Finite Group Databasebeta

← #189440

Subgroups — #189440

Order 512.

849 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60211C211yesmaximal, characteristic
2#20819C211yesmaximal, characteristic, complemented
3#17797C211yesmaximal, characteristic, complemented
4#17586C211yesmaximal, characteristic
5#17585C211yesmaximal, characteristic
6#8697C212yesmaximal, complemented
7#8696C212yesmaximal, complemented
8#8695C212yesmaximal, complemented
9#8694C212yesmaximal, complemented
10#7954C211yesmaximal, characteristic, complemented
11#7952C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2971C411yescharacteristic, complemented
14#2971C411yescharacteristic, complemented
15#2970—21no—
16#2970—21no—
17#2970C411yescharacteristic, complemented
18#2970C411yescharacteristic, complemented
19#2967—22no—
20#2966C2212yes—
21#2966C2212yes—
22#2962C412yescomplemented
23#2935C412yescomplemented
24#2933—22no—
25#2923—21no—
26#2920C2211yescharacteristic
27C2×#432C2211yescharacteristic
28C2×#432C2211yescharacteristic
29#2875—21no—
30C2×C4×M16C2211yescharacteristic
31#2285C2211yescharacteristic
32#2283C2211yescharacteristic
33#1847—21no—
34#1836—21no—
35#1809—22no—
36C4×#107C2212yes—
37C4×#106C2212yes—
38#1569—21no—
39#1569—21no—
40#1568C2212yes—
41#1568C2212yes—
42#1565—22no—
43#1518—21no—
44#1517—21no—
45#1516—21no—
46#1516—21no—
47#1515—21no—
48#1514—21no—
49#1513—21no—
50#1513—21no—
51#1509C2212yes—
52#1509C2212yes—
53#1509C2211yescharacteristic
54#1509C2211yescharacteristic
55#1509C2211yescharacteristic
56#1509C2211yescharacteristic
57#1285C2212yes—
58#1285C2212yes—
59#1285C2212yes—
60#1285C2212yes—
61C2×#131—22no—
62C22×M16D412yes—
63#533C2×C411yescharacteristic
64#533D411yescharacteristic
65#533Q811yescharacteristic
66#532C2×C411yescharacteristic
67#532D411yescharacteristic
68#532Q811yescharacteristic
69#529—21no—
70#529—21no—
71#529—21no—
72#529—21no—
73#529Q811yescharacteristic
74#529Q811yescharacteristic
75#529C2311yescharacteristic
76#529C2311yescharacteristic
77#525—21no—
78#525—21no—
79#525D411yescharacteristic
80#525D411yescharacteristic
81C4×#41—21no—
82C4×#41—21no—
83C4×#41C2×C411yescharacteristic
84C4×#41C2×C411yescharacteristic
85C2×#117—21no—
86C22×C42C2311yescharacteristic
87#446—22no—
88#446—22no—
89#446C2×C412yes—
90#446C2×C412yes—
91C8×Q8C2×C412yes—
92C8×Q8C2×C412yes—
93#436—22no—
94#436—22no—
95#436C2×C412yes—
96#436C2×C412yes—
97#435—22no—
98#435—22no—
99#435—22no—
100#435—22no—
101#435—22no—
102#435—22no—
103C8×D4—22no—
104C8×D4—22no—
105C8×D4C2×C412yes—
106C8×D4C2×C412yes—
107#433—22no—
108#433—22no—
109#432—21no—
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#432—21no—
119#432C2×C411yescharacteristic
120#432C2×C411yescharacteristic
121#432C2311yescharacteristic
122#432C2311yescharacteristic
123#432C2312yes—
124#432C2312yes—
125#431—21no—
126#431—21no—
127C2×#107—22no—
128C2×#106—22no—
129#424—22no—
130#423—21no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8C2×C412yes—
133#408—22no—
134#407—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137#405—22no—
138#405—22no—
139#405—22no—
140C4×M16—21no—
141C4×M16—21no—
142C4×M16—21no—
143C4×M16C2×C412yes—
144C4×M16C2×C411yescharacteristic
145C4×M16C2×C411yescharacteristic
146C4×M16D412yes—
147C4×M16D411yescharacteristic
148C4×M16D411yescharacteristic
149C2×#97—21no—
150C2×#97—21no—
151C2×#97—21no—
152C2×#97—21no—
153C2×#97D411yescharacteristic
154C2×#97D411yescharacteristic
155C2×C4×C8C2312yes—
156#384—21no—
