Finite Group Databasebeta

← #6344000

Subgroups — #6344000

Order 512.

3092 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62800C211yesmaximal, characteristic
2C2×#2988C211yesmaximal, characteristic, complemented
3#60127C211yesmaximal, characteristic, complemented
4#17727C216yesmaximal, complemented
5#17724C216yesmaximal, complemented
6#17723C212yesmaximal, complemented
7#17720C212yesmaximal, complemented
8#17719C213yesmaximal, complemented
9#17717C213yesmaximal, complemented
10#17700C213yesmaximal, complemented
11C2×#1540C213yesmaximal, complemented
12#3522C412yes—
13#3519C416yes—
14#3517—21no—
15#3514—23no—
16#3511C2213yes—
17#3508C2213yes—
18#3503C2211yescharacteristic
19#3498C2211yescharacteristic
20#3494—23no—
21#3488—21no—
22#3483C412yes—
23#3481C416yes—
24#3480—21no—
25D4×#41—23no—
26#3460—23no—
27#3444C2213yes—
28C2×#532C2213yes—
29C4×H(5,𝔽2)—21no—
30C2×C4×#41C2211yescharacteristic
31#2988—23no—
32#2988—21no—
33#2988C2212yes—
34#2988C2216yescomplemented
35C2×C8×Q8—21no—
36C2×C8×D4—23no—
37#2925C2216yescomplemented
38#2924—23no—
39#2922—23no—
40#2920C2213yescomplemented
41C2×#431C2213yescomplemented
42C2×#431C2213yescomplemented
43#2906C2212yes—
44C22×C4⋊C8C2213yescomplemented
45#2880—21no—
46C4×#131—21no—
47#1553C22112yescomplemented
48#1552C2216yescomplemented
49#1551C2216yescomplemented
50#1550C2216yescomplemented
51#1549C2216yescomplemented
52#1547C2216yescomplemented
53#1545C2216yescomplemented
54#1544C2216yescomplemented
55#1543C2216yescomplemented
56#1542C2211yescharacteristic, complemented
57#1542C2213yescomplemented
58#1542C2216yescomplemented
59#1540C2213yescomplemented
60#1540C2216yescomplemented
61#1540C22112yescomplemented
62#1540C2213yescomplemented
63#1540C2216yescomplemented
64C2×#332C2216yescomplemented
65C2×#331C2216yescomplemented
66C2×H(5,𝔽2)—21no—
67C22×#41—23no—
68C22×#41D411yescharacteristic
69C2×#131—22no—
70C23×C8—23no—
71#559—23no—
72#559C2×C416yes—
73Q82C2×C412yes—
74#557—21no—
75#556—23no—
76#555—26no—
77#555C2×C416yes—
78#554C2×C416yes—
79#554C2×C416yes—
80#553—26no—
81#553—26no—
82#553C2×C4112yes—
83#551—212no—
84#550—22no—
85#550—21no—
86#550—23no—
87#550C2×C416yes—
88#550C2×C412yes—
89D4×Q8—23no—
90#548—22no—
91#548—26no—
92#547—23no—
93#547—26no—
94#546—26no—
95#546—26no—
96#546—212no—
97#546C2×C4112yes—
98D42—26no—
99D42—23no—
100D42C2×C416yes—
101#544—23no—
102#542—26no—
103#542—26no—
104#540—26no—
105#540—26no—
106#538—212no—
107#537—23no—
108#535—23no—
109#535—23no—
110#533C2311yescharacteristic
111#533C2313yes—
112#533C2313yes—
113#532—26no—
114#532—23no—
115#532—23no—
116#532—23no—
117#532—23no—
118#532C2×C416yes—
119#532C2×C416yes—
120#532C2313yes—
121#532C2313yes—
122#532C2313yes—
123#532C2316yes—
