Finite Group Databasebeta

← #493701

Subgroups — #493701

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34201C211yesmaximal, characteristic
2C2×#2120C212yesmaximal
3#20640C211yesmaximal, characteristic, complemented
4#20634C211yesmaximal, characteristic, complemented
5#12795C214yesmaximal, complemented
6#12794C214yesmaximal, complemented
7#12575C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C2×#520C2211yescharacteristic
10C22×#117C411yescharacteristic, complemented
11#2905—21no—
12C2×#413C2212yes—
13C2×#410C2212yes—
14#2363—22no—
15#2362—22no—
16#2354—21no—
17#2354—21no—
18#2352—21no—
19#2352—21no—
20#2321C412yescomplemented
21#2321C2212yes—
22#2317C411yescharacteristic, complemented
23#2316—21no—
24#2311C2211yescharacteristic
25C4×#123C412yescomplemented
26C4×#123C2212yes—
27#2280C411yescharacteristic, complemented
28#2274C411yescharacteristic, complemented
29#2120—28no—
30C2×#352C2214yes—
31C2×AΣL(1,𝔽16)C2214yes—
32#1801—22no—
33#1800C2214yes—
34#1799—22no—
35#1799C2214yes—
36C2×#343—21no—
37#1740—21no—
38#1390C2218yes—
39C22×#41C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C22×M16—22no—
42#525—22no—
43#525—22no—
44#525D414yes—
45#525Q814yes—
46C2×#123C2×C411yescharacteristic
47C2×#123C2311yescharacteristic
48C2×C22⋊Q8—21no—
49C2×#121—21no—
50C2×#120C2×C411yescharacteristic
51C2×#120C2311yescharacteristic
52#520—44no—
53C2×C4×Q8D412yes—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4C2312yes—
59C2×#117—21no—
60C2×#117—22no—
61C2×#117—21no—
62C2×#117—22no—
63C2×#117—44no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C22⋊C4C2×C412yes—
74C22×C42C2×C412yes—
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—22no—
88#390—22no—
89#389—22no—
90#389—44no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#386—22no—
100#386—22no—
101#386C2×C412yes—
102#386C2312yes—
103#385C2×C412yes—
104#385C2×C412yes—
105#384—21no—
106#382—21no—
107#381C2×C412yes—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—24no—
113#379C2×C412yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4C2×C412yes—
124#352—28no—
125AΣL(1,𝔽16)—28no—
126#343—24no—
127#342—24no—
128C25C22×C411yescharacteristic
129C2×#41—42no—
130C2×#41—42no—
131C2×#41C2×D412yes—
132C2×#41C2×Q812yes—
133C2×#41#4112yes—
134C2×#41#4112yes—
135C22×Q8C2×D411yescharacteristic
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C22×C411yescharacteristic
139C22×D4C2×D411yescharacteristic
140C22×D4C2411yescharacteristic
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4C22⋊C412yes—
147C23×C4C4⋊C414yescomplemented
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C2×M16—24no—
156C2×M16—42no—
157C2×M16—44no—
158C2×M16—44no—
159C22×C8—42no—
160#123—22no—
161#123—21no—
162#123—21no—
163#123C4⋊C414yescomplemented
164#123C2×D412yes—
165#123C2×Q812yes—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168#121—24no—
169#121—24no—
170#120—22no—
171#120—21no—
172#120—21no—
173#120C4⋊C414yescomplemented
174#120C2×D412yes—
175#120C2×Q812yes—
176C4×Q8—44no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—44no—
182C4×D4—44no—
183C4×D4—44no—
184#117—22no—
185#117—24no—
186#117—42no—
187#117—42no—
188#117—44no—
189#117—48no—
190#117—44no—
191#117—42no—
192#117—44no—
193#117—44no—
194#117C4⋊C414yescomplemented
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C2×D412yes—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22×C411yescharacteristic