157#383—21no—
158#382—21no—
159#382—21no—
160#381—21no—
161#380—21no—
162#380—21no—
163#380—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4C2311yescharacteristic
166C4×C4⋊C4C2311yescharacteristic
167C4×C4⋊C4C2312yes—
168C4×C22⋊C4—21no—
169#376—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—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41C4⋊C411yescharacteristic
191C2×#41C4⋊C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4#4111yescharacteristic
195#131—48no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—44no—
199C2×M16—41no—
200C2×M16C22⋊C412yes—
201C2×M16C22⋊C412yes—
202C2×M16C22⋊C412yes—
203C2×M16C22⋊C412yes—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—41no—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C4⋊Q8C4⋊C411yescharacteristic
211C4⋊Q8C4⋊C411yescharacteristic
212C4⋊Q8C4⋊C411yescharacteristic
213#127C4⋊C411yescharacteristic
214#126—21no—
215#126—21no—
216#126—21no—
217#126—21no—
218#125—21no—
219#125—21no—
220#125C2×Q811yescharacteristic
221#125C2×Q811yescharacteristic
222#124—21no—
223#124—21no—
224#124C2×Q811yescharacteristic
225#124C2×Q811yescharacteristic
226#123—21no—
227#123—21no—
228#123—21no—
229#123—21no—
230#123—21no—
231#123—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8C4⋊C411yescharacteristic
237C22⋊Q8C4⋊C411yescharacteristic
238C22⋊Q8C2×D411yescharacteristic
239C22⋊Q8C2×D411yescharacteristic
240#121—21no—
241#121—21no—
242#121—21no—
243#121—21no—
244#121C4⋊C411yescharacteristic
245#121C4⋊C411yescharacteristic
246#121C2×D411yescharacteristic
247#121C2×D411yescharacteristic
248#120—21no—
249#120—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C4⋊C411yescharacteristic
255C4×Q8C4⋊C411yescharacteristic
256C4×Q8C22×C411yescharacteristic
257C4×Q8C22×C411yescharacteristic
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×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4C4⋊C411yescharacteristic
275C4×D4C4⋊C411yescharacteristic
276C4×D4C22×C411yescharacteristic
277C4×D4C22×C411yescharacteristic
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—82no—
287#117C4⋊C411yescharacteristic
288#117C4⋊C411yescharacteristic
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×Q811yescharacteristic
308C2×C42C2×Q811yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42#4111yescharacteristic
311C2×C42#4111yescharacteristic
312C2×C42C2411yescharacteristic
313#107—44no—
314#106—44no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—22no—
323C4⋊C8—41no—
324C4⋊C8—41no—
325C4⋊C8C22×C412yes—
326C4⋊C8C22×C412yes—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—22no—
330C22⋊C8—22no—
331C22⋊C8—22no—
332C22⋊C8—22no—
333C22⋊C8—22no—
334C22⋊C8—21no—
335C22⋊C8—21no—
336C22⋊C8—22no—
337C22⋊C8—41no—
338C22⋊C8—41no—
339#97—22no—
340#97—22no—
341#97—21no—
342#97—21no—
343#97—22no—
344#97—21no—
345#97—21no—
346#97—22no—
347#97—41no—
348#97C22⋊C411yescharacteristic
349#97C22⋊C411yescharacteristic
350#97C22⋊C411yescharacteristic
351#97C22⋊C411yescharacteristic
352#97C22⋊C412yes—
353#97C22⋊C412yes—
354C4×C8—22no—
355C4×C8—22no—
356C4×C8—41no—
357C4×C8C22×C412yes—
358C4×C8C22×C412yes—
359C4×C8C2×D412yes—
360C4×C8C2×D412yes—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—41no—
365#95—41no—
366#95—41no—
367#95—41no—
368C24#12111yescharacteristic
369#41—44no—
370#41—81no—
371#41—81no—
372#41—81no—
373#41—81no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8C2×C4⋊C411yescharacteristic
377C2×Q8C2×C4⋊C411yescharacteristic
