Finite Group Databasebeta

← #189450

Subgroups — #189450

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60294C211yesmaximal, characteristic
2#21362C211yesmaximal, characteristic, complemented
3#17816C211yesmaximal, characteristic, complemented
4#17585C211yesmaximal, characteristic
5#17543C211yesmaximal, characteristic
6#8697C212yesmaximal, complemented
7#8696C212yesmaximal, complemented
8#8695C212yesmaximal, complemented
9#8694C212yesmaximal, complemented
10#7952C211yesmaximal, characteristic, complemented
11#7949C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2990—21no—
14#2990—21no—
15#2989C411yescharacteristic, complemented
16#2989C411yescharacteristic, complemented
17#2988C411yescharacteristic, complemented
18#2988C411yescharacteristic, complemented
19#2985C412yescomplemented
20#2981C412yescomplemented
21#2977—22no—
22#2968—22no—
23#2966C2212yes—
24#2966C2212yes—
25#2925—21no—
26#2925—21no—
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2285C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1948—21no—
34#1947—21no—
35#1854—22no—
36C4×#107C2212yes—
37C4×#106C2212yes—
38#1572—22no—
39#1571—21no—
40#1570—21no—
41#1568C2212yes—
42#1568C2212yes—
43#1518—21no—
44#1517—21no—
45#1513—21no—
46#1513—21no—
47#1509C2211yescharacteristic
48#1509C2211yescharacteristic
49#1509C2211yescharacteristic
50#1509C2211yescharacteristic
51#1509C2212yes—
52#1509C2212yes—
53#1506—21no—
54#1505—21no—
55#1501—21no—
56#1501—21no—
57#1285C2212yes—
58#1285C2212yes—
59#1285C2212yes—
60#1285C2212yes—
61#533C2×C411yescharacteristic
62#533D411yescharacteristic
63#533Q811yescharacteristic
64#532C2×C411yescharacteristic
65#532D411yescharacteristic
66#532Q811yescharacteristic
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71#529Q811yescharacteristic
72#529Q811yescharacteristic
73#529C2311yescharacteristic
74#529C2311yescharacteristic
75#525—21no—
76#525—21no—
77#525D411yescharacteristic
78#525D411yescharacteristic
79C4×#41—21no—
80C4×#41—21no—
81C4×#41C2×C411yescharacteristic
82C4×#41C2×C411yescharacteristic
83C2×#117—21no—
84C2×#117—21no—
85C22×C42C2311yescharacteristic
86#446—22no—
87#446—22no—
88#446C2×C412yes—
89#446C2×C412yes—
90C8×Q8C2×C412yes—
91C8×Q8C2×C412yes—
92#436C2×C412yes—
93#436C2×C412yes—
94#435—22no—
95#435—22no—
96#435—22no—
97#435—22no—
98#435—22no—
99#435—22no—
100C8×D4—22no—
101C8×D4—22no—
102C8×D4—22no—
103C8×D4—22no—
104C8×D4C2×C412yes—
105C8×D4C2×C412yes—
106#433—22no—
107#433—22no—
108#433—22no—
109#433—22no—
110#432—21no—
111#432—21no—
112#432—21no—
113#432—21no—
114#432—21no—
115#432—21no—
116#432—21no—
117#432—21no—
118#432C2×C412yes—
119#432C2311yescharacteristic
120#432C2311yescharacteristic
121#432C2312yes—
122#432C2312yes—
123#431—21no—
124#431—21no—
125#431—21no—
126#431—21no—
127#431C2×C412yes—
128#431C2×C411yescharacteristic
129#431C2×C411yescharacteristic
130#431D411yescharacteristic
131#431D411yescharacteristic
132#431D412yes—
133C2×#107—22no—
134C2×#106—22no—
135#423—21no—
136#423—21no—
137#423—21no—
138C2×C4⋊C8—21no—
139C2×C4⋊C8—21no—
140C2×C4⋊C8—21no—
141C2×C4⋊C8C2×C411yescharacteristic
142C2×C4⋊C8C2×C411yescharacteristic
143C2×C4⋊C8D411yescharacteristic
144C2×C4⋊C8D411yescharacteristic
145#408—22no—
146#408—22no—
147#407—21no—
148C2×C22⋊C8—21no—
149C2×C22⋊C8—21no—
150C2×C22⋊C8D412yes—
151#405—22no—
152#405—22no—
153C4×M16—21no—
154C2×#97—21no—
155C2×#97—21no—
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#335—22no—
173#335—22no—
