Finite Group Databasebeta

← #493703

Subgroups — #493703

Order 512.

1325 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34265C211yesmaximal, characteristic
2C2×#2122C212yesmaximal
3#20640C211yesmaximal, characteristic, complemented
4#20634C211yesmaximal, characteristic, complemented
5#12795C214yesmaximal, complemented
6#12793C214yesmaximal, complemented
7#12573C212yesmaximal, complemented
8C2×#535C2211yescharacteristic
9C2×#520C2211yescharacteristic
10C22×C4×D4C411yescharacteristic, complemented
11#2905—21no—
12C2×#413C2212yes—
13C2×#410C2212yes—
14#2367—22no—
15#2366—22no—
16#2363—21no—
17#2363—21no—
18#2361—21no—
19#2361—21no—
20#2321C412yescomplemented
21#2321C2212yes—
22#2316—21no—
23#2309C411yescharacteristic, complemented
24C4×#120C412yescomplemented
25C4×#120C2212yes—
26#2297C2211yescharacteristic
27#2296C411yescharacteristic, complemented
28#2274C411yescharacteristic, complemented
29#2122—28no—
30C2×AΣL(1,𝔽16)C2218yes—
31#1801—22no—
32#1800C2214yes—
33#1799—22no—
34#1799C2214yes—
35C2×#343—21no—
36#1740—21no—
37#1390C2214yes—
38#1390C2214yes—
39C22×#41C2311yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C22×M16—22no—
43#535—22no—
44#535—22no—
45#535D414yes—
46#535Q814yes—
47C2×#127—21no—
48C2×#124—21no—
49C2×#121—21no—
50C2×#121—21no—
51C2×#120C2×C412yes—
52C2×#120C2312yes—
53#520—44no—
54C2×C4×Q8D412yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—44no—
61C2×C4×D4C2×C412yes—
62C2×C4×D4D412yes—
63C2×C4×D4C2312yes—
64C2×#117—22no—
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117D411yescharacteristic
69C2×#117D411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C22⋊C4C2×C412yes—
74C22×C42C2×C411yescharacteristic
75#424—44no—
76#423—24no—
77#413—24no—
78#413—24no—
79C2×#101—22no—
80C2×#100C2314yes—
81#410—24no—
82#410—24no—
83C2×#99—22no—
84C2×#99C2314yes—
85#391—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—22no—
90#390—22no—
91#390—22no—
92#389—21no—
93#389—21no—
94#389—44no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#386C2×C411yescharacteristic
104#386C2×C411yescharacteristic
105#386C2311yescharacteristic
106#386C2311yescharacteristic
107#385—22no—
108#385—22no—
109#385C2×C412yes—
110#385C2×C412yes—
111#380—24no—
112#379C2×C412yes—
113#379C2×C412yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4C2×C412yes—
124AΣL(1,𝔽16)—216no—
125#343—24no—
126#342—24no—
127C25C22×C412yes—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41C2×D412yes—
131C2×#41C2×Q812yes—
132C2×#41#4112yes—
133C2×#41#4112yes—
134C22×Q8C2×D411yescharacteristic
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4C4⋊C414yescomplemented
143C22×D4C22×C411yescharacteristic
144C22×D4C2×D411yescharacteristic
145C22×D4C2411yescharacteristic
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—84no—
150C23×C4C22⋊C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C2×M16—24no—
159C2×M16—42no—
160C2×M16—44no—
161C2×M16—44no—
162C22×C8—42no—
163#127—24no—
164#127—24no—
165#124—24no—
166#124—24no—
167#121—24no—
168#121—24no—
169#121—24no—
170#121—24no—
171#120—24no—
172#120—22no—
173#120—22no—
174#120C4⋊C418yescomplemented
175#120C2×D414yes—
176#120C2×Q814yes—
177C4×Q8—44no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4—48no—
182C4×D4—44no—
183C4×D4—44no—
184C4×D4—84no—
185C4×D4—84no—
186#117—22no—
187#117—24no—
188#117—42no—
189#117—42no—
190#117—42no—
191#117—44no—
192#117—44no—
193#117C4⋊C414yescomplemented
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C2×D412yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22×C412yes—
226C2×C22⋊C4C22×C411yescharacteristic
227C2×C22⋊C4C22×C411yescharacteristic
228C2×C22⋊C4C22×C412yes—
229C2×C22⋊C4C2×D412yes—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22⋊C412yes—
239C2×C42C22⋊C412yes—
240C2×C42C2×D412yes—
241C4⋊C8—48no—
242#101—44no—
243#100—24no—
244#100—24no—
245#99—24no—
246#99—24no—
