Finite Group Databasebeta

← #480524

Subgroups — #480524

Order 512.

1009 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#17361C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#11568C214yesmaximal, complemented
5#11564C214yesmaximal, complemented
6#11434C214yesmaximal, complemented
7C2×#536C412yes—
8C22×C4⋊Q8C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2920—21no—
11C2×#431—21no—
12#2857—21no—
13#2857—21no—
14#2621—21no—
15#2621—21no—
16#2547C412yes—
17#2547C2212yes—
18#2546C412yes—
19#2546C2212yes—
20#2486—21no—
21#2486—21no—
22#2486—21no—
23#2486—21no—
24#2310C2212yes—
25#2285C411yescharacteristic
26#1467C2212yes—
27#1466—21no—
28#1465—21no—
29#1465C2212yes—
30#1461C2212yes—
31#1460—21no—
32#1459—21no—
33#1459C2212yes—
34#1458—21no—
35#1458—21no—
36#1458C2212yes—
37#1458C2212yes—
38#1319C22116yes—
39C22×#41C2×C411yescharacteristic
40C23×Q8C2×C412yes—
41#536—48no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2×C414yes—
51C2×#124C2×C414yes—
52C2×#123—22no—
53C2×#123—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117C2×C412yes—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77#432—22no—
78#432—41no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431—41no—
83#431—41no—
84#431—41no—
85#423—41no—
86C2×C4⋊C8—41no—
87#407—22no—
88C2×C22⋊C8—22no—
89C4×M16—41no—
90C2×C4×C8—41no—
91#398—22no—
92#398—22no—
93#398—22no—
94#398—22no—
95#397—22no—
96#397—22no—
97#397—22no—
98#397—22no—
99#391—22no—
100#391—22no—
101#391C2×C414yes—
102#391C2314yes—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#384—21no—
108#382—21no—
109#381—22no—
110#381—22no—
111#380—22no—
112#380—22no—
113#380—21no—
114#380—21no—
115#380—22no—
116#380—22no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119#324—22no—
120#324—22no—
121#324C2314yes—
122#324C2314yes—
123#323—22no—
124#323—22no—
125#323—22no—
126#323—22no—
127C2×#41—24no—
128C2×#41—24no—
129C2×#41—41no—
130C2×#41—41no—
131C22×Q8—22no—
132C22×Q8—28no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—44no—
136C22×Q8C22×C414yes—
137C22×Q8C22×C411yescharacteristic
138C22×D4C22⋊C411yescharacteristic
139C22×D4C22⋊C411yescharacteristic
140C22×D4C22×C411yescharacteristic
141C23×C4—22no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—42no—
152C22×C8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—48no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164#124—48no—
165#123—48no—
166#123—48no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—41no—
191#117—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4C22⋊C411yescharacteristic
215C2×C22⋊C4C22⋊C411yescharacteristic
216C2×C22⋊C4C2×D411yescharacteristic
217C2×C22⋊C4C2×D411yescharacteristic
218C2×C22⋊C4C2×D411yescharacteristic
219C2×C22⋊C4C2×D411yescharacteristic
220C2×C42—21no—
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×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C414yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C22⋊C8—24no—
248C22⋊C8—24no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251#97—42no—
252C4×C8—42no—
253C4×C8—42no—
254C4×C8—42no—
255#95—22no—
256#95—22no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C411yescharacteristic
263C24C2×C22⋊C411yescharacteristic
264#41—48no—
265#41—48no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—28no—
269C2×Q8—41no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—44no—
276C2×Q8—88no—
277C2×Q8—88no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4#10014yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
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×C4#13114yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334M16—84no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—84no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—48no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—44no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—44no—
383C42#9911yescharacteristic
384C42#9911yescharacteristic
385C42#9911yescharacteristic
386C42#9911yescharacteristic
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23#40812yes—
401C23C2×#10012yes—
402C23C22×C22⋊C411yescharacteristic
403Q8—44no—
404Q8—88no—
405Q8—84no—
406Q8—84no—
407D4—44no—
408D4—44no—
409D4—44no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—88no—
456C2×C4#40812yes—
457C2×C4C2×#9911yescharacteristic
458C2×C4C2×#9911yescharacteristic
459C2×C4C2×#9911yescharacteristic
460C2×C4C2×#9911yescharacteristic
461C2×C4C2×#9911yescharacteristic
462C2×C4C2×#9911yescharacteristic
463C2×C4C2×#10012yes—
464C2×C4C2×#10012yes—
465C2×C4C2×#13112yes—
466C8—84no—
467C8—84no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—22no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—81no—
484C22—81no—
485C22C2×#40811yescharacteristic
486C22C22×#9911yescharacteristic
487C22C22×#10011yescharacteristic
488C4—21no—
489C4—21no—
490C4—22no—
491C4—22no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—84no—
500C4—84no—
501C4#147112yes—
502C4#147112yes—
503C4#212912yes—
504C4#213512yes—
505C2—21no—
506C2—21no—
507C2—42no—
508C2—42no—
509C2—81no—
510C2C2×#147111yescharacteristic
511C2C2×#212911yescharacteristic
512C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11568C2semidirect
5. #11564C2semidirect
6. #11434C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#173612
#173492
#115682
#115642
#114342