Finite Group Databasebeta

← #478413

Subgroups — #478413

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15195C211yesmaximal, characteristic
2#15178C211yesmaximal, characteristic
3#11316C212yesmaximal, complemented
4#11190C212yesmaximal, complemented
5#11147C211yesmaximal, characteristic
6#11050C212yesmaximal, complemented
7#11010C212yesmaximal, complemented
8#10970C212yesmaximal
9#10970C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#391C2212yes—
19C2×#384C2211yescharacteristic
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28#1444—21no—
29#1444—21no—
30#1442—21no—
31#1442—21no—
32#1439—21no—
33#1439—21no—
34#1439—21no—
35#1439—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C23×Q8C2312yes—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—42no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—42no—
50C2×C22⋊Q8—42no—
51C2×C22⋊Q8—42no—
52C2×C22⋊Q8—42no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#103—24no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—41no—
82C2×C22⋊C8—41no—
83#391—24no—
84#391—42no—
85#384—41no—
86#384—41no—
87#380—42no—
88#380—42no—
89C4×C22⋊C4—41no—
90C4×C22⋊C4—41no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D411yescharacteristic
101C2×#95D411yescharacteristic
102#340—24no—
103#340—42no—
104#340—42no—
105#340—42no—
106#336—42no—
107#336—42no—
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—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
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—22no—
153C22×C8—22no—
154C22×C8—22no—
155C4⋊Q8—44no—
156C4⋊Q8—48no—
157C4⋊Q8—44no—
158C22⋊Q8—48no—
159C22⋊Q8—48no—
160C22⋊Q8—48no—
161C22⋊Q8—48no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
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—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4#4114yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4Q1614yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211#103—48no—
212#103—48no—
213#103—48no—
214#103—48no—
215C22⋊C8—44no—
216C22⋊C8—44no—
217#95—41no—
218#95—41no—
219#95—44no—
220#95—44no—
221#95—42no—
222#95—42no—
223#95—44no—
224#95—42no—
225#95—42no—
226#95—82no—
227#95—82no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—22no—
232C24—22no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—22no—
238C24C2×Q1612yes—
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—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
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—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
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—42no—
333C22×C4C2×Q1612yes—
334C22×C4C2×Q1612yes—
335C22×C4#13712yes—
336C22×C4#13712yes—
337C22×C4C2×#4112yes—
338C22×C4C2×#4112yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)11yescharacteristic
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—416no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
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—84no—
375C4⋊C4—84no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—416no—
379C22⋊C4—88no—
380C42—42no—
381C42—42no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23—24no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—42no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23#53411yescharacteristic
410C23#53511yescharacteristic
411C23#53511yescharacteristic
412C23#54612yes—
413C23#54612yes—
414C23C22×Q1611yescharacteristic
415C23C2×#13711yescharacteristic
416C23C2×#13711yescharacteristic
417C23#57511yescharacteristic
418C23#57712yes—
419C23#57712yes—
420Q8—44no—
421Q8—44no—
422Q8—44no—
423Q8—44no—
424Q8—82no—
425Q8—82no—
426Q8—82no—
427Q8—82no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—41no—
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—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#53812yes—
486C2×C4#53812yes—
487C2×C4#53812yes—
488C2×C4#53812yes—
489C2×C4#54012yes—
490C2×C4#54012yes—
491C2×C4#55111yescharacteristic
492C2×C4#55112yes—
493C2×C4#55312yes—
494C2×C4#55911yescharacteristic
495C2×C4#56011yescharacteristic
496C2×C4#56012yes—
497C2×C4#56012yes—
498C2×C4#56012yes—
499C2×C4#56012yes—
500C2×C4H(3,𝔽4)11yescharacteristic
501C8—82no—
502C8—82no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—42no—
508C22—81no—
509C22—81no—
510C22—82no—
511C22—82no—
512C22—81no—
513C22—81no—
514C22#268411yescharacteristic
515C22#268512yes—
516C22#270312yes—
517C22#270911yescharacteristic
518C22#281511yescharacteristic
519C22#318911yescharacteristic
520C22#319511yescharacteristic
521C22#320011yescharacteristic
522C22#320011yescharacteristic
523C22#320911yescharacteristic
524C22#320911yescharacteristic
525C22#321611yescharacteristic
526C22#321611yescharacteristic
527C22#322112yes—
528C22#322212yes—
529C22#329912yes—
530C22#329912yes—
531C22#330712yes—
532C22#331612yes—
533C22#332012yes—
534C22#332112yes—
535C22#332912yes—
536C22#333112yes—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—84no—
542C4—81no—
543C4—84no—
544C4—162no—
545C4—162no—
546C4#274912yes—
547C4#279712yes—
548C4#282512yes—
549C4#282812yes—
550C2—82no—
551C2#1714711yescharacteristic
552C2#3126311yescharacteristic
553C2#3127211yescharacteristic
554C2#3129212yes—
555C2#3130512yes—
556C2#3184012yes—
557C2#3185212yes—
558C2#3202911yescharacteristic
559C2#3203711yescharacteristic
560C2#3276211yescharacteristic
561C2#3276611yescharacteristic
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
3. #11316C2semidirect
4. #11190C2semidirect
6. #11050C2semidirect
7. #11010C2semidirect

Maximal subgroups

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

subgroupindex
#151952
#151782
#113162
#111902
#111472
#110502
#110102
#109702