Finite Group Databasebeta

← #224882

Subgroups — #224882

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36579C211yesmaximal, characteristic
2Q8×C22⋊C8C211yesmaximal, characteristic, complemented
3#13931C211yesmaximal, characteristic, complemented
4#9716C212yesmaximal
5#9711C211yesmaximal, characteristic, complemented
6#9681C212yesmaximal
7#9679C211yesmaximal, characteristic, complemented
8#8574C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic
10#8221C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic
12C2×#556C411yescharacteristic, complemented
13C22×C4×Q8C2211yescharacteristic
14C2×C8×Q8—21no—
15#2857C2211yescharacteristic
16#2850C2212yes—
17#2592C412yescomplemented
18#2581C412yescomplemented
19#2568C411yescharacteristic, complemented
20#2538C411yescharacteristic, complemented
21#2481—22no—
22#2469—22no—
23#2421—21no—
24#2409—21no—
25#2362—21no—
26#2355C411yescharacteristic, complemented
27#2352—21no—
28C4×C22⋊Q8C2212yes—
29C4×#121C2211yescharacteristic
30C2×#391—21no—
31#1984—21no—
32#1984—22no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35C4×C22⋊C8C2211yescharacteristic
36C4×C22⋊C8C2212yes—
37C2×#328C2212yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#327C2211yescharacteristic
42C2×#327C2211yescharacteristic
43C2×#327C2212yes—
44#1445—22no—
45C4⋊#95—21no—
46C4⋊#95C2211yescharacteristic
47#1434C2212yes—
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#556—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—22no—
57C2×#124C2×C412yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8C2312yes—
61C2×#121C2311yescharacteristic
62C2×C4×Q8—44no—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D412yes—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C8×Q8—44no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8C2311yescharacteristic
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2311yescharacteristic
85C2×C4×C8—21no—
86C2×C4×C8—22no—
87#400—22no—
88#400—22no—
89#397—22no—
90#396—22no—
91#395—22no—
92#393—22no—
93#391—21no—
94#391—21no—
95#391—44no—
96#391C2×C412yes—
97#391C2×C412yes—
98#391C2×C412yes—
99#390C2×C411yescharacteristic
100#390C2×C412yes—
101#389—21no—
102#389—21no—
103#389C2×C411yescharacteristic
104#389C2×C412yes—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—21no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#386—22no—
115#386—22no—
116#386—21no—
117#385—22no—
118#384C2×C411yescharacteristic
119#382—21no—
120#382C2×C412yes—
121#380—21no—
122#380—21no—
123#380—21no—
124#380C2311yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C412yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—21no—
142#336—22no—
143#336—21no—
144#328—24no—
145#328—24no—
146#328—42no—
147#328—42no—
148#327—22no—
149#327—22no—
150#327—42no—
151C22×Q8—24no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8C22⋊C411yescharacteristic
156C22×Q8C22⋊C412yes—
157C22×Q8C22⋊C411yescharacteristic
158C22×Q8C22×C412yes—
159C22×D4—21no—
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C4⋊Q8—42no—
174#126—48no—
175#124—44no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—44no—
181C22⋊Q8—44no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—42no—
189C4×Q8—44no—
190C4×Q8—42no—
191C4×Q8—44no—
192C4×Q8—44no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196#117—44no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C412yes—
241C2×C42C22×C412yes—
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D411yescharacteristic
247C4⋊C8—44no—
248C4⋊C8—42no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—21no—
254C22⋊C8#4112yes—
255C4×C8—44no—
256C4×C8—42no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—21no—
275#95—42no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—82no—
280#95—81no—
281#95#4112yes—
282C24—21no—
283C24C22×D411yescharacteristic
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—42no—
294C2×Q8—48no—
295C2×Q8—48no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8#9914yes—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
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—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4#13512yes—
363C22×C4#13512yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C42−1+411yescharacteristic
370C2×C8—22no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—42no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—81no—
401C4⋊C4—84no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—81no—
414C22⋊C4—82no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—24no—
419C42—41no—
420C42—42no—
421C42—41no—
422C42—42no—
423C42—44no—
424C42—42no—
425C42—82no—
426C42—84no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23C22×C22⋊C411yescharacteristic
439C23#54811yescharacteristic
440C23#55011yescharacteristic
441C23C2×#13511yescharacteristic
442C23C2×#13511yescharacteristic
443C23#57711yescharacteristic
444C23#57811yescharacteristic
445Q8—28no—
446Q8—48no—
447Q8—44no—
448Q8—82no—
449Q8—82no—
450D4—81no—
451D4—81no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—81no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—82no—
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—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4C2×#9914yes—
529C2×C4C2×#9912yes—
530C2×C4#41612yes—
531C2×C4#41612yes—
532C2×C4#42112yes—
533C2×C4C4×#4111yescharacteristic
534C2×C4#52011yescharacteristic
535C2×C4#54811yescharacteristic
536C2×C4#55011yescharacteristic
537C8—42no—
538C8—81no—
539C8—82no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22#234111yescharacteristic
560C22C22×#9911yescharacteristic
561C22#288411yescharacteristic
562C22C2×#41611yescharacteristic
563C22#289811yescharacteristic
564C22C2×#42111yescharacteristic
565C22#290111yescharacteristic
566C22#336011yescharacteristic
567C22#336411yescharacteristic
568C22#339311yescharacteristic
569C22#339411yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—81no—
585C4—82no—
586C4—81no—
587C4—81no—
588C4—162no—
589C4#151614yes—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—161no—
595C2C2×#151611yescharacteristic
596C2#1760111yescharacteristic
597C2#1760611yescharacteristic
598C2#1761611yescharacteristic
599C2#2253411yescharacteristic
600C2#2254111yescharacteristic
601C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×C22⋊C8C2semidirect
3. #13931C2semidirect
5. #9711C2semidirect
7. #9679C2semidirect
8. #8574C2semidirect
10. #8221C2semidirect
12. C2×#556C4semidirect
17. #2592C4semidirect
18. #2581C4semidirect
19. #2568C4semidirect
20. #2538C4semidirect
26. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#365792
Q8×C22⋊C82
#139312
#97162
#97112
#96812
#96792
#85742
#85712
#82212
C4×#3272