Finite Group Databasebeta

← #6469479

Subgroups — #6469479

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2C4×#541C211yesmaximal, characteristic
3#29403C214yesmaximal, complemented
4#29274C212yesmaximal, complemented
5#28205C212yesmaximal, complemented
6#27649C212yesmaximal
7#27621C212yesmaximal
8#24301C212yesmaximal
9#24072C212yesmaximal
10#23845C212yesmaximal, complemented
11#23829C212yesmaximal, complemented
12#21615C212yesmaximal, complemented
13C4×#475C212yesmaximal, complemented
14#21591C214yesmaximal
15#20786C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#485C2214yes—
19C2×#475C2212yes—
20C2×#475C2212yes—
21#2369—22no—
22#2365—22no—
23#2345—22no—
24Q8×C22⋊C4—22no—
25#2336—22no—
26#2334—22no—
27#2331C2212yes—
28#2331C2212yes—
29#2330C2211yescharacteristic
30#2327C2211yescharacteristic
31#2323C2212yes—
32#2320C2212yes—
33#2316C2212yes—
34#2314C2211yescharacteristic
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2211yescharacteristic
38C4×#125—22no—
39C4×#125C2211yescharacteristic
40C4×#126—22no—
41C4×C22⋊Q8C2212yes—
42#2285C2211yescharacteristic
43#2274C2212yes—
44C42×Q8C2212yes—
45C4×#117C2211yescharacteristic
46#2101C2214yes—
47#2085C2214yes—
48#2084C2214yes—
49#2074C2214yes—
50#2059C2214yes—
51#2058C2214yes—
52#2043C2214yes—
53#2024C2214yes—
54#1988C2212yes—
55#1988C2212yes—
56#1948C2212yes—
57#1947C2212yes—
58#1927C2214yes—
59#1927C2214yes—
60#1921C2214yes—
61#1920C2214yes—
62#1919C2214yes—
63#1919C2214yes—
64#1872C2214yes—
65#1865C2214yes—
66#1863C2214yes—
67#1853C2214yes—
68#1840C2212yes—
69#1831C2212yes—
70#1825C2214yes—
71#1818C2212yes—
72#1803C2212yes—
73#1803C2212yes—
74#1797C2212yes—
75#1776C2214yes—
76C4×#106C2214yes—
77#1767C2211yescharacteristic
78C4×C4⋊C8C2211yescharacteristic
79#1763C2214yes—
80C4×#103C2214yes—
81#1750C2212yes—
82#1746C2212yes—
83C23×Q8—21no—
84#541—28no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#126—22no—
92C2×#125—22no—
93C2×#125C2311yescharacteristic
94C2×C22⋊Q8C2312yes—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×#117C2312yes—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2312yes—
105C22×C42C2311yescharacteristic
106#485—216no—
107#475—28no—
108#475—28no—
109C2×#107C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2312yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×C22⋊C8C2314yes—
120C2×#97C2312yes—
121C2×C4×C8C2312yes—
122#391—22no—
123#391—22no—
124#391—24no—
125#391—24no—
126#391C2312yes—
127#391C2312yes—
128#389—22no—
129#389—24no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389C2312yes—
135#389C2314yes—
136#389C2314yes—
137#389C2312yes—
138#388—22no—
139#388—22no—
140#388—22no—
141#387—22no—
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#387—24no—
152#385—22no—
153#384C2312yes—
154#384C2311yescharacteristic
155#383—22no—
156#382—22no—
157#382C2311yescharacteristic
158#382C2312yes—
159#380—21no—
160#380—21no—
161#380—24no—
162#380C2312yes—
163C4×C4⋊C4—24no—
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⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—24no—
183C4×C22⋊C4D412yes—
184C4×C22⋊C4D412yes—
185#376C2311yescharacteristic
186#376C2311yescharacteristic
187C43—22no—
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2312yes—
