Finite Group Databasebeta

← #189249

Subgroups — #189249

Order 512.

880 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60280C211yesmaximal, characteristic
2#21900C211yesmaximal, characteristic
3#21856C211yesmaximal, characteristic
4#21362C211yesmaximal, characteristic, complemented
5#17817C211yesmaximal, characteristic, complemented
6#8669C212yesmaximal, complemented
7#8668C212yesmaximal, complemented
8#8667C212yesmaximal, complemented
9#8666C212yesmaximal, complemented
10#8095C211yesmaximal, characteristic, complemented
11#8092C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2994—21no—
14#2994—21no—
15#2994C412yescomplemented
16#2994C412yescomplemented
17#2986C411yescharacteristic, complemented
18#2986C411yescharacteristic, complemented
19#2979C411yescharacteristic, complemented
20#2979C411yescharacteristic, complemented
21#2977—22no—
22#2965—21no—
23#2965C2212yes—
24C8×#41—21no—
25C8×#41C2212yes—
26#2922—22no—
27#2920C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31#2285C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1994—21no—
34#1986—21no—
35#1980—21no—
36#1962—21no—
37#1948—21no—
38#1947—21no—
39#1897—22no—
40#1871—22no—
41#1854—22no—
42#1806C2212yes—
43#1806C2212yes—
44#1806C2212yes—
45#1806C2212yes—
46C4×#107C2212yes—
47C4×#106C2212yes—
48#1574—21no—
49#1574—21no—
50#1573—22no—
51#1567C2212yes—
52#1566C2212yes—
53#1303C2211yescharacteristic
54#1303C2211yescharacteristic
55#1303C2211yescharacteristic
56#1303C2211yescharacteristic
57#1303C2212yes—
58#1303C2212yes—
59C22×M16—21no—
60#533C2×C411yescharacteristic
61#533D411yescharacteristic
62#533Q811yescharacteristic
63#532C2×C411yescharacteristic
64#532D411yescharacteristic
65#532Q811yescharacteristic
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529C2×C412yes—
71#529Q812yes—
72#525—21no—
73#525—21no—
74#525D412yes—
75C4×#41—21no—
76C4×#41—21no—
77C4×#41C2312yes—
78C2×#117—21no—
79C2×#117—21no—
80C22×C42C2311yescharacteristic
81#446—21no—
82#446—21no—
83#446C2×C412yes—
84#446C2×C412yes—
85C8×Q8—21no—
86C8×Q8—21no—
87C8×Q8C2×C412yes—
88C8×Q8C2×C412yes—
89#436—21no—
90#436—21no—
91#436C2×C412yes—
92#436C2×C412yes—
93#435—22no—
94#435—22no—
95#435—22no—
96#435—22no—
97C8×D4—22no—
98C8×D4—22no—
99C8×D4—21no—
100C8×D4—21no—
101C8×D4—22no—
102C8×D4—22no—
103C8×D4C2×C412yes—
104C8×D4C2×C412yes—
105#433—22no—
106#433—22no—
107#433—22no—
108#433—22no—
109#432—21no—
110#432—21no—
111#431—21no—
112#431—21no—
113#431—22no—
114#431—22no—
115#431—21no—
116#431—21no—
117#431C2×C411yescharacteristic
118#431C2×C411yescharacteristic
119#431C2×C411yescharacteristic
120#431C2×C411yescharacteristic
121#431C2×C412yes—
122#431D412yes—
123#431D411yescharacteristic
124#431D411yescharacteristic
125#431C2311yescharacteristic
126#431C2311yescharacteristic
127C2×#107—22no—
128C2×#106—22no—
129#424—22no—
130#424—22no—
131#423—21no—
132#423—21no—
133#423—21no—
134#423C2311yescharacteristic
135#423C2312yes—
136#423C2311yescharacteristic
137C2×C4⋊C8—21no—
138C2×C4⋊C8—21no—
139C2×C4⋊C8C2×C411yescharacteristic
140C2×C4⋊C8C2×C411yescharacteristic
141C2×C4⋊C8D411yescharacteristic
142C2×C4⋊C8D411yescharacteristic
143C2×#104—22no—
144C2×#104—22no—
145C2×#104—22no—
146C2×#104—22no—
147#408—22no—
148#408—22no—
149#407—21no—
150C2×C22⋊C8—21no—
151C2×C22⋊C8D412yes—
152C4×M16—22no—
153C4×M16—21no—
154C2×C4×C8—21no—