124C2×C4⋊Q8C2313yes—
125C2×#127C2313yes—
126#529—26no—
127#529—23no—
128#529—23no—
129#529—23no—
130#529—23no—
131#529C2×C416yes—
132#529C2×C416yes—
133C2×#124—23no—
134#525—23no—
135#525—23no—
136#525—212no—
137#525C2×C416yes—
138C2×#121—23no—
139C2×#121C2×C416yes—
140#520—21no—
141#520—22no—
142#520—23no—
143#518—26no—
144#518—23no—
145C4×#41—22no—
146C4×#41—23no—
147C4×#41—23no—
148C4×#41C2×C416yes—
149C4×#41C2×C413yes—
150C4×#41C2×C411yescharacteristic
151C4×#41D413yes—
152C4×#41D411yescharacteristic
153C4×#41D411yescharacteristic
154C4×#41D413yes—
155C4×#41C2311yescharacteristic
156C4×#41C2313yes—
157C2×C4×Q8C2×C411yescharacteristic
158C2×C4×Q8D411yescharacteristic
159C2×C4×D4—26no—
160C2×C4×D4—23no—
161C2×C4×D4C2×C416yes—
162C2×C4×D4C2×C413yes—
163C2×C4×D4D413yes—
164C2×#117D413yes—
165C22×C42C2313yes—
166C8×Q8—22no—
167C8×Q8—26no—
168C8×D4—26no—
169C8×D4—212no—
170C8×D4—26no—
171#433—212no—
172#432—23no—
173#432C2316yes—
174#431—23no—
175#431—23no—
176#431—23no—
177#431—23no—
178#431—23no—
179#431C2311yescharacteristic
180#431C2313yes—
181#431C2316yes—
182#431C2316yes—
183#431C2316yes—
184#424—26no—
185#423C2316yes—
186C2×C4⋊C8—23no—
187C2×C4⋊C8—23no—
188C2×C4⋊C8—23no—
189C2×C4⋊C8C2316yes—
190C2×C4⋊C8C2316yes—
191C2×C4⋊C8C2313yes—
192C2×C4⋊C8C2311yescharacteristic
193C2×C4⋊C8C2316yes—
194#408—26no—
195#407—23no—
196C2×C22⋊C8—23no—
197C2×C22⋊C8—23no—
198#405—26no—
199C4×M16—23no—
200C2×C4×C8—23no—
201C2×C4×C8—23no—
202#333C2312yes—
203#333C2316yes—
204#332C2316yes—
205#332C23112yes—
206#332C2312yes—
207#332C2316yes—
208#332C2316yes—
209#331C2316yes—
210#331C23112yes—
211#331C2316yes—
212H(5,𝔽2)—48no—
213C2×#41—28no—
214C2×#41—22no—
215C2×#41—26no—
216C2×#41—26no—
217C2×#41—23no—
218C2×#41—23no—
219C2×#41—26no—
220C2×#41—86no—
221C2×#41—86no—
222C2×#41C22⋊C413yes—
223C2×#41C22⋊C411yescharacteristic
224C2×#41C22×C416yes—
225C2×#41C2×D413yes—
226C2×#41C2×D411yescharacteristic
227C22×Q8—23no—
228C22×Q8C22⋊C411yescharacteristic
229C22×D4—26no—
230C22×D4—26no—
231C22×D4—23no—
232C22×D4—23no—
233C22×D4C22⋊C413yes—
234C22×D4C22×C416yes—
235C23×C4—26no—
236C23×C4C22×C413yes—
237C23×C4C2×D413yes—
238C23×C4C2×D413yes—
239#131—48no—
240C2×M16—26no—
241C22×C8—23no—
242C22×C8—26no—
243C22×C8—21no—
244C22×C8—26no—
245C22×C8—26no—
246C22×C8—86no—
247C4⋊Q8—26no—
248C4⋊Q8—23no—
249C4⋊Q8—23no—
250C4⋊Q8C22×C416yes—
251C4⋊Q8C2411yescharacteristic
252C4⋊Q8C2413yes—
253C4⋊Q8C2411yescharacteristic
254#127—26no—
255#127—23no—
256#127—23no—
257#127C22×C416yes—
258#127C2413yes—
259#126—212no—
260#126—26no—
261#126—26no—
262#125—26no—
263#125—21no—
264#125—23no—
265#125C22×C416yes—
266#124—26no—
267#124—212no—
268#124—26no—
269#124—26no—
270#124—23no—
271#124—26no—
