Finite Group Databasebeta

← #493715

Subgroups — #493715

Order 512.

1343 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2C2×#2127C212yesmaximal
3#20633C212yesmaximal, complemented
4#12800C212yesmaximal, complemented
5#12799C212yesmaximal, complemented
6#12798C212yesmaximal, complemented
7#12797C212yesmaximal, complemented
8C2×#1393C212yesmaximal, complemented
9C2×#535C2211yescharacteristic
10C2×#520C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12C22×#99C2214yes—
13#2363—22no—
14#2361—22no—
15#2358—22no—
16#2358—22no—
17#2328C411yescharacteristic, complemented
18#2328C2211yescharacteristic
19#2326C411yescharacteristic, complemented
20#2326C2211yescharacteristic
21#2317C411yescharacteristic, complemented
22#2315—22no—
23C4×#127C411yescharacteristic, complemented
24C4×#127C2211yescharacteristic
25C4×#124C411yescharacteristic, complemented
26C4×#124C2211yescharacteristic
27#2297C411yescharacteristic, complemented
28C2×#390C2211yescharacteristic
29C4×#117C411yescharacteristic, complemented
30#2127—28no—
31C2×#354C2214yes—
32C2×#353C2214yes—
33#1799—22no—
34#1799—22no—
35#1799C2214yes—
36#1799C2214yes—
37C2×#342—22no—
38#1393C2214yes—
39#1393C2214yes—
40C22×#41C2×C411yescharacteristic
41C23×D4C2311yescharacteristic
42#535—24no—
43#535D414yes—
44#535Q814yes—
45C2×#127C2×C411yescharacteristic
46C2×#127C2311yescharacteristic
47C2×#124C2×C411yescharacteristic
48C2×#124C2311yescharacteristic
49C2×#121—22no—
50C2×#120—22no—
51#520—44no—
52#518—44no—
53C2×C4×Q8C2×C411yescharacteristic, complemented
54C2×C4×Q8C2×C411yescharacteristic, complemented
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic, complemented
62C2×C4×D4C2×C411yescharacteristic, complemented
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×C4×D4C2311yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—22no—
74C22×C22⋊C4—24no—
75C22×C22⋊C4D412yes—
76C22×C22⋊C4D412yes—
77C22×C42C2×C411yescharacteristic
78C2×#99—28no—
79C2×#99—28no—
80C2×#99—24no—
81C2×#99C2318yes—
82#391C2×C411yescharacteristic, complemented
83#391C2×C411yescharacteristic, complemented
84#390—22no—
85#390—22no—
86#390—22no—
87#390C2×C411yescharacteristic
88#390C2×C411yescharacteristic
89#390C2×C411yescharacteristic, complemented
90#390C2×C411yescharacteristic, complemented
91#390C2311yescharacteristic
92#390C2311yescharacteristic
93#389—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—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—48no—
107#384C2×C411yescharacteristic, complemented
108#384C2×C411yescharacteristic, complemented
109#382C2×C411yescharacteristic, complemented
110#382C2×C411yescharacteristic, complemented
111#380—24no—
112#380—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—22no—
120#376C2×C411yescharacteristic, complemented
121#376C2×C411yescharacteristic, complemented
122C43C2×C411yescharacteristic, complemented
123C43C2×C411yescharacteristic, complemented
124#354—28no—
125#353—28no—
126#342—24no—
127#342—24no—
128C25C2×D412yes—
129C2×#41—22no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41C4⋊C414yescomplemented
133C22×Q8C22×C411yescharacteristic
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4C22×C411yescharacteristic
140C22×D4C22×C411yescharacteristic
141C22×D4C2×D412yes—
142C22×D4C2×D412yes—
143C22×D4C2×Q812yes—
144C22×D4#4114yes—
145C22×D4C2411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—44no—
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157#127—22no—
158#127—22no—
159#127C4⋊C414yescomplemented
160#127C2×D412yes—
161#127C2×Q812yes—
162#124—22no—
163#124—22no—
164#124C4⋊C414yescomplemented
165#124C2×D412yes—
166#124C2×Q812yes—
167#121—28no—
168#121—28no—
169#120—28no—
170#120—28no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4—44no—
182C4×D4—44no—
183C4×D4—84no—
184C4×D4—84no—
185#117—22no—
186#117—24no—
187#117—42no—
188#117—44no—
189#117C4⋊C414yescomplemented
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—84no—
224C2×C22⋊C4C22⋊C411yescharacteristic
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C411yescharacteristic
229C2×C22⋊C4C2×D412yes—
230C2×C22⋊C4C2×D412yes—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—42no—
239C2×C42C22⋊C412yes—
