Finite Group Databasebeta

← #469674

Subgroups — #469674

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2001C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic
3#14258C212yesmaximal
4#11315C212yesmaximal, complemented
5#11190C212yesmaximal, complemented
6#11124C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic
8#11024C212yesmaximal, complemented
9#10970C212yesmaximal
10C23×Q16C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×#117C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#380C2212yes—
22#2001—24no—
23#1818—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1441—22no—
32#1441—22no—
33#1440—22no—
34#1440—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C23×Q8C2312yes—
38C24×C4C2311yescharacteristic
39C22×Q16—24no—
40C22×Q16—22no—
41C22×Q16—42no—
42C22×Q16—41no—
43C22×Q16—41no—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—42no—
52C2×C22⋊Q8—42no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—41no—
56C2×#117—41no—
57C2×#117—41no—
58C2×#117—41no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D412yes—
68C22×C42C2312yes—
69C2×#107—42no—
70C2×#107—41no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—41no—
82C2×C22⋊C8—41no—
83C2×C22⋊C8—41no—
84C2×C22⋊C8—41no—
85C2×C22⋊C8D414yes—
86#391—42no—
87#391—42no—
88#391—42no—
89#391—42no—
90#380—42no—
91#380—42no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101#340—22no—
102#340—24no—
103#340—42no—
104#340—42no—
105#340—41no—
106#336—24no—
107#336—24no—
108C25C2×D411yescharacteristic
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8#4114yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C2×Q16—48no—
142C2×Q16—44no—
143C2×Q16—44no—
144C2×Q16—48no—
145C2×Q16—44no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22⋊Q8—48no—
164C22⋊Q8—48no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167#117—44no—
168#117—44no—
169#117—44no—
170#117—44no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4#4114yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213#107—44no—
214#103—48no—
215#103—48no—
216#103—48no—
217C22⋊C8—44no—
218C22⋊C8—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—82no—
225#95—82no—
226#95—82no—
227#95—82no—
228C24—22no—
229C24—21no—
230C24—21no—
231C24—22no—
232C24—21no—
233C24—22no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24#12112yes—
239C24C22×D411yescharacteristic
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—42no—
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—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
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—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4#12112yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4112yes—
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)12yes—
343Q16—416no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—416no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C22⋊C4—48no—
377C22⋊C4—416no—
378C22⋊C4—88no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C23—22no—
384C23—24no—
385C23—22no—
386C23—22no—
387C23—24no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—41no—
406C23C2×#12111yescharacteristic
407C23#52512yes—
408C23#53411yescharacteristic
409C23#53411yescharacteristic
410C23#53512yes—
411C23#57611yescharacteristic
412C23#57712yes—
413C23#57811yescharacteristic
414C23#57811yescharacteristic
415C23#57812yes—
416C23#57811yescharacteristic
417Q8—44no—
418Q8—44no—
419Q8—44no—
420Q8—44no—
421Q8—82no—
422Q8—82no—
423Q8—82no—
424Q8—82no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
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—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#53411yescharacteristic
479C2×C4#53511yescharacteristic
480C2×C4#53812yes—
481C2×C4#53812yes—
482C2×C4#53811yescharacteristic
483C2×C4#53811yescharacteristic
484C2×C4#54012yes—
485C2×C4#54012yes—
486C2×C4#54612yes—
487C2×C4#54612yes—
488C2×C4#55111yescharacteristic
489C2×C4#55112yes—
490C2×C4#55112yes—
491C2×C4#55312yes—
492C2×C4#55911yescharacteristic
493C2×C4#56011yescharacteristic
494C2×C4#56012yes—
495C2×C4#56011yescharacteristic
496C8—42no—
497C8—42no—
498C8—82no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—44no—
503C22—44no—
504C22—81no—
505C22—81no—
506C22—82no—
507C22—82no—
508C22—81no—
509C22—81no—
510C22#241112yes—
511C22#260011yescharacteristic
512C22#261711yescharacteristic
513C22#263512yes—
514C22#281511yescharacteristic
515C22#305911yescharacteristic
516C22#306111yescharacteristic
517C22#306312yes—
518C22#312012yes—
519C22#312112yes—
520C22#312412yes—
521C22#312712yes—
522C22#320112yes—
523C22#320211yescharacteristic
524C22#320211yescharacteristic
525C22#320211yescharacteristic
526C22#320311yescharacteristic
527C22#320311yescharacteristic
528C22#320411yescharacteristic
529C22#322312yes—
530C22#322312yes—
531C22#322412yes—
532C22#322512yes—
533C4—42no—
534C4—84no—
535C4—82no—
536C4—82no—
537C4—84no—
538C4—162no—
539C4—162no—
540C4#269711yescharacteristic
541C4#270312yes—
542C4#270911yescharacteristic
543C4#274912yes—
544C4#275211yescharacteristic
545C4#276011yescharacteristic
546C2—82no—
547C2#1666011yescharacteristic
548C2#2451412yes—
549C2#2451512yes—
550C2#2942611yescharacteristic
551C2#2942711yescharacteristic
552C2#2973911yescharacteristic
553C2#2974611yescharacteristic
554C2#3012112yes—
555C2#3012412yes—
556C2#3276811yescharacteristic
557C2#3276911yescharacteristic
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
4. #11315C2semidirect
5. #11190C2semidirect
6. #11124C2semidirect
8. #11024C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
C2×#18182
#142582
#113152
#111902
#111242
#110962
#110242
#109702