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4C2×C4⋊C411yescharacteristic
387C2×D4C2×C4⋊C411yescharacteristic
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—81no—
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—24no—
420M16—24no—
421M16—42no—
422M16—42no—
423M16—44no—
424M16—44no—
425M16—82no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—42no—
430C2×C8—44no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—42no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—82no—
437C2×C8C2×C22⋊C412yes—
438C2×C8C2×C22⋊C412yes—
439C2×C8C2×C22⋊C412yes—
440C2×C8C2×C22⋊C412yes—
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—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4C2×C4⋊C411yescharacteristic
464C4⋊C4C2×C4⋊C411yescharacteristic
465C4⋊C4C2×C4⋊C411yescharacteristic
466C4⋊C4C2×C4⋊C411yescharacteristic
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—21no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—81no—
494C42—81no—
495C42—82no—
496C42C2×C22⋊C411yescharacteristic
497C42C2×C22⋊C411yescharacteristic
498C42C2×C4⋊C411yescharacteristic
499C42C2×C4⋊C411yescharacteristic
500C42#11711yescharacteristic
501C42#11711yescharacteristic
502C42#12111yescharacteristic
503C42#12111yescharacteristic
504C42C22⋊Q811yescharacteristic
505C42C22⋊Q811yescharacteristic
506C42C22⋊Q811yescharacteristic
507C42C22⋊Q811yescharacteristic
508C42C23×C411yescharacteristic
509C42C22×D411yescharacteristic
510C42C22×Q811yescharacteristic
511C42C2×#4111yescharacteristic
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—41no—
517C23—41no—
518C23#38011yescharacteristic
519C23#38011yescharacteristic
520C23C2×#12111yescharacteristic
521Q8—41no—
522Q8—41no—
523Q8—42no—
524Q8—81no—
525D4—41no—
526D4—41no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—81no—
535D4—81no—
536D4—81no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
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—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4#38011yescharacteristic
579C2×C4#38011yescharacteristic
580C2×C4#38011yescharacteristic
581C2×C4#38011yescharacteristic
582C2×C4#38011yescharacteristic
583C2×C4#38011yescharacteristic
584C2×C4C22×C22⋊C411yescharacteristic
585C2×C4C22×C4⋊C411yescharacteristic
586C2×C4C2×#11711yescharacteristic
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×C22⋊Q811yescharacteristic
589C2×C4C2×C22⋊Q811yescharacteristic
590C8—22no—
591C8—22no—
592C8—22no—
593C8—42no—
594C8—42no—
595C8—81no—
596C8—81no—
597C8—82no—
598C8#42114yes—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—81no—
614C22—81no—
615C22C2×#38011yescharacteristic
616C22#306811yescharacteristic
617C22#306911yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—161no—
636C4—162no—
637C4—161no—
638C4C2×#42112yes—
639C2—21no—
640C2—21no—
641C2—41no—
642C2—41no—
643C2—81no—
644C2—81no—
645C2#2065411yescharacteristic
646C2#2067111yescharacteristic
647C2#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. #20819C2semidirect
3. #17797C2semidirect
6. #8697C2semidirect
7. #8696C2semidirect
8. #8695C2semidirect
9. #8694C2semidirect
10. #7954C2semidirect
11. #7952C2semidirect
13. #2971C4semidirect
14. #2971C4semidirect
17. #2970C4semidirect
18. #2970C4semidirect
22. #2962C4semidirect
23. #2935C4semidirect

Maximal subgroups

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

subgroupindex
#602112
#208192
#177972
#175862
#175852
#86972
#86962
#86952
#86942
#79542
#79522