Finite Group Databasebeta

← #189447

Subgroups — #189447

Order 512.

847 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60290C211yesmaximal, characteristic
2#21361C211yesmaximal, characteristic, complemented
3#17816C211yesmaximal, characteristic, complemented
4#17580C211yesmaximal, characteristic
5C2×#1509C211yesmaximal, characteristic
6#8697C212yesmaximal, complemented
7#8696C212yesmaximal, complemented
8#8695C212yesmaximal, complemented
9#8694C212yesmaximal, complemented
10#7951C211yesmaximal, characteristic, complemented
11#7950C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2994—22no—
14#2993C412yescomplemented
15#2993C412yescomplemented
16#2983C411yescharacteristic, complemented
17#2982C411yescharacteristic, complemented
18#2975C411yescharacteristic, complemented
19#2974—21no—
20#2974—21no—
21#2972C411yescharacteristic, complemented
22#2969—22no—
23#2966C2212yes—
24#2966C2212yes—
25#2924—21no—
26#2924—21no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2283C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1949—22no—
34#1853—21no—
35#1852—21no—
36C4×#107C2212yescomplemented
37C4×#106C2212yescomplemented
38#1572—22no—
39#1571—21no—
40#1570—21no—
41#1568C2212yescomplemented
42#1568C2212yescomplemented
43#1521—21no—
44#1521—21no—
45#1511—21no—
46#1510—21no—
47#1509—21no—
48#1509—21no—
49#1509C2212yes—
50#1509C2212yes—
51#1509C2211yescharacteristic
52#1509C2211yescharacteristic
53#1509C2211yescharacteristic
54#1509C2211yescharacteristic
55C2×#328—21no—
56C2×#327—21no—
57#1285C2212yescomplemented
58#1285C2212yescomplemented
59#1285C2212yescomplemented
60#1285C2212yescomplemented
61#529—21no—
62#529—21no—
63#529—21no—
64#529—21no—
65#529C2×C411yescharacteristic
66#529C2×C411yescharacteristic
67#529D411yescharacteristic
68#529D411yescharacteristic
69#529Q811yescharacteristic
70#529Q811yescharacteristic
71#529C2311yescharacteristic
72#529C2311yescharacteristic
73C2×#126—21no—
74C2×#125Q811yescharacteristic
75C2×#124Q811yescharacteristic
76C2×#123—21no—
77C2×C22⋊Q8D411yescharacteristic
78C2×#121D411yescharacteristic
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#446—22no—
87#446C2×C412yescomplemented
88#446C2×C412yescomplemented
89C8×Q8—22no—
90C8×Q8C2×C412yescomplemented
91C8×Q8C2×C412yescomplemented
92#436—22no—
93#436—22no—
94#436—22no—
95#436C2×C412yescomplemented
96#436C2×C412yescomplemented
97#435—22no—
98#435—22no—
99#435—22no—
100#435—22no—
101C8×D4—22no—
102C8×D4—22no—
103C8×D4—22no—
104C8×D4C2×C412yescomplemented
105C8×D4C2×C412yescomplemented
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#432C2312yes—
118#432C2311yescharacteristic
119#432C2311yescharacteristic
120#432C2312yes—
121#431—21no—
122#431—21no—
123#431—21no—
124#431—21no—
125#431—21no—
126#431C2×C411yescharacteristic
127#431C2×C411yescharacteristic
128#431C2×C411yescharacteristic
129#431C2×C411yescharacteristic
130#431C2×C412yes—
131#431D411yescharacteristic
132#431D411yescharacteristic
133#431D412yes—
134C2×#107—22no—
135C2×#106—22no—
136#424—22no—
137#424—22no—
138#423—21no—
139#423C2×C412yes—
140C2×C4⋊C8—21no—
141C2×C4⋊C8D412yes—
142#407—21no—
143#407—21no—
144#407—21no—
145#407—21no—
146C2×C22⋊C8—21no—
147C2×C22⋊C8—21no—
148C2×C22⋊C8—21no—
149C2×C22⋊C8D411yescharacteristic
150C2×C22⋊C8D411yescharacteristic
151#405—22no—
152#405—22no—
153C4×M16—21no—
154C4×M16—21no—
155C2×#97—21no—
156C2×C4×C8C2312yes—
157#383—21no—
158#382—21no—
159#381—21no—
160#380—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4C2312yes—
165C4×C4⋊C4C2311yescharacteristic
166C4×C4⋊C4C2311yescharacteristic
167C4×C22⋊C4—21no—
168#376—21no—
169C2×#95—21no—
170#340—22no—
171#340—22no—
172#335—22no—
173#335—22no—
174#334—22no—
175#334—22no—
176#328—22no—
177#328—22no—
178#328—21no—
179#328—21no—
