Finite Group Databasebeta

← #6393845

Subgroups — #6393845

Order 512.

3277 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62800C211yesmaximal, characteristic
2#60263C211yesmaximal, characteristic, complemented
3#34125C211yesmaximal, characteristic, complemented
4#22550C214yesmaximal, complemented
5#22435C2112yesmaximal, complemented
6#21857C213yesmaximal, complemented
7#21816C213yesmaximal, complemented
8#21416C213yesmaximal, complemented
9#21209C213yesmaximal, complemented
10#3522—21no—
11#3519—23no—
12#3517C412yescomplemented
13#3514C416yescomplemented
14#3511C413yescomplemented
15#3508C413yescomplemented
16#3503C411yescharacteristic, complemented
17#3498C411yescharacteristic, complemented
18#3494—23no—
19#3488—21no—
20#3483—21no—
21#3481—23no—
22#3480C2212yes—
23D4×#41C2216yes—
24#3460—23no—
25#3444C2213yes—
26C2×#532C2213yes—
27C4×H(5,𝔽2)—21no—
28C2×C4×#41C2211yescharacteristic
29#2991—21no—
30#2990—23no—
31#2989—23no—
32#2988—21no—
33Q8×M16C2212yescomplemented
34D4×M16C2216yescomplemented
35#2920C2213yescomplemented
36C2×#423C2213yescomplemented
37C2×#421C22112yescomplemented
38#2357—23no—
39#2351—21no—
40#2347—23no—
41#2341—21no—
42#2293C2216yescomplemented
43#2283C2213yescomplemented
44C42×Q8C2212yescomplemented
45C4×#117C2213yescomplemented
46#1995C2213yescomplemented
47#1995C2211yescharacteristic, complemented
48#1980C22112yescomplemented
49#1963C2213yescomplemented
50#1963C2216yescomplemented
51#1963C2213yescomplemented
52#1962C22112yescomplemented
53#1897C22112yescomplemented
54#1871C22112yescomplemented
55#1806C22112yescomplemented
56#1789C22112yescomplemented
57C4×#104C22112yescomplemented
58C2×H(5,𝔽2)—21no—
59C22×#41D411yescharacteristic
60C22×#41C2316yes—
61C22×M16C2316yes—
62#559—23no—
63#559C2×C416yes—
64Q82—21no—
65#557C2×C412yes—
66#556C2×C416yes—
67#555—23no—
68#555—26no—
69#554—23no—
70#554—23no—
71#553—26no—
72#553—26no—
73#553—26no—
74#551—212no—
75#550—22no—
76#550—21no—
77#550—23no—
78#550C2×C416yes—
79#550C2×C412yes—
80D4×Q8C2×C416yes—
81#548—22no—
82#548C2×C4112yes—
83#547—26no—
84#547C2×C416yes—
85#546—26no—
86#546—26no—
87#546—212no—
88#546C2×C4112yes—
89D42—26no—
90D42—23no—
91D42C2×C416yes—
92#544—23no—
93#542—26no—
94#542—26no—
95#540—26no—
96#540—26no—
97#538—212no—
98#537—23no—
99#535—23no—
100#535—23no—
101#533C2×C413yes—
102#533C2×C413yes—
103#533C2311yescharacteristic
104#532—26no—
105#532—23no—
106#532—23no—
107#532—23no—
108#532—23no—
109#532C2×C416yes—
110#532C2×C416yes—
111#532C2×C413yes—
112#532C2×C413yes—
113#532C2313yes—
114#532C2316yes—
115C2×C4⋊Q8C2×C413yes—
116C2×#127C2×C413yes—
117#529—26no—
118#529—23no—
119#529—23no—
120#529—23no—
121#529—23no—
122#529C2×C416yes—
123#529C2×C416yes—
124C2×#124C2×C416yes—
