Finite Group Databasebeta

← #6344105

Subgroups — #6344105

Order 512.

3137 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62800C211yesmaximal, characteristic
2#60283C211yesmaximal, characteristic, complemented
3#60099C211yesmaximal, characteristic, complemented
4#17765C216yesmaximal, complemented
5#17762C216yesmaximal, complemented
6#17753C213yesmaximal, complemented
7#17745C213yesmaximal, complemented
8#17737C213yesmaximal, complemented
9C2×#1557C213yesmaximal, complemented
10#17723C212yesmaximal, complemented
11#17720C212yesmaximal, complemented
12C2×#568—21no—
13#3522C412yes—
14#3519C416yes—
15#3517—21no—
16#3514—23no—
17#3511C2213yes—
18#3508C2213yes—
19#3503C2211yescharacteristic
20#3498C2211yescharacteristic
21#3494—23no—
22#3488—21no—
23#3483C412yes—
24#3481C416yes—
25#3480—21no—
26D4×#41—23no—
27#3460—23no—
28#3444C2213yes—
29C2×#532C2213yes—
30C4×H(5,𝔽2)—21no—
31C2×C4×#41C2211yescharacteristic
32#2989C2216yescomplemented
33#2988C2212yes—
34#2984—21no—
35#2974—23no—
36#2971—21no—
37#2970—23no—
38#2920C2213yescomplemented
39C2×#432C2213yescomplemented
40#2906—21no—
41#2906—23no—
42#2906C2212yes—
43#2906C2216yescomplemented
44#2905—23no—
45C2×#423C2213yescomplemented
46C2×#423C2213yescomplemented
47#1561C2216yescomplemented
48#1560C2216yescomplemented
49#1559C2211yescharacteristic, complemented
50#1559C2213yescomplemented
51#1557C2213yescomplemented
52#1557C2216yescomplemented
53#1557C22112yescomplemented
54#1557C2213yescomplemented
55#1552C2216yescomplemented
56#1551C2216yescomplemented
57#1550C2216yescomplemented
58#1549C2216yescomplemented
59#1547C2216yescomplemented
60#1547C22112yescomplemented
61#1545C2216yescomplemented
62#1542C2216yescomplemented
63#1540C2216yescomplemented
64C2×#332C2216yescomplemented
65C2×#331C2216yescomplemented
66C2×H(5,𝔽2)—21no—
67C22×#41—23no—
68C22×#41D411yescharacteristic
69#568—48no—
70C2×#131—22no—
71C2×#131—26no—
72C22×M16—23no—
73C22×M16—23no—
74#559—23no—
75#559C2×C416yes—
76Q82C2×C412yes—
77#557—21no—
78#556—23no—
79#555—26no—
80#555C2×C416yes—
81#554C2×C416yes—
82#554C2×C416yes—
83#553—26no—
84#553—26no—
85#553C2×C4112yes—
86#551—212no—
87#550—22no—
88#550—21no—
89#550—23no—
90#550C2×C416yes—
91#550C2×C412yes—
92D4×Q8—23no—
93#548—22no—
94#548—26no—
95#547—23no—
96#547—26no—
97#546—26no—
98#546—26no—
99#546—212no—
100#546C2×C4112yes—
101D42—26no—
102D42—23no—
103D42C2×C416yes—
104#544—23no—
105#542—26no—
106#542—26no—
107#540—26no—
108#540—26no—
109#538—212no—
110#537—23no—
111#535—23no—
112#535—23no—
113#533C2313yes—
114#533C2311yescharacteristic
115#533C2313yes—
116#532—26no—
117#532—23no—
118#532—23no—
119#532—23no—
120#532—23no—
121#532C2×C416yes—
122#532C2×C416yes—
123#532C2313yes—
124#532C2313yes—
125#532C2316yes—