174#334—22no—
175#334—22no—
176#328—21no—
177#328—21no—
178#328—21no—
179#328—21no—
180#328—22no—
181#328—22no—
182#327—21no—
183#327—21no—
184#327—21no—
185#327—21no—
186#327—22no—
187#327—22no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41C4⋊C411yescharacteristic
191C2×#41C4⋊C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4#4111yescharacteristic
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—41no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—41no—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C4⋊Q8C4⋊C411yescharacteristic
206C4⋊Q8C4⋊C411yescharacteristic
207C4⋊Q8C4⋊C411yescharacteristic
208#127C4⋊C411yescharacteristic
209#126—21no—
210#126—21no—
211#126—21no—
212#126—21no—
213#125—21no—
214#125—21no—
215#125C2×Q811yescharacteristic
216#125C2×Q811yescharacteristic
217#124—21no—
218#124—21no—
219#124C2×Q811yescharacteristic
220#124C2×Q811yescharacteristic
221#123—21no—
222#123—21no—
223#123—21no—
224#123—21no—
225#123—21no—
226#123—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8C4⋊C411yescharacteristic
232C22⋊Q8C4⋊C411yescharacteristic
233C22⋊Q8C2×D411yescharacteristic
234C22⋊Q8C2×D411yescharacteristic
235#121—21no—
236#121—21no—
237#121—21no—
238#121—21no—
239#121C4⋊C411yescharacteristic
240#121C4⋊C411yescharacteristic
241#121C2×D411yescharacteristic
242#121C2×D411yescharacteristic
243#120—21no—
244#120—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C4⋊C411yescharacteristic
250C4×Q8C4⋊C411yescharacteristic
251C4×Q8C22×C411yescharacteristic
252C4×Q8C22×C411yescharacteristic
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4C4⋊C411yescharacteristic
270C4×D4C4⋊C411yescharacteristic
271C4×D4C22×C411yescharacteristic
272C4×D4C22×C411yescharacteristic
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—42no—
282#117—42no—
283#117—82no—
284#117C4⋊C411yescharacteristic
285#117C4⋊C411yescharacteristic
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—41no—
299C2×C42—41no—
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×Q811yescharacteristic
307C2×C42C2×Q811yescharacteristic
308C2×C42#4111yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42#4111yescharacteristic
311C2×C42C2411yescharacteristic
312#107—44no—
313#106—44no—
314C4⋊C8—22no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—22no—
323C4⋊C8—41no—
324C4⋊C8—41no—
325C4⋊C8C22⋊C411yescharacteristic
326C4⋊C8C22⋊C411yescharacteristic
327C4⋊C8C22⋊C411yescharacteristic
328C4⋊C8C22⋊C411yescharacteristic
329C4⋊C8C22⋊C412yes—
330C4⋊C8C22⋊C412yes—
331C4⋊C8C22×C412yes—
332C4⋊C8C22×C412yes—
333C22⋊C8—22no—
334C22⋊C8—22no—
335C22⋊C8—22no—
336C22⋊C8—22no—
337C22⋊C8—22no—
338C22⋊C8—22no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—22no—
342C22⋊C8—41no—
343C22⋊C8—41no—
344C22⋊C8C22⋊C412yes—
345C22⋊C8C22⋊C412yes—
346C22⋊C8C22⋊C412yes—
347C22⋊C8C22⋊C412yes—
348#97—22no—
349#97—22no—
350#97—22no—
351#97—22no—
352#97—21no—
353#97—21no—
354#97—22no—
355#97—41no—
356C4×C8—22no—
357C4×C8—22no—
358C4×C8—41no—
359C4×C8C22×C412yes—
360C4×C8C22×C412yes—
361C4×C8C2×D412yes—
362C4×C8C2×D412yes—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—41no—
367#95—41no—
368C24#12111yescharacteristic
369#41—42no—
370#41—42no—
371#41—81no—
372#41—81no—
373#41—81no—
374#41—81no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8C2×C4⋊C411yescharacteristic
378C2×Q8C2×C4⋊C411yescharacteristic
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4C2×C4⋊C411yescharacteristic
