Finite Group Databasebeta

← #495167

Subgroups — #495167

Order 512.

1275 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35633C211yesmaximal, characteristic
2C2×#2123C212yesmaximal, complemented
3C2×#2120C212yesmaximal
4#20640C211yesmaximal, characteristic, complemented
5#20393C211yesmaximal, characteristic, complemented
6#13222C214yesmaximal, complemented
7#13220C214yesmaximal, complemented
8C2×#536C2211yescharacteristic
9C2×#525C2211yescharacteristic
10C22×C22⋊Q8C411yescharacteristic, complemented
11C2×#520C2211yescharacteristic
12#2905—21no—
13C2×#413C2212yes—
14C2×#410C2212yes—
15#2534C412yescomplemented
16#2534C2212yes—
17#2486C412yescomplemented
18#2486C2212yes—
19#2429—22no—
20#2427—22no—
21#2417—21no—
22#2417—21no—
23#2415—21no—
24#2415—21no—
25#2316C411yescharacteristic, complemented
26#2310C411yescharacteristic, complemented
27#2280C411yescharacteristic, complemented
28#2274—21no—
29#2123—28no—
30#2120—28no—
31C2×#352C2214yes—
32C2×#352C2218yes—
33C2×AΣL(1,𝔽16)C2214yes—
34#1801—22no—
35#1800C2214yes—
36C4×#99—22no—
37C4×#99C2214yes—
38C2×#343—21no—
39#1740—21no—
40C22×#41C2311yescharacteristic
41C23×Q8C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C22×M16—22no—
44#536—24no—
45#536—24no—
46C2×C4⋊Q8—21no—
47C2×#124—21no—
48#525—24no—
49#525—24no—
50C2×#123C2×C411yescharacteristic
51C2×#123C2×C412yes—
52C2×#123C2311yescharacteristic
53C2×#123C2312yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—44no—
61C2×#121—21no—
62C2×#120C2×C411yescharacteristic
63C2×#120C2311yescharacteristic
64#520—44no—
65C2×C4×Q8D412yes—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4D412yes—
70C2×C4×D4C2312yes—
71C2×#117—22no—
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C412yes—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C22⋊C4C2×C412yes—
82#424—44no—
83#423—24no—
84#413—24no—
85#413—24no—
86C2×#101—22no—
87C2×#100C2314yes—
88#410—24no—
89#410D418yes—
90C2×#99—22no—
91C2×#99C2314yes—
92#398—22no—
93#398—22no—
94#397—22no—
95#397—22no—
96#396—22no—
97#396—22no—
98#394—22no—
99#394—22no—
100#391—21no—
101#391—21no—
102#391—22no—
103#389—21no—
104#389—22no—
105#389—21no—
106#389—22no—
107#389—22no—
108#388—22no—
109#388—22no—
110#386—22no—
111#386—22no—
112#381C2×C412yes—
113#381C2×C412yes—
114#381C2×C412yes—
115#380—22no—
116#380—22no—
117#380—21no—
118#380—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121#379C2×C412yes—
122C4×C4⋊C4—44no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—24no—
125#352—28no—
126#352—216no—
127AΣL(1,𝔽16)—28no—
128#343—24no—
129#342—24no—
130C25C22×C411yescharacteristic
131C2×#41—42no—
132C2×#41—42no—
133C2×#41C2×D414yes—
134C2×#41#4114yes—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—24no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8C2×D411yescharacteristic
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C22×C411yescharacteristic
144C22×D4C2×D411yescharacteristic
145C22×D4C2411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—24no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C22⋊C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C2×M16—24no—
160C2×M16—42no—
161C2×M16—44no—
162C2×M16—44no—
163C22×C8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166#124—44no—
167#123—24no—
168#123—28no—
169#123—22no—
170#123—24no—
171#123—22no—
172#123—24no—
173C22⋊Q8—44no—
174C22⋊Q8—48no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—44no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—44no—
181#121—44no—
182#120—24no—
183#120—22no—
184#120—22no—
185C4×Q8—44no—
186C4×D4—44no—
187C4×D4—44no—
