Finite Group Databasebeta

← #481477

Subgroups — #481477

Order 512.

1591 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#35691C211yesmaximal, characteristic
2#20354C211yesmaximal, characteristic, complemented
3#17350C211yesmaximal, characteristic, complemented
4#11693C212yesmaximal, complemented
5#11692C214yesmaximal, complemented
6#11691C212yesmaximal, complemented
7C2×#1343C214yesmaximal, complemented
8C2×D42—21no—
9C2×D42C411yescharacteristic, complemented
10C2×#518—21no—
11C2×#518C2211yescharacteristic
12#2972—21no—
13#2619C2211yescharacteristic
14#2617C2212yes—
15#2543C412yescomplemented
16#2541C411yescharacteristic, complemented
17#2535C411yescharacteristic, complemented
18#2428—21no—
19#2424—22no—
20#2414—21no—
21#2410—22no—
22#2408C412yescomplemented
23#2392C2211yescharacteristic
24C2×#392C2212yes—
25#2349C411yescharacteristic, complemented
26D4×C22⋊C4—21no—
27#1740—21no—
28#1740C2212yescomplemented
29C2×#342—22no—
30C2×#342C2214yescomplemented
31#1460—21no—
32#1459—22no—
33#1458C2212yescomplemented
34C2×#323C2214yescomplemented
35#1344C2214yescomplemented
36#1343C2214yes—
37#1343C2218yescomplemented
38C23×D4—22no—
39C23×D4—21no—
40C23×D4C2×C412yescomplemented
41C23×D4D411yescharacteristic
42D42—44no—
43D42—44no—
44D42—44no—
45C2×#127—21no—
46C2×#127C2×C411yescharacteristic, complemented
47C2×#121—22no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—21no—
53C2×#121C2×C412yescomplemented
54C2×#120—22no—
55C2×#120—22no—
56C2×#120—22no—
57#518—28no—
58#518—44no—
59#518—44no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—44no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic, complemented
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×#117—22no—
71C2×#117D411yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C22⋊C4—24no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C22⋊C4C2312yes—
77C8×D4—44no—
78#431—22no—
79C2×C22⋊C8—24no—
80#398C2×C411yescharacteristic, complemented
81#397C2×C412yescomplemented
82#397C2311yescharacteristic
83#396—21no—
84#394—21no—
85#394—22no—
86#394—22no—
87#394—21no—
88#394C2×C412yescomplemented
89#394C2×C411yescharacteristic, complemented
90#392—21no—
91#392—22no—
92#392—22no—
93#392C2×C412yescomplemented
94#392C2×C412yescomplemented
95#392C2×C412yescomplemented
96#392C2312yes—
97#392C2311yescharacteristic
98#390—21no—
99#390—22no—
100#390—21no—
101#389—21no—
102#388—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#385—22no—
108#382C2×C411yescharacteristic, complemented
109#380—21no—
110#380C2×C412yescomplemented
111#379—22no—
112C4×C4⋊C4C2×C411yescharacteristic, complemented
113C4×C22⋊C4—21no—
114C2×#95C2×C412yescomplemented
115#342—22no—
116#342—24no—
117#342C2314yes—
118#324—22no—
119#323—24no—
120#323C2314yes—
121C25—22no—
122C25C2×D412yes—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—44no—
139C22×D4—44no—
140C22×D4—84no—
141C22×D4—84no—
142C22×D4—84no—
143C22×D4C22⋊C412yes—
144C22×D4C22⋊C411yescharacteristic
145C22×D4C22⋊C411yescharacteristic
146C22×D4D814yes—
147C22×D4SD1614yes—
148C22×D4C22×C411yescharacteristic
149C22×D4C22×C412yes—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—44no—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C412yes—
155C23×C4C22×C412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C22×C8—44no—
159#127—42no—
160#127—44no—
161#121—44no—
162#121—44no—
163#121—44no—
164#121—48no—
165#121—42no—
166#121—42no—
167#121—84no—
168#121—84no—
169#120—48no—
170#120—44no—
171#120—44no—
172#120—48no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4—432no—
181C4×D4—44no—
182C4×D4—42no—
183C4×D4—42no—
184#117—28no—
185#117—42no—
186#117—42no—
187#117—44no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—216no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—48no—
218C2×C22⋊C4—84no—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22×C412yes—
222C2×C22⋊C4C2×D411yescharacteristic
223C2×C22⋊C4C2×D412yes—
224C2×C22⋊C4C2×D412yes—
225C2×C22⋊C4C2411yescharacteristic
226C2×C42—21no—
227C2×C42—42no—
228C2×C42C22⋊C412yes—
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C4⋊C8—42no—
232C22⋊C8—24no—
233C22⋊C8—44no—
234C4×C8—42no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95C22×C411yescharacteristic
241#95C22×C412yes—