126#532C2313yes—
127C2×C4⋊Q8C2313yes—
128C2×#127C2313yes—
129#529—26no—
130#529—23no—
131#529—23no—
132#529—23no—
133#529—23no—
134#529C2×C416yes—
135#529C2×C416yes—
136C2×#124—23no—
137#525—23no—
138#525—23no—
139#525—212no—
140#525C2×C416yes—
141C2×#121—23no—
142C2×#121C2×C416yes—
143#520—21no—
144#520—22no—
145#520—23no—
146#518—26no—
147#518—23no—
148C4×#41—22no—
149C4×#41—23no—
150C4×#41—23no—
151C4×#41C2×C416yes—
152C4×#41C2×C413yes—
153C4×#41C2×C411yescharacteristic
154C4×#41D413yes—
155C4×#41D411yescharacteristic
156C4×#41D411yescharacteristic
157C4×#41D413yes—
158C4×#41C2311yescharacteristic
159C4×#41C2313yes—
160C2×C4×Q8C2×C411yescharacteristic
161C2×C4×Q8D411yescharacteristic
162C2×C4×D4—26no—
163C2×C4×D4—23no—
164C2×C4×D4C2×C416yes—
165C2×C4×D4C2×C413yes—
166C2×C4×D4D413yes—
167C2×#117D413yes—
168C22×C42C2313yes—
169#446—26no—
170C8×Q8—22no—
171#436—26no—
172#435—212no—
173C8×D4—26no—
174#432—23no—
175#432—23no—
176#432—23no—
177#432C2316yes—
178#432C2313yes—
179#432C2311yescharacteristic
180#431—23no—
181#431C2316yes—
182#424—26no—
183#424—26no—
184#424—212no—
185#423—23no—
186#423—23no—
187#423—23no—
188#423—23no—
189#423C2316yes—
190#423C2316yes—
191#423C2311yescharacteristic
192#423C2313yes—
193#423C2316yes—
194C2×C4⋊C8—23no—
195C2×C4⋊C8—23no—
196C2×C4⋊C8C2316yes—
197C2×C4⋊C8C2316yes—
198C2×C4⋊C8C2316yes—
199#407—23no—
200C2×C22⋊C8—23no—
201C4×M16—23no—
202C2×#97—23no—
203#333C2312yes—
204#333C2316yes—
205#332C2316yes—
206#332C23112yes—
207#332C2312yes—
208#332C2316yes—
209#332C2316yes—
210#331C2316yes—
211#331C23112yes—
212#331C2316yes—
213H(5,𝔽2)—44no—
214H(5,𝔽2)—44no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—22no—
218C2×#41—26no—
219C2×#41—26no—
220C2×#41—23no—
221C2×#41—23no—
222C2×#41—26no—
223C2×#41—812no—
224C2×#41C22⋊C413yes—
225C2×#41C22⋊C411yescharacteristic
226C2×#41C22×C416yes—
227C2×#41C2×D413yes—
228C2×#41C2×D411yescharacteristic
229C22×Q8—23no—
230C22×Q8C22⋊C411yescharacteristic
231C22×D4—26no—
232C22×D4—26no—
233C22×D4—23no—
234C22×D4—23no—
235C22×D4C22⋊C413yes—
236C22×D4C22×C416yes—
237C23×C4—26no—
238C23×C4C22×C413yes—
239C23×C4C2×D413yes—
240C23×C4C2×D413yes—
241#131—424no—
242#131—48no—
243C2×M16—21no—
244C2×M16—23no—
245C2×M16—26no—
246C2×M16—26no—
247C2×M16—26no—
248C2×M16—412no—
249C2×M16—412no—
250C22×C8—26no—
251C22×C8—26no—
252C22×C8—26no—
253C4⋊Q8—26no—
254C4⋊Q8—23no—
255C4⋊Q8—23no—
256C4⋊Q8C22×C416yes—
257C4⋊Q8C2411yescharacteristic
258C4⋊Q8C2413yes—
259C4⋊Q8C2411yescharacteristic
260#127—26no—
261#127—23no—
262#127—23no—
263#127C22×C416yes—
264#127C2413yes—
265#126—212no—
266#126—26no—
267#126—26no—
268#125—26no—
269#125—21no—
270#125—23no—
271#125C22×C416yes—
272#124—26no—
273#124—212no—
274#124—26no—
275#124—26no—
276#124—23no—
