Finite Group Databasebeta

← #492995

Subgroups — #492995

Order 512.

1417 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34265C211yesmaximal, characteristic
2C2×#2122C212yesmaximal
3#20397C211yesmaximal, characteristic, complemented
4#20393C211yesmaximal, characteristic, complemented
5#12622C214yesmaximal, complemented
6C4×AΣL(1,𝔽16)C214yesmaximal, complemented
7#12573C212yesmaximal, complemented
8C2×#535C2211yescharacteristic
9C2×#520C2211yescharacteristic
10C22×C4×D4C411yescharacteristic, complemented
11C2×#413C2212yes—
12C2×#410C2212yes—
13#2873—21no—
14#2367—22no—
15#2366—22no—
16#2363—21no—
17#2363—21no—
18#2361—21no—
19#2361—21no—
20#2321C412yescomplemented
21#2321C2212yes—
22#2309C411yescharacteristic, complemented
23C4×#120C412yescomplemented
24C4×#120C2212yes—
25#2297C2211yescharacteristic
26#2296C411yescharacteristic, complemented
27#2274—21no—
28#2274C411yescharacteristic, complemented
29#2122C4116yescomplemented
30C2×AΣL(1,𝔽16)C2218yes—
31C4×#101—22no—
32C4×#100C2214yes—
33C4×#99—22no—
34C4×#99C2214yes—
35C2×#343—21no—
36#1740—21no—
37#1390C2214yes—
38#1390C2214yes—
39C22×#41C2311yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C22×M16—22no—
43#535—24no—
44#535C2×C418yes—
45C2×#127—21no—
46C2×#124—21no—
47C2×#121—21no—
48C2×#121—21no—
49C2×#120C2×C412yes—
50C2×#120C2312yes—
51#520—44no—
52C2×C4×Q8D412yes—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—21no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—44no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4C2312yes—
62C2×#117—22no—
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C22⋊C4C2×C412yes—
72C22×C42C2×C411yescharacteristic
73#413—24no—
74#413C2×C418yescomplemented
75C2×#101—22no—
76C2×#100C2314yes—
77#410—24no—
78#410C2×C418yescomplemented
79C2×#99—22no—
80C2×#99C2314yes—
81#405—44no—
82C4×M16—24no—
83#391—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—22no—
88#390—22no—
89#390—22no—
90#389—21no—
91#389—21no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#386C2×C411yescharacteristic
101#386C2×C411yescharacteristic
102#386C2311yescharacteristic
103#386C2311yescharacteristic
104#385—22no—
105#385—22no—
106#385C2×C412yes—
107#385C2×C412yes—
108#379C2×C412yes—
109#379C2×C412yes—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—44no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—24no—
120C4×C22⋊C4C2×C412yes—
121C4×C22⋊C4C2×C412yes—
122AΣL(1,𝔽16)C2×C4132yescomplemented
123#343—24no—
124#342—24no—
125C25C22×C412yes—
126C2×#41—42no—
127C2×#41—42no—
128C2×#41C22×C414yes—
129C2×#41#4114yes—
130C22×Q8C2×D411yescharacteristic
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4C4214yescomplemented
139C22×D4C22×C411yescharacteristic
140C22×D4C2×D411yescharacteristic
141C22×D4C2411yescharacteristic
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4—84no—
146C23×C4C22⋊C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C2×M16—28no—
155C2×M16—24no—
156C2×M16—44no—
157C2×M16—42no—
158C22×C8—42no—
159#127—28no—
160#124—28no—
161#121—28no—
162#121—28no—
163#120—24no—
164#120—24no—
165#120C4218yescomplemented
166#120C22×C418yes—
167C4×Q8—44no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—44no—
171C4×D4—48no—
172C4×D4—44no—
173C4×D4—44no—
174C4×D4—84no—
175C4×D4—84no—
176#117—22no—
177#117—24no—
178#117—28no—
179#117—42no—
180#117—42no—
181#117—44no—
182#117—42no—
183#117C4214yescomplemented
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22×C412yes—
215C2×C22⋊C4C22×C411yescharacteristic
216C2×C22⋊C4C22×C411yescharacteristic
217C2×C22⋊C4C22×C412yes—
218C2×C22⋊C4C2×D412yes—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—44no—
228C2×C42C22⋊C412yes—
229C2×C42C22⋊C412yes—
230C2×C42C2×D412yes—
231#101—28no—
232#100—24no—
233#100C22×C418yes—
234#99—28no—
235#99—24no—
236#99C22×C418yes—
237#97—44no—