180#328—21no—
181#328—21no—
182#327—22no—
183#327—22no—
184#327—21no—
185#327—21no—
186#327—21no—
187#327—21no—
188C22×Q8C4⋊C411yescharacteristic
189C22×D4—21no—
190C22×D4C4⋊C411yescharacteristic
191C23×C4—21no—
192C23×C4—41no—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4#4111yescharacteristic
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—41no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—41no—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205#126—21no—
206#126—21no—
207#126—21no—
208#126—21no—
209#126—21no—
210#126—21no—
211#126—21no—
212#126—21no—
213#125—21no—
214#125—21no—
215#125C4⋊C411yescharacteristic
216#125C4⋊C411yescharacteristic
217#125C2×Q811yescharacteristic
218#125C2×Q811yescharacteristic
219#124—21no—
220#124—21no—
221#124C4⋊C411yescharacteristic
222#124C4⋊C411yescharacteristic
223#124C2×Q811yescharacteristic
224#124C2×Q811yescharacteristic
225#123—21no—
226#123—21no—
227#123—21no—
228#123—21no—
229#123—21no—
230#123—21no—
231#123—21no—
232#123—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8C4⋊C411yescharacteristic
236C22⋊Q8C4⋊C411yescharacteristic
237C22⋊Q8C2×D411yescharacteristic
238C22⋊Q8C2×D411yescharacteristic
239#121—21no—
240#121—21no—
241#121C4⋊C411yescharacteristic
242#121C4⋊C411yescharacteristic
243#121C2×D411yescharacteristic
244#121C2×D411yescharacteristic
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8C4⋊C411yescharacteristic
248C4×Q8C4⋊C411yescharacteristic
249C4×Q8C22×C411yescharacteristic
250C4×Q8C22×C411yescharacteristic
251C4×D4—21no—
252C4×D4—21no—
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×D4C4⋊C411yescharacteristic
262C4×D4C4⋊C411yescharacteristic
263C4×D4C22×C411yescharacteristic
264C4×D4C22×C411yescharacteristic
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4C4⋊C411yescharacteristic
283C2×C4⋊C4C4⋊C411yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—41no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—41no—
295C2×C42—82no—
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×Q811yescharacteristic
303C2×C42C2×Q811yescharacteristic
304C2×C42#4111yescharacteristic
305C2×C42#4111yescharacteristic
306C2×C42#4111yescharacteristic
307C2×C42C2411yescharacteristic
308#107—44no—
309#106—44no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315C4⋊C8—22no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—41no—
319C4⋊C8—41no—
320C4⋊C8C22⋊C412yes—
321C4⋊C8C22⋊C412yes—
322C4⋊C8C22⋊C412yes—
323C4⋊C8C22⋊C412yes—
324C4⋊C8C22×C412yes—
325C4⋊C8C22×C412yes—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—22no—
329C22⋊C8—22no—
330C22⋊C8—22no—
331C22⋊C8—22no—
332C22⋊C8—22no—
333C22⋊C8—22no—
334C22⋊C8—21no—
335C22⋊C8—21no—
336C22⋊C8—41no—
337C22⋊C8—41no—
338C22⋊C8C22⋊C411yescharacteristic
339C22⋊C8C22⋊C411yescharacteristic
340C22⋊C8C22⋊C411yescharacteristic
341C22⋊C8C22⋊C411yescharacteristic
342C22⋊C8C22⋊C412yes—
343C22⋊C8C22⋊C412yes—
344#97—22no—
345#97—22no—
346#97—22no—
347#97—22no—
348#97—21no—
349#97—21no—
350#97—41no—
351C4×C8—22no—
352C4×C8—22no—
353C4×C8—21no—
354C4×C8—21no—
355C4×C8—41no—
356C4×C8C22×C412yes—
357C4×C8C22×C412yes—
358C4×C8C2×D412yes—
359C4×C8C2×D412yes—
360#95—22no—
361#95—21no—
362#95—21no—
363#95—41no—
364#95—41no—
365#95—41no—
366#95—41no—
367C24—21no—
368C24#12111yescharacteristic
369C2×Q8—21no—
370C2×Q8—42no—
371C2×Q8C2×C4⋊C411yescharacteristic
372C2×Q8C2×C4⋊C411yescharacteristic
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—42no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4C2×C4⋊C411yescharacteristic
382C2×D4C2×C4⋊C411yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C4⋊C411yescharacteristic
404C22×C4C2×C4⋊C411yescharacteristic
405C22×C4#11711yescharacteristic
406C22×C4#11711yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4C22⋊Q811yescharacteristic
