Finite Group Databasebeta

← #410873

Subgroups — #410873

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2C2×#1854C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4#14264C211yesmaximal, characteristic
5#11328C212yesmaximal, complemented
6#11327C212yesmaximal, complemented
7#11220C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2212yes—
17C22×#106C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2316—24no—
21#2316—24no—
22C2×#389—21no—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#380—22no—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30#1854—24no—
31#1853—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1444—21no—
42#1442—21no—
43#1439—22no—
44C4⋊#95C2212yes—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—42no—
54C2×#117—42no—
55C2×#117—42no—
56C2×#117—42no—
57C2×#117D414yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D414yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42C2312yes—
80C2×#107—22no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—22no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—42no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8Q814yes—
92C2×C22⋊C8Q814yes—
93#389—28no—
94#389—28no—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—24no—
99#389—42no—
100#389—42no—
101#380—216no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380Q814yes—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—24no—
117#340—24no—
118#340—42no—
119#340—42no—
120#336—42no—
121#336—42no—
122C25C2×D411yescharacteristic
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—28no—
129C23×C4—24no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—81no—
136C23×C4—82no—
137C23×C4—81no—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×Q812yes—
144C23×C4C2×Q812yes—
145C23×C4C2×Q812yes—
146C23×C4#4112yes—
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8C2×Q812yes—
163C22×C8C2×Q814yes—
164C22×C8C2×Q812yes—
165#117—44no—
166#117—44no—
167#117—48no—
168#117—44no—
169#117—44no—
170#117—48no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—216no—
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—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4#4114yes—
216C2×C4⋊C4#4114yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—81no—
225C2×C22⋊C4—81no—
226C2×C22⋊C4—82no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—44no—
235#107—44no—
236#106—44no—
237#106—48no—
238C22⋊C8—28no—
239C22⋊C8—28no—
240#95—28no—
241#95—28no—
242#95—24no—
243#95—44no—
244#95—42no—
245#95—82no—
246#95—82no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—21no—
253C24—22no—
254C24—21no—
255C24#12012yes—
256C24C22⋊Q812yes—
257C24C4⋊Q812yes—
258C24C22×D411yescharacteristic
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—216no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
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—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4#12012yes—
329C22×C4C22⋊Q814yes—
330C22×C4C22⋊Q814yes—
331C22×C4C22⋊Q812yes—
332C22×C4C4⋊Q814yes—
333C22×C4C4⋊Q814yes—
334C22×C4C4⋊Q812yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C22×Q811yescharacteristic
339C22×C4C22×Q811yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)12yes—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—216no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—88no—
385C4⋊C4—88no—
386C4⋊C4—88no—
387C4⋊C4#12318yes—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C42—48no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23#39514yes—
420C23C2×#12011yescharacteristic
421C23C2×C22⋊Q811yescharacteristic
422C23#52512yes—
423C23C2×C4⋊Q811yescharacteristic
424C23#53412yes—
425C23#57611yescharacteristic
426C23#57711yescharacteristic
427C23#57711yescharacteristic
428C23#57712yes—
429C23#57712yes—
430C23#57811yescharacteristic
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—816no—
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—81no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4#39514yes—
491C2×C4C2×C22⋊Q811yescharacteristic
492C2×C4C2×C22⋊Q811yescharacteristic
493C2×C4C2×#12312yes—
494C2×C4#52511yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4C2×C4⋊Q811yescharacteristic
497C2×C4C2×C4⋊Q811yescharacteristic
498C2×C4#53411yescharacteristic
499C2×C4#53411yescharacteristic
500C2×C4#53512yes—
501C2×C4#55312yes—
502C2×C4#55312yes—
503C2×C4#55412yes—
504C2×C4#55412yes—
505C2×C4#55412yes—
506C2×C4#55412yes—
507C2×C4#55412yes—
508C2×C4#55412yes—
509C8—42no—
510C8—42no—
511C8—44no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22C2×#39511yescharacteristic
524C22#257912yes—
525C22#258312yes—
526C22#261311yescharacteristic
527C22#284911yescharacteristic
528C22#300911yescharacteristic
529C22#301011yescharacteristic
530C22#301211yescharacteristic
531C22#301311yescharacteristic
532C22#308111yescharacteristic
533C22#308211yescharacteristic
534C22#308412yes—
535C22#311812yes—
536C22#312112yes—
537C22#312312yes—
538C22#312612yes—
539C22#316312yes—
540C22#316311yescharacteristic
541C22#316411yescharacteristic
542C22#319812yes—
543C22#320112yes—
544C22#320412yes—
545C22#320412yes—
546C4—24no—
547C4—48no—
548C4—84no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—161no—
554C4—161no—
555C4#248712yes—
556C4#250212yes—
557C4#257911yescharacteristic
558C4#257911yescharacteristic
559C4#258311yescharacteristic
560C4#258311yescharacteristic
561C2—44no—
562C2#1509211yescharacteristic
563C2#2355611yescharacteristic
564C2#2355711yescharacteristic
565C2#2924812yes—
566C2#2925012yes—
567C2#2930712yes—
568C2#2930812yes—
569C2#2960611yescharacteristic
570C2#2960911yescharacteristic
571C2#3301511yescharacteristic
572C2#3301811yescharacteristic
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
5. #11328C2semidirect
6. #11327C2semidirect
7. #11220C2semidirect
8. #11220C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C2×#18542
C2×#18532
#142642
#113282
#113272
#112202
#107312
#107302
#107132