230C2×C22⋊C4C22×C411yescharacteristic
231C2×C22⋊C4C22×C412yes—
232C2×C22⋊C4C22×C412yes—
233C2×C22⋊C4C2×D412yes—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—44no—
241C2×C42—42no—
242C2×C42C22⋊C412yes—
243C2×C42C22⋊C412yes—
244C2×C42C2×D412yes—
245C4⋊C8—48no—
246#101—44no—
247#100—24no—
248#100—24no—
249#99—24no—
250#99—24no—
251#99—44no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—44no—
261C24—21no—
262C24—21no—
263C24—41no—
264C24—41no—
265C24—81no—
266C24—81no—
267C24C2×C22⋊C411yescharacteristic
268C24C2×C4⋊C412yes—
269C24C23×C411yescharacteristic
270C24C22×D411yescharacteristic
271#41—44no—
272#41—44no—
273#41—44no—
274#41—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8#12112yes—
278C2×Q8C22⋊Q812yes—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—81no—
289C2×D4—81no—
290C2×D4—81no—
291C2×D4—81no—
292C2×D4C2×C4⋊C412yes—
293C2×D4#11712yes—
294C2×D4#12112yes—
295C2×D4C22⋊Q812yes—
296C2×D4C22×D411yescharacteristic
297C2×D4C22×Q811yescharacteristic
298C2×D4C2×#4111yescharacteristic
299C2×D4C2×#4111yescharacteristic
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C4⋊C412yes—
339C22×C4C2×C4⋊C412yes—
340C22×C4#11712yes—
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4C22⋊Q812yes—
344C22×C4C22⋊Q812yes—
345C22×C4C22×D411yescharacteristic
346M16—44no—
347M16—44no—
348M16—84no—
349C2×C8—44no—
350C2×C8—84no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—84no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—164no—
368C4⋊C4#9914yes—
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4C2×C4⋊C412yes—
390C22⋊C4C2×C4⋊C412yes—
391C42—24no—
392C42—24no—
393C42—42no—
394C42—42no—
395C42—44no—
396C42—42no—
397C42—44no—
398C42—84no—
399C42—82no—
400C42—82no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—84no—
427C23#38012yes—
428C23C22×C22⋊C411yescharacteristic
429C23C22×C4⋊C411yescharacteristic
430C23C2×#11711yescharacteristic
431C23C2×#12111yescharacteristic
432C23C2×C22⋊Q811yescharacteristic
433Q8—42no—
434Q8—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—82no—
444D4—82no—
445D4—84no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—48no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—162no—
491C2×C4#38012yes—
492C2×C4#38012yes—
493C2×C4#38012yes—
494C2×C4C2×#9914yes—
495C2×C4C2×#9912yes—
496C2×C4#41012yes—
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×C22⋊Q811yescharacteristic
499C8—84no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—84no—
527C22—84no—
528C22—81no—
529C22—81no—
530C22C2×#38011yescharacteristic
531C22C22×#9911yescharacteristic
532C22C2×#41011yescharacteristic
533C4—24no—
534C4—21no—
535C4—22no—
536C4—21no—
537C4—22no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—82no—
546C4—164no—
547C4#179914yes—
548C2—21no—
549C2—21no—
550C2—41no—
551C2—41no—
552C2—41no—
553C2—41no—
554C2—42no—
555C2—82no—
556C2C2×#179911yescharacteristic
557C2#3335011yescharacteristic
558C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#525C22
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. #12794C2semidirect
7. #12575C2semidirect
10. C22×#117C4semidirect
20. #2321C4semidirect
22. #2317C4semidirect
25. C4×#123C4semidirect
27. #2280C4semidirect
28. #2274C4semidirect
147. C23×C4C4⋊C4semidirect
163. #123C4⋊C4semidirect
173. #120C4⋊C4semidirect
194. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342012
C2×#21202
#206402
#206342
#127952
#127942
#125752