242#95C22×C411yescharacteristic
243C24—24no—
244C24—24no—
245C24—21no—
246C24—22no—
247C24—44no—
248C24—44no—
249C24—44no—
250C24—44no—
251C24—82no—
252C24—82no—
253C24—82no—
254C24—82no—
255C24—82no—
256C24C2×C22⋊C412yes—
257C24C2×C22⋊C412yes—
258C24C2×D814yes—
259C24C2×SD1614yes—
260C24C22×D411yescharacteristic
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—216no—
264C2×D4—216no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—82no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—88no—
300C2×D4—88no—
301C2×D4—816no—
302C2×D4—816no—
303C2×D4—816no—
304C2×D4#9914yes—
305C2×D4#10218yes—
306C2×D4#10214yes—
307C2×D4#10214yes—
308C2×D4#10414yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—416no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—164no—
332C22×C4—164no—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×D812yes—
340C22×C4C2×SD1612yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C2×C8—44no—
344C2×C8—164no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—82no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—28no—
358C22⋊C4—41no—
359C22⋊C4—42no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—44no—
365C22⋊C4—42no—
366C22⋊C4—44no—
367C22⋊C4—416no—
368C22⋊C4—42no—
369C22⋊C4—44no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C42—22no—
381C42—22no—
382C42—42no—
383C42—48no—
384C42—42no—
385C23—21no—
386C23—22no—
387C23—21no—
388C23—22no—
389C23—28no—
390C23—28no—
391C23—22no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—84no—
412C23—84no—
413C23—88no—
414C23—84no—
415C23—84no—
416C23—88no—
417C23—88no—
418C23—88no—
419C23—164no—
420C23—164no—
421C23—164no—
422C23—164no—
423C23C2×#9914yes—
424C23C2×#10214yes—
425C23C2×#10214yes—
426C23C2×#10414yes—
427C23C22×C22⋊C411yescharacteristic
428C23C22×D811yescharacteristic
429C23C22×SD1611yescharacteristic
430D4—216no—
431D4—28no—
432D4—42no—
433D4—44no—
434D4—44no—
435D4—42no—
436D4—88no—
437D4—88no—
438D4—84no—
439D4—88no—
440D4—84no—
441D4—84no—
442D4—88no—
443D4—84no—
444D4—162no—
445D4—164no—
446D4#327116yes—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—48no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—88no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—162no—
483C2×C4—162no—
484C2×C4—1616no—
485C2×C4C2×#9912yes—
486C2×C4C2×#10212yes—
487C2×C4C2×#10212yes—
488C2×C4C2×#10412yes—
489C8—84no—
490C22—21no—
491C22—24no—
492C22—21no—
493C22—24no—
494C22—21no—
495C22—21no—
496C22—24no—
497C22—44no—
498C22—44no—
499C22—41no—
500C22—41no—
501C22—44no—
502C22—42no—
503C22—42no—
504C22—41no—
505C22—42no—
506C22—41no—
507C22—42no—
508C22—42no—
509C22—88no—
510C22—82no—
511C22—81no—
512C22—88no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—81no—
520C22—81no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—1616no—
525C22C2×#32718yes—
526C22C22×#9911yescharacteristic
527C22C22×#10211yescharacteristic
528C22C22×#10411yescharacteristic
529C4—22no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—41no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—82no—
538C4—81no—
539C4—88no—
540C4—81no—
541C4C2×#32714yes—
542C2—21no—
543C2—21no—
544C2—28no—
545C2—41no—
546C2—44no—
547C2—41no—
548C2—82no—
549C2—162no—
550C2C22×#32711yescharacteristic
551C2#3334811yescharacteristic
552C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35691C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20354C2semidirect
3. #17350C2semidirect
4. #11693C2semidirect
5. #11692C2semidirect
6. #11691C2semidirect
7. C2×#1343C2semidirect
9. C2×D42C4semidirect
15. #2543C4semidirect
16. #2541C4semidirect
17. #2535C4semidirect
22. #2408C4semidirect
25. #2349C4semidirect
28. #1740C22semidirect
30. C2×#342C22semidirect
33. #1458C22semidirect
34. C2×#323C22semidirect
35. #1344C22semidirect
37. #1343C22semidirect
40. C23×D4C2×C4semidirect
46. C2×#127C2×C4semidirect
53. C2×#121C2×C4semidirect
66. C2×C4×D4C2×C4semidirect
80. #398C2×C4semidirect
81. #397C2×C4semidirect
88. #394C2×C4semidirect
89. #394C2×C4semidirect
93. #392C2×C4semidirect
94. #392C2×C4semidirect
95. #392C2×C4semidirect
108. #382C2×C4semidirect
110. #380C2×C4semidirect
112. C4×C4⋊C4C2×C4semidirect
114. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#356912
#203542
#173502
#116932
#116922
#116912
C2×#13432