Finite Group Databasebeta

← #421079

Subgroups — #421079

Order 512.

1742 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1933C212yesmaximal, complemented
2C2×#1787C212yesmaximal
3#15223C211yesmaximal, characteristic
4#13922C211yesmaximal, characteristic
5#11199C212yesmaximal
6#11190C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#11052C212yesmaximal, complemented
9#10921C211yesmaximal, characteristic
10C24×Q8C2211yescharacteristic
11C22×#103C2212yes—
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#391C2212yes—
18C2×#391C2212yes—
19C2×#389C2212yes—
20C2×#382C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1933—28no—
25#1787—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32#1446—22no—
33#1445—22no—
34#1439—22no—
35C4⋊#95C2211yescharacteristic
36C22⋊#95—22no—
37C23×Q8—24no—
38C23×Q8—21no—
39C23×Q8—24no—
40C23×Q8—24no—
41C23×Q8—24no—
42C23×Q8—41no—
43C23×Q8—41no—
44C23×Q8—42no—
45C23×Q8—42no—
46C23×Q8C2312yes—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D412yes—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—42no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103D418yes—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—42no—
75C2×C4×C8—24no—
76C2×C4×C8—41no—
77#391—24no—
78#391—24no—
79#391—42no—
80#391—42no—
81#389—24no—
82#389—42no—
83#382—41no—
84#382—41no—
85#380—24no—
86#380—24no—
87#380—42no—
88C4×C22⋊C4—41no—
89C4×C22⋊C4—41no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#336—41no—
105#336—42no—
106C25C2×D411yescharacteristic
107C22×Q8—24no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—216no—
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—44no—
121C22×Q8—44no—
122C22×Q8—48no—
123C22×Q8—48no—
124C22×Q8—48no—
125C22×Q8—48no—
126C22×Q8—42no—
127C22×Q8—48no—
128C22×Q8—48no—
129C22×Q8—48no—
130C22×Q8—44no—
131C22×Q8—44no—
132C22×Q8—48no—
133C22×Q8—48no—
134C22×Q8—48no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—48no—
138C22×Q8—42no—
139C22×Q8C2×D414yes—
140C22×Q8#4114yes—
141C22×Q8#4114yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4#4114yes—
162C23×C4C2411yescharacteristic
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C2×D414yes—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C2×D414yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—82no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214#103—216no—
215#103—48no—
216#103—48no—
217#103—48no—
218C22⋊C8—48no—
219C4×C8—44no—
220#95—42no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—42no—
229#95—82no—
230C24—21no—
231C24—21no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24#12412yes—
238C24#13514yes—
239C24C22×D411yescharacteristic
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—48no—
256C2×Q8—416no—
257C2×Q8—416no—
258C2×Q8—432no—
259C2×Q8—432no—
260C2×Q8—432no—
261C2×Q8—432no—
262C2×Q8—48no—
263C2×Q8—48no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8#12318yes—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—416no—
305C22×C4—48no—
306C22×C4—416no—
307C22×C4—48no—
308C22×C4—48no—
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—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
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—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4#12412yes—
344C22×C4#13714yes—
345C22×C4C22×D412yes—
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4112yes—
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C4H(5,𝔽2)12yes—
351C22×C4H(5,𝔽2)12yes—
352C22×C4H(5,𝔽2)11yescharacteristic
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—88no—
380C22⋊C4—48no—
381C22⋊C4—88no—
382C22⋊C4—84no—
383C42—42no—
384C42—42no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C23—22no—
390C23—24no—
391C23—24no—
392C23—22no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—42no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23#48814yes—
410C23#52511yescharacteristic
411C23C2×#12411yescharacteristic
412C23#53411yescharacteristic
413C23D4212yes—
414C23#54612yes—
415C23C2×#13512yes—
416C23C2×#13712yes—
417C23#57712yes—
418C23#57812yes—
419Q8—28no—
420Q8—28no—
421Q8—432no—
422Q8—44no—
423Q8—44no—
424Q8—44no—
425Q8—44no—
426Q8—44no—
427Q8—44no—
428C2×C4—22no—
429C2×C4—28no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—432no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
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—82no—
479C2×C4—82no—
480C2×C4C2×#12312yes—
481C2×C4#52512yes—
482C2×C4#53512yes—
483C2×C4#53812yes—
484C2×C4#53812yes—
485C2×C4#53812yes—
486C2×C4D4211yescharacteristic
487C2×C4#54612yes—
488C2×C4#54612yes—
489C2×C4#55112yes—
490C2×C4#55112yes—
491C2×C4#55112yes—
492C2×C4#55112yes—
493C2×C4#55111yescharacteristic
494C2×C4#56011yescharacteristic
495C2×C4#56011yescharacteristic
496C8—44no—
497C8—82no—
498C22—28no—
499C22—24no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22—44no—
504C22—42no—
505C22—44no—
506C22—42no—
507C22#241812yes—
508C22#247212yes—
509C22#256111yescharacteristic
510C22#266911yescharacteristic
511C22#269711yescharacteristic
512C22#310812yes—
513C22#312611yescharacteristic
514C22#312711yescharacteristic
515C22C2×#48811yescharacteristic
516C22#313712yes—
517C22#313911yescharacteristic
518C22#319012yes—
519C22#320112yes—
520C22#326612yes—
521C22#327012yes—
522C22#328212yes—
523C22#328512yes—
524C22#330012yes—
525C22#330112yes—
526C22#332812yes—
527C22#332912yes—
528C4—44no—
529C4—42no—
530C4—48no—
531C4—84no—
532C4—82no—
533C4—162no—
534C4—162no—
535C4#241812yes—
536C4#248212yes—
537C4#268512yes—
538C4#273512yes—
539C2—44no—
540C2#1528411yescharacteristic
541C2#2489412yes—
542C2#2490212yes—
543C2#2726012yes—
544C2#2727712yes—
545C2#2910611yescharacteristic
546C2#2911711yescharacteristic
547C2#3050812yes—
548C2#3167312yes—
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×#1933C2semidirect
6. #11190C2semidirect
7. #11067C2semidirect
8. #11052C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19332
C2×#17872
#152232
#139222
#111992
#111902
#110672
#110522
#109212