188C4×D4—44no—
189C4×D4—44no—
190C4×D4—42no—
191C4×D4—42no—
192#117—24no—
193#117—24no—
194#117—24no—
195#117—28no—
196#117—42no—
197#117—42no—
198#117—44no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C2×D412yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22×C411yescharacteristic
235C2×C22⋊C4C22×C411yescharacteristic
236C2×C22⋊C4C22×C412yes—
237C2×C22⋊C4C22×C412yes—
238C2×C22⋊C4C2×D412yes—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—42no—
242C2×C42—44no—
243C2×C42C22⋊C412yes—
244C2×C42C22⋊C412yes—
245C2×C42C2×D412yes—
246C4⋊C8—48no—
247#101—44no—
248#100—24no—
249#100—24no—
250#99—28no—
251#99—24no—
252#99C2×D418yes—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—44no—
262C24—21no—
263C24—22no—
264C24—41no—
265C24—42no—
266C24—41no—
267C24—81no—
268C24C2×C22⋊C411yescharacteristic
269C24C23×C411yescharacteristic
270C24C22×D411yescharacteristic
271#41—44no—
272#41—44no—
273#41—44no—
274#41—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×Q8#12412yes—
286C2×Q8#12712yes—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—81no—
298C2×D4—81no—
299C2×D4C4×D414yes—
300C2×D4#12114yes—
301C2×D4C22×D412yes—
302C2×D4C2×#4112yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C4×D414yes—
343C22×C4#12114yes—
344C22×C4#12412yes—
345C22×C4#12712yes—
346C22×C4C22×D411yescharacteristic
347M16—44no—
348M16—44no—
349M16—84no—
350C2×C8—44no—
351C2×C8—84no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—48no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4#9914yes—
372C22⋊C4—24no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—24no—
376C22⋊C4—28no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—42no—
385C22⋊C4—44no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—82no—
391C42—24no—
392C42—42no—
393C42—42no—
394C42—48no—
395C42—42no—
396C42—42no—
397C42—164no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—42no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—82no—
420C23—84no—
421C23—84no—
422C23—82no—
423C23#39012yes—
424C23C22×C22⋊C411yescharacteristic
425C23C2×C4×D412yes—
426C23C2×#12112yes—
427Q8—42no—
428Q8—42no—
429Q8—84no—
430Q8—84no—
431D4—44no—
432D4—44no—
433D4—42no—
434D4—42no—
435D4—84no—
436D4—84no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—82no—
479C2×C4—84no—
480C2×C4—162no—
481C2×C4#39012yes—
482C2×C4#39012yes—
483C2×C4#39012yes—
484C2×C4C2×#9914yes—
485C2×C4C2×#9912yes—
486C2×C4#41012yes—
487C2×C4C2×#12411yescharacteristic
488C2×C4C2×#12711yescharacteristic
489C8—84no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—42no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—84no—
513C22—84no—
514C22—81no—
515C22—81no—
516C22C2×#39011yescharacteristic
517C22C22×#9911yescharacteristic
518C22C2×#41011yescharacteristic
519C4—24no—
520C4—21no—
521C4—22no—
522C4—21no—
523C4—22no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—81no—
529C4—81no—
530C4—88no—
531C4—81no—
532C4—82no—
533C4#196414yes—
534C2—21no—
535C2—21no—
536C2—42no—
537C2—42no—
538C2—42no—
539C2—82no—
540C2C2×#196411yescharacteristic
541C2#3334911yescharacteristic
542C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#525C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2123C2semidirect
4. #20640C2semidirect
5. #20393C2semidirect
6. #13222C2semidirect
7. #13220C2semidirect
10. C22×C22⋊Q8C4semidirect
15. #2534C4semidirect
17. #2486C4semidirect
25. #2316C4semidirect
26. #2310C4semidirect
27. #2280C4semidirect

Maximal subgroups

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

subgroupindex
#356332
C2×#21232
C2×#21202
#206402
#203932
#132222
#132202