125#525—23no—
126#525—23no—
127#525—212no—
128#525C2316yes—
129C2×#121—23no—
130C2×#121—23no—
131#520—21no—
132#520—22no—
133#520—23no—
134#518—26no—
135#518—23no—
136C4×#41—22no—
137C4×#41—23no—
138C4×#41—23no—
139C4×#41C2×C411yescharacteristic
140C4×#41C2×C413yes—
141C4×#41C2×C413yes—
142C4×#41C2×C411yescharacteristic
143C4×#41D411yescharacteristic
144C4×#41D413yes—
145C4×#41D413yes—
146C4×#41D411yescharacteristic
147C4×#41C2316yes—
148C2×C4×Q8—22no—
149C2×C4×Q8D411yescharacteristic
150C2×C4×Q8C2311yescharacteristic
151C2×C4×D4—26no—
152C2×C4×D4—23no—
153C2×C4×D4—23no—
154C2×C4×D4D413yes—
155C2×C4×D4C2313yes—
156C2×#117D413yes—
157C2×#117C2316yes—
158C22×C42C2313yes—
159#446—26no—
160C8×Q8—22no—
161#436—26no—
162#435—212no—
163C8×D4—26no—
164#432—23no—
165#432—23no—
166#431—23no—
167#431—23no—
168#423—23no—
169#423—23no—
170#423C2316yes—
171#423C2313yes—
172#423C2311yescharacteristic
173C2×C4⋊C8—23no—
174C2×C4⋊C8—23no—
175#421—424no—
176C2×#104C23112yes—
177C2×#104C23124yes—
178C2×#104C23112yes—
179#407C2316yes—
180C4×M16C2316yes—
181#390—26no—
182#389—26no—
183#388—212no—
184#383—23no—
185#382—23no—
186#381—23no—
187#380—23no—
188C4×C4⋊C4—26no—
189C4×C4⋊C4—23no—
190C4×C4⋊C4—23no—
191C4×C4⋊C4C2316yes—
192C4×C22⋊C4—23no—
193C4×C22⋊C4—23no—
194C4×C22⋊C4C2316yes—
195#376C2311yescharacteristic
196#376C2313yes—
197C43C2316yes—
198#339C23112yes—
199#339C2314yes—
200H(5,𝔽2)—48no—
201C2×#41—22no—
202C2×#41—23no—
203C2×#41—23no—
204C2×#41—26no—
205C2×#41—26no—
206C2×#41—412no—
207C2×#41—412no—
208C2×#41—44no—
209C2×#41C22⋊C411yescharacteristic
210C2×#41C22⋊C411yescharacteristic
211C2×#41C22⋊C413yes—
212C2×#41C22⋊C413yes—
213C2×#41C22×C4112yes—
214C2×#41C2416yes—
215C22×Q8C22×C416yes—
216C22×Q8C2×D411yescharacteristic
217C22×D4—26no—
218C22×D4—26no—
219C22×D4—23no—
220C22×D4—23no—
221C22×D4C22×C416yes—
222C22×D4C2×D413yes—
223C23×C4—26no—
224C23×C4C2×D413yes—
225C23×C4C2×D413yes—
226C23×C4C2413yes—
227C2×M16—26no—
228C2×M16—412no—
229C2×M16C2416yes—
230C2×M16C2412yes—
231C22×C8—26no—
232C4⋊Q8—26no—
233C4⋊Q8—23no—
234C4⋊Q8—23no—
235C4⋊Q8C22×C413yes—
236C4⋊Q8C22×C416yes—
237C4⋊Q8C22×C411yescharacteristic
238C4⋊Q8C22×C411yescharacteristic
239#127—26no—
240#127—23no—
241#127—23no—
242#127C22×C413yes—
243#127C22×C416yes—
244#126—212no—
245#126—26no—
246#126—26no—
247#125—26no—
248#125—23no—
249#125—23no—
250#125C22×C412yes—
251#124—26no—
252#124—212no—
253#124—26no—
254#124—26no—
255#124—23no—
256#124—23no—
257#124C22×C416yes—
258#124C22×C4112yes—
259#123—26no—
260#123—26no—
261#123—26no—
262#123—212no—
263#123—212no—
264#123—212no—
265C22⋊Q8—26no—
266C22⋊Q8—23no—
267C22⋊Q8—23no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—212no—
271C22⋊Q8—212no—
