Finite Group Databasebeta

← #480301

Subgroups — #480301

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2#17357C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#11520C214yesmaximal, complemented
5#11517C214yesmaximal
6#11454C214yesmaximal, complemented
7C22×#117C411yescharacteristic
8C22×#117C2211yescharacteristic
9C23×C4⋊C4C411yescharacteristic
10#2920—21no—
11#2920—21no—
12#2317—21no—
13#2317—21no—
14#2317—21no—
15#2317—21no—
16#2317C412yes—
17#2317C412yes—
18#2317C2212yes—
19#2317C2212yes—
20C2×#389—21no—
21C2×#389—21no—
22C2×#385C412yes—
23C2×#385C2212yes—
24C2×#384—21no—
25C2×#382—21no—
26#1464—21no—
27#1464—21no—
28#1464C2212yes—
29#1464C2212yes—
30#1463—21no—
31#1463—21no—
32#1462C2212yes—
33#1462C2212yes—
34#1458—21no—
35#1458—21no—
36#1458C2212yes—
37#1458C2212yes—
38#1326C22116yes—
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41C2×#117—21no—
42C2×#117—21no—
43C2×#117—28no—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117—41no—
47C2×#117—41no—
48C2×#117C2×C412yes—
49C2×#117C2×C412yes—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117D411yescharacteristic
53C2×#117D411yescharacteristic
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—41no—
66C22×C4⋊C4—41no—
67C22×C4⋊C4—41no—
68C22×C4⋊C4—41no—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#432—22no—
80#432—22no—
81#432—41no—
82#432—41no—
83#431—22no—
84#431—22no—
85#431—41no—
86#431—41no—
87#423—41no—
88#423—41no—
89#407—22no—
90#407—22no—
91C4×M16—41no—
92C4×M16—41no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385C2×C414yes—
106#385C2×C414yes—
107#385C2×C414yes—
108#385C2314yes—
109#384—22no—
110#384—22no—
111#384—22no—
112#384—22no—
113#382—22no—
114#382—22no—
115#382—22no—
116#382—22no—
117C2×#95—22no—
118C2×#95—22no—
119#324—22no—
120#324—22no—
121#324—22no—
122#324—22no—
123#323—22no—
124#323—22no—
125#323C2314yes—
126#323C2314yes—
127C25C22×C411yescharacteristic
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—84no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C414yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—42no—
152C2×M16—42no—
153#117—28no—
154#117—28no—
155#117—24no—
156#117—28no—
157#117—41no—
158#117—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162#117—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170#117—41no—
171#117—41no—
172#117—41no—
173#117M1618yescomplemented
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—84no—
204C2×C4⋊C4—84no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4C22⋊C411yescharacteristic
220C2×C22⋊C4C22⋊C411yescharacteristic
221C2×C22⋊C4C22×C414yes—
222C2×C22⋊C4C22×C411yescharacteristic
223C2×C22⋊C4C22×C411yescharacteristic
224C2×C22⋊C4C22×C411yescharacteristic
225C2×C22⋊C4C2×D411yescharacteristic
226C2×C22⋊C4C2×D411yescharacteristic
227C2×C22⋊C4C2411yescharacteristic
228C2×C42—28no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C22⋊C8—24no—
252C22⋊C8—24no—
253C22⋊C8—42no—
254C22⋊C8—42no—
255#97—42no—
256#97—42no—
257C4×C8—42no—
258C4×C8—42no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—42no—
268C24—42no—
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271C24C23×C411yescharacteristic
272C22×C4—28no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4—88no—
322C22×C4—88no—
323C22×C4#9911yescharacteristic
324C22×C4#9911yescharacteristic
325C22×C4#9911yescharacteristic
326C22×C4#9911yescharacteristic
327C22×C4#10114yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×M1614yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341M16—84no—
342M16—84no—
343C2×C8—44no—
344C2×C8—44no—
345C4⋊C4—28no—
346C4⋊C4—28no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C22⋊C4—28no—
366C22⋊C4—24no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4C2×M1618yes—
376C42—28no—
377C42—28no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C23—21no—
395C23—21no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—24no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—48no—
410C23—48no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23C2×#9911yescharacteristic
415C23C2×#9911yescharacteristic
416C23C2×#10112yes—
417C23C22×C22⋊C411yescharacteristic
418C23C22×M1612yes—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—44no—
449C2×C4—416no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—84no—
471C2×C4—88no—
472C2×C4—84no—
473C2×C4—88no—
474C2×C4#40712yes—
475C2×C4#40712yes—
476C2×C4C2×#9911yescharacteristic
477C2×C4C2×#9911yescharacteristic
478C2×C4C2×#9911yescharacteristic
479C2×C4C2×#9911yescharacteristic
480C2×C4C2×#10112yes—
481C2×C4C2×#10112yes—
482C8—84no—
483C8—84no—
484C22—24no—
485C22—24no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—48no—
499C22—48no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22#212212yes—
505C22#212312yes—
506C22C2×#40711yescharacteristic
507C22C22×#9911yescharacteristic
508C22C22×#10111yescharacteristic
509C4—22no—
510C4—21no—
511C4—21no—
512C4—22no—
513C4—48no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—84no—
523C4#146312yes—
524C4#146312yes—
525C2—21no—
526C2—21no—
527C2—22no—
528C2—22no—
529C2—42no—
530C2—42no—
531C2C2×#146311yescharacteristic
532C2C2×#212211yescharacteristic
533C2C2×#212311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11520C2semidirect
6. #11454C2semidirect
173. #117M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
#173572
#115202
#115172
#114542