Finite Group Databasebeta

← #480109

Subgroups — #480109

Order 512.

1086 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34070C211yesmaximal, characteristic
2#17349C212yesmaximal, complemented
3#11482C212yesmaximal, complemented
4#11481C212yesmaximal, complemented
5#11480C212yesmaximal, complemented
6#11479C212yesmaximal, complemented
7#11447C212yesmaximal, complemented
8#11431C212yesmaximal, complemented
9C2×#532C411yescharacteristic
10C2×C4×#41C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#431—22no—
13#2328C411yescharacteristic
14#2328C2211yescharacteristic
15#2327C411yescharacteristic
16#2327C2211yescharacteristic
17#2326C411yescharacteristic
18#2326C411yescharacteristic
19#2326C2211yescharacteristic
20#2326C2211yescharacteristic
21#2325—22no—
22#2325—22no—
23#2297C411yescharacteristic
24#2297C2211yescharacteristic
25#2290—22no—
26#2290—22no—
27#2285C2211yescharacteristic
28C2×#376C411yescharacteristic
29#1461C2214yes—
30#1460—22no—
31#1459—22no—
32#1459C2214yes—
33#1458—22no—
34#1458C2214yes—
35#1325C2214yes—
36#1324C2214yes—
37#1319C2214yes—
38#1318C2214yes—
39C22×#41C2×C411yescharacteristic
40C22×#41C2×C411yescharacteristic
41#532—24no—
42#532C818yescomplemented
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C411yescharacteristic
47C2×#121—22no—
48C4×#41—42no—
49C4×#41—42no—
50C4×#41—42no—
51C4×#41—42no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×C4×D4C2311yescharacteristic
65C2×#117—21no—
66C2×#117C2×C411yescharacteristic
67C2×#117D412yes—
68C2×#117C2311yescharacteristic
69C22×C42—21no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42D412yes—
74#431—24no—
75#431—24no—
76#431—42no—
77#431—42no—
78C2×C4⋊C8—42no—
79C2×C22⋊C8—24no—
80C2×C4×C8—42no—
81#391C2×C411yescharacteristic
82#391C2×C411yescharacteristic
83#391C2×C411yescharacteristic
84#391C2311yescharacteristic
85#390C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#390C2×C411yescharacteristic
88#390C2311yescharacteristic
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#384C2×C411yescharacteristic
108#384C2311yescharacteristic
109#382C2×C411yescharacteristic
110#382C2311yescharacteristic
111#380—22no—
112C4×C4⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115#376—21no—
116#376—21no—
117#376C2×C411yescharacteristic
118#376C2×C411yescharacteristic
119#376C2×C411yescharacteristic
120#376C2×C411yescharacteristic
121C2×#95—21no—
122C2×#95—21no—
123#324—24no—
124#324C2318yes—
125#323—24no—
126#323—24no—
127C2×#41—22no—
128C2×#41—41no—
129C2×#41—41no—
130C2×#41—42no—
131C2×#41—41no—
132C2×#41—41no—
133C2×#41—42no—
134C2×#41C2×C814yes—
135C2×#41C2×C814yescomplemented
136C2×#41M1614yes—
137C22×Q8C22×C411yescharacteristic
138C22×Q8C22×C411yescharacteristic
139C22×D4—21no—
140C22×D4—22no—
141C22×D4C22⋊C412yes—
142C22×D4C22×C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D412yes—
155C22×C8—44no—
156C4⋊Q8—22no—
157C4⋊Q8C2×C814yescomplemented
158#127—22no—
159#127C2×C814yes—
160#124—22no—
161#124—22no—
162#124C2×C814yes—
163#124C2×C814yescomplemented
164#121—216no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×D4—216no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—44no—
179C4×D4—44no—
180#117—42no—
181#117—42no—
182#117—41no—
183#117—41no—
184#117—42no—
185#117—42no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2411yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C2×D412yes—
210C2×C22⋊C4C2×D412yes—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C412yes—
228C2×C42C22⋊C411yescharacteristic
229C2×C42C2×C814yescomplemented
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C4⋊C8—44no—
239C4⋊C8—44no—
240C22⋊C8—28no—
241C22⋊C8—44no—
242C4×C8—44no—
243C4×C8—44no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258C24—21no—
259C24C2×C22⋊C412yes—
260C24C2×C22⋊C411yescharacteristic
261#41—44no—
262#41—44no—
263#41—44no—
264#41—44no—
265#41—44no—
266#41—44no—
267#41—44no—
268#41—44no—
269C2×Q8—21no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8C22×C812yes—
275C2×Q8C22×C812yes—
276C2×Q8C2×M1612yes—
277C2×D4—24no—
278C2×D4—28no—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4C22⋊C818yes—
291C2×D4C22×C812yes—
292C2×D4C22×C812yes—
293C2×D4C2×M1612yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4C22⋊C818yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C22×C812yes—
340C22×C4C2×M1612yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D412yes—
343C2×C8—48no—
344C2×C8—88no—
345C4⋊C4—28no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C22⋊C4—28no—
357C22⋊C4—28no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C42—21no—
369C42—21no—
370C42—24no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—21no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—21no—
381C42—41no—
382C42—41no—
383C42—42no—
384C42—42no—
385C42—41no—
386C42—41no—
387C42—42no—
388C42#9912yes—
389C42#9912yes—
390C42C22×C812yes—
391C42C22×C812yes—
392C23—21no—
393C23—22no—
394C23—21no—
395C23—22no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23C2×C22⋊C814yes—
406C23C2×C22⋊C812yes—
407C23C22×C22⋊C411yescharacteristic
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412D4—48no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—48no—
418D4—88no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4C2×C22⋊C814yes—
471C2×C4C2×C22⋊C812yes—
472C2×C4C2×#9912yes—
473C2×C4C2×#9912yes—
474C2×C4C2×#9911yescharacteristic
475C2×C4C2×#9911yescharacteristic
476C2×C4#41312yes—
477C2×C4C23×C811yescharacteristic
478C2×C4C22×M1611yescharacteristic
479C8—88no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—24no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—44no—
488C22—42no—
489C22—41no—
490C22—42no—
491C22—41no—
492C22—42no—
493C22—82no—
494C22—82no—
495C22—81no—
496C22—81no—
497C22C22×C22⋊C811yescharacteristic
498C22C22×#9911yescharacteristic
499C22C2×#41311yescharacteristic
500C4—22no—
501C4—22no—
502C4—21no—
503C4—21no—
504C4—24no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—41no—
509C4—42no—
510C4—41no—
511C4—42no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4#145912yes—
517C4#145912yes—
518C2—21no—
519C2—21no—
520C2—42no—
521C2—44no—
522C2—82no—
523C2C2×#145911yescharacteristic
524C2#3335811yescharacteristic
525C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C23×C8
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC2×#532C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
2. #17349C2semidirect
3. #11482C2semidirect
4. #11481C2semidirect
5. #11480C2semidirect
6. #11479C2semidirect
7. #11447C2semidirect
8. #11431C2semidirect
42. #532C8semidirect
135. C2×#41C2×C8semidirect
157. C4⋊Q8C2×C8semidirect
163. #124C2×C8semidirect
229. C2×C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
#340702
#173492
#114822
#114812
#114802
#114792
#114472
#114312