388C2×D4C2×C4⋊C411yescharacteristic
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C4⋊C411yescharacteristic
406C22×C4C2×C4⋊C411yescharacteristic
407C22×C4#11711yescharacteristic
408C22×C4#11711yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4C22⋊Q811yescharacteristic
415C22×C4C22⋊Q811yescharacteristic
416C22×C4C22⋊Q811yescharacteristic
417C22×C4C22⋊Q811yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421M16—82no—
422M16—82no—
423M16—82no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—82no—
432C2×C8—82no—
433C2×C8—81no—
434C2×C8—81no—
435C2×C8—82no—
436C2×C8C2×C22⋊C412yes—
437C2×C8C2×C22⋊C412yes—
438C2×C8C2×C22⋊C412yes—
439C2×C8C2×C22⋊C412yes—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4C2×C4⋊C411yescharacteristic
465C4⋊C4C2×C4⋊C411yescharacteristic
466C4⋊C4C2×C4⋊C411yescharacteristic
467C4⋊C4C2×C4⋊C411yescharacteristic
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—44no—
487C22⋊C4—82no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—44no—
495C42—81no—
496C42—81no—
497C42—82no—
498C42—82no—
499C42C2×C22⋊C411yescharacteristic
500C42C2×C22⋊C411yescharacteristic
501C42C2×C4⋊C411yescharacteristic
502C42C2×C4⋊C411yescharacteristic
503C42#11711yescharacteristic
504C42#11711yescharacteristic
505C42#12111yescharacteristic
506C42#12111yescharacteristic
507C42C22⋊Q811yescharacteristic
508C42C22⋊Q811yescharacteristic
509C42C22⋊Q811yescharacteristic
510C42C22⋊Q811yescharacteristic
511C42C23×C411yescharacteristic
512C42C22×D411yescharacteristic
513C42C22×Q811yescharacteristic
514C42C2×#4111yescharacteristic
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—41no—
520C23—41no—
521C23#38011yescharacteristic
522C23#38011yescharacteristic
523C23C2×#12111yescharacteristic
524Q8—41no—
525Q8—41no—
526Q8—81no—
527Q8—81no—
528D4—41no—
529D4—41no—
530D4—42no—
531D4—42no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—81no—
537D4—81no—
538D4—81no—
539D4—81no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4#38011yescharacteristic
587C2×C4#38011yescharacteristic
588C2×C4#38011yescharacteristic
589C2×C4#38011yescharacteristic
590C2×C4#38011yescharacteristic
591C2×C4#38011yescharacteristic
592C2×C4C22×C22⋊C411yescharacteristic
593C2×C4C22×C4⋊C411yescharacteristic
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×C22⋊Q811yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C8—22no—
599C8—22no—
600C8—22no—
601C8—22no—
602C8—42no—
603C8—42no—
604C8—81no—
605C8—81no—
606C8—82no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—81no—
622C22—81no—
623C22C2×#38011yescharacteristic
624C22#306811yescharacteristic
625C22#306911yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—82no—
643C4—82no—
644C4—162no—
645C4#181212yes—
646C4#290111yescharacteristic
647C4#290111yescharacteristic
648C2—21no—
649C2—21no—
650C2—41no—
651C2—41no—
652C2—81no—
653C2—81no—
654C2#2065511yescharacteristic
655C2C2×#181211yescharacteristic
656C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21362C2semidirect
3. #17816C2semidirect
6. #8697C2semidirect
7. #8696C2semidirect
8. #8695C2semidirect
9. #8694C2semidirect
10. #7952C2semidirect
11. #7949C2semidirect
15. #2989C4semidirect
16. #2989C4semidirect
17. #2988C4semidirect
18. #2988C4semidirect
19. #2985C4semidirect
20. #2981C4semidirect

Maximal subgroups

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

subgroupindex
#602942
#213622
#178162
#175852
#175432
#86972
#86962
#86952
#86942
#79522
#79492