Finite Group Databasebeta

← #404854

Subgroups — #404854

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic, complemented
3C2×#2074C211yesmaximal, characteristic, complemented
4#14258C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11096C212yesmaximal
8#10970C212yesmaximal
9#10932C212yesmaximal, complemented
10#10907C212yesmaximal, complemented
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18#2314—24no—
19#2314—24no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#388—22no—
23C2×#387—22no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29#2095—24no—
30#2074—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1441—22no—
40#1440—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—42no—
51C2×#117—24no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—24no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4D412yes—
67C22×C42—22no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—24no—
72C2×#107—42no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—42no—
82#391—22no—
83#391—22no—
84#391—24no—
85#388—216no—
86#387—216no—
87#384—22no—
88#384D414yes—
89#382—22no—
90#382D414yes—
91#380—24no—
92#380—24no—
93#380—42no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—22no—
111#336—24no—
112C25C2×D411yescharacteristic
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8#4112yes—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4#4114yes—
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8#4114yes—
157C22⋊Q8—48no—
158C22⋊Q8—48no—
159#117—216no—
160#117—216no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
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—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4#4112yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—28no—
211C2×C42—82no—
212C2×C42—82no—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42#4114yes—
216#107—48no—
217#103—48no—
218#103—48no—
219C22⋊C8—48no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—42no—
224#95—44no—
225#95—44no—
226#95—82no—
227#95—82no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24#12312yes—
239C24C22×D411yescharacteristic
240C2×Q8—24no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4#12014yes—
327C22×C4#12314yes—
328C22×C4#12312yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4112yes—
332C22×C4C2×#4112yes—
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4112yes—
335C22×C4C2×#4111yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+411yescharacteristic
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—216no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—48no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—88no—
370C22⋊C4—216no—
371C22⋊C4—48no—
372C22⋊C4—88no—
373C22⋊C4—84no—
374C42—24no—
375C42—24no—
376C42—24no—
377C42—24no—
378C42—88no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23C2×#12011yescharacteristic
403C23C2×#12311yescharacteristic
404C23C2×#12311yescharacteristic
405C23#52511yescharacteristic
406C23#53211yescharacteristic
407C23#53511yescharacteristic
408C23#53711yescharacteristic
409C23#57612yes—
410C23#57712yes—
411C23#57812yes—
412C23#57812yes—
413Q8—44no—
414Q8—44no—
415Q8—82no—
416Q8—82no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#52512yes—
479C2×C4#52512yes—
480C2×C4#52912yes—
481C2×C4#52912yes—
482C2×C4#54012yes—
483C2×C4#54212yes—
484C2×C4#54611yescharacteristic
485C2×C4#54612yes—
486C2×C4#54711yescharacteristic
487C2×C4#54811yescharacteristic
488C2×C4#55011yescharacteristic
489C2×C4#55012yes—
490C2×C4#55112yes—
491C2×C4#55112yes—
492C2×C4#55112yes—
493C2×C4#55212yes—
494C8—44no—
495C8—82no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—81no—
502C22—81no—
503C22—82no—
504C22—82no—
505C22—81no—
506C22—81no—
507C22#239511yescharacteristic
508C22#240711yescharacteristic
509C22#243111yescharacteristic
510C22#248211yescharacteristic
511C22#249811yescharacteristic
512C22#262411yescharacteristic
513C22#284911yescharacteristic
514C22#301112yes—
515C22#301212yes—
516C22#309712yes—
517C22#309712yes—
518C22#309812yes—
519C22#309812yes—
520C22#312112yes—
521C22#312412yes—
522C22#317712yes—
523C22#317912yes—
524C22#322011yescharacteristic
525C22#322111yescharacteristic
526C22#322311yescharacteristic
527C22#322311yescharacteristic
528C22#325812yes—
529C22#326212yes—
530C4—42no—
531C4—44no—
532C4—44no—
533C4—84no—
534C4—82no—
535C4—84no—
536C4—82no—
537C4—161no—
538C4—161no—
539C4#247112yes—
540C4#247112yes—
541C4#247712yes—
542C4#247912yes—
543C2—82no—
544C2#1490511yescharacteristic
545C2#2376812yes—
546C2#2416812yes—
547C2#2560911yescharacteristic
548C2#2561111yescharacteristic
549C2#2759511yescharacteristic
550C2#2759611yescharacteristic
551C2#2787212yes—
552C2#2993111yescharacteristic
553C2#2993411yescharacteristic
554C2#3298912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
2. C2×#2095C2semidirect
3. C2×#2074C2semidirect
5. #11373C2semidirect
6. #11358C2semidirect
9. #10932C2semidirect
10. #10907C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#20952
C2×#20742
#142582
#113732
#113582
#110962
#109702
#109322
#109072
#107132