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246#99—28no—
247#99—28no—
248#99—48no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—44no—
258#95—44no—
259C24—24no—
260C24—24no—
261C24—21no—
262C24—82no—
263C24—82no—
264C24C2×C22⋊C412yes—
265C24C2×C22⋊C411yescharacteristic
266C24#12114yes—
267C24C22⋊Q814yes—
268C24C22×D412yes—
269#41—48no—
270#41—48no—
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8C2×C4⋊C412yes—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—28no—
279C2×D4—21no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—44no—
285C2×D4—42no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4C2×C4⋊C412yes—
298C2×D4C2×C4⋊C412yes—
299C2×D4#11712yes—
300C2×D4#12114yes—
301C2×D4C22⋊Q814yes—
302C2×D4C22×D411yescharacteristic
303C2×D4C22×Q811yescharacteristic
304C2×D4C2×#4112yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—81no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—88no—
336C22×C4—164no—
337C22×C4—164no—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C4⋊C412yes—
346C22×C4#11712yes—
347C22×C4C23×C411yescharacteristic
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—164no—
361C4⋊C4—164no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—28no—
366C22⋊C4—24no—
367C22⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C22⋊C4—42no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—88no—
383C22⋊C4#9914yes—
384C42—21no—
385C42—21no—
386C42—22no—
387C42—22no—
388C42—42no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—81no—
397C42—81no—
398C42C2×C4⋊C412yes—
399C42C2×C4⋊C412yes—
400C23—22no—
401C23—22no—
402C23—28no—
403C23—28no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—84no—
418C23—84no—
419C23—84no—
420C23—84no—
421C23—164no—
422C23#38014yes—
423C23#38012yes—
424C23C2×#9912yes—
425C23#41012yes—
426C23C22×C22⋊C411yescharacteristic
427C23C2×#12112yes—
428C23C2×C22⋊Q812yes—
429Q8—42no—
430Q8—42no—
431D4—42no—
432D4—42no—
433D4—44no—
434D4—44no—
435D4—41no—
436D4—41no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—48no—
441D4—84no—
442D4—84no—
443D4—88no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—48no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—88no—
484C2×C4—88no—
485C2×C4—88no—
486C2×C4—162no—
487C2×C4—162no—
488C2×C4—162no—
489C2×C4—162no—
490C2×C4—162no—
491C2×C4—1616no—
492C2×C4#38012yes—
493C2×C4C2×#9912yes—
494C2×C4C2×#9912yes—
495C2×C4C22×C4⋊C411yescharacteristic
496C2×C4C2×#11711yescharacteristic
497C22—21no—
498C22—22no—
499C22—21no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—24no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—44no—
509C22—44no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—88no—
516C22—82no—
517C22—82no—
518C22—88no—
519C22—82no—
520C22—81no—
521C22—81no—
522C22#179914yes—
523C22C2×#38011yescharacteristic
524C22C22×#9911yescharacteristic
525C22C2×#41011yescharacteristic
526C4—22no—
527C4—22no—
528C4—22no—
529C4—21no—
530C4—21no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—41no—
536C4—42no—
537C4—41no—
538C4—42no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—168no—
543C2—21no—
544C2—21no—
545C2—24no—
546C2—42no—
547C2—42no—
548C2—44no—
549C2—82no—
550C2C2×#179911yescharacteristic
551C2#3335911yescharacteristic
552C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20633C2semidirect
4. #12800C2semidirect
5. #12799C2semidirect
6. #12798C2semidirect
7. #12797C2semidirect
8. C2×#1393C2semidirect
10. C2×#520C4semidirect
17. #2328C4semidirect
19. #2326C4semidirect
21. #2317C4semidirect
23. C4×#127C4semidirect
25. C4×#124C4semidirect
27. #2297C4semidirect
29. C4×#117C4semidirect
53. C2×C4×Q8C2×C4semidirect
54. C2×C4×Q8C2×C4semidirect
61. C2×C4×D4C2×C4semidirect
62. C2×C4×D4C2×C4semidirect
82. #391C2×C4semidirect
83. #391C2×C4semidirect
89. #390C2×C4semidirect
90. #390C2×C4semidirect
107. #384C2×C4semidirect
108. #384C2×C4semidirect
109. #382C2×C4semidirect
110. #382C2×C4semidirect
120. #376C2×C4semidirect
121. #376C2×C4semidirect
122. C43C2×C4semidirect
123. C43C2×C4semidirect
132. C2×#41C4⋊C4semidirect
159. #127C4⋊C4semidirect
164. #124C4⋊C4semidirect
189. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342752
C2×#21272
#206332
#128002
#127992
#127982
#127972
C2×#13932