413C22×C4C22⋊Q811yescharacteristic
414C22×C4C22⋊Q811yescharacteristic
415C22×C4C22⋊Q811yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419M16—82no—
420M16—82no—
421M16—82no—
422M16—82no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—82no—
431C2×C8—82no—
432C2×C8—82no—
433C2×C8C2×C22⋊C412yes—
434C2×C8C2×C22⋊C412yes—
435C2×C8C2×C22⋊C412yes—
436C2×C8C2×C22⋊C412yes—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
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—24no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4C2×C4⋊C411yescharacteristic
461C4⋊C4C2×C4⋊C411yescharacteristic
462C4⋊C4C2×C4⋊C411yescharacteristic
463C4⋊C4C2×C4⋊C411yescharacteristic
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
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—82no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—81no—
490C42—81no—
491C42—82no—
492C42—82no—
493C42C2×C22⋊C411yescharacteristic
494C42C2×C22⋊C411yescharacteristic
495C42C2×C4⋊C411yescharacteristic
496C42C2×C4⋊C411yescharacteristic
497C42#11711yescharacteristic
498C42#11711yescharacteristic
499C42#12111yescharacteristic
500C42#12111yescharacteristic
501C42C22⋊Q811yescharacteristic
502C42C22⋊Q811yescharacteristic
503C42C22⋊Q811yescharacteristic
504C42C22⋊Q811yescharacteristic
505C42C23×C411yescharacteristic
506C42C22×D411yescharacteristic
507C42C22×Q811yescharacteristic
508C42C2×#4111yescharacteristic
509C23—22no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—41no—
515C23—42no—
516C23—81no—
517C23—81no—
518C23#38011yescharacteristic
519C23#38011yescharacteristic
520C23C2×#12111yescharacteristic
521Q8—41no—
522Q8—41no—
523Q8—42no—
524D4—41no—
525D4—41no—
526D4—42no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—81no—
532D4—81no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—88no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4#38011yescharacteristic
577C2×C4#38011yescharacteristic
578C2×C4#38011yescharacteristic
579C2×C4#38011yescharacteristic
580C2×C4#38011yescharacteristic
581C2×C4#38011yescharacteristic
582C2×C4C22×C22⋊C411yescharacteristic
583C2×C4C22×C4⋊C411yescharacteristic
584C2×C4C2×#11711yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4C2×C22⋊Q811yescharacteristic
588C8—22no—
589C8—22no—
590C8—22no—
591C8—22no—
592C8—42no—
593C8—42no—
594C8—82no—
595C8—81no—
596C8—81no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—42no—
605C22—42no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—162no—
613C22#181212yes—
614C22C2×#38011yescharacteristic
615C22#306811yescharacteristic
616C22#306911yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—84no—
635C4—161no—
636C4—161no—
637C4#290112yes—
638C2—21no—
639C2—21no—
640C2—22no—
641C2—41no—
642C2—81no—
643C2—81no—
644C2#2065511yescharacteristic
645C2C2×#181211yescharacteristic
646C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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. #21361C2semidirect
3. #17816C2semidirect
6. #8697C2semidirect
7. #8696C2semidirect
8. #8695C2semidirect
9. #8694C2semidirect
10. #7951C2semidirect
11. #7950C2semidirect
14. #2993C4semidirect
15. #2993C4semidirect
16. #2983C4semidirect
17. #2982C4semidirect
18. #2975C4semidirect
21. #2972C4semidirect
36. C4×#107C22semidirect
37. C4×#106C22semidirect
41. #1568C22semidirect
42. #1568C22semidirect
57. #1285C22semidirect
58. #1285C22semidirect
59. #1285C22semidirect
60. #1285C22semidirect
87. #446C2×C4semidirect
88. #446C2×C4semidirect
90. C8×Q8C2×C4semidirect
91. C8×Q8C2×C4semidirect
95. #436C2×C4semidirect
96. #436C2×C4semidirect
104. C8×D4C2×C4semidirect
105. C8×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602902
#213612
#178162
#175802
C2×#15092
#86972
#86962
#86952
#86942
#79512
#79502