272C22⋊Q8C22×C416yes—
273C22⋊Q8C22×C416yes—
274#121—212no—
275#121—26no—
276#121—26no—
277#121—26no—
278#121—26no—
279#121—26no—
280#121—23no—
281#121—23no—
282#121—23no—
283#121—23no—
284#121—26no—
285#121—26no—
286#121—26no—
287#121—26no—
288#121—212no—
289#121—212no—
290#121C22×C416yes—
291#121C22×C416yes—
292#120—26no—
293#120—212no—
294#120—212no—
295C4×Q8—22no—
296C4×Q8—23no—
297C4×Q8—23no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—48no—
301C4×Q8C22⋊C411yescharacteristic
302C4×Q8C22⋊C413yes—
303C4×Q8C22⋊C411yescharacteristic
304C4×Q8C22⋊C413yes—
305C4×Q8C22×C416yes—
306C4×Q8C22×C416yes—
307C4×Q8C22×C411yescharacteristic
308C4×Q8C22×C413yes—
309C4×Q8C2×D411yescharacteristic
310C4×Q8C2×D413yes—
311C4×D4—26no—
312C4×D4—26no—
313C4×D4—26no—
314C4×D4—26no—
315C4×D4—26no—
316C4×D4—26no—
317C4×D4—26no—
318C4×D4—26no—
319C4×D4—26no—
320C4×D4—212no—
321C4×D4—26no—
322C4×D4—26no—
323C4×D4—26no—
324C4×D4—26no—
325C4×D4—23no—
326C4×D4—23no—
327C4×D4—26no—
328C4×D4—23no—
329C4×D4—23no—
330C4×D4—26no—
331C4×D4C22⋊C413yes—
332C4×D4C22⋊C416yes—
333C4×D4C22⋊C413yes—
334C4×D4C22⋊C413yes—
335C4×D4C22⋊C416yes—
336C4×D4C22⋊C413yes—
337C4×D4C22×C416yes—
338C4×D4C22×C416yes—
339C4×D4C22×C413yes—
340C4×D4C22×C416yes—
341C4×D4C22×C413yes—
342C4×D4C2×D413yes—
343C4×D4C2×D416yes—
344C4×D4C2×D413yes—
345#117—23no—
346#117—23no—
347#117—26no—
348#117—26no—
349#117—26no—
350#117—26no—
351#117—212no—
352#117—46no—
353#117C22⋊C413yes—
354#117C22⋊C413yes—
355#117C22⋊C416yes—
356#117C22⋊C416yes—
357#117C22⋊C413yes—
358#117C22⋊C413yes—
359#117C2412yes—
360C2×C4⋊C4—23no—
361C2×C4⋊C4—23no—
362C2×C4⋊C4—26no—
363C2×C4⋊C4—26no—
364C2×C4⋊C4—26no—
365C2×C4⋊C4C2×D413yes—
366C2×C4⋊C4C2×D413yes—
367C2×C22⋊C4—26no—
368C2×C22⋊C4—26no—
369C2×C22⋊C4—212no—
370C2×C22⋊C4—26no—
371C2×C22⋊C4—26no—
372C2×C22⋊C4C2×D416yes—
373C2×C42—26no—
374C2×C42—26no—
375C2×C42—26no—
376C2×C42C22×C413yes—
377C2×C42C22×C413yes—
378C2×C42C22×C413yes—
379C2×C42C22×C413yes—
380C2×C42C2×D413yes—
381C2×C42C2×D413yes—
382C2×C42C2×D413yes—
383C2×C42C2×D413yes—
384C2×C42C2×D413yes—
385C2×C42C2×D413yes—
386C2×C42C2416yes—
387C2×C42C2412yes—
388C2×C42C2413yes—
389C2×C42C2411yescharacteristic
390C4⋊C8—23no—
391C4⋊C8—23no—
392C4⋊C8—26no—
393C4⋊C8—26no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396#104—224no—
397#104—412no—
398#104—424no—
399C22⋊C8—26no—
400C22⋊C8—26no—
401#97—26no—
402C4×C8—26no—
403#95—21no—
404#95—21no—
405#95—23no—
406#95—23no—
407#95—26no—
408#95—26no—
409C24—26no—
410C24C22×D413yes—
411#41—24no—
412#41—24no—
413#41—212no—
414#41—212no—
415#41—46no—
416#41—46no—
417#41—412no—
418#41—424no—
419#41—48no—
420#41—86no—
421#41—86no—
422#41—812no—
423#41—86no—
424C2×Q8—24no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—26no—