277#124—26no—
278#124—23no—
279#124C22×C416yes—
280#123—26no—
281#123—26no—
282#123—26no—
283#123—212no—
284#123—212no—
285#123—212no—
286C22⋊Q8—26no—
287C22⋊Q8—21no—
288C22⋊Q8—23no—
289C22⋊Q8—23no—
290C22⋊Q8—23no—
291C22⋊Q8—212no—
292C22⋊Q8—212no—
293C22⋊Q8C22×C412yes—
294C22⋊Q8C22×C416yes—
295#121—212no—
296#121—26no—
297#121—26no—
298#121—26no—
299#121—26no—
300#121—26no—
301#121—23no—
302#121—26no—
303#121—23no—
304#121—23no—
305#121—23no—
306#121—26no—
307#121—26no—
308#121—212no—
309#121—212no—
310#121C22×C416yes—
311#121C22×C4112yes—
312#121C22×C416yes—
313#120—26no—
314#120—212no—
315#120—212no—
316C4×Q8—22no—
317C4×Q8—21no—
318C4×Q8—23no—
319C4×Q8—23no—
320C4×Q8—23no—
321C4×Q8C22⋊C411yescharacteristic
322C4×Q8C22⋊C413yes—
323C4×Q8C22⋊C411yescharacteristic
324C4×Q8C22⋊C413yes—
325C4×Q8C22×C412yes—
326C4×Q8C22×C416yes—
327C4×Q8C22×C411yescharacteristic
328C4×Q8C22×C413yes—
329C4×Q8C2×D411yescharacteristic
330C4×Q8C2×D413yes—
331C4×D4—26no—
332C4×D4—26no—
333C4×D4—26no—
334C4×D4—26no—
335C4×D4—26no—
336C4×D4—26no—
337C4×D4—26no—
338C4×D4—26no—
339C4×D4—26no—
340C4×D4—212no—
341C4×D4—26no—
342C4×D4—26no—
343C4×D4—26no—
344C4×D4—26no—
345C4×D4—23no—
346C4×D4—26no—
347C4×D4—23no—
348C4×D4—23no—
349C4×D4—23no—
350C4×D4C22⋊C413yes—
351C4×D4C22⋊C413yes—
352C4×D4C22⋊C416yes—
353C4×D4C22⋊C413yes—
354C4×D4C22⋊C416yes—
355C4×D4C22⋊C413yes—
356C4×D4C22×C416yes—
357C4×D4C22×C4112yes—
358C4×D4C22×C416yes—
359C4×D4C22×C413yes—
360C4×D4C22×C416yes—
361C4×D4C22×C413yes—
362C4×D4C2×D413yes—
363C4×D4C2×D416yes—
364C4×D4C2×D413yes—
365#117—21no—
366#117—23no—
367#117—26no—
368#117—26no—
369#117—26no—
370#117—26no—
371#117C22⋊C416yes—
372#117C22⋊C413yes—
373#117C22⋊C413yes—
374#117C22×C416yes—
375#117C2×D413yes—
376#117C2×D413yes—
377#117C2×D416yes—
378C2×C4⋊C4—23no—
379C2×C4⋊C4—26no—
380C2×C4⋊C4C22⋊C413yes—
381C2×C4⋊C4C22⋊C413yes—
382C2×C4⋊C4C22×C416yes—
383C2×C22⋊C4—26no—
384C2×C22⋊C4—26no—
385C2×C22⋊C4—212no—
386C2×C22⋊C4C22⋊C416yes—
387C2×C42—26no—
388C2×C42C22×C413yes—
389C2×C42C22×C413yes—
390C2×C42C22×C413yes—
391C2×C42C2×D413yes—
392C2×C42C2×D413yes—
393C2×C42C2×D413yes—
394C2×C42C2×D413yes—
395C2×C42C2×D413yes—
396C2×C42C2×D413yes—
397C2×C42C2413yes—
398C2×C42C2413yes—
399C2×C42C2411yescharacteristic
400C4⋊C8—21no—
401C4⋊C8—23no—
402C4⋊C8—26no—
403C4⋊C8—26no—
404C4⋊C8—26no—
405C4⋊C8—26no—
406C4⋊C8—26no—
407C4⋊C8—26no—
408C4⋊C8C2412yes—
409C4⋊C8C2416yes—
410C4⋊C8C2416yes—
411C4⋊C8C2412yes—
412C22⋊C8—23no—
413C22⋊C8—26no—
414C22⋊C8—21no—
415C22⋊C8—26no—
416#97—23no—
417#97—26no—
418#97—21no—
419C4×C8—26no—
420C24—26no—
421C24C2×C22⋊C413yes—
422#41—24no—
423#41—212no—
424#41—26no—
425#41—26no—
426#41—412no—
427#41—48no—
428#41—46no—
429#41—46no—
430#41—812no—
431#41—812no—
432#41—812no—
