Finite Group Databasebeta

← #398667

Subgroups — #398667

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2087C211yesmaximal, characteristic, complemented
2C2×#1850C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic
5#13951C212yesmaximal
6#11377C211yesmaximal, characteristic, complemented
7#11328C211yesmaximal, characteristic, complemented
8#11292C212yesmaximal, complemented
9#11232C211yesmaximal, characteristic, complemented
10#11227C212yesmaximal, complemented
11#10731C212yesmaximal
12C24×C8C2211yescharacteristic
13C22×#117C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#2087—24no—
24#1850—24no—
25#1808—24no—
26#1735—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1439—22no—
38#1439—22no—
39C4⋊#95—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1434—22no—
43C24×C4C2311yescharacteristic
44C23×C8—24no—
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—42no—
48C23×C8D414yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—42no—
57C2×#117—42no—
58C2×#117Q818yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—42no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—24no—
84#389—24no—
85#389—24no—
86#389—42no—
87#389—42no—
88#380—24no—
89#380—24no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D411yescharacteristic
100C2×#95D411yescharacteristic
101#340—24no—
102#340—22no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340D414yes—
109#336—22no—
110#336—22no—
111C25C2×D411yescharacteristic
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×Q814yes—
129C23×C4#4112yes—
130C23×C4#4114yes—
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—28no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—28no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—44no—
151C22×C8—48no—
152C22×C8—48no—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8#4112yes—
157#117—216no—
158#117—48no—
159#117—48no—
160#117—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×Q814yes—
189C2×C4⋊C4C2×Q814yes—
190C2×C4⋊C4#4114yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C42—28no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—44no—
214C2×C42—44no—
215#107—44no—
216#107—44no—
217C4⋊C8—44no—
218C4⋊C8—44no—
219C22⋊C8—44no—
220#95—22no—
221#95—22no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—82no—
228#95—82no—
229C24—21no—
230C24—22no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—21no—
237C24—21no—
238C24#12112yes—
239C24C22⋊Q814yes—
240C24C22×D411yescharacteristic
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4#12112yes—
308C22×C4C22⋊Q814yes—
309C22×C4#12314yes—
310C22×C4#12518yes—
311C22×C4C4⋊Q814yes—
312C22×C4#13512yes—
313C22×C4#13512yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×Q812yes—
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4112yes—
319C22×C4C2×#4112yes—
320C22×C4C2×#4111yescharacteristic
321C22×C4H(5,𝔽2)12yes—
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C42−1+411yescharacteristic
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—416no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8#12114yes—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—88no—
361C4⋊C4—88no—
362C4⋊C4C22⋊Q818yes—
363C22⋊C4—28no—
364C22⋊C4—416no—
365C22⋊C4—44no—
366C22⋊C4—88no—
367C42—416no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—42no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23#39814yes—
398C23C2×#12111yescharacteristic
399C23C2×C22⋊Q812yes—
400C23D4211yescharacteristic
401C23#54612yes—
402C23#54711yescharacteristic
403C23C2×#13511yescharacteristic
404C23C2×#13511yescharacteristic
405C23#57512yes—
406C23#57612yes—
407C23#57811yescharacteristic
408C23#57811yescharacteristic
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—28no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4#39814yes—
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×C22⋊Q812yes—
472C2×C4C2×#12311yescharacteristic
473C2×C4C2×#12512yes—
474C2×C4C2×C4⋊Q811yescharacteristic
475C2×C4#53211yescharacteristic
476C2×C4#53312yes—
477C2×C4#53411yescharacteristic
478C2×C4#53711yescharacteristic
479C2×C4#54312yes—
480C2×C4#54412yes—
481C2×C4#54612yes—
482C2×C4#54811yescharacteristic
483C2×C4#55112yes—
484C2×C4#55112yes—
485C2×C4#55212yes—
486C2×C4#55311yescharacteristic
487C2×C4#55312yes—
488C8—24no—
489C8—42no—
490C8—81no—
491C8—81no—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—42no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22C2×#39811yescharacteristic
504C22#241611yescharacteristic
505C22#243712yes—
506C22#243812yes—
507C22#254311yescharacteristic
508C22#305011yescharacteristic
509C22#305411yescharacteristic
510C22#305811yescharacteristic
511C22#305911yescharacteristic
512C22#307612yes—
513C22#307612yes—
514C22#308312yes—
515C22#308312yes—
516C22#326812yes—
517C22#328311yescharacteristic
518C22#328511yescharacteristic
519C22#329712yes—
520C22#330512yes—
521C22#331312yes—
522C22#331512yes—
523C22#333611yescharacteristic
524C22#334211yescharacteristic
525C22#334712yes—
526C4—24no—
527C4—48no—
528C4—42no—
529C4—42no—
530C4—84no—
531C4—81no—
532C4—81no—
533C4—162no—
534C4—162no—
535C4#241211yescharacteristic
536C4#244212yes—
537C4#249811yescharacteristic
538C4#256912yes—
539C4#265911yescharacteristic
540C4#285711yescharacteristic
541C2—44no—
542C2#1476611yescharacteristic
543C2#2366611yescharacteristic
544C2#2366711yescharacteristic
545C2#2477911yescharacteristic
546C2#2478411yescharacteristic
547C2#2595612yes—
548C2#2595912yes—
549C2#2598912yes—
550C2#2600312yes—
551C2#2855211yescharacteristic
552C2#2856411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2087C2semidirect
6. #11377C2semidirect
7. #11328C2semidirect
8. #11292C2semidirect
9. #11232C2semidirect
10. #11227C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20872
C2×#18502
C2×#18082
C2×#17352
#139512
#113772
#113282
#112922
#112322
#112272
#107312