Finite Group Databasebeta

← #406718

Subgroups — #406718

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#1825C212yesmaximal
3#13795C211yesmaximal, characteristic
4#11385C212yesmaximal, complemented
5#11293C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11267C212yesmaximal, complemented
8#10730C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×#125C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2629—28no—
16C2×#398—22no—
17C2×#396—22no—
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×#380C2212yes—
21#1825—28no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C4⋊#95—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C22⋊#95—22no—
37C24×C4—21no—
38C24×C4C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×#125—24no—
42C2×#125—24no—
43C2×#125—41no—
44C2×#125—41no—
45C2×#117—24no—
46C2×#117—22no—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—24no—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117Q814yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—41no—
60C22×C4⋊C4—41no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D412yes—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75#398—216no—
76#396—216no—
77#389—24no—
78#389—24no—
79#384—24no—
80#380—24no—
81#380D418yes—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#336—24no—
97#336—41no—
98#336—41no—
99C25C2×D411yescharacteristic
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—82no—
114C23×C4—82no—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D414yes—
119C23×C4C2×Q812yes—
120C23×C4#4114yes—
121C23×C4#4112yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137#125—48no—
138#125—48no—
139#117—216no—
140#117—48no—
141#117—48no—
142#117—48no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
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—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2×Q814yes—
183C2×C4⋊C4#4114yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42#4114yes—
204C22⋊C8—48no—
205#95—28no—
206#95—28no—
207#95—82no—
208#95—82no—
209#95—82no—
210#95—82no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24#12012yes—
220C24C22⋊Q812yes—
221C24#12412yes—
222C24C22×D411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—48no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
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—84no—
284C22×C4—84no—
285C22×C4—88no—
286C22×C4—88no—
287C22×C4#12012yes—
288C22×C4C22⋊Q818yes—
289C22×C4C22⋊Q812yes—
290C22×C4#12314yes—
291C22×C4#12418yes—
292C22×C4#12412yes—
293C22×C4C22×D412yes—
294C22×C4C22×Q811yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4H(5,𝔽2)11yescharacteristic
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C42−1+411yescharacteristic
302C22×C42−1+411yescharacteristic
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—416no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—88no—
337C4⋊C4—88no—
338C22⋊C4—48no—
339C22⋊C4—48no—
340C22⋊C4—88no—
341C42—24no—
342C42—24no—
343C42—42no—
344C42—42no—
345C42—48no—
346C23—22no—
347C23—24no—
348C23—24no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—24no—
357C23—22no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23#39314yes—
369C23C2×#12011yescharacteristic
370C23C2×C22⋊Q811yescharacteristic
371C23C2×#12311yescharacteristic
372C23#52511yescharacteristic
373C23C2×#12411yescharacteristic
374C23#53411yescharacteristic
375C23#53611yescharacteristic
376C23#57612yes—
377C23#57712yes—
378C23#57712yes—
379C23#57812yes—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—816no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#39314yes—
434C2×C4C2×C22⋊Q812yes—
435C2×C4C2×#12412yes—
436C2×C4#52912yes—
437C2×C4#53312yes—
438C2×C4#53911yescharacteristic
439C2×C4#53911yescharacteristic
440C2×C4#54411yescharacteristic
441C2×C4#54411yescharacteristic
442C2×C4#54612yes—
443C2×C4#54612yes—
444C2×C4#54712yes—
445C2×C4#54712yes—
446C2×C4#55112yes—
447C2×C4#55112yes—
448C2×C4#55212yes—
449C2×C4#55212yes—
450C8—82no—
451C8—82no—
452C22—24no—
453C22—28no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—44no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—44no—
463C22C2×#39311yescharacteristic
464C22#240511yescharacteristic
465C22#240611yescharacteristic
466C22#250311yescharacteristic
467C22#256111yescharacteristic
468C22#262711yescharacteristic
469C22#285511yescharacteristic
470C22#301212yes—
471C22#301312yes—
472C22#308312yes—
473C22#308412yes—
474C22#309812yes—
475C22#309912yes—
476C22#312211yescharacteristic
477C22#312311yescharacteristic
478C22#312611yescharacteristic
479C22#312711yescharacteristic
480C22#313612yes—
481C22#313712yes—
482C22#319812yes—
483C22#320112yes—
484C22#324112yes—
485C22#324212yes—
486C4—24no—
487C4—44no—
488C4—84no—
489C4—84no—
490C4—84no—
491C4—82no—
492C4—84no—
493C4#243812yes—
494C4#243812yes—
495C4#248812yes—
496C4#248812yes—
497C2—44no—
498C2#1493411yescharacteristic
499C2#2345812yes—
500C2#2401712yes—
501C2#2407011yescharacteristic
502C2#2407211yescharacteristic
503C2#2797012yes—
504C2#2911511yescharacteristic
505C2#2911711yescharacteristic
506C2#2997912yes—
507C2#3315611yescharacteristic
508C2#3315711yescharacteristic
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
4. #11385C2semidirect
5. #11293C2semidirect
6. #11292C2semidirect
7. #11267C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#18252
#137952
#113852
#112932
#112922
#112672
#107302
#107132