433#41—86no—
434#41AGL(1,ℤ/8)14yes—
435#41#13714yes—
436C2×Q8—24no—
437C2×Q8—26no—
438C2×Q8—21no—
439C2×Q8—23no—
440C2×Q8—86no—
441C2×Q8C2×C22⋊C413yes—
442C2×Q8C2×C22⋊C411yescharacteristic
443C2×Q8C23×C412yes—
444C2×D4—212no—
445C2×D4—26no—
446C2×D4—26no—
447C2×D4—26no—
448C2×D4—212no—
449C2×D4—26no—
450C2×D4—26no—
451C2×D4—26no—
452C2×D4—23no—
453C2×D4—23no—
454C2×D4—424no—
455C2×D4—46no—
456C2×D4—46no—
457C2×D4—86no—
458C2×D4—86no—
459C2×D4—812no—
460C2×D4C2×C22⋊C416yes—
461C2×D4C2×C22⋊C413yes—
462C2×D4C2×C22⋊C413yes—
463C2×D4C23×C416yes—
464C22×C4—26no—
465C22×C4—26no—
466C22×C4—26no—
467C22×C4—26no—
468C22×C4—26no—
469C22×C4—26no—
470C22×C4—212no—
471C22×C4—46no—
472C22×C4—83no—
473C22×C4—812no—
474C22×C4C2×C22⋊C413yes—
475C22×C4C2×C22⋊C413yes—
476C22×C4C2×C22⋊C413yes—
477C22×C4C2×C22⋊C413yes—
478C22×C4C2×C22⋊C413yes—
479C22×C4C2×C22⋊C413yes—
480C22×C4C2×C22⋊C413yes—
481C22×C4C2×C22⋊C413yes—
482C22×C4C2×C22⋊C413yes—
483C22×C4C23×C411yescharacteristic
484C22×C4C23×C413yes—
485C22×C4C23×C413yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D413yes—
488C22×C4C22×D413yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D413yes—
491C22×C4C22×D413yes—
492M16—46no—
493M16—42no—
494M16—412no—
495M16—412no—
496M16—86no—
497C2×C8—22no—
498C2×C8—26no—
499C2×C8—26no—
500C2×C8—22no—
501C2×C8—412no—
502C2×C8—412no—
503C2×C8—86no—
504C4⋊C4—26no—
505C4⋊C4—26no—
506C4⋊C4—26no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—26no—
510C4⋊C4—26no—
511C4⋊C4—26no—
512C4⋊C4—23no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4C2×C22⋊C413yes—
517C4⋊C4C2×C22⋊C413yes—
518C4⋊C4C2×C22⋊C413yes—
519C4⋊C4C2×C22⋊C416yes—
520C4⋊C4C2×C22⋊C413yes—
521C4⋊C4C2×C22⋊C416yes—
522C4⋊C4C23×C416yes—
523C4⋊C4C23×C412yes—
524C22⋊C4—26no—
525C22⋊C4—22no—
526C22⋊C4—26no—
527C22⋊C4—26no—
528C22⋊C4—212no—
529C22⋊C4—26no—
530C22⋊C4—26no—
531C22⋊C4—212no—
532C22⋊C4—26no—
533C22⋊C4—26no—
534C22⋊C4—26no—
535C22⋊C4C2×C22⋊C416yes—
536C22⋊C4C2×C22⋊C416yes—
537C22⋊C4C2×C22⋊C416yes—
538C22⋊C4C2×C22⋊C416yes—
539C42—22no—
540C42—26no—
541C42—26no—
542C42C2×C22⋊C413yes—
543C42C2×C22⋊C413yes—
544C42C2×C22⋊C413yes—
545C42C2×C22⋊C413yes—
546C42C2×C22⋊C413yes—
547C42C2×C22⋊C413yes—
548C42C23×C411yescharacteristic
549C42C23×C413yes—
550C42C23×C413yes—
551C42C22×D413yes—
552C42C22×D411yescharacteristic
553C42C22×D413yes—
554C42C22×D411yescharacteristic
555C42C22×D413yes—
556C42C22×D413yes—
557C42C2511yescharacteristic
558C23—26no—
559C23—26no—
560C23—43no—
561C23—43no—
562C23—412no—
563C23—86no—
564C23—86no—
565C23C22×C22⋊C413yes—
566C23C22×C22⋊C411yescharacteristic
567C23C22×C22⋊C413yes—
568Q8—26no—
569Q8—42no—
570Q8—46no—
571Q8—41no—
572Q8—41no—
573Q8—86no—
574Q8#41714yes—
575D4—212no—
576D4—26no—
577D4—46no—
578D4—412no—
579D4—412no—
