Finite Group Databasebeta

← #215928

Subgroups — #215928

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2#21906C211yesmaximal, characteristic
3#13948C211yesmaximal, characteristic
4#9912C211yesmaximal, characteristic, complemented
5#9911C211yesmaximal, characteristic, complemented
6#9909C211yesmaximal, characteristic, complemented
7#9908C211yesmaximal, characteristic, complemented
8#9887C212yesmaximal, complemented
9#9887C212yesmaximal, complemented
10#8715C212yesmaximal
11#8715C212yesmaximal
12#2857C2211yescharacteristic
13#2857C2211yescharacteristic
14#2851C2212yes—
15#2626—21no—
16#2626—21no—
17#2608—22no—
18#2607—22no—
19#2441C412yescomplemented
20#2441C412yescomplemented
21#2439C412yescomplemented
22#2439C412yescomplemented
23#2329—21no—
24#2329—21no—
25C2×#388—21no—
26#2280C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1924—22no—
29#1924—22no—
30#1797C2211yescharacteristic
31#1797C2211yescharacteristic
32#1779C2212yes—
33#1749C2212yes—
34#1442—21no—
35#1439—21no—
36#1436—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
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×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×#117C2×C412yes—
58C2×#117C2×C412yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8C2312yes—
73C2×C22⋊C8C2312yes—
74C2×#97—22no—
75#399—22no—
76#399—22no—
77#396—22no—
78#396—22no—
79#396—22no—
80#396—22no—
81#393—22no—
82#393—22no—
83#391—21no—
84#391—21no—
85#391—22no—
86#391—22no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389—22no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#388—44no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#382C2×C412yes—
107#381—22no—
108#380—21no—
109#380—21no—
110#380C2×C412yes—
111#380C2×C412yes—
112#380C2312yes—
113#380C2312yes—
114#379—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
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—21no—
135#336—21no—
136#336—22no—
137C25C2×D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—81no—
149C23×C4—81no—
150C23×C4C22×C412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—22no—
160C4⋊Q8—81no—
161C4⋊Q8—81no—
162#125—82no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167#117—82no—
168#117—82no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—81no—
206C2×C22⋊C4—81no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—81no—
216C2×C42—81no—
217C2×C42—81no—
218C2×C42C22×C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C4⋊C8—81no—
225C4⋊C8—81no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—82no—
231C22⋊C8—82no—
232#97—82no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—81no—
256#95—81no—
257#95—81no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24C2×C22⋊C412yes—
273C24C22×D411yescharacteristic
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—81no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×Q8—81no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
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—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
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—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4#13712yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C42−1+411yescharacteristic
356C22×C42−1+411yescharacteristic
357C22×C42−1+411yescharacteristic
358C22×C42−1+411yescharacteristic
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—81no—
374C4⋊C4—81no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—84no—
396C22⋊C4—84no—
397C42—42no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—41no—
402C42—41no—
403C42—82no—
404C42—82no—
405C42—84no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23C22×C22⋊C411yescharacteristic
434C23#53611yescharacteristic
435C23#53611yescharacteristic
436C23C2×AGL(1,ℤ/8)11yescharacteristic
437C23C2×#13711yescharacteristic
438C23#57511yescharacteristic
439C23#57511yescharacteristic
440Q8—162no—
441Q8—162no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
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—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—82no—
496C2×C4—82no—
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—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4#41712yes—
524C2×C4#41712yes—
525C2×C4#51911yescharacteristic
526C2×C4#51911yescharacteristic
527C2×C4#53611yescharacteristic
528C2×C4#53611yescharacteristic
529C8—82no—
530C8—82no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22#231011yescharacteristic
555C22#288311yescharacteristic
556C22#288311yescharacteristic
557C22C2×#41711yescharacteristic
558C22#289611yescharacteristic
559C22#290111yescharacteristic
560C22#290111yescharacteristic
561C22#323511yescharacteristic
562C22#323511yescharacteristic
563C22#323711yescharacteristic
564C22#323711yescharacteristic
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—161no—
583C4—161no—
584C2—21no—
585C2—21no—
586C2—21no—
587C2—21no—
588C2—82no—
589C2#1762211yescharacteristic
590C2#1762511yescharacteristic
591C2#1762911yescharacteristic
592C2#1763011yescharacteristic
593C2#2182711yescharacteristic
594C2#2183911yescharacteristic
595C2#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. #9912C2semidirect
5. #9911C2semidirect
6. #9909C2semidirect
7. #9908C2semidirect
8. #9887C2semidirect
9. #9887C2semidirect
19. #2441C4semidirect
20. #2441C4semidirect
21. #2439C4semidirect
22. #2439C4semidirect

Maximal subgroups

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

subgroupindex
#350332
#219062
#139482
#99122
#99112
#99092
#99082
#98872
#87152