238C4×C8—44no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—44no—
248C24—22no—
249C24—22no—
250C24—42no—
251C24—42no—
252C24—82no—
253C24—82no—
254C24C2×C4214yes—
255C24C2×C22⋊C411yescharacteristic
256C24C23×C411yescharacteristic
257C24C22×D411yescharacteristic
258#41—44no—
259#41—44no—
260#41—44no—
261#41—44no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8C4×D414yes—
265C2×D4—28no—
266C2×D4—28no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—82no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4C2×C4212yes—
286C2×D4#11712yes—
287C2×D4C4×D414yes—
288C2×D4C23×C412yes—
289C2×D4C2×#4112yes—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—41no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—41no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—81no—
317C22×C4—82no—
318C22×C4—88no—
319C22×C4—88no—
320C22×C4C2×C4212yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4#11712yes—
329C22×C4C4×D414yes—
330C22×C4C4×D414yes—
331C22×C4C22×D411yescharacteristic
332M16—28no—
333M16—44no—
334M16—48no—
335C2×C8—44no—
336C2×C8—48no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—82no—
348C4⋊C4#9914yes—
349C22⋊C4—22no—
350C22⋊C4—28no—
351C22⋊C4—28no—
352C22⋊C4—24no—
353C22⋊C4—28no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—81no—
366C22⋊C4—81no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4C2×C4214yes—
370C42—24no—
371C42—24no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—48no—
376C42—42no—
377C42—42no—
378C42—82no—
379C42—82no—
380C42—164no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—24no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—44no—
399C23—44no—
400C23—42no—
401C23—42no—
402C23—84no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23—88no—
407C23C4×C22⋊C412yes—
408C23C22×C4211yescharacteristic
409C23C22×C22⋊C411yescharacteristic
410C23C2×#11711yescharacteristic
411C23C2×C4×D412yes—
412Q8—44no—
413D4—42no—
414D4—42no—
415D4—44no—
416D4—42no—
417D4—42no—
418D4—48no—
419D4—48no—
420D4—88no—
421D4—88no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—48no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—88no—
462C2×C4—88no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—82no—
466C2×C4—162no—
467C2×C4C4×C22⋊C412yes—
468C2×C4C4×C22⋊C412yes—
469C2×C4C4×C22⋊C412yes—
470C2×C4C2×#9914yes—
471C2×C4C2×#9912yes—
472C2×C4#41012yes—
473C2×C4C2×C4×D412yes—
474C8—48no—
475C22—21no—
476C22—21no—
477C22—22no—
478C22—21no—
479C22—21no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—41no—
484C22—41no—
485C22—42no—
486C22—44no—
487C22—44no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—44no—
495C22—81no—
496C22—81no—
497C22—82no—
498C22—82no—
499C22—88no—
500C22—88no—
501C22—82no—
502C22—82no—
503C22C2×C4×C22⋊C411yescharacteristic
504C22C22×#9911yescharacteristic
505C22C2×#41011yescharacteristic
506C4—24no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—44no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—81no—
517C4—88no—
518C4—81no—
519C4C4×#9914yes—
520C2—21no—
521C2—21no—
522C2—42no—
523C2—41no—
524C2—41no—
525C2—42no—
526C2—84no—
527C2C2×C4×#9911yescharacteristic
528C2#3334811yescharacteristic
529C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
prime radicalC2×#535C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20397C2semidirect
4. #20393C2semidirect
5. #12622C2semidirect
6. C4×AΣL(1,𝔽16)C2semidirect
7. #12573C2semidirect
10. C22×C4×D4C4semidirect
20. #2321C4semidirect
22. #2309C4semidirect
23. C4×#120C4semidirect
26. #2296C4semidirect
28. #2274C4semidirect
29. #2122C4semidirect
74. #413C2×C4semidirect
78. #410C2×C4semidirect
122. AΣL(1,𝔽16)C2×C4semidirect
138. C22×D4C42semidirect
165. #120C42semidirect
183. #117C42semidirect

Maximal subgroups

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

subgroupindex
#342652
C2×#21222
#203972
#203932
#126222
C4×AΣL(1,𝔽16)2
#125732