Finite Group Databasebeta

← #459310

Subgroups — #459310

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2043C212yesmaximal, complemented
2C2×#1892C212yesmaximal
3#15223C211yesmaximal, characteristic
4#15178C211yesmaximal, characteristic
5#11292C212yesmaximal, complemented
6#11190C212yesmaximal, complemented
7#11052C212yesmaximal, complemented
8#10970C212yesmaximal
9#10714C211yesmaximal, characteristic
10C24×C8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C2×#391C2212yes—
19C2×#389C2212yes—
20C2×#382C2211yescharacteristic
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#2043—28no—
24#1892—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1445—22no—
32#1444—22no—
33#1439—22no—
34#1439—22no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C23×Q8C2312yes—
38C24×C4C2311yescharacteristic
39C23×C8—22no—
40C23×C8—24no—
41C23×C8—42no—
42C23×C8—41no—
43C23×C8—41no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—42no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—42no—
55C2×C22⋊Q8—42no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103D418yes—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—41no—
81C2×C22⋊C8—41no—
82#391—24no—
83#391—42no—
84#389—24no—
85#389—42no—
86#382—41no—
87#382—41no—
88#380—24no—
89#380—42no—
90C4×C22⋊C4—41no—
91C4×C22⋊C4—41no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#336—24no—
105#336—42no—
106C25C2×D411yescharacteristic
107C22×Q8—22no—
108C22×Q8—22no—
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×Q8C2×D414yes—
121C23×C4—22no—
122C23×C4—22no—
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×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4#4112yes—
137C23×C4#4112yes—
138C23×C4C2411yescharacteristic
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—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—44no—
153C22×C8—48no—
154C22×C8—48no—
155C22×C8—44no—
156C22×C8—44no—
157C22×C8C2×D414yes—
158C4⋊Q8—44no—
159C4⋊Q8—48no—
160C4⋊Q8—44no—
161C22⋊Q8—216no—
162C22⋊Q8—48no—
163C22⋊Q8—48no—
164C22⋊Q8—48no—
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—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4#4114yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—82no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209#103—48no—
210#103—48no—
211#103—48no—
212#103—48no—
213C22⋊C8—44no—
214#95—42no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—82no—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24#12412yes—
230C24#13714yes—
231C24C22×D411yescharacteristic
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
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—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
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—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#12412yes—
324C22×C4#13514yes—
325C22×C4C22×D412yes—
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4111yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)12yes—
331C22×C4H(5,𝔽2)11yescharacteristic
332C22×C4H(5,𝔽2)12yes—
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C42−1+411yescharacteristic
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—416no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—416no—
355C4⋊C4—416no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—88no—
365C22⋊C4—48no—
366C22⋊C4—416no—
367C22⋊C4—88no—
368C42—42no—
369C42—42no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C23—22no—
375C23—24no—
376C23—22no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23#49114yes—
395C23C2×#12411yescharacteristic
396C23#53411yescharacteristic
397C23#53711yescharacteristic
398C23D4212yes—
399C23#54612yes—
400C23C2×#13512yes—
401C23C2×#13712yes—
402C23#57712yes—
403C23#57812yes—
404Q8—44no—
405Q8—44no—
406Q8—44no—
407Q8—44no—
408Q8—82no—
409Q8—82no—
410Q8—82no—
411Q8—82no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
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#52512yes—
466C2×C4#53412yes—
467C2×C4#53512yes—
468C2×C4#53812yes—
469C2×C4#53811yescharacteristic
470C2×C4#53811yescharacteristic
471C2×C4#53911yescharacteristic
472C2×C4#53911yescharacteristic
473C2×C4#54012yes—
474C2×C4#54012yes—
475C2×C4D4211yescharacteristic
476C2×C4#54612yes—
477C2×C4#54712yes—
478C2×C4#55112yes—
479C2×C4#55911yescharacteristic
480C2×C4#56011yescharacteristic
481C2×C4#56011yescharacteristic
482C2×C4#56212yes—
483C8—44no—
484C8—82no—
485C22—28no—
486C22—24no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—42no—
494C22#247212yes—
495C22#263911yescharacteristic
496C22#266911yescharacteristic
497C22#267112yes—
498C22#270911yescharacteristic
499C22#313111yescharacteristic
500C22C2×#49111yescharacteristic
501C22#313712yes—
502C22#319012yes—
503C22#320112yes—
504C22#324812yes—
505C22#326011yescharacteristic
506C22#326111yescharacteristic
507C22#326612yes—
508C22#327012yes—
509C22#328212yes—
510C22#328512yes—
511C22#330212yes—
512C22#330312yes—
513C22#331212yes—
514C22#332012yes—
515C4—44no—
516C4—42no—
517C4—82no—
518C4—84no—
519C4—84no—
520C4—84no—
521C4—162no—
522C4#245511yescharacteristic
523C4#263512yes—
524C4#266811yescharacteristic
525C4#270612yes—
526C4#275211yescharacteristic
527C4#282611yescharacteristic
528C2—44no—
529C2#1633811yescharacteristic
530C2#2726112yes—
531C2#2726812yes—
532C2#3019011yescharacteristic
533C2#3019611yescharacteristic
534C2#3050812yes—
535C2#3060112yes—
536C2#3060412yes—
537C2#3203412yes—
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×#2043C2semidirect
5. #11292C2semidirect
6. #11190C2semidirect
7. #11052C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20432
C2×#18922
#152232
#151782
#112922
#111902
#110522
#109702
#107142