Finite Group Databasebeta

← #440931

Subgroups — #440931

Order 512.

1394 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2C2×#2025C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic, complemented
4#15218C211yesmaximal, characteristic, complemented
5#11385C212yesmaximal, complemented
6#11372C211yesmaximal, characteristic, complemented
7#11207C212yesmaximal, complemented
8#11066C212yesmaximal
9#10925C211yesmaximal, characteristic
10#10874C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2566—24no—
19#2566—24no—
20C2×#396—22no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2212yes—
24C2×#385—22no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#2025—24no—
29#1991—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1444—22no—
40#1441—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—24no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—24no—
53C2×#125—42no—
54C2×#125—42no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—24no—
58C2×C4×D4—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102Q818yes—
82C2×C4×C8—22no—
83C2×C4×C8—42no—
84#396—216no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390D414yes—
89#389—24no—
90#389—42no—
91#385—216no—
92#384—22no—
93#384—22no—
94#382—22no—
95#382D414yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D412yes—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#340—42no—
110#336—22no—
111#336—24no—
112#336—42no—
113C25—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4C2×D412yes—
121C22×D4C2×Q814yes—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—82no—
131C23×C4—82no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2411yescharacteristic
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—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×Q814yes—
151#125—28no—
152#125—48no—
153#125—48no—
154#125—48no—
155#125—44no—
156C4×D4—48no—
157C4×D4—48no—
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—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×Q814yes—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42#4114yes—
221#102—48no—
222#102—48no—
223C4×C8—44no—
224#95—28no—
225#95—24no—
226#95—44no—
227#95—44no—
228#95—82no—
229#95—82no—
230C24—22no—
231C24—24no—
232C24—41no—
233C24—41no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24C22×D411yescharacteristic
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4C4⋊Q814yes—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
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—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4#12014yes—
311C22×C4#12412yes—
312C22×C4#12412yes—
313C22×C4#13514yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×Q812yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4112yes—
319C22×C4H(5,𝔽2)12yes—
320C22×C4H(5,𝔽2)12yes—
321C22×C42−1+412yes—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
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—
335C2×C8—42no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—416no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—88no—
355C4⋊C4—88no—
356C22⋊C4—416no—
357C22⋊C4—48no—
358C22⋊C4—88no—
359C42—22no—
360C42—22no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—41no—
365C42—41no—
366C42—42no—
367C23—24no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—81no—
379C23—82no—
380C23—81no—
381C23—82no—
382C23—81no—
383C23—81no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23C2×#12011yescharacteristic
389C23C2×#12411yescharacteristic
390C23#53511yescharacteristic
391C23#54612yes—
392C23D4×Q812yes—
393C23C2×#13512yes—
394C23#57512yes—
395C23#57612yes—
396C23#57712yes—
397D4—28no—
398D4—44no—
399D4—44no—
400D4—44no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—816no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4#52511yescharacteristic
456C2×C4C2×C4⋊Q811yescharacteristic
457C2×C4#53312yes—
458C2×C4#53511yescharacteristic
459C2×C4#53611yescharacteristic
460C2×C4#54212yes—
461C2×C4#54212yes—
462C2×C4#54312yes—
463C2×C4#54612yes—
464C2×C4#54612yes—
465C2×C4#54612yes—
466C2×C4D4×Q812yes—
467C2×C4#55412yes—
468C2×C4#55412yes—
469C2×C4#55612yes—
470C2×C4#56212yes—
471C8—44no—
472C8—82no—
473C22—44no—
474C22—48no—
475C22—82no—
476C22—82no—
477C22—84no—
478C22—84no—
479C22—84no—
480C22—84no—
481C22—84no—
482C22#240411yescharacteristic
483C22#243411yescharacteristic
484C22#243511yescharacteristic
485C22#249212yes—
486C22#269212yes—
487C22#300212yes—
488C22#301212yes—
489C22#313111yescharacteristic
490C22#313612yes—
491C22#314211yescharacteristic
492C22#321212yes—
493C22#322011yescharacteristic
494C22#322211yescharacteristic
495C22#329712yes—
496C22#330112yes—
497C22#330712yes—
498C22#330912yes—
499C22#338012yes—
500C22#338112yes—
501C22#338812yes—
502C22#338912yes—
503C4—42no—
504C4—42no—
505C4—44no—
506C4—44no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—84no—
511C4—84no—
512C4#246512yes—
513C4#254912yes—
514C4#269212yes—
515C4#270012yes—
516C2—84no—
517C2#1567111yescharacteristic
518C2#2396212yes—
519C2#2584312yes—
520C2#2585811yescharacteristic
521C2#2586711yescharacteristic
522C2#2781712yes—
523C2#2782912yes—
524C2#3153012yes—
525C2#3153212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1991C2semidirect
4. #15218C2semidirect
5. #11385C2semidirect
6. #11372C2semidirect
7. #11207C2semidirect
10. #10874C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#20252
C2×#19912
#152182
#113852
#113722
#112072
#110662
#109252
#108742
#107882