272#124—23no—
273#124C22×C416yes—
274#123—26no—
275#123—26no—
276#123—26no—
277#123—212no—
278#123—212no—
279#123—212no—
280C22⋊Q8—26no—
281C22⋊Q8—21no—
282C22⋊Q8—23no—
283C22⋊Q8—23no—
284C22⋊Q8—23no—
285C22⋊Q8—212no—
286C22⋊Q8—212no—
287C22⋊Q8C22×C412yes—
288C22⋊Q8C22×C416yes—
289#121—212no—
290#121—26no—
291#121—26no—
292#121—26no—
293#121—26no—
294#121—26no—
295#121—23no—
296#121—26no—
297#121—23no—
298#121—23no—
299#121—23no—
300#121—26no—
301#121—26no—
302#121—212no—
303#121—212no—
304#121C22×C416yes—
305#121C22×C4112yes—
306#121C22×C416yes—
307#120—26no—
308#120—212no—
309#120—212no—
310C4×Q8—22no—
311C4×Q8—21no—
312C4×Q8—23no—
313C4×Q8—23no—
314C4×Q8—23no—
315C4×Q8C22⋊C411yescharacteristic
316C4×Q8C22⋊C413yes—
317C4×Q8C22⋊C411yescharacteristic
318C4×Q8C22⋊C413yes—
319C4×Q8C22×C412yes—
320C4×Q8C22×C416yes—
321C4×Q8C22×C411yescharacteristic
322C4×Q8C22×C413yes—
323C4×Q8C2×D411yescharacteristic
324C4×Q8C2×D413yes—
325C4×D4—26no—
326C4×D4—26no—
327C4×D4—26no—
328C4×D4—26no—
329C4×D4—26no—
330C4×D4—26no—
331C4×D4—26no—
332C4×D4—26no—
333C4×D4—26no—
334C4×D4—212no—
335C4×D4—26no—
336C4×D4—26no—
337C4×D4—26no—
338C4×D4—26no—
339C4×D4—23no—
340C4×D4—26no—
341C4×D4—23no—
342C4×D4—23no—
343C4×D4—23no—
344C4×D4C22⋊C413yes—
345C4×D4C22⋊C413yes—
346C4×D4C22⋊C416yes—
347C4×D4C22⋊C413yes—
348C4×D4C22⋊C416yes—
349C4×D4C22⋊C413yes—
350C4×D4C22×C416yes—
351C4×D4C22×C4112yes—
352C4×D4C22×C416yes—
353C4×D4C22×C413yes—
354C4×D4C22×C416yes—
355C4×D4C22×C413yes—
356C4×D4C2×D413yes—
357C4×D4C2×D416yes—
358C4×D4C2×D413yes—
359#117—21no—
360#117—23no—
361#117—26no—
362#117—26no—
363#117—26no—
364#117—26no—
365#117C22⋊C416yes—
366#117C22⋊C413yes—
367#117C22⋊C413yes—
368#117C22×C416yes—
369#117C2×D413yes—
370#117C2×D413yes—
371#117C2×D416yes—
372C2×C4⋊C4—23no—
373C2×C4⋊C4—26no—
374C2×C4⋊C4C22⋊C413yes—
375C2×C4⋊C4C22⋊C413yes—
376C2×C4⋊C4C22×C416yes—
377C2×C22⋊C4—26no—
378C2×C22⋊C4—26no—
379C2×C22⋊C4—212no—
380C2×C22⋊C4C22⋊C416yes—
381C2×C42—26no—
382C2×C42C22×C413yes—
383C2×C42C22×C413yes—
384C2×C42C22×C413yes—
385C2×C42C2×D413yes—
386C2×C42C2×D413yes—
387C2×C42C2×D413yes—
388C2×C42C2×D413yes—
389C2×C42C2×D413yes—
390C2×C42C2×D413yes—
391C2×C42C2413yes—
392C2×C42C2413yes—
393C2×C42C2411yescharacteristic
394C4⋊C8—23no—
395C4⋊C8—26no—
396C4⋊C8—26no—
397C4⋊C8—26no—
398C4⋊C8—21no—
399C4⋊C8—26no—
400C4⋊C8C2416yes—
401C4⋊C8C2412yes—
402C4⋊C8C2416yes—
403C4⋊C8C2412yes—
404C22⋊C8—21no—
405C22⋊C8—23no—
406C22⋊C8—26no—
407C22⋊C8—26no—
408C22⋊C8—26no—
409C22⋊C8—26no—
410#97—26no—
411C4×C8—21no—
412C4×C8—23no—
413C4×C8—26no—
414C4×C8—26no—
415C4×C8—26no—
416C24—26no—
417C24C2×C22⋊C413yes—
418#41—24no—
419#41—212no—
420#41—212no—
421#41—48no—
422#41—412no—
423#41—46no—
424#41—46no—
