Finite Group Databasebeta

← #188971

Subgroups — #188971

Order 512.

844 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#60265C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic, complemented
3#21962C211yesmaximal, characteristic, complemented
4#17594C211yesmaximal, characteristic
5#17563C211yesmaximal, characteristic
6#8629C212yesmaximal, complemented
7#8628C212yesmaximal, complemented
8#8627C212yesmaximal, complemented
9#8626C212yesmaximal, complemented
10#8109C211yesmaximal, characteristic, complemented
11#8106C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2994—21no—
14#2994—21no—
15#2992—22no—
16#2990—22no—
17#2989—21no—
18#2988—21no—
19#2962C411yescharacteristic, complemented
20#2962C411yescharacteristic, complemented
21#2935C411yescharacteristic, complemented
22#2935C411yescharacteristic, complemented
23#2929C2212yes—
24#2929C2212yes—
25#2925C412yescomplemented
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29#2906C412yescomplemented
30C22×C4⋊C8C2211yescharacteristic
31#2285C2211yescharacteristic
32#2283C2211yescharacteristic
33#1950—21no—
34#1950—21no—
35#1950—22no—
36#1949—22no—
37#1948—21no—
38#1947—21no—
39#1810C2212yes—
40#1810C2212yes—
41#1808C2212yes—
42#1807C2212yes—
43#1521—22no—
44#1513C2212yes—
45#1513C2212yes—
46#1513C2212yes—
47#1513C2212yes—
48#1512—22no—
49#1506—21no—
50#1505—21no—
51#1503—21no—
52#1502—21no—
53#1305C2211yescharacteristic
54#1305C2211yescharacteristic
55#1305C2211yescharacteristic
56#1305C2211yescharacteristic
57#1305C2212yes—
58#1305C2212yes—
59C2×#131C2×C412yes—
60C2×#131C2×C412yes—
61C23×C8C2312yes—
62#533C2×C411yescharacteristic
63#533D411yescharacteristic
64#533Q811yescharacteristic
65#532C2×C411yescharacteristic
66#532D411yescharacteristic
67#532Q811yescharacteristic
68#529—21no—
69#529—21no—
70#529—21no—
71#529—21no—
72#529D412yes—
73#529Q812yes—
74#525—21no—
75#525—21no—
76#525C2312yes—
77C4×#41—21no—
78C4×#41—21no—
79C4×#41C2×C412yes—
80C2×#117—21no—
81C22×C42C2311yescharacteristic
82#446—21no—
83#446—21no—
84#446—22no—
85#446—22no—
86C8×Q8—21no—
87C8×Q8—21no—
88#436—21no—
89#436—21no—
90#435—22no—
91#435—22no—
92#435—22no—
93#435—22no—
94#435—22no—
95#435—22no—
96C8×D4—22no—
97C8×D4—22no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—21no—
101C8×D4—21no—
102#433—22no—
103#433—22no—
104#432—21no—
105#432—21no—
106#432—22no—
107#432—21no—
108#432—21no—
109#432—21no—
110#432—21no—
111#431—21no—
112#431—21no—
113#431—22no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118C2×#108—22no—
119C2×#108—22no—
120C2×#107—22no—
121C2×#106—22no—
122#424—22no—
123#424—22no—
124#424C2×C412yes—
125#424C2×C412yes—
126#423—22no—
127#423—21no—
128#423—21no—
129#423C2×C411yescharacteristic
130#423C2×C411yescharacteristic
131C2×C4⋊C8—22no—
132C2×C4⋊C8C2×C412yes—
133C2×C4⋊C8C2×C411yescharacteristic
134C2×C4⋊C8C2×C411yescharacteristic
135C2×C4⋊C8D411yescharacteristic
136C2×C4⋊C8D411yescharacteristic
137C2×C4⋊C8D412yes—
138C2×C4⋊C8D411yescharacteristic
139C2×C4⋊C8D411yescharacteristic
140C2×C4⋊C8D412yes—
141#408C2×C412yes—
142#408C2×C412yes—
143#407C2311yescharacteristic
144#407C2312yes—
145#407C2311yescharacteristic
146#407C2311yescharacteristic
147#407C2311yescharacteristic
148#405C2×C412yes—
149#405C2×C412yes—
150C4×M16—21no—
151C4×M16—21no—
