Finite Group Databasebeta

← #6490389

Subgroups — #6490389

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34629C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#33104C214yesmaximal
4#30101C212yesmaximal, complemented
5#30070C212yesmaximal, complemented
6#29526C214yesmaximal, complemented
7#29341C212yesmaximal
8#29306C212yesmaximal
9#29275C212yesmaximal, complemented
10#28206C212yesmaximal, complemented
11#27682C212yesmaximal
12#24302C212yesmaximal
13#22119C212yesmaximal, complemented
14#22113C212yesmaximal, complemented
15#20786C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#477C2212yes—
18C2×#477C2212yes—
19#2655—22no—
20#2652—22no—
21#2647C2211yescharacteristic
22#2645C2212yes—
23#2642C2211yescharacteristic
24#2629C2211yescharacteristic
25#2625C2212yes—
26#2621C2211yescharacteristic
27#2402—22no—
28#2401—22no—
29#2369—22no—
30#2365—22no—
31#2345—22no—
32Q8×C22⋊C4—22no—
33#2331C2212yes—
34#2331C2212yes—
35#2316C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2211yescharacteristic
39C4×#125C2211yescharacteristic
40C4×C22⋊Q8C2212yes—
41#2295C2212yes—
42#2285C2211yescharacteristic
43#2274C2212yes—
44C4×#117C2211yescharacteristic
45#2101C2214yes—
46#2101C2214yes—
47#2097C2214yes—
48#2083C2214yes—
49#2059C2214yes—
50#2057C2214yes—
51#2043C2214yes—
52#2024C2214yes—
53#1987C2212yes—
54#1987C2212yes—
55#1948C2212yes—
56#1947C2212yes—
57#1943C2214yes—
58#1942C2214yes—
59#1935C2214yes—
60#1933C2214yes—
61#1928C2214yes—
62#1928C2214yes—
63#1928C2214yes—
64#1927C2214yes—
65#1921C2214yes—
66#1920C2214yes—
67#1918C2214yes—
68#1918C2214yes—
69#1873C2214yes—
70#1872C2214yes—
71#1854C2214yes—
72#1852C2214yes—
73#1840C2212yes—
74#1831C2212yes—
75#1818C2212yes—
76#1803C2212yes—
77#1803C2212yes—
78#1797C2212yes—
79#1767C2211yescharacteristic
80C4×C4⋊C8C2211yescharacteristic
81#1750C2212yes—
82#1746C2212yes—
83C23×Q8—21no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#125C2311yescharacteristic
91C2×C22⋊Q8C2312yes—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8C2312yes—
95C2×#117C2312yes—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2312yes—
100C22×C42C2311yescharacteristic
101#477—28no—
102#477—28no—
103C2×#107C2314yes—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#106C2314yes—
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2312yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×C22⋊C8C2314yes—
114C2×#97C2312yes—
115C2×C4×C8C2312yes—
116#401—22no—
117#400—22no—
118#400—22no—
119#400—22no—
120#400—22no—
121#400—22no—
122#400—22no—
123#400—22no—
124#398—22no—
125#398—22no—
126#398—22no—
127#398—22no—
128#397—22no—
129#397—22no—
130#397—22no—
131#396—22no—
132#391—22no—
133#391—22no—
134#391—24no—
135#391—24no—
136#391C2312yes—
137#391C2312yes—
138#391C2312yes—
139#391C2312yes—
140#389—24no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389C2312yes—
145#389C2314yes—
146#389C2314yes—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—24no—
155#387—24no—
156#385—22no—
157#384C2312yes—
158#384C2311yescharacteristic
159#382C2311yescharacteristic
160#382C2312yes—
161#380—21no—
162#380—21no—
163#380—24no—
164#380C2312yes—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—22no—
180C4×C22⋊C4—24no—
181C4×C22⋊C4D412yes—
182C4×C22⋊C4D412yes—
183#376—22no—
184#376C2311yescharacteristic
185#376C2311yescharacteristic
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201C22×Q8—28no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—84no—
205C22×Q8C2412yes—
206C23×C4—22no—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
210C22×C8C2414yes—
211C22×C8C2414yes—
212C4⋊Q8—28no—
213C4⋊Q8—28no—
214C4⋊Q8—28no—
215C4⋊Q8—24no—
216C4⋊Q8—41no—
217C4⋊Q8—41no—
218C4⋊Q8#4118yes—
219#125—41no—
220#125—41no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—84no—
226C4×Q8—84no—
227#117—24no—
228#117—41no—
229#117—41no—
230#117—42no—
231#117—42no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
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—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280#107—28no—
281#107—28no—
282#107—44no—
283#106—44no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286#103—28no—
287#103—28no—
288#103—44no—
289#103—44no—
290C22⋊C8—28no—
291#97—42no—
292C4×C8—42no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95C2×D412yes—
315#95C2×D412yes—
316#95C2×D412yes—
317#95C2×D412yes—
318C24C22×D411yescharacteristic
319C2×Q8—28no—
320C2×Q8—28no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—84no—
330C2×Q8—816no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—84no—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C42−1+414yes—
356C22×C4C2511yescharacteristic
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C4⋊C4—28no—
364C4⋊C4—28no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4C2×#4118yes—
391C22⋊C4—24no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—82no—
396C42—22no—
397C42—22no—
398C42—24no—
399C42—24no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—42no—
408C42—42no—
409C42—44no—
410C42—44no—
411C42—44no—
412C42—84no—
413C42—84no—
414C42C2×#4114yes—
415C23—22no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23#53612yes—
420C23C23×D411yescharacteristic
421Q8—416no—
422Q8—42no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426Q8—88no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—28no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—88no—
469C2×C4#52512yes—
470C2×C4#52512yes—
471C2×C4#53612yes—
472C2×C4#57814yes—
473C2×C4C22×#4112yes—
474C2×C4C2×2−1+412yes—
475C2×C4#58511yescharacteristic
476C2×C4#58511yescharacteristic
477C8—44no—
478C8—44no—
479C22—22no—
480C22—24no—
481C22—42no—
482C22—41no—
483C22—41no—
484C22C2×#52511yescharacteristic
485C22C2×#53611yescharacteristic
486C22#344911yescharacteristic
487C22C2×#57812yes—
488C22#358512yes—
489C4—22no—
490C4—22no—
491C4—42no—
492C4—42no—
493C4—41no—
494C4—41no—
495C4—48no—
496C4—42no—
497C4—42no—
498C4—84no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4#312714yes—
504C4#351612yes—
505C4#351612yes—
506C2—42no—
507C2#3524511yescharacteristic
508C2C2×#312711yescharacteristic
509C2#6056411yescharacteristic
510C2#6089312yes—
511C2#6099312yes—
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. #30101C2semidirect
5. #30070C2semidirect
6. #29526C2semidirect
9. #29275C2semidirect
10. #28206C2semidirect
13. #22119C2semidirect
14. #22113C2semidirect

Maximal subgroups

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

subgroupindex
#346292
#342162
#331042
#301012
#300702
#295262
#293412
#293062
#292752
#282062
#276822
#243022
#221192
#221132
#207862