Finite Group Databasebeta

← #493704

Subgroups — #493704

Order 512.

1165 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34266C211yesmaximal, characteristic
2C2×#2123C212yesmaximal
3#20640C211yesmaximal, characteristic
4#20634C211yesmaximal, characteristic
5#12796C214yesmaximal, complemented
6#12794C214yesmaximal, complemented
7#12573C212yesmaximal
8C2×#536C2211yescharacteristic
9C2×#520C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic, complemented
11#2905—21no—
12C2×#413C2212yes—
13C2×#410C2212yes—
14#2367—22no—
15#2365—22no—
16#2362—21no—
17#2362—21no—
18#2360—21no—
19#2360—21no—
20#2322C412yescomplemented
21#2322C2212yes—
22#2316—21no—
23#2310C411yescharacteristic, complemented
24C4×#123C412yescomplemented
25C4×#123C2212yes—
26#2297C2211yescharacteristic
27#2296C411yescharacteristic, complemented
28#2274C411yescharacteristic, complemented
29#2123—28no—
30C2×#352C2218yes—
31#1801—22no—
32#1800C2214yes—
33#1799—22no—
34#1799C2214yes—
35C2×#343—21no—
36#1740—21no—
37#1390C2214yes—
38#1390C2214yes—
39C22×#41C2311yescharacteristic
40C23×Q8C2×C411yescharacteristic
41C22×M16—22no—
42#536—22no—
43#536—22no—
44#536D414yes—
45#536Q814yes—
46C2×C4⋊Q8—21no—
47C2×#124—21no—
48C2×#123C2×C412yes—
49C2×#123C2312yes—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52#520—44no—
53C2×C4×Q8—21no—
54C2×C4×Q8—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8—44no—
57C2×C4×Q8D412yes—
58C2×C4×D4C2×C412yes—
59C2×C4×D4D412yes—
60C2×C4×D4C2312yes—
61C2×#117—22no—
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C412yes—
73#424—44no—
74#423—24no—
75#413—24no—
76#413—24no—
77C2×#101—22no—
78C2×#100C2314yes—
79#410—24no—
80#410—24no—
81C2×#99—22no—
82C2×#99C2314yes—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—22no—
87#390—21no—
88#389—22no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—44no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#386C2×C411yescharacteristic
102#386C2×C411yescharacteristic
103#386C2311yescharacteristic
104#386C2311yescharacteristic
105#385—22no—
106#385—22no—
107#385C2×C412yes—
108#385C2×C412yes—
109#381C2×C412yes—
110#381C2×C412yes—
111#380—24no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4C2×C412yes—
124#352—216no—
125#343—24no—
126#342—24no—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41C2×D412yes—
130C2×#41C2×Q812yes—
131C2×#41#4112yes—
132C2×#41#4112yes—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8C4⋊C414yescomplemented
139C22×Q8C2×D411yescharacteristic
140C22×D4C22×C411yescharacteristic
141C22×D4C2×D411yescharacteristic
142C22×D4C2411yescharacteristic
143C23×C4C22⋊C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C22×C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—44no—
155C2×M16—44no—
156C22×C8—42no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159#124—24no—
160#124—24no—
161#123—24no—
162#123—22no—
163#123—22no—
164#123C4⋊C418yescomplemented
165#123C2×D414yes—
166#123C2×Q814yes—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C4×Q8—44no—
172C4×Q8—44no—
173C4×Q8—48no—
174C4×Q8—44no—
175C4×D4—44no—
176C4×D4—44no—
177C4×D4—44no—
178#117—22no—
179#117—24no—
180#117—42no—
181#117—42no—
182#117—42no—
183#117—44no—
184#117—44no—
185#117C4⋊C414yescomplemented
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C2×D412yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22×C412yes—
216C2×C22⋊C4C22×C411yescharacteristic
217C2×C22⋊C4C22×C411yescharacteristic
218C2×C22⋊C4C22×C412yes—
219C2×C22⋊C4C2×D412yes—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—42no—
229C2×C42—44no—
230C2×C42—42no—
231C2×C42C22⋊C412yes—
232C2×C42C22⋊C412yes—
233C2×C42C2×D412yes—
234C4⋊C8—48no—
235#101—44no—
236#100—24no—
237#100—24no—
238#99—24no—
239#99—24no—
240#99—44no—
241#95—22no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—44no—
254C24C2×C22⋊C411yescharacteristic
255C24C23×C411yescharacteristic
256C24C22×D411yescharacteristic
257#41—44no—
258#41—44no—
259#41—44no—
260#41—44no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—48no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—48no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8#12112yes—
272C2×Q8C22⋊Q812yes—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4C2×C4⋊C412yes—
280C2×D4#11712yes—
281C2×D4#12112yes—
282C2×D4C22⋊Q812yes—
283C2×D4C22×D411yescharacteristic
284C2×D4C22×Q811yescharacteristic
285C2×D4C2×#4111yescharacteristic
286C2×D4C2×#4111yescharacteristic
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—44no—
309C22×C4—82no—
310C22×C4—81no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C4⋊C412yes—
321C22×C4C2×C4⋊C414yes—
322C22×C4#11712yes—
323C22×C4#12112yes—
324C22×C4#12112yes—
325C22×C4C22⋊Q812yes—
326C22×C4C22⋊Q812yes—
327C22×C4C22×D411yescharacteristic
328M16—44no—
329M16—44no—
330M16—84no—
331C2×C8—44no—
332C2×C8—84no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—84no—
348C4⋊C4—82no—
349C4⋊C4—164no—
350C4⋊C4#9914yes—
351C22⋊C4—22no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—28no—
355C22⋊C4—24no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—81no—
365C22⋊C4—81no—
366C22⋊C4C2×C4⋊C414yes—
367C42—24no—
368C42—24no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—44no—
373C42—44no—
374C42—42no—
375C42—42no—
376C42—84no—
377C42—82no—
378C42—82no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23#38012yes—
392C23C22×C22⋊C411yescharacteristic
393C23C22×C4⋊C411yescharacteristic
394C23C2×#11711yescharacteristic
395C23C2×#12111yescharacteristic
396C23C2×C22⋊Q811yescharacteristic
397Q8—42no—
398Q8—42no—
399Q8—48no—
400Q8—48no—
401D4—42no—
402D4—42no—
403D4—42no—
404D4—42no—
405D4—41no—
406D4—41no—
407D4—41no—
408D4—41no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—82no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—162no—
457C2×C4#38012yes—
458C2×C4#38012yes—
459C2×C4#38012yes—
460C2×C4C2×#9914yes—
461C2×C4C2×#9912yes—
462C2×C4#41012yes—
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×C22⋊Q811yescharacteristic
465C8—84no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—42no—
478C22—42no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—81no—
486C22—81no—
487C22C2×#38011yescharacteristic
488C22C22×#9911yescharacteristic
489C22C2×#41011yescharacteristic
490C4—24no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—22no—
495C4—44no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—44no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—164no—
505C4#179914yes—
506C2—21no—
507C2—21no—
508C2—41no—
509C2—41no—
510C2—41no—
511C2—41no—
512C2—42no—
513C2C2×#179911yescharacteristic
514C2#3334911yescharacteristic
515C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
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
5. #12796C2semidirect
6. #12794C2semidirect
10. C22×C4×Q8C4semidirect
20. #2322C4semidirect
23. #2310C4semidirect
24. C4×#123C4semidirect
27. #2296C4semidirect
28. #2274C4semidirect
138. C22×Q8C4⋊C4semidirect
164. #123C4⋊C4semidirect
185. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342662
C2×#21232
#206402
#206342
#127962
#127942
#125732