Finite Group Databasebeta

← #224881

Subgroups — #224881

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2#22625C211yesmaximal, characteristic, complemented
3#13936C211yesmaximal, characteristic, complemented
4#9716C212yesmaximal
5#9711C211yesmaximal, characteristic, complemented
6#9681C212yesmaximal
7#9679C211yesmaximal, characteristic, complemented
8#8576C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic
10#8221C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic
12#2857C2211yescharacteristic
13#2850C2212yes—
14#2596—22no—
15#2571C411yescharacteristic, complemented
16#2547C411yescharacteristic, complemented
17#2514C412yescomplemented
18#2510—22no—
19#2494—21no—
20#2487—21no—
21#2448C412yescomplemented
22#2365C411yescharacteristic, complemented
23#2354C411yescharacteristic, complemented
24#2352—21no—
25#2347—21no—
26C4×C22⋊Q8C2212yes—
27C4×#121C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—21no—
30#1916—22no—
31#1797C2211yescharacteristic
32#1797C2212yes—
33C8×C4⋊C4—21no—
34C4×C22⋊C8C2211yescharacteristic
35C4×C22⋊C8C2212yes—
36C2×#328C2212yes—
37C2×#328C2212yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#327C2211yescharacteristic
41C2×#327C2211yescharacteristic
42C2×#327C2212yes—
43#1443—22no—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2211yescharacteristic
47#1434C2212yes—
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#124C2×C412yes—
55C2×C22⋊Q8C2312yes—
56C2×#121C2311yescharacteristic
57C2×C4×Q8C2×C412yes—
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×#117—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8C2311yescharacteristic
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2311yescharacteristic
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74#401—22no—
75#400—22no—
76#400—22no—
77#399—22no—
78#397—22no—
79#397—22no—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2×C412yes—
83#390C2×C412yes—
84#390C2×C411yescharacteristic
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389C2×C412yes—
90#389C2×C411yescharacteristic
91#388—22no—
92#388—22no—
93#388—22no—
94#388—21no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#386—22no—
101#386—22no—
102#386—21no—
103#385—22no—
104#384C2×C411yescharacteristic
105#383—22no—
106#382—21no—
107#382C2×C412yes—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380C2311yescharacteristic
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—22no—
125C4×C22⋊C4—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95D412yes—
133C2×#95C2311yescharacteristic
134C2×#95C2312yes—
135C2×#95C2311yescharacteristic
136#336—21no—
137#336—22no—
138#336—21no—
139#336—22no—
140#328—24no—
141#328—24no—
142#328—42no—
143#328—42no—
144#327—22no—
145#327—22no—
146#327—42no—
147C22×Q8C22×C412yes—
148C22×D4—21no—
149C22×D4C22×C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C4⋊Q8—42no—
163#124—82no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168#121—21no—
169#121—21no—
170#121—21no—
171#121—21no—
172C4×Q8—82no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176#117—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—82no—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C412yes—
221C2×C42C22×C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C4⋊C8—42no—
228C4⋊C8—82no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—21no—
234C22⋊C8#4112yes—
235C4×C8—42no—
236C4×C8—82no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—21no—
255#95—42no—
256#95—44no—
257#95—82no—
258#95—81no—
259#95—82no—
260#95—82no—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95#4112yes—
266C24—21no—
267C24C22×D411yescharacteristic
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
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—42no—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4#13512yes—
336C22×C4#13512yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C42−1+411yescharacteristic
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—42no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—82no—
371C4⋊C4—81no—
372C4⋊C4—82no—
373C4⋊C4—84no—
374C22⋊C4—44no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—42no—
383C22⋊C4—81no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C42—22no—
388C42—22no—
389C42—42no—
390C42—42no—
391C42—41no—
392C42—41no—
393C42—42no—
394C42—42no—
395C42—44no—
396C42—42no—
397C42—82no—
398C42—84no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23C22×C22⋊C411yescharacteristic
411C23#54811yescharacteristic
412C23#55011yescharacteristic
413C23C2×#13511yescharacteristic
414C23C2×#13511yescharacteristic
415C23#57711yescharacteristic
416C23#57811yescharacteristic
417Q8—82no—
418Q8—82no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—81no—
423D4—81no—
424D4—81no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—84no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4#41012yes—
499C2×C4#41612yes—
500C2×C4#41612yes—
501C2×C4#42112yes—
502C2×C4C4×#4111yescharacteristic
503C2×C4#52011yescharacteristic
504C2×C4#54811yescharacteristic
505C2×C4#55011yescharacteristic
506C8—42no—
507C8—81no—
508C8—82no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22#234111yescharacteristic
529C22C2×#41011yescharacteristic
530C22#288311yescharacteristic
531C22C2×#41611yescharacteristic
532C22#289811yescharacteristic
533C22C2×#42111yescharacteristic
534C22#290111yescharacteristic
535C22#336011yescharacteristic
536C22#336411yescharacteristic
537C22#339311yescharacteristic
538C22#339411yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—42no—
545C4—42no—
546C4—41no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—81no—
554C4—82no—
555C4—81no—
556C4—81no—
557C4—162no—
558C4#152114yes—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—161no—
564C2C2×#152111yescharacteristic
565C2#1760011yescharacteristic
566C2#1760511yescharacteristic
567C2#1761711yescharacteristic
568C2#2253511yescharacteristic
569C2#2254111yescharacteristic
570C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22625C2semidirect
3. #13936C2semidirect
5. #9711C2semidirect
7. #9679C2semidirect
8. #8576C2semidirect
10. #8221C2semidirect
15. #2571C4semidirect
16. #2547C4semidirect
17. #2514C4semidirect
21. #2448C4semidirect
22. #2365C4semidirect
23. #2354C4semidirect

Maximal subgroups

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

subgroupindex
#365862
#226252
#139362
#97162
#97112
#96812
#96792
#85762
#85712
#82212
C4×#3272