428C2×Q8—412no—
429C2×Q8C2×C22⋊C413yes—
430C2×Q8C2×C22⋊C411yescharacteristic
431C2×Q8C23×C416yes—
432C2×D4—26no—
433C2×D4—26no—
434C2×D4—26no—
435C2×D4—212no—
436C2×D4—23no—
437C2×D4—23no—
438C2×D4—26no—
439C2×D4—26no—
440C2×D4—26no—
441C2×D4—46no—
442C2×D4—412no—
443C2×D4—412no—
444C2×D4—424no—
445C2×D4—86no—
446C2×D4—812no—
447C2×D4C2×C22⋊C413yes—
448C2×D4C2×C22⋊C413yes—
449C2×D4C2×C22⋊C416yes—
450C2×D4C23×C416yes—
451C22×C4—26no—
452C22×C4—26no—
453C22×C4—22no—
454C22×C4—26no—
455C22×C4—26no—
456C22×C4—26no—
457C22×C4—26no—
458C22×C4—212no—
459C22×C4—26no—
460C22×C4—46no—
461C22×C4—86no—
462C22×C4—86no—
463C22×C4—83no—
464C22×C4C2×C22⋊C413yes—
465C22×C4C2×C22⋊C413yes—
466C22×C4C2×C22⋊C413yes—
467C22×C4C2×C22⋊C413yes—
468C22×C4C2×C22⋊C413yes—
469C22×C4C2×C22⋊C413yes—
470C22×C4C2×C22⋊C413yes—
471C22×C4C2×C22⋊C413yes—
472C22×C4C23×C413yes—
473C22×C4C23×C413yes—
474C22×C4C22×D413yes—
475C22×C4C22×D413yes—
476C22×C4C22×D413yes—
477C22×C4C22×D413yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D413yes—
481C22×C4C2511yescharacteristic
482M16—212no—
483M16—24no—
484M16—86no—
485C2×C8—22no—
486C2×C8—26no—
487C2×C8—86no—
488C4⋊C4—23no—
489C4⋊C4—23no—
490C4⋊C4—26no—
491C4⋊C4—26no—
492C4⋊C4—26no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—26no—
498C4⋊C4—26no—
499C4⋊C4—26no—
500C4⋊C4—412no—
501C4⋊C4—412no—
502C4⋊C4C2×C22⋊C413yes—
503C4⋊C4C2×C22⋊C413yes—
504C4⋊C4C2×C22⋊C413yes—
505C4⋊C4C2×C22⋊C416yes—
506C4⋊C4C2×C22⋊C413yes—
507C4⋊C4C2×C22⋊C416yes—
508C4⋊C4C23×C412yes—
509C4⋊C4C23×C412yes—
510C22⋊C4—26no—
511C22⋊C4—26no—
512C22⋊C4—22no—
513C22⋊C4—212no—
514C22⋊C4—26no—
515C22⋊C4—26no—
516C22⋊C4—26no—
517C22⋊C4—212no—
518C22⋊C4—26no—
519C22⋊C4—26no—
520C22⋊C4—26no—
521C22⋊C4—412no—
522C22⋊C4—86no—
523C22⋊C4C2×C22⋊C416yes—
524C22⋊C4C2×C22⋊C416yes—
525C22⋊C4C2×C22⋊C416yes—
526C22⋊C4C2×C22⋊C416yes—
527C42—26no—
528C42—26no—
529C42—22no—
530C42—22no—
531C42—26no—
532C42—26no—
533C42—412no—
534C42—86no—
535C42C2×C22⋊C413yes—
536C42C2×C22⋊C413yes—
537C42C2×C22⋊C413yes—
538C42C2×C22⋊C413yes—
539C42C2×C22⋊C413yes—
540C42C2×C22⋊C413yes—
541C42C2×C22⋊C413yes—
542C42C2×C22⋊C413yes—
543C42C23×C411yescharacteristic
544C42C23×C411yescharacteristic
545C42C23×C413yes—
546C42C23×C413yes—
547C42C22×D413yes—
548C42C22×D413yes—
549C42C22×D411yescharacteristic
550C42C22×D411yescharacteristic
551C42H(5,𝔽2)14yescomplemented
552C23—26no—
553C23—26no—
554C23—43no—
555C23—43no—
556C23—412no—
557C23—86no—
558C23—86no—
559C23C22×C22⋊C413yes—
560C23C22×C22⋊C413yes—
561C23C23×D411yescharacteristic
562Q8—26no—
563Q8—212no—
564Q8—42no—
565Q8—42no—
566Q8—86no—
567Q8#42114yescomplemented
568D4—26no—
569D4—26no—
570D4—212no—
571D4—46no—
572D4—46no—
573D4—46no—
