Finite Group Databasebeta

← #6526722

Subgroups — #6526722

Order 512.

1878 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2C2×#3113C211yesmaximal, characteristic
3C2×#2567C211yesmaximal, characteristic
4#34235C211yesmaximal, characteristic
5#28124C214yesmaximal
6#27647C214yesmaximal
7#25453C214yesmaximal, complemented
8#25324C214yesmaximal, complemented
9#24831C214yesmaximal, complemented
10#24698C214yesmaximal, complemented
11#21953C211yesmaximal, characteristic
12#21840C211yesmaximal, characteristic
13#20735C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17#3229—28no—
18C2×#498C2212yes—
19C2×#487C2212yes—
20#3113—28no—
21C2×#483C2212yes—
22C2×#483C2214yes—
23C2×#475C2212yes—
24C2×#475C2214yes—
25C2×#462C2212yes—
26C2×#451C2212yes—
27C2×#441C2214yes—
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30#2567—21no—
31#2567—22no—
32#2567—21no—
33#2567C2214yes—
34#2567C2214yes—
35C2×#395—22no—
36#2365C2212yes—
37#2365C2212yes—
38#2360—22no—
39#2356C2212yes—
40#2351C2212yes—
41#2343—22no—
42#2336—22no—
43#2320—21no—
44C4×C22⋊Q8C2214yes—
45C2×#391C2212yes—
46C2×#389—22no—
47C2×#384C2212yes—
48#2283C2211yescharacteristic
49#2276—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2084C2218yes—
52#2072C2218yes—
53#2066C2218yes—
54#2056C2218yes—
55#1993C2212yes—
56#1993C2212yes—
57#1987C2214yes—
58#1948C2218yes—
59#1940C2214yes—
60#1939C2214yes—
61#1927C2218yes—
62#1878C2218yes—
63#1823C2214yes—
64#1823C2212yes—
65#1805C2212yes—
66#1805C2212yes—
67#1803C2218yes—
68#1797C2214yes—
69#1787C2212yes—
70C4×#103C2214yes—
71C23×Q8—21no—
72C23×Q8C2311yescharacteristic
73C22×Q16C2314yes—
74#533—24no—
75#533D418yes—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2311yescharacteristic
84C2×#125—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8C2312yes—
87C2×C4×Q8—22no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2314yes—
97C2×#117—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4D412yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C42—21no—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107#498—28no—
108#487—28no—
109#483—28no—
110#483—216no—
111#475—28no—
112#475—216no—
113#462—28no—
114#451—28no—
115#441—216no—
116#432—24no—
117#431—24no—
118C2×#106C2318yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2318yes—
123C2×#103C2318yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×#97C2312yes—
131C2×C4×C8C2312yes—
132#395—24no—
133#395—24no—
134#395—28no—
135#391—22no—
136#391—22no—
137#391—22no—
138#391C2318yes—
139#391C2314yes—
140#389—28no—
141#389—24no—
142#389—24no—
143#389—22no—
144#389—24no—
145#389C2314yes—
146#387—24no—
147#387—24no—
148#387—24no—
149#387—24no—
150#385—24no—
151#384—22no—
152#384—22no—
153#384C2318yes—
154#383—21no—
155#383—22no—
156#382—22no—
157#382C2311yescharacteristic
158#382C2311yescharacteristic
159#381—21no—
160#380C2312yes—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D412yes—
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2312yes—
169C4×C4⋊C4C2312yes—
170C4×C4⋊C4C2312yes—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—21no—
173#376—21no—
174#376—22no—
175C2×#95—24no—
176C2×#95D412yes—
177#340C2318yes—
178#340C2314yes—
179#340C2318yes—
180#340C2314yes—
181#336C2318yes—
182C22×Q8—24no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8—44no—
191C22×Q8—84no—
192C22×Q8C2412yes—
193C22×Q8C2412yes—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4C2×D414yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2411yescharacteristic