425#41—812no—
426#41—812no—
427#41—86no—
428#41—86no—
429#41—86no—
430#41#13518yes—
431C2×Q8—24no—
432C2×Q8—26no—
433C2×Q8—21no—
434C2×Q8—23no—
435C2×Q8—83no—
436C2×Q8—83no—
437C2×Q8C2×C22⋊C413yes—
438C2×Q8C2×C22⋊C411yescharacteristic
439C2×Q8C23×C412yes—
440C2×D4—212no—
441C2×D4—26no—
442C2×D4—26no—
443C2×D4—26no—
444C2×D4—212no—
445C2×D4—26no—
446C2×D4—26no—
447C2×D4—26no—
448C2×D4—23no—
449C2×D4—23no—
450C2×D4—424no—
451C2×D4—412no—
452C2×D4—83no—
453C2×D4—83no—
454C2×D4—86no—
455C2×D4—812no—
456C2×D4C2×C22⋊C416yes—
457C2×D4C2×C22⋊C413yes—
458C2×D4C2×C22⋊C413yes—
459C2×D4C23×C416yes—
460C22×C4—212no—
461C22×C4—26no—
462C22×C4—26no—
463C22×C4—26no—
464C22×C4—26no—
465C22×C4—26no—
466C22×C4—26no—
467C22×C4—212no—
468C22×C4—812no—
469C22×C4—83no—
470C22×C4C2×C22⋊C413yes—
471C22×C4C2×C22⋊C413yes—
472C22×C4C2×C22⋊C413yes—
473C22×C4C2×C22⋊C413yes—
474C22×C4C2×C22⋊C413yes—
475C22×C4C2×C22⋊C413yes—
476C22×C4C2×C22⋊C413yes—
477C22×C4C2×C22⋊C413yes—
478C22×C4C2×C22⋊C413yes—
479C22×C4C23×C411yescharacteristic
480C22×C4C23×C413yes—
481C22×C4C23×C413yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D413yes—
484C22×C4C22×D413yes—
485C22×C4C22×D413yes—
486C22×C4C22×D413yes—
487C22×C4C22×D411yescharacteristic
488M16—412no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—26no—
492C2×C8—26no—
493C2×C8—412no—
494C2×C8—46no—
495C2×C8—42no—
496C2×C8—86no—
497C2×C8—86no—
498C4⋊C4—26no—
499C4⋊C4—26no—
500C4⋊C4—26no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—26no—
504C4⋊C4—26no—
505C4⋊C4—26no—
506C4⋊C4—23no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4C2×C22⋊C413yes—
511C4⋊C4C2×C22⋊C413yes—
512C4⋊C4C2×C22⋊C413yes—
513C4⋊C4C2×C22⋊C416yes—
514C4⋊C4C2×C22⋊C413yes—
515C4⋊C4C2×C22⋊C416yes—
516C4⋊C4C23×C416yes—
517C4⋊C4C23×C412yes—
518C22⋊C4—26no—
519C22⋊C4—22no—
520C22⋊C4—26no—
521C22⋊C4—26no—
522C22⋊C4—212no—
523C22⋊C4—26no—
524C22⋊C4—26no—
525C22⋊C4—212no—
526C22⋊C4—26no—
527C22⋊C4—26no—
528C22⋊C4—26no—
529C22⋊C4C2×C22⋊C416yes—
530C22⋊C4C2×C22⋊C416yes—
531C22⋊C4C2×C22⋊C416yes—
532C22⋊C4C2×C22⋊C416yes—
533C42—26no—
534C42—26no—
535C42—22no—
536C42C2×C22⋊C413yes—
537C42C2×C22⋊C413yes—
538C42C2×C22⋊C413yes—
539C42C2×C22⋊C413yes—
540C42C2×C22⋊C413yes—
541C42C2×C22⋊C413yes—
542C42C23×C413yes—
543C42C23×C413yes—
544C42C23×C411yescharacteristic
545C42C22×D413yes—
546C42C22×D413yes—
547C42C22×D413yes—
548C42C22×D411yescharacteristic
549C42C22×D413yes—
550C42C22×D411yescharacteristic
551C42C2511yescharacteristic
552C23—26no—
553C23—26no—
554C23—26no—
555C23—43no—
556C23—412no—
557C23—86no—
558C23—86no—
559C23C22×C22⋊C413yes—
560C23C22×C22⋊C413yes—
561C23C22×C22⋊C411yescharacteristic
562Q8—26no—
563Q8—42no—
564Q8—42no—
565Q8—43no—
566Q8—43no—
567Q8—86no—