247#99—44no—
248#95—21no—
249#95—22no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—21no—
255#95—21no—
256#95—44no—
257C24—22no—
258C24—22no—
259C24—42no—
260C24—42no—
261C24—82no—
262C24—82no—
263C24C2×C22⋊C411yescharacteristic
264C24C2×C4⋊C414yes—
265C24C23×C411yescharacteristic
266C24C22×D411yescharacteristic
267#41—44no—
268#41—44no—
269#41—44no—
270#41—44no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8#12112yes—
274C2×Q8C22⋊Q812yes—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—28no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—48no—
289C2×D4—48no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4C2×C4⋊C412yes—
297C2×D4#11712yes—
298C2×D4#12112yes—
299C2×D4C22⋊Q812yes—
300C2×D4C22×D411yescharacteristic
301C2×D4C22×Q811yescharacteristic
302C2×D4C2×#4111yescharacteristic
303C2×D4C2×#4111yescharacteristic
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—41no—
323C22×C4—44no—
324C22×C4—41no—
325C22×C4—44no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—81no—
331C22×C4—82no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C4⋊C412yes—
342C22×C4#11712yes—
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4C22⋊Q812yes—
346C22×C4C22⋊Q812yes—
347C22×C4C22×D411yescharacteristic
348M16—44no—
349M16—44no—
350M16—84no—
351C2×C8—44no—
352C2×C8—84no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—84no—
363C4⋊C4—82no—
364C4⋊C4—164no—
365C4⋊C4#9914yes—
366C22⋊C4—22no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—28no—
370C22⋊C4—24no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—81no—
384C22⋊C4—81no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4C2×C4⋊C414yes—
388C42—24no—
389C42—24no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—84no—
396C42—82no—
397C42—82no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—24no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—84no—
420C23—84no—
421C23—84no—
422C23—84no—
423C23—88no—
424C23#38012yes—
425C23C22×C22⋊C411yescharacteristic
426C23C22×C4⋊C411yescharacteristic
427C23C2×#11711yescharacteristic
428C23C2×#12111yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430Q8—42no—
431Q8—42no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—48no—
441D4—48no—
442D4—84no—
443D4—84no—
444D4—88no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—88no—
484C2×C4—88no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—162no—
490C2×C4#38012yes—
491C2×C4#38012yes—
492C2×C4#38012yes—
493C2×C4C2×#9914yes—
494C2×C4C2×#9912yes—
495C2×C4#41012yes—
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×C22⋊Q811yescharacteristic
498C8—84no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—41no—
507C22—41no—
508C22—42no—
509C22—41no—
510C22—41no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—44no—
521C22—81no—
522C22—81no—
523C22—82no—
524C22—82no—
525C22—88no—
526C22—88no—
527C22—82no—
528C22—82no—
529C22C2×#38011yescharacteristic
530C22C22×#9911yescharacteristic
531C22C2×#41011yescharacteristic
532C4—24no—
533C4—21no—
534C4—21no—
535C4—22no—
536C4—22no—
537C4—44no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—81no—
543C4—81no—
544C4—81no—
545C4—164no—
546C4#179914yes—
547C2—21no—
548C2—21no—
549C2—41no—
550C2—41no—
551C2—41no—
552C2—41no—
553C2—42no—
554C2—84no—
555C2C2×#179911yescharacteristic
556C2#3334811yescharacteristic
557C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#535C22
LCSG ▷ C24 ▷ 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. #20640C2semidirect
4. #20634C2semidirect
5. #12795C2semidirect
6. #12793C2semidirect
7. #12573C2semidirect
10. C22×C4×D4C4semidirect
20. #2321C4semidirect
23. #2309C4semidirect
24. C4×#120C4semidirect
27. #2296C4semidirect
28. #2274C4semidirect
142. C22×D4C4⋊C4semidirect
174. #120C4⋊C4semidirect
193. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342652
C2×#21222
#206402
#206342
#127952
#127932
#125732