Finite Group Databasebeta

← #482121

Subgroups — #482121

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#11901C214yesmaximal, complemented
5#11899C214yesmaximal, complemented
6#11718C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic, complemented
8C2×C4×#41C2211yescharacteristic
9C2×#431—21no—
10C2×#410—21no—
11#2857—21no—
12#2857—21no—
13#2628C412yescomplemented
14#2628C2212yes—
15#2568C412yescomplemented
16#2567—21no—
17#2563—21no—
18#2542C2212yes—
19#2487C412yescomplemented
20#2487C2212yes—
21#2409—21no—
22#2409—21no—
23#2317—21no—
24#2311—21no—
25#2285—21no—
26#2285C411yescharacteristic, complemented
27#1799—21no—
28#1799—21no—
29#1799C2212yes—
30#1799C2212yes—
31#1740C2212yes—
32#1461—21no—
33#1460C2212yes—
34#1459—21no—
35#1459C2212yes—
36#1458—21no—
37#1458C2212yes—
38#1352C22116yes—
39C22×#41D411yescharacteristic
40C23×Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C4×#41—41no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117—22no—
65C2×#117C2×C411yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C412yes—
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73#431—22no—
74#431—22no—
75#431—41no—
76#431—41no—
77C2×C4⋊C8—41no—
78#410—44no—
79C2×#99—22no—
80C2×#99—22no—
81C2×C22⋊C8—22no—
82C2×C4×C8—41no—
83#400C2×C414yes—
84#400C2×C414yes—
85#399C2×C414yes—
86#399C2314yes—
87#395—22no—
88#395—22no—
89#393—22no—
90#393—22no—
91#391—22no—
92#391—22no—
93#391—22no—
94#390—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#386—22no—
100#386—22no—
101#385—22no—
102#385—22no—
103#384—22no—
104#384—21no—
105#384—41no—
106#382—22no—
107#382—41no—
108#382C2×C412yes—
109#380—22no—
110#380—22no—
111#380—21no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380C2×C412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—41no—
119C4×C4⋊C4—41no—
120C4×C4⋊C4C2×C412yes—
121#342—22no—
122#342C2314yes—
123#324—22no—
124#324—22no—
125#323—22no—
126#323C2314yes—
127C2×#41—41no—
128C2×#41—41no—
129C2×#41—42no—
130C2×#41SD1614yes—
131C2×#41Q1614yes—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—84no—
137C22×Q8C22⋊C411yescharacteristic
138C22×D4C22⋊C411yescharacteristic
139C22×D4C2×D411yescharacteristic
140C22×D4C2×D411yescharacteristic
141C23×C4—22no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C22×C8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C22⋊Q8—84no—
163C22⋊Q8—84no—
164C4×Q8—41no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—41no—
180#117—41no—
181#117—41no—
182#117—41no—
183#117—41no—
184#117—41no—
185#117—41no—
186#117—44no—
187#117—44no—
188#117—44no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4C22⋊C411yescharacteristic
216C2×C22⋊C4C22⋊C411yescharacteristic
217C2×C22⋊C4C22×C411yescharacteristic
218C2×C22⋊C4C2×D411yescharacteristic
219C2×C22⋊C4C2×D411yescharacteristic
220C2×C22⋊C4C2411yescharacteristic
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C4⋊C8—42no—
246C4⋊C8—42no—
247#99—44no—
248#99—44no—
249C22⋊C8—24no—
250C22⋊C8—42no—
251C4×C8—42no—
252C4×C8—42no—
253#95—22no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—42no—
259#95—42no—
260#95C22×C414yes—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C411yescharacteristic
263C24C22×D411yescharacteristic
264#41—44no—
265#41—44no—
266#41—44no—
267#41—44no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—84no—
276C2×Q8—88no—
277C2×Q8—88no—
278C2×Q8#10314yes—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4#10314yes—
286C2×D4C2×SD1612yes—
287C2×D4C2×SD1612yes—
288C2×D4C2×Q1612yes—
289C2×D4C2×Q1612yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—84no—
322C22×C4#10314yes—
323C22×C4#10314yes—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×SD1612yes—
335C22×C4C2×Q1612yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C2×C8—44no—
339C2×C8—84no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—84no—
370C22⋊C4—84no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—44no—
390C42—84no—
391C42#9911yescharacteristic
392C42#9911yescharacteristic
393C42#9911yescharacteristic
394C42#9911yescharacteristic
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23C2×#10312yes—
408C23C2×#10312yes—
409C23C22×C22⋊C411yescharacteristic
410C23C22×SD1611yescharacteristic
411C23C22×Q1611yescharacteristic
412Q8—44no—
413Q8—84no—
414Q8—88no—
415Q8—84no—
416D4—44no—
417D4—42no—
418D4—42no—
419D4—42no—
420D4—42no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—88no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4C2×#9911yescharacteristic
472C2×C4C2×#9911yescharacteristic
473C2×C4C2×#9911yescharacteristic
474C2×C4C2×#9911yescharacteristic
475C2×C4C2×#9911yescharacteristic
476C2×C4C2×#9911yescharacteristic
477C2×C4C2×#10312yes—
478C2×C4C2×#10312yes—
479C2×C4C2×#10312yes—
480C2×C4C2×#10312yes—
481C2×C4#42112yes—
482C8—84no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—81no—
501C22—81no—
502C22C22×#9911yescharacteristic
503C22C22×#10311yescharacteristic
504C22C2×#42111yescharacteristic
505C4—21no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—41no—
516C4—84no—
517C4—84no—
518C4—164no—
519C4#152012yes—
520C4#152012yes—
521C4#212312yes—
522C4#212812yes—
523C2—21no—
524C2—21no—
525C2—41no—
526C2—42no—
527C2—41no—
528C2—42no—
529C2C2×#152011yescharacteristic
530C2C2×#212311yescharacteristic
531C2C2×#212811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11901C2semidirect
5. #11899C2semidirect
6. #11718C2semidirect
7. C22×C4⋊Q8C4semidirect
13. #2628C4semidirect
15. #2568C4semidirect
19. #2487C4semidirect
26. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#352792
#206322
#173492
#119012
#118992
#117182