580D4—46no—
581D4—412no—
582D4—43no—
583D4—43no—
584D4—86no—
585D4—86no—
586D4—812no—
587D4—86no—
588D4#417112yes—
589C2×C4—26no—
590C2×C4—26no—
591C2×C4—26no—
592C2×C4—22no—
593C2×C4—26no—
594C2×C4—26no—
595C2×C4—23no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—23no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—26no—
602C2×C4—43no—
603C2×C4—43no—
604C2×C4—43no—
605C2×C4—41no—
606C2×C4—43no—
607C2×C4—43no—
608C2×C4—43no—
609C2×C4—43no—
610C2×C4—43no—
611C2×C4—412no—
612C2×C4—412no—
613C2×C4—812no—
614C2×C4—86no—
615C2×C4—86no—
616C2×C4C22×C22⋊C411yescharacteristic
617C2×C4C22×C22⋊C413yes—
618C2×C4C22×C22⋊C411yescharacteristic
619C2×C4C22×C22⋊C411yescharacteristic
620C2×C4C22×C22⋊C413yes—
621C2×C4C22×C22⋊C413yes—
622C2×C4C22×C22⋊C413yes—
623C2×C4C22×C22⋊C413yes—
624C2×C4C22×C22⋊C413yes—
625C2×C4C2×AGL(1,ℤ/8)16yes—
626C2×C4C2×#13716yes—
627C2×C4C24×C411yescharacteristic
628C2×C4C23×D411yescharacteristic
629C2×C4C23×D411yescharacteristic
630C8—46no—
631C8—42no—
632C8—46no—
633C8—42no—
634C22—23no—
635C22—23no—
636C22—21no—
637C22—23no—
638C22—21no—
639C22—23no—
640C22—21no—
641C22—23no—
642C22—46no—
643C22—46no—
644C22—46no—
645C22—86no—
646C22—1612no—
647C22C2×#41716yes—
648C22C23×C22⋊C411yescharacteristic
649C4—21no—
650C4—21no—
651C4—23no—
652C4—23no—
653C4—21no—
654C4—23no—
655C4—21no—
656C4—23no—
657C4—23no—
658C4—42no—
659C4—42no—
660C4—46no—
661C4C2×#41716yes—
662C4C22×AGL(1,ℤ/8)11yescharacteristic
663C4C22×#13711yescharacteristic
664C4#358512yes—
665C2—26no—
666C2—43no—
667C2—41no—
668C2—86no—
669C2#6007611yescharacteristic
670C2C22×#41711yescharacteristic
671C2#6008711yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
prime radical#62800C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60283C2semidirect
3. #60099C2semidirect
4. #17765C2semidirect
5. #17762C2semidirect
6. #17753C2semidirect
7. #17745C2semidirect
8. #17737C2semidirect
9. C2×#1557C2semidirect
10. #17723C2semidirect
11. #17720C2semidirect
32. #2989C22semidirect
38. #2920C22semidirect
39. C2×#432C22semidirect
43. #2906C22semidirect
45. C2×#423C22semidirect
46. C2×#423C22semidirect
47. #1561C22semidirect
48. #1560C22semidirect
49. #1559C22semidirect
50. #1559C22semidirect
51. #1557C22semidirect
52. #1557C22semidirect
53. #1557C22semidirect
54. #1557C22semidirect
55. #1552C22semidirect
56. #1551C22semidirect
57. #1550C22semidirect
58. #1549C22semidirect
59. #1547C22semidirect
60. #1547C22semidirect
61. #1545C22semidirect
62. #1542C22semidirect
63. #1540C22semidirect
64. C2×#332C22semidirect
65. C2×#331C22semidirect

Maximal subgroups

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

subgroupindex
#628002
#602832
#600992
#177652
#177622
#177532
#177452
#177372
C2×#15572
#177232
#177202