201#336C2312yes—
202#336C2314yes—
203C22×Q8—28no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—84no—
207C22×Q8C2412yes—
208C23×C4—22no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2411yescharacteristic
212C22×C8C2414yes—
213C22×C8C2414yes—
214C4⋊Q8—28no—
215C4⋊Q8—28no—
216C4⋊Q8—24no—
217C4⋊Q8—28no—
218C4⋊Q8—24no—
219C4⋊Q8#4118yes—
220#126—216no—
221#125—216no—
222#125—24no—
223C22⋊Q8—24no—
224C22⋊Q8#4118yes—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—84no—
232C4×Q8—84no—
233#117—24no—
234#117—22no—
235#117—22no—
236#117—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4C2×D412yes—
266C2×C22⋊C4C2×D412yes—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289#107—28no—
290#106—28no—
291#106—28no—
292#106—28no—
293C4⋊C8—24no—
294C4⋊C8—24no—
295#103—28no—
296#103—28no—
297#103—28no—
298#103—28no—
299C22⋊C8—28no—
300#97—24no—
301C4×C8—24no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95C2×D412yes—
320#95C2×D412yes—
321#95C2×D412yes—
322#95C2×D412yes—
323C24C22×D411yescharacteristic
324C2×Q8—28no—
325C2×Q8—28no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—84no—
334C2×Q8—816no—
335C2×Q8C2×#4114yes—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—42no—
350C22×C4—84no—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4114yes—
361C22×C4C2511yescharacteristic
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4C2×#4118yes—
395C4⋊C4C2×#4114yes—
396C22⋊C4—28no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—44no—
400C22⋊C4—82no—
401C42—28no—
402C42—22no—
403C42—22no—
404C42—24no—
405C42—24no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—21no—
410C42—21no—
411C42—42no—
412C42—42no—
413C42—44no—
414C42—44no—
415C42—44no—
416C42—44no—
417C42—82no—
418C42—82no—
419C42—84no—
420C42—84no—
421C42C2×#4114yes—
422C422−1+412yes—
423C422−1+412yes—
424C23—22no—
425C23—21no—
426C23—21no—
427C23—22no—
428C23#52512yes—
429C23C23×D411yescharacteristic
430Q8—24no—
431Q8—24no—
432Q8—416no—
433Q8—42no—
434Q8—42no—
435Q8—88no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—88no—
477C2×C4#52512yes—
478C2×C4#52512yes—
479C2×C4#52512yes—
480C2×C4#53614yes—
481C2×C4#57714yes—
482C2×C4C22×#4112yes—
483C2×C4C22×#4112yes—
484C2×C4C2×2−1+411yescharacteristic
485C2×C4C2×2−1+411yescharacteristic
486C8—44no—
487C8—44no—
488C22—22no—
489C22—24no—
490C22—41no—
491C22—41no—
492C22—42no—
493C22C2×#52511yescharacteristic
494C22C2×#52511yescharacteristic
495C22C2×#53611yescharacteristic
496C22C2×#57712yes—
497C22#358612yes—
498C4—22no—
499C4—22no—
500C4—24no—
501C4—41no—
502C4—48no—
503C4—41no—
504C4—44no—
505C4—42no—
506C4—42no—
507C4—44no—
508C4—42no—
509C4—82no—
510C4—84no—
511C4—82no—
512C4#312314yes—
513C4#348212yes—
514C4#348212yes—
515C2—42no—
516C2#3497111yescharacteristic
517C2C2×#312311yescharacteristic
518C2#6056611yescharacteristic
519C2#6059912yes—
520C2#6088912yes—
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. #29403C2semidirect
4. #29274C2semidirect
5. #28205C2semidirect
10. #23845C2semidirect
11. #23829C2semidirect
12. #21615C2semidirect
13. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#342162
C4×#5412
#294032
#292742
#282052
#276492
#276212
#243012
#240722
#238452
#238292
#216152
C4×#4752
#215912
#207862