Finite Group Databasebeta

← #460091

Subgroups — #460091

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2026C212yesmaximal, complemented
2C2×#1894C212yesmaximal
3#15216C211yesmaximal, characteristic
4#15178C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11190C212yesmaximal, complemented
7#11010C212yesmaximal, complemented
8#10970C212yesmaximal
9#10715C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×#125C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#2026—28no—
23#1894—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1444—22no—
31#1442—22no—
32#1439—22no—
33#1439—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C23×Q8C2312yes—
38C24×C4C2311yescharacteristic
39C22×M16—22no—
40C22×M16—24no—
41C22×M16—42no—
42C22×M16—41no—
43C22×M16—41no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×#125—24no—
51C2×#125—41no—
52C2×#125—41no—
53C2×#125—42no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D412yes—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—41no—
85C2×C22⋊C8—41no—
86#391—24no—
87#391—42no—
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—42no—
103#340D418yes—
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×Q8#4114yes—
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
139C2×M16—44no—
140C2×M16—48no—
141C2×M16—48no—
142C2×M16—44no—
143C2×M16—44no—
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—22no—
157C22×C8C2×D414yes—
158C4⋊Q8—44no—
159C4⋊Q8—48no—
160C4⋊Q8—44no—
161#125—44no—
162#125—48no—
163#125—44no—
164C22⋊Q8—216no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167C22⋊Q8—48no—
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—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×D414yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—82no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212#103—48no—
213#103—48no—
214#103—48no—
215#103—48no—
216C22⋊C8—44no—
217#95—42no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—82no—
224#95—82no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24#12412yes—
234C24C22×D411yescharacteristic
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
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—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C22×C4—21no—
264C22×C4—22no—
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—24no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
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—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4#12412yes—
327C22×C4#13714yes—
328C22×C4C22×D412yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4111yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C4H(5,𝔽2)12yes—
335C22×C4H(5,𝔽2)12yes—
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C42−1+411yescharacteristic
338M16—416no—
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—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—416no—
358C4⋊C4—416no—
359C4⋊C4—416no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C22⋊C4—48no—
369C22⋊C4—416no—
370C22⋊C4—88no—
371C42—42no—
372C42—42no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C23—24no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—24no—
382C23—41no—
383C23—42no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23C2×#12411yescharacteristic
399C23#53411yescharacteristic
400C23#53711yescharacteristic
401C23D4212yes—
402C23#54612yes—
403C23C2×#13712yes—
404C23#57512yes—
405C23#57712yes—
406C23#57812yes—
407Q8—44no—
408Q8—44no—
409Q8—44no—
410Q8—44no—
411Q8—82no—
412Q8—82no—
413Q8—82no—
414Q8—82no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
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—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4#52512yes—
469C2×C4#53512yes—
470C2×C4#53512yes—
471C2×C4#53812yes—
472C2×C4#53811yescharacteristic
473C2×C4#53812yes—
474C2×C4#53812yes—
475C2×C4#53811yescharacteristic
476C2×C4#53911yescharacteristic
477C2×C4#53911yescharacteristic
478C2×C4D4211yescharacteristic
479C2×C4#54612yes—
480C2×C4#54812yes—
481C2×C4#55112yes—
482C2×C4#55111yescharacteristic
483C2×C4#56011yescharacteristic
484C2×C4H(3,𝔽4)11yescharacteristic
485C2×C4#56212yes—
486C8—44no—
487C8—82no—
488C22—44no—
489C22—44no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22#247212yes—
498C22#263911yescharacteristic
499C22#266911yescharacteristic
500C22#267612yes—
501C22#268411yescharacteristic
502C22#313712yes—
503C22#313911yescharacteristic
504C22#314211yescharacteristic
505C22#319612yes—
506C22#320112yes—
507C22#325212yes—
508C22#326011yescharacteristic
509C22#326111yescharacteristic
510C22#326712yes—
511C22#326912yes—
512C22#327612yes—
513C22#327812yes—
514C22#331212yes—
515C22#332012yes—
516C22#332612yes—
517C22#332712yes—
518C4—44no—
519C4—42no—
520C4—82no—
521C4—84no—
522C4—84no—
523C4—84no—
524C4—162no—
525C4#246311yescharacteristic
526C4#263112yes—
527C4#267011yescharacteristic
528C4#269612yes—
529C4#276011yescharacteristic
530C4#282411yescharacteristic
531C2—82no—
532C2#1635711yescharacteristic
533C2#2727212yes—
534C2#2728312yes—
535C2#3019611yescharacteristic
536C2#3020211yescharacteristic
537C2#3051412yes—
538C2#3092412yes—
539C2#3093212yes—
540C2#3127712yes—
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×#2026C2semidirect
5. #11316C2semidirect
6. #11190C2semidirect
7. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20262
C2×#18942
#152162
#151782
#113162
#111902
#110102
#109702
#107152