201C2×Q16—28no—
202C2×Q16—28no—
203C2×Q16—44no—
204C22×C8C2414yes—
205C22×C8C2414yes—
206C4⋊Q8—24no—
207C4⋊Q8—24no—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—44no—
213C4⋊Q8—44no—
214C4⋊Q8C2×D414yes—
215C4⋊Q8C2×D414yes—
216#125—28no—
217C22⋊Q8—28no—
218C22⋊Q8—24no—
219C22⋊Q8C2×D418yes—
220C4×Q8—28no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—48no—
227C4×Q8—48no—
228C4×Q8—44no—
229C4×Q8—44no—
230C4×Q8C2×D418yes—
231#117—28no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—88no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D414yes—
278C2×C42#4114yes—
279C2×C42C2414yes—
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282#106—28no—
283#106—216no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286C4⋊C8—48no—
287#103—28no—
288#103—216no—
289#103—28no—
290#103—28no—
291#103—44no—
292C22⋊C8—24no—
293C22⋊C8—24no—
294#97—24no—
295C4×C8—24no—
296#95—28no—
297#95—28no—
298#95—21no—
299#95—22no—
300#95—21no—
301#95—24no—
302#95—24no—
303#95—21no—
304#95—21no—
305#95—24no—
306#95—22no—
307#95—24no—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D414yes—
311C24C22×D411yescharacteristic
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8—816no—
332C2×Q8C22×D414yes—
333C2×Q82−1+414yes—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—48no—
355C22×C4#12114yes—
356C22×C4#12114yes—
357C22×C4C22×D414yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D414yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C2×#4112yes—
366C22×C4C2511yescharacteristic
367Q16—416no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4—84no—
398C4⋊C4#13718yes—
399C4⋊C4C22×D414yes—
400C22⋊C4—28no—
401C22⋊C4—44no—
402C42—28no—
403C42—28no—
404C42—42no—
405C42—42no—
406C42—44no—
407C42—44no—
408C42—48no—
409C42—44no—
410C42—84no—
411C42#12114yes—
412C42#12114yes—
413C42C22×D412yes—
414C42C22×D412yes—
415C42C2×#4112yes—
416C42C2×#4112yes—
417C23—21no—
418C23—21no—
419C23—22no—
420C23C2×#12112yes—
421C23#57714yes—
422C23C23×D411yescharacteristic
423Q8—24no—
424Q8—28no—
425Q8—24no—
426Q8—48no—
427Q8—42no—
428Q8—44no—
429Q8—42no—
430Q8—48no—
431Q8—88no—
432Q8—88no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—28no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—416no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—84no—
468C2×C4—88no—
469C2×C4C2×#12112yes—
470C2×C4C2×#12112yes—
471C2×C4C2×#12112yes—
472C2×C4C2×#12112yes—
473C2×C4C2×#12112yes—
474C2×C4C2×#13718yes—
475C2×C4C2×#13714yes—
476C2×C4C23×D411yescharacteristic
477C2×C4C22×#4111yescharacteristic
478C2×C4C2×2−1+412yes—
479C2×C4#58512yes—
480C8—44no—
481C8—44no—
482C22—24no—
483C22—22no—
484C22—24no—
485C22—22no—
486C22#304314yes—
487C22C22×#12111yescharacteristic
488C22#348511yescharacteristic
489C22#348611yescharacteristic
490C22C22×#13712yes—
491C22C2×#57712yes—
492C4—22no—
493C4—22no—
494C4—28no—
495C4—22no—
496C4—22no—
497C4—44no—
498C4—44no—
499C4—44no—
500C4—44no—
501C4—82no—
502C4—84no—
503C4—82no—
504C4—82no—
505C4#303914yes—
506C4#348912yes—
507C4#349512yes—
508C2—24no—
509C2#3563311yescharacteristic
510C2C2×#303911yescharacteristic
511C2C2×#304311yescharacteristic
512C2#6173512yes—
513C2#6183812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #25453C2semidirect
8. #25324C2semidirect
9. #24831C2semidirect
10. #24698C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
C2×#31132
C2×#25672
#342352
#281242
#276472
#254532
#253242
#248312
#246982
#219532
#218402
#207352