Finite Group Databasebeta

← #405501

Subgroups — #405501

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3#13756C211yesmaximal, characteristic
4#11385C212yesmaximal, complemented
5#11325C212yesmaximal, complemented
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#10731C212yesmaximal
9#10712C211yesmaximal, characteristic
10C22×#125C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17#2851—28no—
18C2×#389C2212yes—
19C2×#387—22no—
20C2×#385—22no—
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23#1854—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C22⋊#95—22no—
36C4×#95—22no—
37C24×C4D412yes—
38C24×C4C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×#125—24no—
42C2×#125—24no—
43C2×#125—41no—
44C2×#125—41no—
45C2×#117—22no—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—41no—
49C2×#117—41no—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117—41no—
53C2×#117—41no—
54C2×#117Q814yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C42—24no—
65C22×C42—21no—
66C22×C42—21no—
67C22×C42—24no—
68C22×C42—41no—
69C22×C42—41no—
70C22×C42D412yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—24no—
75C2×#107—42no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—42no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8Q818yes—
82#389—24no—
83#389—24no—
84#387—216no—
85#385—216no—
86#382—24no—
87#380—24no—
88#380D418yes—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—24no—
103#336—41no—
104#336—41no—
105C25C2×D411yescharacteristic
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D414yes—
127C23×C4C2×Q814yes—
128C23×C4C2×Q812yes—
129C23×C4#4112yes—
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
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×C8C2×Q814yes—
144C22×C8C2×Q814yes—
145#125—48no—
146#125—48no—
147#117—44no—
148#117—44no—
149#117—48no—
150#117—48no—
151#117—44no—
152#117—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
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—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×Q814yes—
182C2×C4⋊C4#4114yes—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—48no—
204C2×C42—48no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42#4114yes—
211#107—48no—
212#106—48no—
213C22⋊C8—216no—
214#95—28no—
215#95—28no—
216#95—28no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24#12012yes—
230C24C22⋊Q812yes—
231C24C4⋊Q812yes—
232C24C22×D411yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
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#12012yes—
299C22×C4C22⋊Q818yes—
300C22×C4C22⋊Q812yes—
301C22×C4#12314yes—
302C22×C4C4⋊Q818yes—
303C22×C4C4⋊Q812yes—
304C22×C4#13514yes—
305C22×C4C22×D412yes—
306C22×C4C22×Q812yes—
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4112yes—
311C22×C4H(5,𝔽2)11yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—416no—
338C4⋊C4—416no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C22⋊C4—416no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C42—24no—
351C42—24no—
352C42—48no—
353C42—48no—
354C42—48no—
355C42—42no—
356C42—42no—
357C23—24no—
358C23—24no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—24no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23#39514yes—
380C23C2×#12011yescharacteristic
381C23C2×C22⋊Q811yescharacteristic
382C23C2×#12311yescharacteristic
383C23#52511yescharacteristic
384C23C2×C4⋊Q811yescharacteristic
385C23#53411yescharacteristic
386C23#53511yescharacteristic
387C23C2×#13512yes—
388C23#57512yes—
389C23#57512yes—
390C23#57512yes—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—432no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
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#39514yes—
445C2×C4C2×C22⋊Q812yes—
446C2×C4#52912yes—
447C2×C4C2×C4⋊Q812yes—
448C2×C4#53312yes—
449C2×C4#53811yescharacteristic
450C2×C4#53811yescharacteristic
451C2×C4#54311yescharacteristic
452C2×C4#54311yescharacteristic
453C2×C4#54612yes—
454C2×C4#54612yes—
455C2×C4#54612yes—
456C2×C4#54612yes—
457C2×C4#55412yes—
458C2×C4#55412yes—
459C2×C4#55412yes—
460C2×C4#55412yes—
461C8—44no—
462C8—44no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—44no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—44no—
472C22—44no—
473C22—44no—
474C22C2×#39511yescharacteristic
475C22#240411yescharacteristic
476C22#240511yescharacteristic
477C22#250211yescharacteristic
478C22#258311yescharacteristic
479C22#262611yescharacteristic
480C22#285511yescharacteristic
481C22#300212yes—
482C22#300512yes—
483C22#307612yes—
484C22#308012yes—
485C22#309112yes—
486C22#309312yes—
487C22#310212yes—
488C22#311112yes—
489C22#316011yescharacteristic
490C22#316211yescharacteristic
491C22#316611yescharacteristic
492C22#316611yescharacteristic
493C22#319112yes—
494C22#319712yes—
495C22#321212yes—
496C22#321912yes—
497C4—24no—
498C4—44no—
499C4—44no—
500C4—84no—
501C4—84no—
502C4—84no—
503C4—84no—
504C4#245012yes—
505C4#245012yes—
506C4#254912yes—
507C4#254912yes—
508C2—44no—
509C2#1491411yescharacteristic
510C2#2355111yescharacteristic
511C2#2355311yescharacteristic
512C2#2393812yes—
513C2#2400812yes—
514C2#2795111yescharacteristic
515C2#2795411yescharacteristic
516C2#2930011yescharacteristic
517C2#2930511yescharacteristic
518C2#2995812yes—
519C2#3315212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #11385C2semidirect
5. #11325C2semidirect
6. #11220C2semidirect
7. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#18542
#137562
#113852
#113252
#112202
#112192
#107312
#107122