152C4×M16C2×C411yescharacteristic
153C4×M16C2×C411yescharacteristic
154#384—21no—
155#383—21no—
156#382—21no—
157#382—21no—
158#381—21no—
159#380—21no—
160#380C2311yescharacteristic
161#380C2312yes—
162#380C2311yescharacteristic
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C22⋊C4—21no—
167#376—21no—
168#341—22no—
169#341—22no—
170#340—22no—
171#340—22no—
172#328—22no—
173#328—22no—
174#328—22no—
175#328—22no—
176#327—22no—
177#327—22no—
178#327—22no—
179#327—22no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41C22×C412yes—
183C23×C4#4111yescharacteristic
184C23×C4#4111yescharacteristic
185C23×C4C2411yescharacteristic
186#131—82no—
187#131—82no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—21no—
191C2×M16—22no—
192C2×M16—21no—
193C2×M16C22×C411yescharacteristic
194C2×M16C22×C411yescharacteristic
195C2×M16C22×C411yescharacteristic
196C2×M16C22×C411yescharacteristic
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—82no—
200C22×C8C22×C412yes—
201C22×C8C22×C412yes—
202C22×C8C2×D412yes—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C4⋊Q8C4⋊C411yescharacteristic
207C4⋊Q8C4⋊C411yescharacteristic
208C4⋊Q8C4⋊C411yescharacteristic
209#127C4⋊C411yescharacteristic
210#126—22no—
211#126—22no—
212#125—21no—
213#125—21no—
214#125C4⋊C412yes—
215#124—21no—
216#124—21no—
217#124C4⋊C412yes—
218#123—22no—
219#123—22no—
220#123—22no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8C2×D412yes—
226C22⋊Q8C2×Q812yes—
227#121—21no—
228#121—21no—
229#121—21no—
230#121—21no—
231#121C2×D412yes—
232#121C2×Q812yes—
233#120—22no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8C4⋊C412yes—
239C4×Q8C4⋊C412yes—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4C4⋊C412yes—
251C4×D4C4⋊C412yes—
252#117—21no—
253#117—21no—
254#117—22no—
255#117—22no—
256#117—22no—
257#117—82no—
258#117C22×C412yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C42—22no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×Q811yescharacteristic
279C2×C42C2×Q811yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42#4111yescharacteristic
282#108—82no—
283#108—82no—
284#107—82no—
285#106—82no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—41no—
295C4⋊C8—41no—
296C4⋊C8—41no—
297C4⋊C8—41no—
298C4⋊C8C22⋊C412yes—
299C4⋊C8C22⋊C412yes—
300C4⋊C8C22⋊C412yes—
301C4⋊C8C22⋊C412yes—
302C4⋊C8C22⋊C411yescharacteristic
303C4⋊C8C22⋊C411yescharacteristic
304C4⋊C8C22⋊C412yes—
305C4⋊C8C22⋊C412yes—
306C4⋊C8C22⋊C411yescharacteristic
307C4⋊C8C22⋊C411yescharacteristic
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8—21no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—22no—
319C22⋊C8—21no—
320C22⋊C8—22no—
321C22⋊C8—21no—
322#97—22no—
323#97—22no—
324#97—21no—
325#97—21no—
326#97—41no—
327#97—41no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—41no—
333C4×C8—41no—
334#95—21no—
335#95—21no—
336#95—22no—
337#95—41no—
338#95—41no—
339#95—41no—
340#95—41no—
341C24C2×#4111yescharacteristic
342#41—44no—
343#41—42no—
344#41—42no—
345#41—81no—
346#41—81no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8C2×C4⋊C412yes—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4C2×C4⋊C412yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4#11711yescharacteristic
368C22×C4#11711yescharacteristic
369C22×C4#11711yescharacteristic