574D4—412no—
575D4—412no—
576D4—412no—
577D4—86no—
578D4—86no—
579D4—812no—
580D4—86no—
581C2×C4—26no—
582C2×C4—26no—
583C2×C4—26no—
584C2×C4—26no—
585C2×C4—26no—
586C2×C4—26no—
587C2×C4—23no—
588C2×C4—23no—
589C2×C4—23no—
590C2×C4—23no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—26no—
594C2×C4—24no—
595C2×C4—43no—
596C2×C4—41no—
597C2×C4—43no—
598C2×C4—43no—
599C2×C4—41no—
600C2×C4—43no—
601C2×C4—43no—
602C2×C4—43no—
603C2×C4—43no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—412no—
607C2×C4—46no—
608C2×C4—46no—
609C2×C4—46no—
610C2×C4—46no—
611C2×C4—46no—
612C2×C4—812no—
613C2×C4—86no—
614C2×C4—86no—
615C2×C4C22×C22⋊C413yes—
616C2×C4C22×C22⋊C411yescharacteristic
617C2×C4C22×C22⋊C411yescharacteristic
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×C4C24×C411yescharacteristic
626C2×C4C23×D411yescharacteristic
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4#58512yes—
629C8—46no—
630C8—42no—
631C22—21no—
632C22—21no—
633C22—23no—
634C22—23no—
635C22—23no—
636C22—23no—
637C22—23no—
638C22—23no—
639C22—46no—
640C22—46no—
641C22—46no—
642C22—86no—
643C22—1612no—
644C22C23×C22⋊C411yescharacteristic
645C22#348812yes—
646C4—21no—
647C4—21no—
648C4—23no—
649C4—21no—
650C4—21no—
651C4—23no—
652C4—23no—
653C4—23no—
654C4—23no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—46no—
659C4—86no—
660C4C2×#42116yes—
661C4C4×H(5,𝔽2)12yes—
662C4#349811yescharacteristic
663C4#350011yescharacteristic
664C2—22no—
665C2—43no—
666C2—43no—
667C2—86no—
668C2#3425811yescharacteristic
669C2C22×#42111yescharacteristic
670C2#6009311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C22⋊C4
prime radical#62800C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60263C2semidirect
3. #34125C2semidirect
4. #22550C2semidirect
5. #22435C2semidirect
6. #21857C2semidirect
7. #21816C2semidirect
8. #21416C2semidirect
9. #21209C2semidirect
12. #3517C4semidirect
13. #3514C4semidirect
14. #3511C4semidirect
15. #3508C4semidirect
16. #3503C4semidirect
17. #3498C4semidirect
33. Q8×M16C22semidirect
34. D4×M16C22semidirect
35. #2920C22semidirect
36. C2×#423C22semidirect
37. C2×#421C22semidirect
42. #2293C22semidirect
43. #2283C22semidirect
44. C42×Q8C22semidirect
45. C4×#117C22semidirect
46. #1995C22semidirect
47. #1995C22semidirect
48. #1980C22semidirect
49. #1963C22semidirect
50. #1963C22semidirect
51. #1963C22semidirect
52. #1962C22semidirect
53. #1897C22semidirect
54. #1871C22semidirect
55. #1806C22semidirect
56. #1789C22semidirect
57. C4×#104C22semidirect
551. C42H(5,𝔽2)semidirect
567. Q8#421semidirect

Maximal subgroups

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

subgroupindex
#628002
#602632
#341252
#225502
#224352
#218572
#218162
#214162
#212092