Finite Group Databasebeta

← #215922

Subgroups — #215922

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2#21902C211yesmaximal, characteristic
3#13947C211yesmaximal, characteristic
4#9913C212yesmaximal, complemented
5#9910C212yesmaximal, complemented
6#9886C212yesmaximal, complemented
7#9886C212yesmaximal, complemented
8#8713C211yesmaximal, characteristic
9#8713C211yesmaximal, characteristic
10#8712C211yesmaximal, characteristic
11#8712C211yesmaximal, characteristic
12#2851C2212yes—
13#2851C2212yes—
14#2646—21no—
15#2646—21no—
16#2569—22no—
17#2569—22no—
18#2442C412yescomplemented
19#2442C412yescomplemented
20#2442C412yescomplemented
21#2442C412yescomplemented
22#2315—21no—
23#2315—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1923—22no—
28#1923—22no—
29#1797C2212yes—
30#1779C2211yescharacteristic
31#1779C2211yescharacteristic
32C4×C22⋊C8C2212yes—
33#1443—21no—
34#1440—21no—
35#1437—21no—
36#1434—21no—
37#1434C2211yescharacteristic
38#1434C2211yescharacteristic
39#1434C2211yescharacteristic
40C22⋊#95C2211yescharacteristic
41C22⋊#95C2211yescharacteristic
42C4×#95C2211yescharacteristic
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C24×C4C2311yescharacteristic
52C2×#125C2×C412yes—
53C2×#125C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—22no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8C2312yes—
73C2×C22⋊C8C2312yes—
74C2×C4×C8—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#400—22no—
79#398—22no—
80#398—22no—
81#398—22no—
82#398—22no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#385—22no—
92#385—22no—
93#385—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#382C2×C412yes—
100#382C2×C412yes—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380C2×C412yes—
110#380C2×C412yes—
111#380C2312yes—
112#380C2312yes—
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4D412yes—
116C4×C22⋊C4D412yes—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95D411yescharacteristic
129C2×#95D411yescharacteristic
130C2×#95C2311yescharacteristic
131C2×#95C2311yescharacteristic
132C2×#95C2311yescharacteristic
133C2×#95C2311yescharacteristic
134#336—22no—
135#336—22no—
136C25C2×D411yescharacteristic
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—81no—
148C23×C4—81no—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161#125—82no—
162#125—82no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—81no—
176C2×C4⋊C4—81no—
177C2×C4⋊C4—81no—
178C2×C4⋊C4—81no—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—81no—
200C2×C22⋊C4—81no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—81no—
208C2×C42—81no—
209C2×C42—81no—
210C2×C42C22×C412yes—
211C2×C42C22×C412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C4⋊C8—82no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—81no—
220C22⋊C8—81no—
221C22⋊C8—81no—
222C22⋊C8—81no—
223C4×C8—82no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—81no—
245#95—81no—
246#95—81no—
247#95—81no—
248#95—81no—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C411yescharacteristic
263C24#13512yes—
264C24#13512yes—
265C24C22×D411yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
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—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C42−1+412yes—
355C22×C42−1+412yes—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—44no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—84no—
389C22⋊C4—84no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—84no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23#41612yes—
428C23#41612yes—
429C23C22×C22⋊C411yescharacteristic
430C23#53611yescharacteristic
431C23#53611yescharacteristic
432C23C2×#13511yescharacteristic
433C23C2×#13511yescharacteristic
434C23#57511yescharacteristic
435C23#57511yescharacteristic
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—162no—
515C2×C4—162no—
516C2×C4—162no—
517C2×C4—162no—
518C2×C4#51912yes—
519C2×C4#53612yes—
520C8—82no—
521C8—82no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22#231011yescharacteristic
553C22#288311yescharacteristic
554C22#288311yescharacteristic
555C22C2×#41611yescharacteristic
556C22#289611yescharacteristic
557C22#290111yescharacteristic
558C22#290111yescharacteristic
559C22#323211yescharacteristic
560C22#323211yescharacteristic
561C22#323211yescharacteristic
562C22#323211yescharacteristic
563C4—22no—
564C4—22no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—161no—
578C4—161no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—21no—
583C2—42no—
584C2—42no—
585C2#1761511yescharacteristic
586C2#1761511yescharacteristic
587C2#1762311yescharacteristic
588C2#1762511yescharacteristic
589C2#2182711yescharacteristic
590C2#2183611yescharacteristic
591C2#2184611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9913C2semidirect
5. #9910C2semidirect
6. #9886C2semidirect
7. #9886C2semidirect
18. #2442C4semidirect
19. #2442C4semidirect
20. #2442C4semidirect
21. #2442C4semidirect

Maximal subgroups

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

subgroupindex
#350202
#219022
#139472
#99132
#99102
#98862
#87132
#87122