Finite Group Databasebeta

← #469866

Subgroups — #469866

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15125C211yesmaximal, characteristic
2#13924C211yesmaximal, characteristic
3#11384C212yesmaximal, complemented
4#11378C212yesmaximal, complemented
5#11315C212yesmaximal, complemented
6#11231C212yesmaximal, complemented
7#11095C212yesmaximal
8#11042C211yesmaximal, characteristic
9#10730C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C2×#393—21no—
16C2×#380C2212yes—
17C2×#380C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2212yes—
28#1446—21no—
29#1445—21no—
30#1445—21no—
31#1442—21no—
32#1439—21no—
33C4⋊#95C2212yes—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C22⋊#95—21no—
38C22⋊#95—21no—
39C23×Q8—21no—
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—41no—
52C22×C4⋊C4—41no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4Q814yescomplemented
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#106—24no—
67C2×#106—42no—
68C2×#106—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—41no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—42no—
74#393—44no—
75#380—24no—
76#380—42no—
77#380—41no—
78#380—41no—
79C4×C22⋊C4—41no—
80C4×C22⋊C4—41no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D411yescharacteristic
92C2×#95D411yescharacteristic
93#340—24no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—42no—
101#336—41no—
102C25C2×D411yescharacteristic
103C22×Q8—44no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—82no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q812yes—
131C23×C4#4112yes—
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8#4114yes—
147C4⋊Q8—44no—
148C4⋊Q8—48no—
149C4⋊Q8—44no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
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—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4C2×Q814yes—
182C2×C4⋊C4#4114yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4SD1614yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203#106—48no—
204C4⋊C8—44no—
205C22⋊C8—48no—
206#95—41no—
207#95—41no—
208#95—44no—
209#95—42no—
210#95—44no—
211#95—42no—
212#95—42no—
213#95—82no—
214#95—82no—
215#95—82no—
216#95—82no—
217#95—82no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—22no—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24C22⋊Q812yes—
227C24C2×SD1612yes—
228C24#13712yes—
229C24C22×D411yescharacteristic
230C2×Q8—48no—
231C2×Q8—88no—
232C2×Q8—84no—
233C2×Q8—88no—
234C2×Q8—84no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—48no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
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—88no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4—88no—
307C22×C4C22⋊Q812yes—
308C22×C4C2×SD1612yes—
309C22×C4C2×SD1612yes—
310C22×C4#13712yes—
311C22×C4C22×Q811yescharacteristic
312C22×C4C2×#4112yes—
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C4H(5,𝔽2)12yes—
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C42−1+412yes—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—88no—
356C22⋊C4—84no—
357C42—42no—
358C42—42no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—24no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23#47514yes—
386C23C2×C22⋊Q811yescharacteristic
387C23#53411yescharacteristic
388C23#53511yescharacteristic
389C23#54612yes—
390C23#54712yes—
391C23C22×SD1611yescharacteristic
392C23C2×#13711yescharacteristic
393C23C2×#13711yescharacteristic
394C23#57511yescharacteristic
395C23#57612yes—
396C23#57712yes—
397Q8—816no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—816no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
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#53312yes—
459C2×C4#53812yes—
460C2×C4#53812yes—
461C2×C4#53811yescharacteristic
462C2×C4#53811yescharacteristic
463C2×C4#54212yes—
464C2×C4#54212yes—
465C2×C4#54311yescharacteristic
466C2×C4#54311yescharacteristic
467C2×C4#54412yes—
468C2×C4#55111yescharacteristic
469C2×C4#55111yescharacteristic
470C2×C4#55112yes—
471C2×C4#55212yes—
472C2×C4#55912yes—
473C2×C4#56011yescharacteristic
474C2×C4#56212yes—
475C2×C4#56311yescharacteristic
476C8—44no—
477C8—82no—
478C22—28no—
479C22—24no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—42no—
488C22—42no—
489C22#243812yes—
490C22#262611yescharacteristic
491C22#269711yescharacteristic
492C22#271012yes—
493C22#271711yescharacteristic
494C22C2×#47511yescharacteristic
495C22#307811yescharacteristic
496C22#308112yes—
497C22#318711yescharacteristic
498C22#319511yescharacteristic
499C22#319911yescharacteristic
500C22#320011yescharacteristic
501C22#320611yescharacteristic
502C22#321511yescharacteristic
503C22#322412yes—
504C22#329512yes—
505C22#331612yes—
506C22#332012yes—
507C22#332212yes—
508C22#333512yes—
509C22#334612yes—
510C22#335012yes—
511C22#335712yes—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—84no—
517C4—81no—
518C4—84no—
519C4—162no—
520C4—162no—
521C4#260711yescharacteristic
522C4#266512yes—
523C4#275112yes—
524C4#276011yescharacteristic
525C4#276411yescharacteristic
526C4#284111yescharacteristic
527C2—44no—
528C2#1666611yescharacteristic
529C2#2599212yes—
530C2#2601012yes—
531C2#2995111yescharacteristic
532C2#2995911yescharacteristic
533C2#3166711yescharacteristic
534C2#3168211yescharacteristic
535C2#3204812yes—
536C2#3206612yes—
537C2#3226011yescharacteristic
538C2#3226811yescharacteristic
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. #11384C2semidirect
4. #11378C2semidirect
5. #11315C2semidirect
6. #11231C2semidirect
55. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#151252
#139242
#113842
#113782
#113152
#112312
#110952
#110422
#107302