Finite Group Databasebeta

← #496415

Subgroups — #496415

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#20723C211yesmaximal, characteristic
4#13297C214yesmaximal, complemented
5#13293C212yesmaximal, complemented
6#13292C212yesmaximal, complemented
7#12853C214yesmaximal
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10#2920—21no—
11#2582—21no—
12#2581—21no—
13#2580—21no—
14#2579C412yescomplemented
15#2579C2212yes—
16#2538—21no—
17#2487C411yescharacteristic, complemented
18#2487C2211yescharacteristic
19#2409C411yescharacteristic, complemented
20#2409C2211yescharacteristic
21#2396—21no—
22#2396—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25#2297C2212yes—
26#2296C412yescomplemented
27#2285C411yescharacteristic, complemented
28#2283—21no—
29#1822—21no—
30#1820—21no—
31#1801—21no—
32#1800C2212yes—
33#1799—21no—
34#1799C2212yes—
35#1785C2212yes—
36#1782C2212yes—
37C4×#101—21no—
38C4×#100C2212yes—
39C4×#99—21no—
40C4×#99C2212yes—
41#1390C2218yes—
42#1390C2218yes—
43C22×#41C2311yescharacteristic
44C23×Q8C2×C411yescharacteristic
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C4×#41—41no—
48C4×#41—41no—
49C4×#41—41no—
50C4×#41—41no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—41no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117C2×C412yes—
68C2×#117C2×C411yescharacteristic
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C412yes—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77#432—41no—
78#432—41no—
79#431—41no—
80#431—41no—
81#423—22no—
82C2×#101—22no—
83C2×#100C2314yes—
84C2×#99—22no—
85C2×#99C2314yes—
86#407—22no—
87C4×M16—22no—
88#400—21no—
89#400—21no—
90#399—21no—
91#399—21no—
92#398—22no—
93#398—22no—
94#396—22no—
95#396—22no—
96#395—21no—
97#395—21no—
98#393—21no—
99#393—21no—
100#391—22no—
101#391—21no—
102#391—21no—
103#390—22no—
104#389—22no—
105#389—22no—
106#389—21no—
107#389—21no—
108#387—22no—
109#387—22no—
110#386C2×C412yes—
111#386C2×C412yes—
112#386C2312yes—
113#386C2312yes—
114#385—22no—
115#385—22no—
116#385C2×C412yes—
117#385C2×C412yes—
118#385C2×C412yes—
119#385C2×C412yes—
120#384—21no—
121#383—41no—
122#382—21no—
123#382—41no—
124#381—22no—
125#380—21no—
126#380—21no—
127#380—22no—
128#380C2×C412yes—
129#380C2×C411yescharacteristic
130#380C2×C411yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—41no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139#376—41no—
140#343—22no—
141#343—22no—
142#342—22no—
143#342—22no—
144C2×#41—24no—
145C2×#41—41no—
146C2×#41—41no—
147C2×#41—42no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8—44no—
153C22×Q8C2×D411yescharacteristic
154C22×D4C22×C411yescharacteristic
155C22×D4C2×D411yescharacteristic
156C22×D4C2411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C412yes—
163C23×C4C22×C412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C2×M16—24no—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197#117—41no—
198#117—41no—
199#117—41no—
200#117—41no—
201#117—41no—
202#117—41no—
203#117—44no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4C22⋊C411yescharacteristic
233C2×C22⋊C4C22⋊C411yescharacteristic
234C2×C22⋊C4C22×C412yes—
235C2×C22⋊C4C22×C412yes—
236C2×C22⋊C4C22×C411yescharacteristic
237C2×C22⋊C4C22×C411yescharacteristic
238C2×C22⋊C4C2×D411yescharacteristic
239C2×C22⋊C4C2×D411yescharacteristic
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C4⋊C8—42no—
269C4⋊C8—42no—
270#101—44no—
271#100—28no—
272#99—28no—
273#99—44no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276#97—42no—
277C4×C8—42no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—42no—
293C24C2×C22⋊C411yescharacteristic
294C24C23×C411yescharacteristic
295C24C22×D411yescharacteristic
296#41—48no—
297#41—48no—
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—48no—
307C2×Q8—44no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D42−1+414yes—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—82no—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370M16—44no—
371M16—44no—
372C2×C8—44no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—44no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—84no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—44no—
427C42—44no—
428C42—42no—
429C42—42no—
430C42—84no—
431C42#9911yescharacteristic
432C42#9911yescharacteristic
433C42#9911yescharacteristic
434C42#9911yescharacteristic
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23C22×C22⋊C411yescharacteristic
449C23#51912yes—
450C23#53612yes—
451Q8—44no—
452Q8—44no—
453Q8—48no—
454Q8—44no—
455D4—44no—
456D4—44no—
457D4—44no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4C2×#9911yescharacteristic
518C2×C4C2×#9911yescharacteristic
519C2×C4C2×#9911yescharacteristic
520C2×C4C2×#9911yescharacteristic
521C2×C4C2×#9911yescharacteristic
522C2×C4C2×#9911yescharacteristic
523C2×C4#41012yes—
524C2×C4#53612yes—
525C8—84no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—81no—
542C22—81no—
543C22#231011yescharacteristic
544C22C22×#9911yescharacteristic
545C22C2×#41011yescharacteristic
546C4—21no—
547C4—22no—
548C4—21no—
549C4—21no—
550C4—22no—
551C4—21no—
552C4—21no—
553C4—44no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—164no—
565C4#212012yes—
566C2—21no—
567C2—21no—
568C2—42no—
569C2—42no—
570C2—81no—
571C2#2182511yescharacteristic
572C2C2×#212011yescharacteristic
573C2#3334711yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13297C2semidirect
5. #13293C2semidirect
6. #13292C2semidirect
9. C22×C4×Q8C4semidirect
14. #2579C4semidirect
17. #2487C4semidirect
19. #2409C4semidirect
26. #2296C4semidirect
27. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#349522
#207282
#207232
#132972
#132932
#132922
#128532