Finite Group Databasebeta

← #497583

Subgroups — #497583

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39594C211yesmaximal, characteristic
2#22515C211yesmaximal, characteristic, complemented
3#22513C211yesmaximal, characteristic, complemented
4#13296C211yesmaximal, characteristic, complemented
5#13295C212yesmaximal, complemented
6#13293C211yesmaximal, characteristic, complemented
7#13292C212yesmaximal, complemented
8#12795C212yesmaximal
9#12793C211yesmaximal, characteristic
10#12622C212yesmaximal
11C4×AΣL(1,𝔽16)C211yesmaximal, characteristic
12C2×#554C2211yescharacteristic
13C22×C22⋊Q8C411yescharacteristic, complemented
14Q8×M16—21no—
15#2848C411yescharacteristic, complemented
16#2720—22no—
17#2692—22no—
18#2626C412yescomplemented
19#2542—21no—
20#2502C412yescomplemented
21#2492—21no—
22#2487C2212yes—
23#2434C411yescharacteristic, complemented
24#2429—21no—
25#2417—21no—
26#2409C2211yescharacteristic
27#2345—21no—
28#2345C411yescharacteristic, complemented
29#2321C2212yes—
30C4×#120C2211yescharacteristic
31C2×AΣL(1,𝔽16)C2211yescharacteristic
32C2×AΣL(1,𝔽16)C2212yes—
33C2×AΣL(1,𝔽16)C2211yescharacteristic
34#1998—22no—
35#1997—21no—
36#1996—22no—
37#1996—21no—
38#1800C2212yes—
39#1800C2211yescharacteristic
40#1799C2212yes—
41#1799C2211yescharacteristic
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
54C24×C4C2×C411yescharacteristic
55#554—41no—
56#554—42no—
57#554—41no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8D411yescharacteristic
60C2×#125D412yes—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—42no—
67C2×C22⋊Q8—41no—
68C2×C22⋊Q8—41no—
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×C22⋊Q8C2×C412yes—
71C2×C22⋊Q8D412yes—
72C2×C22⋊Q8D411yescharacteristic
73C2×#120C2311yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4C2312yes—
77C2×C4×D4C2311yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C412yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C22⋊C4C2×C412yes—
82#446—42no—
83#446—41no—
84C8×Q8—41no—
85#423—21no—
86#423—22no—
87C2×#100C2311yescharacteristic
88C2×#100C2312yes—
89C2×#100C2311yescharacteristic
90C2×#99C2311yescharacteristic
91C2×#99C2312yes—
92C2×#99C2311yescharacteristic
93C4×M16—21no—
94C4×M16—22no—
95#400—22no—
96#399—22no—
97#399—22no—
98#398—22no—
99#398—22no—
100#396—22no—
101#396—22no—
102#395—22no—
103#395—21no—
104#393—22no—
105#393—21no—
106#391—22no—
107#391—21no—
108#391—21no—
109#390—22no—
110#389—21no—
111#389—22no—
112#389—41no—
113#388—22no—
114#388—22no—
115#387—22no—
116#387—42no—
117#386—22no—
118#386—21no—
119#386C2312yes—
120#386C2311yescharacteristic
121#385—22no—
122#385—21no—
123#385C2×C412yes—
124#385C2×C412yes—
125#385C2×C412yes—
126#385C2×C412yes—
127#385C2×C411yescharacteristic
128#384—22no—
129#380—22no—
130#380—21no—
131#380—22no—
132#380—21no—
133#380C2×C411yescharacteristic
134#380C2×C412yes—
135#379C2×C412yes—
136#379C2×C411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—41no—
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—22no—
148#343—21no—
149#343—22no—
150#343—21no—
151#342—21no—
152#342—22no—
153#342—21no—
154#342—22no—
155C25C22×C411yescharacteristic
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—82no—
161C22×Q8C22⋊C411yescharacteristic
162C22×Q8C22⋊C412yes—
163C22×D4—21no—
164C22×D4C2411yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—82no—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—22no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—82no—
182#125—42no—
183#125—42no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196C22⋊Q8—42no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C22⋊Q8—82no—
200C22⋊Q8—82no—
201#120—21no—
202#120—21no—
203#120C2×Q812yes—
204#120#4112yes—
205C4×Q8—21no—
206C4×Q8—22no—
207C4×Q8—21no—
208C4×D4—42no—
209C4×D4—41no—
210C4×D4—42no—
211C4×D4—41no—
212C4×D4—82no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C412yes—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22⋊C412yes—
238C2×C4⋊C4C22⋊C412yes—
