes—
198#336C2312yes—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8C2412yes—
202C23×C4C2×D412yes—
203C23×C4C2411yescharacteristic
204C2×M16—24no—
205C2×M16—42no—
206C2×M16—42no—
207C22×C8—24no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8C2414yes—
211C22×C8C2414yes—
212C4⋊Q8—22no—
213C4⋊Q8—44no—
214C4⋊Q8—44no—
215C4⋊Q8—84no—
216C4⋊Q8C2×Q814yes—
217#126—216no—
218#125—216no—
219#125—22no—
220#125C2×Q814yes—
221C22⋊Q8—24no—
222C22⋊Q8—84no—
223C22⋊Q8C2×Q818yes—
224C4×Q8—24no—
225C4×Q8—44no—
226C4×Q8—44no—
227C4×Q8—84no—
228C4×Q8C2×Q818yes—
229#117—42no—
230#117C2×Q814yes—
231#117#4114yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4C2×D411yescharacteristic
260C2×C22⋊C4C2×D411yescharacteristic
261C2×C22⋊C4C2×D412yes—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—42no—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281#107—44no—
282#106—44no—
283#106—44no—
284#106—44no—
285C4⋊C8—28no—
286C4⋊C8—28no—
287#103—44no—
288#103—44no—
289#103—44no—
290#103—44no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—24no—
308#95—22no—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312#95C2×D412yes—
313C24C22×D411yescharacteristic
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—84no—
322C2×Q8C22×Q814yes—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D412yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×Q812yes—
344C22×C4C2×#4112yes—
345C22×C4C2511yescharacteristic
346M16—84no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—84no—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4C22⋊Q818yes—
380C4⋊C4C22×Q814yes—
381C22⋊C4—28no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—82no—
385C22⋊C4—84no—
386C22⋊C4C22⋊Q818yes—
387C42—28no—
388C42—24no—
389C42—24no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—44no—
398C42—44no—
399C42—82no—
400C42—82no—
401C42—82no—
402C42—84no—
403C42C22×Q812yes—
404C42C22×Q812yes—
405C42C2×#4112yes—
406C42C2×#4112yes—
407C23—22no—
408C23—21no—
409C23—21no—
410C23—22no—
411C23C2×C22⋊Q812yes—
412C23C23×D411yescharacteristic
413Q8—28no—
414Q8—44no—
415Q8—88no—
416Q8—164no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—84no—
457C2×C4C2×C22⋊Q814yes—
458C2×C4C2×C22⋊Q814yes—
459C2×C4C2×C22⋊Q812yes—
460C2×C4#57714yes—
461C2×C4C23×Q811yescharacteristic
462C2×C4C22×#4111yescharacteristic
463C2×C4#58512yes—
464C2×C4#58512yes—
465C8—44no—
466C8—44no—
467C22—22no—
468C22—24no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22C22×C22⋊Q811yescharacteristic
473C22#344911yescharacteristic
474C22#344911yescharacteristic
475C22C2×#57712yes—
476C22#358612yes—
477C4—22no—
478C4—24no—
479C4—24no—
480C4—24no—
481C4—44no—
482C4—42no—
483C4—44no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—82no—
489C4—84no—
490C4—84no—
491C4—82no—
492C4#350412yes—
493C4#351012yes—
494C2—42no—
495C2#3533411yescharacteristic
496C2#6049712yes—
497C2#6097712yes—
498C2#6101211yescharacteristic
499C2#6102011yescharacteristic
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
3. #33075C2semidirect
4. #29985C2semidirect
5. #29983C2semidirect
6. #29284C2semidirect
7. #29280C2semidirect
8. #29264C2semidirect
9. #29255C2semidirect
86. #541Q8semidirect

Maximal subgroups

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

subgroupindex
#352832
#342812
#330752
#299852
#299832
#292842
#292802
#292642
#292552
#291702
#291652
#291592
#219672
#218982
#207542