Finite Group Databasebeta

← #393311

Subgroups — #393311

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3C2×#1989C212yesmaximal, complemented
4C2×#1798C211yesmaximal, characteristic
5#10885C212yesmaximal, complemented
6#10884C212yesmaximal, complemented
7#10849C212yesmaximal, complemented
8#10731C212yesmaximal
9#10730C212yesmaximal
10C22×#125C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16#2629—24no—
17#2629C418yescomplemented
18C2×#398—22no—
19C2×#396—22no—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#380C2212yes—
23#2274—28no—
24C2×C4×C4⋊C4—22no—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4—22no—
28#1989—28no—
29#1798—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#125—22no—
45C2×#125—24no—
46C2×#125—24no—
47C2×#125C2×C414yescomplemented
48C2×#117—22no—
49C2×#117—24no—
50C2×#117—24no—
51C2×#117—24no—
52C2×#117—24no—
53C2×#117C2×C414yes—
54C2×#117D414yes—
55C2×#117Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4Q814yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D412yes—
72C22×C42—21no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83#398—216no—
84#396—216no—
85#389—24no—
86#389C2×C418yescomplemented
87#384—22no—
88#384C2×C414yescomplemented
89#380—24no—
90#380C2×C418yes—
91C4×C4⋊C4—216no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4D418yes—
96C4×C22⋊C4—216no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336—22no—
110#336—22no—
111C25C2×D411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C22×C412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×Q812yes—
131C23×C4C2×Q812yes—
132C23×C4#4112yes—
133C23×C4#4112yes—
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149#125—216no—
150#125—48no—
151#117—216no—
152#117—216no—
153#117—216no—
154#117—216no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—216no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
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—216no—
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—28no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C22⋊C418yes—
188C2×C4⋊C4C22×C414yes—
189C2×C4⋊C4C22×C414yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×Q814yes—
193C2×C4⋊C4C2×Q814yes—
194C2×C4⋊C4#4114yes—
195C2×C4⋊C4#4114yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4C22⋊C418yes—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—28no—
211C2×C42—28no—
212C2×C42—82no—
213C2×C42—82no—
214C2×C42—82no—
215C2×C42—82no—
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D414yes—
220C2×C42C2×D412yes—
221C2×C42#4114yes—
222C4⋊C8—48no—
223C22⋊C8—48no—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—28no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24C2×C22⋊C412yes—
237C24C22⋊Q812yes—
238C24C22⋊Q812yes—
239C24C22×D411yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—216no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
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—88no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—88no—
299C22×C4C2×C22⋊C414yes—
300C22×C4C2×C22⋊C414yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C4×Q814yes—
303C22×C4C4×Q814yes—
304C22×C4#12014yes—
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4#12314yes—
308C22×C4#12414yes—
309C22×C4C4⋊Q814yes—
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D412yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×Q811yescharacteristic
314C22×C4C22×Q811yescharacteristic
315C22×C4C2×#4112yes—
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4112yes—
319C22×C4C2×#4112yes—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—216no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—216no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—416no—
351C4⋊C4—88no—
352C4⋊C4—88no—
353C4⋊C4C22⋊Q818yes—
354C4⋊C4C22⋊Q818yes—
355C22⋊C4—216no—
356C22⋊C4—216no—
357C22⋊C4—88no—
358C42—216no—
359C42—24no—
360C42—24no—
361C42—88no—
362C42—88no—
363C42C4×D418yes—
364C42#12118yes—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—24no—
375C23—22no—
376C23—22no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23#39114yes—
388C23C22×C22⋊C411yescharacteristic
389C23C2×#12011yescharacteristic
390C23C2×C22⋊Q811yescharacteristic
391C23C2×C22⋊Q811yescharacteristic
392C23C2×#12311yescharacteristic
393C23#52511yescharacteristic
394C23#52511yescharacteristic
395C23#57612yes—
396C23#57712yes—
397C23#57712yes—
398C23#57812yes—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—816no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4#38618yes—
452C2×C4#39114yes—
453C2×C4C2×C4×D412yes—
454C2×C4C2×C4×Q811yescharacteristic
455C2×C4C2×C4×Q811yescharacteristic
456C2×C4C4×#4112yes—
457C2×C4C2×#12112yes—
458C2×C4C2×C22⋊Q812yes—
459C2×C4C2×C22⋊Q812yes—
460C2×C4#52512yes—
461C2×C4#52512yes—
462C2×C4C2×#12411yescharacteristic
463C2×C4#52912yes—
464C2×C4#52912yes—
465C2×C4C2×C4⋊Q811yescharacteristic
466C2×C4#53212yes—
467C2×C4#53312yes—
468C2×C4#53312yes—
469C8—82no—
470C8—82no—
471C22—28no—
472C22—24no—
473C22—42no—
474C22—42no—
475C22—44no—
476C22—42no—
477C22—42no—
478C22—44no—
479C22—42no—
480C22—42no—
481C22—44no—
482C22C2×#38611yescharacteristic
483C22C2×#39111yescharacteristic
484C22#229711yescharacteristic
485C22#284811yescharacteristic
486C22#284811yescharacteristic
487C22#285511yescharacteristic
488C22#285511yescharacteristic
489C22#289712yes—
490C22#289812yes—
491C22#301212yes—
492C22#301312yes—
493C22#308112yes—
494C22#308212yes—
495C22#308312yes—
496C22#308412yes—
497C22#309812yes—
498C22#309912yes—
499C22#311912yes—
500C22#312012yes—
501C22#312211yescharacteristic
502C22#312311yescharacteristic
503C22#312611yescharacteristic
504C22#312711yescharacteristic
505C4—24no—
506C4—28no—
507C4—44no—
508C4—48no—
509C4—48no—
510C4—84no—
511C4—84no—
512C4C4×C22⋊Q812yes—
513C4C4×C22⋊Q812yes—
514C4#285012yes—
515C4#285712yes—
516C2—44no—
517C2#1470211yescharacteristic
518C2#2091512yes—
519C2#2131512yes—
520C2#2139811yescharacteristic
521C2#2139911yescharacteristic
522C2#3296611yescharacteristic
523C2#3296711yescharacteristic
524C2#3296812yes—
525C2#3315611yescharacteristic
526C2#3315711yescharacteristic
527C2#3315812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1989C2semidirect
5. #10885C2semidirect
6. #10884C2semidirect
7. #10849C2semidirect
17. #2629C4semidirect
47. C2×#125C2×C4semidirect
86. #389C2×C4semidirect
88. #384C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#22742
C2×#19892
C2×#17982
#108852
#108842
#108492
#107312
#107302