370C22×C4#11711yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4C22⋊Q811yescharacteristic
376C22×C4C22⋊Q811yescharacteristic
377C22×C4C22⋊Q811yescharacteristic
378C22×C4C22⋊Q811yescharacteristic
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4111yescharacteristic
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—42no—
391M16—41no—
392M16—41no—
393M16—42no—
394M16—41no—
395M16—82no—
396M16—82no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—82no—
411C2×C8—82no—
412C2×C8C2×C22⋊C412yes—
413C2×C8C2×C22⋊C412yes—
414C2×C8C2×C22⋊C412yes—
415C2×C8C2×C22⋊C412yes—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4C2×C4⋊C412yes—
435C4⋊C4C2×C4⋊C412yes—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—81no—
446C22⋊C4—81no—
447C22⋊C4—82no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—82no—
452C42—82no—
453C42C2×C22⋊C411yescharacteristic
454C42C2×C22⋊C411yescharacteristic
455C42C2×C22⋊C411yescharacteristic
456C42C2×C22⋊C411yescharacteristic
457C42C2×C4⋊C411yescharacteristic
458C42C2×C4⋊C411yescharacteristic
459C42C2×C4⋊C411yescharacteristic
460C42C2×C4⋊C411yescharacteristic
461C42#12111yescharacteristic
462C42#12111yescharacteristic
463C42#12111yescharacteristic
464C42#12111yescharacteristic
465C42C22⋊Q811yescharacteristic
466C42C22⋊Q811yescharacteristic
467C42C22⋊Q811yescharacteristic
468C42C22⋊Q811yescharacteristic
469C23—21no—
470C23—21no—
471C23—22no—
472C23—41no—
473C23—41no—
474C23C2×#11711yescharacteristic
475C23C2×#12111yescharacteristic
476C23C2×C22⋊Q811yescharacteristic
477Q8—42no—
478Q8—42no—
479Q8—81no—
480D4—42no—
481D4—44no—
482D4—44no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—81no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4#38011yescharacteristic
527C2×C4#38011yescharacteristic
528C2×C4#38011yescharacteristic
529C2×C4#38011yescharacteristic
530C2×C4#38011yescharacteristic
531C2×C4#38011yescharacteristic
532C2×C4#38011yescharacteristic
533C2×C4#38011yescharacteristic
534C2×C4C22×C22⋊C411yescharacteristic
535C2×C4C22×C4⋊C411yescharacteristic
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×C22⋊Q811yescharacteristic
538C8—22no—
539C8—22no—
540C8—22no—
541C8—22no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—81no—
547C8—81no—
548C8—82no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22C2×#38011yescharacteristic
563C22#306811yescharacteristic
564C22#306911yescharacteristic
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—161no—
579C4—162no—
580C4—161no—
581C4#181412yes—
582C4#288311yescharacteristic
583C4#288411yescharacteristic
584C2—21no—
585C2—21no—
586C2—41no—
587C2—41no—
588C2—82no—
589C2#2063411yescharacteristic
590C2C2×#181411yescharacteristic
591C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#3444C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21969C2semidirect
3. #21962C2semidirect
6. #8629C2semidirect
7. #8628C2semidirect
8. #8627C2semidirect
9. #8626C2semidirect
10. #8109C2semidirect
11. #8106C2semidirect
19. #2962C4semidirect
20. #2962C4semidirect
21. #2935C4semidirect
22. #2935C4semidirect
25. #2925C4semidirect
29. #2906C4semidirect

Maximal subgroups

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

subgroupindex
#602652
#219692
#219622
#175942
#175632
#86292
#86282
#86272
#86262
#81092
#81062