Finite Group Databasebeta

← #497582

Subgroups — #497582

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39595C211yesmaximal, characteristic
2Q8×#100C211yesmaximal, characteristic, complemented
3Q8×#99C211yesmaximal, characteristic, complemented
4#13297C212yesmaximal, complemented
5#13295C211yesmaximal, characteristic, complemented
6#13294C212yesmaximal, complemented
7#13292C211yesmaximal, characteristic, complemented
8#12795C212yesmaximal
9#12793C211yesmaximal, characteristic
10#12622C212yesmaximal
11C4×AΣL(1,𝔽16)C211yesmaximal, characteristic
12C2×D4×Q8C2211yescharacteristic
13C22×C22⋊Q8C411yescharacteristic, complemented
14Q8×M16—21no—
15#2848C411yescharacteristic, complemented
16#2732—22no—
17#2729—22no—
18#2626C412yescomplemented
19#2583C412yescomplemented
20#2579C2212yes—
21#2538—21no—
22#2502C411yescharacteristic, complemented
23#2487C2211yescharacteristic
24#2429—21no—
25#2423—21no—
26#2409—21no—
27Q8×C22⋊C4—21no—
28Q8×C22⋊C4C411yescharacteristic, complemented
29#2321C2212yes—
30C4×#120C2211yescharacteristic
31C2×AΣL(1,𝔽16)C2211yescharacteristic
32C2×AΣL(1,𝔽16)C2212yes—
33C2×AΣL(1,𝔽16)C2211yescharacteristic
34#1997—21no—
35#1997—22no—
36#1996—22no—
37#1996—21no—
38#1800C2211yescharacteristic
39#1800C2212yes—
40#1799C2211yescharacteristic
41#1799C2212yes—
42C4×#100C2211yescharacteristic
43C4×#100C2212yes—
44C4×#99C2211yescharacteristic
45C4×#99C2212yes—
46#1390C2211yescharacteristic
47#1390C2211yescharacteristic
48#1390C2212yes—
49#1390C2212yes—
50#1390C2212yes—
51#1390C2212yes—
52#1390C2212yes—
53C23×Q8C2×C411yescharacteristic
54C23×Q8D411yescharacteristic
55C24×C4C2×C411yescharacteristic
56D4×Q8—41no—
57D4×Q8—41no—
58D4×Q8—42no—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D412yes—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—41no—
66C2×C22⋊Q8—41no—
67C2×C22⋊Q8—42no—
68C2×C22⋊Q8C2×C412yes—
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×C22⋊Q8D412yes—
71C2×C22⋊Q8D411yescharacteristic
72C2×#120C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—41no—
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4C2312yes—
78C2×C4×D4C2311yescharacteristic
79C22×C4⋊C4C2×C412yes—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C412yes—
82#446—41no—
83#446—42no—
84C8×Q8—41no—
85#423—21no—
86#423—22no—
87C2×#100C2312yes—
88C2×#100C2311yescharacteristic
89C2×#100C2311yescharacteristic
90C2×#99C2312yes—
91C2×#99C2311yescharacteristic
92C2×#99C2311yescharacteristic
93C4×M16—21no—
94C4×M16—22no—
95#400—21no—
96#399—22no—
97#399—21no—
98#398—22no—
99#398—22no—
100#398—22no—
101#396—22no—
102#396—22no—
103#396—22no—
104#395—22no—
105#393—22no—
106#391—22no—
107#391—21no—
108#391—21no—
109#391—41no—
110#391—42no—
111#390—22no—
112#389—22no—
113#389—22no—
114#389—21no—
115#388—22no—
116#388—22no—
117#387—22no—
118#386—22no—
119#386—21no—
120#386C2312yes—
121#386C2311yescharacteristic
122#385—22no—
123#385—22no—
124#385—21no—
125#385C2×C412yes—
126#385C2×C412yes—
127#385C2×C412yes—
128#385C2×C411yescharacteristic
129#385C2×C412yes—
130#384—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380—22no—
135#380C2×C412yes—
136#380C2×C411yescharacteristic
137#379C2×C412yes—
138#379C2×C411yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4C2×C412yes—
143C4×C22⋊C4C2×C411yescharacteristic
144AΣL(1,𝔽16)—22no—
145AΣL(1,𝔽16)—24no—
146AΣL(1,𝔽16)Q814yescomplemented
147#343—21no—
148#343—22no—
149#343—21no—
150#343—22no—
151#342—21no—
152#342—22no—
153#342—21no—
154#342—22no—
155C25C22×C411yescharacteristic
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—22no—
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—82no—
163C22×Q8—82no—
164C22×Q8C22⋊C411yescharacteristic
165C22×Q8C22⋊C411yescharacteristic
166C22×Q8C22⋊C412yes—
167C22×Q8C22⋊C412yes—
168C22×Q8C22⋊C411yescharacteristic
169C22×D4—21no—
170C22×D4C2411yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—82no—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16—21no—
185C4⋊Q8—41no—
186C4⋊Q8—42no—
187C4⋊Q8—41no—
188C4⋊Q8—42no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C22⋊Q8—41no—
194C22⋊Q8—42no—
195C22⋊Q8—41no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—42no—
204C22⋊Q8—82no—
205#120—21no—
206#120—21no—
207#120C2×Q812yes—
208#120#4112yes—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—22no—
212C4×Q8—82no—
213C4×Q8—84no—
214C4×D4—41no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—41no—
218C4×D4—82no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C412yes—