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D412yes—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4C22×C411yescharacteristic
257C2×C22⋊C4C22×C412yes—
258C2×C22⋊C4C22×C411yescharacteristic
259C2×C22⋊C4C22×C411yescharacteristic
260C2×C22⋊C4C22×C412yes—
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D412yes—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—41no—
266C2×C42—42no—
267C2×C42C2×D412yes—
268C2×C42C2×D411yescharacteristic
269C4⋊C8—42no—
270C4⋊C8—41no—
271C4⋊C8—42no—
272C4⋊C8—41no—
273#100—21no—
274#100—21no—
275#100—22no—
276#100—21no—
277#100—22no—
278#100C2×Q812yes—
279#99—21no—
280#99—21no—
281#99—22no—
282#99—21no—
283#99—22no—
284#99C2×Q812yes—
285#97—42no—
286#97—41no—
287C4×C8—42no—
288C4×C8—41no—
289#95—21no—
290#95—21no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—22no—
297#95—22no—
298#95—21no—
299#95—22no—
300#95—41no—
301#95—42no—
302#95—42no—
303#95—41no—
304C24—21no—
305C24—21no—
306C24—41no—
307C24—41no—
308C24—81no—
309C24—81no—
310C24C4×Q812yes—
311C24C23×C411yescharacteristic
312C24C22×D411yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—81no—
331C2×D4—81no—
332C2×D4C22×Q811yescharacteristic
333C2×D4C2×#4111yescharacteristic
334C2×D42−1+411yescharacteristic
335C2×D42−1+411yescharacteristic
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C412yes—
377C22×C4C22×D411yescharacteristic
378M16—22no—
379M16—41no—
380M16—41no—
381M16—42no—
382M16—41no—
383M16—42no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—41no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—82no—
406C4⋊C4—81no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—162no—
413C4⋊C4#9912yes—
414C4⋊C4#9911yescharacteristic
415C4⋊C4#9911yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—41no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—42no—
434C22⋊C4—82no—
435C22⋊C4—81no—
436C22⋊C4—81no—
437C22⋊C4—81no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4C4×Q812yes—
441C42—22no—
442C42—21no—
443C42—22no—
444C42—21no—
445C42—42no—
446C42—82no—
447C42—82no—
448C42—162no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—84no—
471C23C22×C22⋊C411yescharacteristic
472C23C2×C4×Q811yescharacteristic
473C23#51911yescharacteristic
474C23#54811yescharacteristic
475C23D4×Q811yescharacteristic
476Q8—42no—
477Q8—81no—
478Q8—82no—
479Q8—84no—
480Q8—82no—
481Q8—162no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—82no—
487D4—82no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—84no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4C2×#9912yes—
542C2×C4C2×#9912yes—
543C2×C4C2×#9911yescharacteristic
544C2×C4C2×#9911yescharacteristic
545C2×C4#54811yescharacteristic
546C2×C4D4×Q811yescharacteristic
547C8—42no—
548C8—82no—
549C8—81no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—84no—
572C22—84no—
573C22—81no—
574C22—81no—
575C22Q8×C22⋊C411yescharacteristic
576C22C22×#9911yescharacteristic
577C22#288411yescharacteristic
578C4—21no—
579C4—22no—
580C4—22no—
581C4—21no—
582C4—21no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—81no—
591C4—82no—
592C4—82no—
593C4—81no—
594C4—82no—
595C4—161no—
596C4—162no—
597C4#212212yes—
598C2—21no—
599C2—21no—
600C2—41no—
601C2—41no—
602C2—41no—
603C2—41no—
604C2—82no—
605C2Q8×#9911yescharacteristic
606C2C2×#212211yescharacteristic
607C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#120C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#554 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22515C2semidirect
3. #22513C2semidirect
4. #13296C2semidirect
5. #13295C2semidirect
6. #13293C2semidirect
7. #13292C2semidirect
13. C22×C22⋊Q8C4semidirect
15. #2848C4semidirect
18. #2626C4semidirect
20. #2502C4semidirect
23. #2434C4semidirect
28. #2345C4semidirect
146. AΣL(1,𝔽16)Q8semidirect

Maximal subgroups

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

subgroupindex
#395942
#225152
#225132
#132962
#132952
#132932
#132922
#127952
#127932
#126222
C4×AΣL(1,𝔽16)2