155C2×C4×C8—22no—
156C2×C4×C8C2312yes—
157#384—21no—
158#382—21no—
159#380—21no—
160#380—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4C2311yescharacteristic
164C4×C4⋊C4C2311yescharacteristic
165C4×C4⋊C4C2312yes—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168#376—21no—
169C43—21no—
170#340—22no—
171#340—22no—
172#339—22no—
173#339—22no—
174#339—22no—
175#339—22no—
176#335—22no—
177#335—22no—
178#334—22no—
179#334—22no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4#4111yescharacteristic
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—41no—
192C2×M16—81no—
193C2×M16—81no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—41no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C4⋊Q8C4⋊C411yescharacteristic
202C4⋊Q8C4⋊C411yescharacteristic
203C4⋊Q8C4⋊C411yescharacteristic
204#127C4⋊C411yescharacteristic
205#126—22no—
206#126—22no—
207#125—21no—
208#125—21no—
209#125C4⋊C412yes—
210#124—21no—
211#124—21no—
212#124C4⋊C412yes—
213#123—22no—
214#123—22no—
215#123—22no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8C4⋊C412yes—
221C22⋊Q8C4⋊C412yes—
222#121—21no—
223#121—21no—
224#121—21no—
225#121—21no—
226#121C4⋊C412yes—
227#121C4⋊C412yes—
228#120—22no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8C22×C412yes—
234C4×Q8C2×Q812yes—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4C22×C412yes—
246C4×D4C2×Q812yes—
247#117—21no—
248#117—21no—
249#117—22no—
250#117—22no—
251#117—22no—
252#117—42no—
253#117—42no—
254#117—81no—
255#117—81no—
256#117C2×D412yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—21no—
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×Q811yescharacteristic
277C2×C42C2×Q811yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42C2411yescharacteristic
282#107—44no—
283#106—44no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—22no—
292C4⋊C8—21no—
293C4⋊C8—22no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—41no—
297C4⋊C8—41no—
298C4⋊C8C22⋊C412yes—
299C4⋊C8C22⋊C412yes—
300C4⋊C8C22⋊C411yescharacteristic
301C4⋊C8C22⋊C411yescharacteristic
302C4⋊C8C22⋊C411yescharacteristic
303C4⋊C8C22⋊C411yescharacteristic
304C4⋊C8C22×C411yescharacteristic
305C4⋊C8C22×C411yescharacteristic
306C4⋊C8C22×C411yescharacteristic
307C4⋊C8C22×C411yescharacteristic
308#104—82no—
309#104—82no—
310#104—82no—
311#104—82no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—41no—
321C22⋊C8—41no—
322C22⋊C8C22⋊C412yes—
323C22⋊C8C22⋊C412yes—
324C22⋊C8C22⋊C412yes—
325C22⋊C8C22⋊C412yes—
326#97—22no—
327#97—22no—
328#97—41no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—41no—
336C4×C8C22×C412yes—
337C4×C8C22×C412yes—
338C4×C8C2×D412yes—
339C4×C8C2×D412yes—
340#95—21no—
341#95—21no—
342#95—22no—
343#95—41no—
344#95—41no—
345C24#12111yescharacteristic
346#41—22no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—44no—
351#41—81no—
352#41—81no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8C2×C4⋊C412yes—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4C2×C4⋊C412yes—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C4⋊C411yescharacteristic
378C22×C4C2×C4⋊C411yescharacteristic
379C22×C4#11711yescharacteristic
380C22×C4#11711yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4#12111yescharacteristic
386C22×C4C22⋊Q811yescharacteristic
387C22×C4C22⋊Q811yescharacteristic
388C22×C4C22⋊Q811yescharacteristic
