Finite Group Databasebeta

← #224833

Subgroups — #224833

Order 512.

941 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic, complemented
3#13933C211yesmaximal, characteristic, complemented
4#9716C212yesmaximal, complemented
5#9712C211yesmaximal, characteristic
6#9682C212yesmaximal, complemented
7#9680C211yesmaximal, characteristic
8#8577C212yesmaximal, complemented
9#8572C211yesmaximal, characteristic, complemented
10#8216C212yesmaximal, complemented
11C4×#328C211yesmaximal, characteristic, complemented
12#2856C2211yescharacteristic
13#2850C2212yes—
14#2597—22no—
15#2571C411yescharacteristic, complemented
16#2548C411yescharacteristic, complemented
17#2513—21no—
18#2513C412yescomplemented
19#2504—21no—
20#2447C412yescomplemented
21#2444—22no—
22#2366C411yescharacteristic, complemented
23#2361—21no—
24#2357—21no—
25#2355C411yescharacteristic, complemented
26C4×C22⋊Q8C2211yescharacteristic
27C4×#121C2212yes—
28C2×#389—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1939—21no—
31#1917—22no—
32#1797C2211yescharacteristic
33#1797C2212yes—
34#1770—21no—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#328C2211yescharacteristic
38C2×#328C2211yescharacteristic
39C2×#328C2212yes—
40C2×#327C2212yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44#1442—22no—
45C4⋊#95—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C2×#127C2×C411yescharacteristic
54C2×#124C2×C412yes—
55C2×C22⋊Q8C2311yescharacteristic
56C2×#121—21no—
57C2×#121—21no—
58C2×#121C2312yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2312yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8C2311yescharacteristic
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2311yescharacteristic
75C2×#97—22no—
76C2×C4×C8—21no—
77#400—22no—
78#397—22no—
79#397—22no—
80#397—22no—
81#397—22no—
82#394—22no—
83#391C2×C412yes—
84#391C2×C411yescharacteristic
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#389—21no—
94#389—44no—
95#389C2×C412yes—
96#389C2×C411yescharacteristic
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#387—21no—
103#387—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#385—21no—
109#385—22no—
110#384C2×C411yescharacteristic
111#382—21no—
112#382—21no—
113#382C2×C412yes—
114#381—22no—
115#380C2311yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95D412yes—
134C2×#95C2311yescharacteristic
135C2×#95C2312yes—
136C2×#95C2311yescharacteristic
137#336—21no—
138#336—22no—
139#336—21no—
140#336—22no—
141#328—22no—
142#328—22no—
143#328—42no—
144#327—24no—
145#327—24no—
146#327—42no—
147#327—42no—
148C22×Q8C22×C411yescharacteristic
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4C22×C412yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165#127—42no—
166#124—82no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171#121—22no—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—42no—
176#121—42no—
177#121—81no—
178#121—81no—
179C4×Q8—42no—
180C4×D4—44no—
181C4×D4—82no—
182C4×D4—82no—
183C4×D4—82no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—82no—
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C412yes—
231C2×C42C22×C412yes—
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C4⋊C8—42no—
238C4⋊C8—82no—
239C22⋊C8—21no—
240C22⋊C8—22no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—22no—
244C22⋊C8#4112yes—
245#97—82no—
246C4×C8—42no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—21no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—42no—
264#95—44no—
265#95—82no—
266#95—81no—
267#95—82no—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272#95#4112yes—
273C24—22no—
274C24C22×D411yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—81no—
294C2×D4—81no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—44no—
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—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4AGL(1,ℤ/8)12yes—
349C22×C4#13712yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C42−1+411yescharacteristic
356C2×C8—22no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—81no—
382C4⋊C4—82no—
383C4⋊C4—81no—
384C4⋊C4—84no—
385C4⋊C4—88no—
386C22⋊C4—44no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C42—22no—
402C42—22no—
403C42—42no—
404C42—42no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—82no—
412C42—84no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23C22×C22⋊C411yescharacteristic
425C23#54811yescharacteristic
426C23#55011yescharacteristic
427C23C2×AGL(1,ℤ/8)11yescharacteristic
428C23C2×#13711yescharacteristic
429C23#57611yescharacteristic
430C23#57711yescharacteristic
431Q8—81no—
432Q8—81no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—84no—
440D4—162no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—82no—
495C2×C4—82no—
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—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—164no—
514C2×C4#41012yes—
515C2×C4#41712yes—
516C2×C4#41712yes—
517C2×C4#42112yes—
518C2×C4C4×#4111yescharacteristic
519C2×C4#52011yescharacteristic
520C2×C4#54811yescharacteristic
521C2×C4#55011yescharacteristic
522C8—42no—
523C8—81no—
524C8—82no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22#234111yescharacteristic
545C22C2×#41011yescharacteristic
546C22#288311yescharacteristic
547C22C2×#41711yescharacteristic
548C22#289711yescharacteristic
549C22C2×#42111yescharacteristic
550C22#290111yescharacteristic
551C22#337311yescharacteristic
552C22#337511yescharacteristic
553C22#339911yescharacteristic
554C22#340011yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—22no—
560C4—42no—
561C4—42no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—82no—
567C4—82no—
568C4—81no—
569C4—82no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—162no—
574C4#151214yes—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—162no—
580C2C2×#151211yescharacteristic
581C2#1759911yescharacteristic
582C2#1760711yescharacteristic
583C2#1763011yescharacteristic
584C2#2253511yescharacteristic
585C2#2254611yescharacteristic
586C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
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. #22629C2semidirect
3. #13933C2semidirect
4. #9716C2semidirect
6. #9682C2semidirect
8. #8577C2semidirect
9. #8572C2semidirect
10. #8216C2semidirect
11. C4×#328C2semidirect
15. #2571C4semidirect
16. #2548C4semidirect
18. #2513C4semidirect
20. #2447C4semidirect
22. #2366C4semidirect
25. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#365872
#226292
#139332
#97162
#97122
#96822
#96802
#85772
#85722
#82162
C4×#3282