Finite Group Databasebeta

← #946569

Subgroups — #946569

Order 512.

2434 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3#17227C214yesmaximal, complemented
4#16668C214yesmaximal, complemented
5#16470C214yesmaximal, complemented
6#15463C214yesmaximal
7#15234C212yesmaximal, complemented
8#15195C212yesmaximal, complemented
9#15150C212yesmaximal, complemented
10#14276C212yesmaximal, complemented
11#13881C212yesmaximal, complemented
12#13881C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14C22×#123C2212yes—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#2487—28no—
19#2486—28no—
20C2×#400C2212yes—
21C2×#399C2212yes—
22C2×#398C2212yes—
23C2×#397C2212yes—
24C2×#397C2218yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#386C2212yes—
28C2×#385C2212yes—
29C2×#384C2212yes—
30C2×#382C2212yes—
31C2×#382C2212yes—
32C2×#381C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×#376C2211yescharacteristic
41#1445C2212yes—
42#1445C2218yes—
43#1442C2218yes—
44#1442C2218yes—
45#1442C2218yes—
46#1442C2218yes—
47#1441C2218yes—
48#1441C2218yes—
49#1441C2218yes—
50#1439C2212yes—
51#1439C2212yes—
52#1439C2218yes—
53#1438C2212yes—
54C4⋊#95C2212yes—
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C22⋊#95C2218yes—
60C22⋊#95C2218yes—
61C23×D4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2312yes—
65C24×C4C2312yes—
66C2×#123—28no—
67C2×#123—48no—
68C2×C4×D4—42no—
69C2×C4×D4—42no—
70C2×C4×D4D418yes—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2318yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2311yescharacteristic
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98C22×C42C2312yes—
99#400—28no—
100#399—28no—
101#398—28no—
102#397—28no—
103#397—416no—
104#389—28no—
105#389—28no—
106#386—28no—
107#385—28no—
108#384—28no—
109#382—44no—
110#382—44no—
111#381—28no—
112#380—28no—
113#380—24no—
114#380—28no—
115#380—24no—
116C4×C4⋊C4—42no—
117C4×C22⋊C4—28no—
118C4×C22⋊C4—28no—
119#376—42no—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C2×#95C2318yes—
133C2×#95C2318yes—
134C2×#95C2318yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C25C2411yescharacteristic
139C25C2411yescharacteristic
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—41no—
143C22×D4—41no—
144C22×D4C2×D414yes—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—28no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—48no—
162C23×C4C2×D414yes—
163C23×C4C2×D414yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2411yescharacteristic
170C23×C4C2411yescharacteristic
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2414yes—
174C23×C4C2411yescharacteristic
175#123—432no—
176C4×D4—416no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—216no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—416no—
204C2×C4⋊C4—416no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4C2×D414yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—216no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—48no—
227C2×C22⋊C4C2×D414yes—
228C2×C22⋊C4C2×D414yes—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42C2×D414yes—
237C2×C42#4118yes—
238#95—28no—
239#95—28no—
240#95—24no—
241#95—24no—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—48no—
247#95—48no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—48no—
252#95—48no—
253#95—48no—
254#95—48no—
255#95—48no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—24no—
267C24—24no—
268C24C22×D412yes—
269C24C22×D412yes—
270C24H(5,𝔽2)14yes—
271C242−1+414yes—
272C24C2511yescharacteristic
273C2×D4—28no—
274C2×D4—28no—
275C2×D4—44no—
276C2×D4—48no—
277C2×D4—48no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—216no—
302C22×C4—216no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—28no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4C22×D412yes—
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C2×#4118yes—
351C22×C4C2×#4114yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C4H(5,𝔽2)12yes—
354C22×C42−1+414yes—
355C4⋊C4—44no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—416no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—816no—
365C4⋊C4—88no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—416no—
369C22⋊C4—416no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—416no—
373C22⋊C4—88no—
374C42—28no—
375C42—216no—
376C42—48no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—24no—
382C23—24no—
383C23—28no—
384C23—28no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—28no—
392C23—28no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—44no—
398C23—44no—
399C23—41no—
400C23—41no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23#53514yes—
408C23#53614yes—
409C23C23×D411yescharacteristic
410C23C22×#4112yes—
411C23C2×H(5,𝔽2)11yescharacteristic
412C23C2×H(5,𝔽2)11yescharacteristic
413C23C2×H(5,𝔽2)12yes—
414C23C2×2−1+412yes—
415C23C2×2−1+412yes—
416C23#58514yes—
417D4—416no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—416no—
445C2×C4—416no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—816no—
474C2×C4—816no—
475C2×C4#53414yes—
476C2×C4#53918yes—
477C2×C4#54718yes—
478C2×C4#55914yes—
479C2×C4#56314yes—
480C22—24no—
481C22—24no—
482C22—28no—
483C22—28no—
484C22—42no—
485C22—42no—
486C22—44no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—48no—
491C22—48no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22C2×#53411yescharacteristic
497C22C2×#53511yescharacteristic
498C22C2×#53611yescharacteristic
499C22C2×#53912yes—
500C22C2×#54712yes—
501C22C2×#55911yescharacteristic
502C22#349414yes—
503C22#349514yes—
504C22#350012yes—
505C22C2×#56311yescharacteristic
506C22#353612yes—
507C22#353612yes—
508C22#354012yes—
509C22#355112yes—
510C22#355914yes—
511C22#356314yes—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—48no—
517C4—84no—
518C4—84no—
519C4—84no—
520C4—84no—
521C4—84no—
522C4#272418yes—
523C2—44no—
524C2—44no—
525C2#3741911yescharacteristic
526C2#3745111yescharacteristic
527C2#3786412yes—
528C2#3786512yes—
529C2#4295614yes—
530C2#4543312yes—
531C2C2×#272411yescharacteristic
532C2#5299112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17227C2semidirect
4. #16668C2semidirect
5. #16470C2semidirect
7. #15234C2semidirect
8. #15195C2semidirect
9. #15150C2semidirect
10. #14276C2semidirect
11. #13881C2semidirect
12. #13881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#24862
#172272
#166682
#164702
#154632
#152342
#151952
#151502
#142762
#138812
#138472