568Q8#41614yes—
569D4—212no—
570D4—26no—
571D4—46no—
572D4—412no—
573D4—412no—
574D4—412no—
575D4—46no—
576D4—43no—
577D4—43no—
578D4—86no—
579D4—86no—
580D4—812no—
581D4—86no—
582D4#416112yes—
583C2×C4—26no—
584C2×C4—26no—
585C2×C4—26no—
586C2×C4—26no—
587C2×C4—26no—
588C2×C4—26no—
589C2×C4—22no—
590C2×C4—26no—
591C2×C4—23no—
592C2×C4—23no—
593C2×C4—23no—
594C2×C4—23no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—26no—
598C2×C4—41no—
599C2×C4—43no—
600C2×C4—43no—
601C2×C4—43no—
602C2×C4—43no—
603C2×C4—43no—
604C2×C4—412no—
605C2×C4—412no—
606C2×C4—812no—
607C2×C4—86no—
608C2×C4—86no—
609C2×C4C22×C22⋊C413yes—
610C2×C4C22×C22⋊C411yescharacteristic
611C2×C4C22×C22⋊C411yescharacteristic
612C2×C4C22×C22⋊C413yes—
613C2×C4C22×C22⋊C413yes—
614C2×C4C22×C22⋊C413yes—
615C2×C4C22×C22⋊C413yes—
616C2×C4C22×C22⋊C413yes—
617C2×C4C22×C22⋊C411yescharacteristic
618C2×C4C2×#135112yes—
619C2×C4#57612yes—
620C2×C4#57712yes—
621C2×C4C24×C411yescharacteristic
622C2×C4C23×D411yescharacteristic
623C2×C4C23×D411yescharacteristic
624C8—42no—
625C8—46no—
626C8—46no—
627C8—42no—
628C22—21no—
629C22—23no—
630C22—21no—
631C22—23no—
632C22—23no—
633C22—23no—
634C22—23no—
635C22—412no—
636C22—46no—
637C22—46no—
638C22—46no—
639C22—1612no—
640C22C2×#41616yes—
641C22#289712yes—
642C22C23×C22⋊C411yescharacteristic
643C4—21no—
644C4—21no—
645C4—23no—
646C4—23no—
647C4—21no—
648C4—23no—
649C4—23no—
650C4—23no—
651C4—42no—
652C4—42no—
653C4—46no—
654C4C2×#41616yes—
655C4#289712yes—
656C4C22×#13512yes—
657C4C2×#57611yescharacteristic
658C4C2×#57711yescharacteristic
659C2—22no—
660C2—26no—
661C2—43no—
662C2—86no—
663C2#6007611yescharacteristic
664C2C22×#41611yescharacteristic
665C2C2×#289711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C22⋊C4
prime radical#62800C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2988C2semidirect
3. #60127C2semidirect
4. #17727C2semidirect
5. #17724C2semidirect
6. #17723C2semidirect
7. #17720C2semidirect
8. #17719C2semidirect
9. #17717C2semidirect
10. #17700C2semidirect
11. C2×#1540C2semidirect
34. #2988C22semidirect
37. #2925C22semidirect
40. #2920C22semidirect
41. C2×#431C22semidirect
42. C2×#431C22semidirect
44. C22×C4⋊C8C22semidirect
47. #1553C22semidirect
48. #1552C22semidirect
49. #1551C22semidirect
50. #1550C22semidirect
51. #1549C22semidirect
52. #1547C22semidirect
53. #1545C22semidirect
54. #1544C22semidirect
55. #1543C22semidirect
56. #1542C22semidirect
57. #1542C22semidirect
58. #1542C22semidirect
59. #1540C22semidirect
60. #1540C22semidirect
61. #1540C22semidirect
62. #1540C22semidirect
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
C2×#29882
#601272
#177272
#177242
#177232
#177202
#177192
#177172
#177002
C2×#15402