389C22×C4C22⋊Q811yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393M16—41no—
394M16—42no—
395M16—41no—
396M16—42no—
397M16—41no—
398M16—41no—
399M16—82no—
400M16—82no—
401M16—82no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8—82no—
412C2×C8—82no—
413C2×C8—81no—
414C2×C8—81no—
415C2×C8—82no—
416C2×C8C2×C22⋊C412yes—
417C2×C8C2×C22⋊C412yes—
418C2×C8C2×C22⋊C412yes—
419C2×C8C2×C22⋊C412yes—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4C2×C4⋊C412yes—
439C4⋊C4C2×C4⋊C412yes—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—44no—
455C42—41no—
456C42—42no—
457C42—41no—
458C42—42no—
459C42—41no—
460C42—41no—
461C42—82no—
462C42C2×C22⋊C411yescharacteristic
463C42C2×C22⋊C411yescharacteristic
464C42C2×C4⋊C411yescharacteristic
465C42C2×C4⋊C411yescharacteristic
466C42#11711yescharacteristic
467C42#11711yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42C22⋊Q811yescharacteristic
471C42C22⋊Q811yescharacteristic
472C42C22⋊Q811yescharacteristic
473C42C22⋊Q811yescharacteristic
474C42C23×C411yescharacteristic
475C42C22×D411yescharacteristic
476C42C22×Q811yescharacteristic
477C42C2×#4111yescharacteristic
478C23—21no—
479C23—21no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23#38011yescharacteristic
484C23#38011yescharacteristic
485C23C2×#12111yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—42no—
489Q8—81no—
490D4—22no—
491D4—22no—
492D4—44no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—81no—
497D4—82no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4#38011yescharacteristic
541C2×C4#38011yescharacteristic
542C2×C4#38011yescharacteristic
543C2×C4#38011yescharacteristic
544C2×C4#38011yescharacteristic
545C2×C4#38011yescharacteristic
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4C22×C4⋊C411yescharacteristic
548C2×C4C2×#11711yescharacteristic
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4C2×C22⋊Q811yescharacteristic
552C2×C4#57611yescharacteristic
553C2×C4#57711yescharacteristic
554C2×C4#57711yescharacteristic
555C2×C4#57811yescharacteristic
556C8—22no—
557C8—22no—
558C8—22no—
559C8—22no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—82no—
565C8—81no—
566C8—81no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—81no—
579C22—81no—
580C22C2×#38011yescharacteristic
581C22#306211yescharacteristic
582C22#306311yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—82no—
595C4—82no—
596C4—162no—
597C4#181412yes—
598C4#290111yescharacteristic
599C4#290111yescharacteristic
600C4#291211yescharacteristic
601C4#291311yescharacteristic
602C4#306211yescharacteristic
603C4#306311yescharacteristic
604C2—21no—
605C2—21no—
606C2—41no—
607C2—41no—
608C2—82no—
609C2#2065511yescharacteristic
610C2#2066411yescharacteristic
611C2C2×#181411yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#380
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21362C2semidirect
5. #17817C2semidirect
6. #8669C2semidirect
7. #8668C2semidirect
8. #8667C2semidirect
9. #8666C2semidirect
10. #8095C2semidirect
11. #8092C2semidirect
15. #2994C4semidirect
16. #2994C4semidirect
17. #2986C4semidirect
18. #2986C4semidirect
19. #2979C4semidirect
20. #2979C4semidirect

Maximal subgroups

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

subgroupindex
#602802
#219002
#218562
#213622
#178172
#86692
#86682
#86672
#86662
#80952
#80922