238C2×C4⋊C4C22⋊C412yes—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C22⋊C412yes—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D412yes—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4C22×C411yescharacteristic
258C2×C22⋊C4C22×C412yes—
259C2×C22⋊C4C22×C411yescharacteristic
260C2×C22⋊C4C22×C412yes—
261C2×C22⋊C4C22×C411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D412yes—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—42no—
267C2×C42—41no—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D412yes—
270C4⋊C8—42no—
271C4⋊C8—41no—
272C4⋊C8—42no—
273C4⋊C8—41no—
274#100—21no—
275#100—21no—
276#100—22no—
277#100—22no—
278#100—21no—
279#100C2×Q812yes—
280#99—21no—
281#99—21no—
282#99—22no—
283#99—22no—
284#99—21no—
285#99C2×Q812yes—
286#97—42no—
287#97—41no—
288C4×C8—42no—
289C4×C8—41no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—21no—
300#95—22no—
301#95—41no—
302#95—42no—
303#95—41no—
304#95—42no—
305C24—21no—
306C24—21no—
307C24—41no—
308C24—41no—
309C24—81no—
310C24—81no—
311C24C4×Q812yescomplemented
312C24C23×C411yescharacteristic
313C24C22×D411yescharacteristic
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—21no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—44no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8#9911yescharacteristic
334C2×Q8#9911yescharacteristic
335C2×Q8#9912yes—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—81no—
341C2×D4—81no—
342C2×D4C22×Q811yescharacteristic
343C2×D4C2×#4111yescharacteristic
344C2×D42−1+411yescharacteristic
345C2×D42−1+411yescharacteristic
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—82no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C22×D411yescharacteristic
388M16—22no—
389M16—41no—
390M16—42no—
391M16—41no—
392M16—41no—
393M16—42no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—41no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—81no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—84no—
420C4⋊C4—82no—
421C22⋊C4—21no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C22⋊C4—42no—
428C22⋊C4—41no—
429C22⋊C4—44no—
430C22⋊C4—41no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—41no—
435C22⋊C4—42no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—42no—
439C22⋊C4—82no—
440C22⋊C4—81no—
441C22⋊C4—82no—
442C22⋊C4—81no—
443C22⋊C4—81no—
444C22⋊C4—82no—
445C22⋊C4C4×Q812yescomplemented
446C42—22no—
447C42—21no—
448C42—22no—
449C42—21no—
450C42—42no—
451C42—82no—
452C42—84no—
453C42—82no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—84no—
476C23C22×C22⋊C411yescharacteristic
477C23C2×C4×Q811yescharacteristic
478C23#51911yescharacteristic
479C23#54811yescharacteristic
480C23D4×Q811yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—42no—
484Q8—42no—
485Q8—44no—
486Q8—42no—
487Q8—42no—
488Q8—81no—
489Q8—82no—
490Q8—82no—
491Q8—82no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—82no—
497D4—82no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—84no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4C2×#9912yes—
552C2×C4C2×#9912yes—
553C2×C4C2×#9911yescharacteristic
554C2×C4C2×#9911yescharacteristic
555C2×C4#54811yescharacteristic
556C2×C4D4×Q811yescharacteristic
557C8—42no—
558C8—82no—
559C8—81no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—84no—
582C22—84no—
583C22—81no—
584C22—81no—
585C22Q8×C22⋊C411yescharacteristic
586C22C22×#9911yescharacteristic
587C22#288411yescharacteristic
588C4—21no—
589C4—22no—
590C4—22no—
591C4—21no—
592C4—21no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—82no—
602C4—82no—
603C4—81no—
604C4—82no—
605C4—162no—
606C4—161no—
607C4#212112yes—
608C2—21no—
609C2—21no—
610C2—41no—
611C2—41no—
612C2—41no—
613C2—41no—
614C2—82no—
615C2Q8×#9911yescharacteristic
616C2C2×#212111yescharacteristic
617C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#120C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×D4×Q8 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#100C2semidirect
3. Q8×#99C2semidirect
4. #13297C2semidirect
5. #13295C2semidirect
6. #13294C2semidirect
7. #13292C2semidirect
13. C22×C22⋊Q8C4semidirect
15. #2848C4semidirect
18. #2626C4semidirect
19. #2583C4semidirect
22. #2502C4semidirect
28. Q8×C22⋊C4C4semidirect
146. AΣL(1,𝔽16)Q8semidirect
311. C24C4×Q8semidirect
445. C22⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#395952
Q8×#1002
Q8×#992
#132972
#132952
#132942
#132922
#127952
#127932
#126222
C4×AΣL(1,𝔽16)2