Finite Group Databasebeta

← #480820

Subgroups — #480820

Order 512.

819 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2#17365C211yesmaximal, characteristic
3C2×#1467C211yesmaximal, characteristic
4#11573C212yesmaximal, complemented
5#11568C212yesmaximal, complemented
6#11551C214yesmaximal
7#11536C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11C2×#432—21no—
12#2855—21no—
13#2855—21no—
14#2634—21no—
15#2634—21no—
16#2581—21no—
17#2581—21no—
18#2568C411yescharacteristic
19#2568C2211yescharacteristic
20#2546C411yescharacteristic
21#2546C2211yescharacteristic
22#2503C2212yes—
23#2484—21no—
24#2484—21no—
25#2434C412yes—
26#2327C412yes—
27#2327C2212yes—
28#2283C411yescharacteristic
29#1467—21no—
30#1467—21no—
31#1467—21no—
32#1467C2212yes—
33#1467C2212yes—
34#1467C2212yes—
35#1463—21no—
36#1463C2212yes—
37#1461—21no—
38#1461C2212yes—
39C2×#324—21no—
40C2×#324C2212yes—
41#1324C2218yes—
42#1319C2218yes—
43C23×Q8C2×C411yescharacteristic
44#533—41no—
45#533—41no—
46#533—41no—
47#533—41no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C4×Q8—21no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×#117—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76#432—22no—
77#432—22no—
78#432—22no—
79#432—41no—
80#432—41no—
81#432—41no—
82#431—22no—
83#431—41no—
84#423—41no—
85C2×C4⋊C8—41no—
86#407—22no—
87C2×C22⋊C8—22no—
88C4×M16—41no—
89C2×#97—41no—
90#400—21no—
91#400—21no—
92#400—21no—
93#400—21no—
94#400—22no—
95#400—22no—
96#398—21no—
97#398—21no—
98#398—21no—
99#398—21no—
100#395—22no—
101#395—22no—
102#391—21no—
103#391—21no—
104#391—21no—
105#391—21no—
106#391C2×C412yes—
107#391C2×C412yes—
108#391C2×C412yes—
109#391C2312yes—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#387—22no—
121#387—22no—
122#385—22no—
123#385—22no—
124#383—21no—
125#382—21no—
126#382C2×C412yes—
127#382C2312yes—
128#381—21no—
129#381—22no—
130#381—22no—
131#380—21no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C22⋊C4C2×C412yes—
135#376C2×C412yes—
136#324—22no—
137#324—22no—
138#324—22no—
139#324—22no—
140#324—22no—
141#324—22no—
142#324C2314yes—
143#324C2314yes—
144C22×Q8—21no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8C22⋊C411yescharacteristic
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22×C412yes—
152C22×Q8C22×C411yescharacteristic
153C22×Q8C22×C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—42no—
162C22×C8—42no—
163C4⋊Q8—44no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—44no—
169#125—42no—
170#125—42no—
171C22⋊Q8—41no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C22⋊Q8—41no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197#117—42no—
198#117—42no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2411yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22×C412yes—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C4⋊C8—42no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C22⋊C8—24no—
263C22⋊C8—24no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266#97—42no—
267#97—42no—
268#97—42no—
269C4×C8—42no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287C24C2×C22⋊C411yescharacteristic
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—44no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—41no—
307C2×Q8—41no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4#9911yescharacteristic
335C22×C4#9911yescharacteristic
336C22×C4#9911yescharacteristic
337C22×C4#9911yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352M16—84no—
353C2×C8—44no—
354C2×C8—44no—
355C2×C8—84no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—24no—
410C42—42no—
411C42—44no—
412C23—21no—
413C23—21no—
414C23—22no—
415C23—41no—
416C23C2×#9911yescharacteristic
417C23C2×#9911yescharacteristic
418C23C22×C22⋊C411yescharacteristic
419Q8—44no—
420Q8—44no—
421Q8—44no—
422Q8—44no—
423Q8—84no—
424Q8—82no—
425Q8—82no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4C2×#9911yescharacteristic
484C2×C4C2×#9911yescharacteristic
485C2×C4C2×#9911yescharacteristic
486C2×C4C2×#9911yescharacteristic
487C2×C4#41312yes—
488C2×C4#56812yes—
489C8—84no—
490C8—84no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—42no—
500C22#212912yes—
501C22#287911yescharacteristic
502C22C22×#9911yescharacteristic
503C22C2×#41311yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—41no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4#213312yes—
524C2—21no—
525C2—21no—
526C2—22no—
527C2—41no—
528C2#1740011yescharacteristic
529C2C2×#212911yescharacteristic
530C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11573C2semidirect
5. #11568C2semidirect
7. #11536C2semidirect

Maximal subgroups

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

subgroupindex
#352482
#173652
C2×#14672
#115732
#115682
#115512
#115362