Finite Group Databasebeta

← #368945

Subgroups — #368945

Order 512.

1424 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic
6#11217C214yesmaximal
7#10899C214yesmaximal
8#10857C211yesmaximal, characteristic
9C4×#339C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—24no—
13#3165C418yescomplemented
14C2×#501—22no—
15C2×#500—21no—
16C2×#498—21no—
17C2×#428C2214yes—
18C2×#423C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2360—22no—
21#2360—22no—
22#2355—21no—
23#2355—21no—
24Q8×C4⋊C4—21no—
25Q8×C4⋊C4—21no—
26#2330—22no—
27#2317C2212yes—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C2×#391—22no—
31C2×#384C2211yescharacteristic
32#2274C2212yes—
33C4×#117C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#1994—24no—
36#1988—21no—
37#1988—21no—
38#1987—21no—
39#1987—21no—
40C2×#343C2214yes—
41#1736C2212yes—
42#1736C2212yes—
43C2×#339C2214yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C4⋊#95—22no—
47C4⋊#95C2214yes—
48C23×Q8—21no—
49C22×M16C2312yes—
50C22×M16C2312yes—
51#533—22no—
52#533—22no—
53#533C2×C414yes—
54#533D414yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yes—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8D411yescharacteristic
65C2×#125D412yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×#117C2311yescharacteristic
73C2×#117C2312yes—
74C2×#117C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#501—216no—
85#500—28no—
86#498—28no—
87#428—28no—
88#428C2×C4116yes—
89C2×#107—24no—
90C2×#106—24no—
91#423—22no—
92#423—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C4×M16—22no—
96C4×M16C2×C414yes—
97C2×#97—21no—
98C2×C4×C8—21no—
99#391—22no—
100#391—22no—
101#391—48no—
102#389—22no—
103#389—22no—
104#389—24no—
105#389—24no—
106#389—22no—
107#389—22no—
108#387—24no—
109#387—24no—
110#385—24no—
111#384—22no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116#382—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—24no—
130#376—21no—
131C2×#95—28no—
132C43—21no—
133C43—22no—
134C43—22no—
135#343—28no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#339—24no—
141#339—28no—
142#339Q818yescomplemented
143C22×Q8—24no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—44no—
147C22×Q8—44no—
148C22×Q8—41no—
149C22×Q8—41no—
150C22×Q8—42no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D414yes—
156C23×C4C2411yescharacteristic
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—24no—
162C2×M16—22no—
163C2×M16—24no—
164C2×M16C22×C414yes—
165C2×M16C2×Q814yes—
166C2×M16#4114yes—
167C22×C8—22no—
168C22×C8—22no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8C22⋊C414yes—
180C4⋊Q8C22⋊C414yes—
181#125—24no—
182#125C22⋊C418yes—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×Q8—84no—
192C4×Q8—84no—
193#117—22no—
194#117—22no—
195#117—22no—
196#117—22no—
197#117—22no—
198#117C22×C414yes—
199#117C2×D414yes—
200#117C2×Q814yes—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C22×C414yes—
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C22×C412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×Q814yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265#107—216no—
266#106—216no—
267C4⋊C8—42no—
268C4⋊C8—42no—
269#97—24no—
270C4×C8—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—416no—
280C24C22×D411yescharacteristic
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—416no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—82no—
291C2×Q8—82no—
292C2×Q8—88no—
293C2×Q8—82no—
294C2×Q8—82no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—28no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4C4×D412yes—
322C22×C4C4×D412yes—
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×Q812yes—
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4111yescharacteristic
337C22×C42−1+412yes—
338C22×C42−1+412yes—
339M16—24no—
340M16—24no—
341M16—24no—
342M16—44no—
343M16—44no—
344M16C4×Q818yescomplemented
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4C2×C22⋊C414yes—
379C4⋊C4C2×C22⋊C414yes—
380C4⋊C4C4⋊Q812yes—
381C4⋊C4C4⋊Q812yes—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—48no—
392C42—48no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—84no—
401C42—84no—
402C42C2×C22⋊C412yes—
403C42C2×C22⋊C412yes—
404C42C4×D412yes—
405C42C4×D412yes—
406C42#12112yes—
407C42#12112yes—
408C42#12312yes—
409C42#12312yes—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23C2×C4×D411yescharacteristic
415C23C2×#12011yescharacteristic
416C23C2×#12111yescharacteristic
417C23#54712yes—
418C23D4×Q812yes—
419Q8—48no—
420Q8—48no—
421Q8—88no—
422Q8—88no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—48no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—84no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—88no—
465C2×C4#38612yes—
466C2×C4#38612yes—
467C2×C4#38612yes—
468C2×C4#38612yes—
469C2×C4#45312yes—
470C2×C4#45412yes—
471C2×C4#45512yes—
472C2×C4#45612yes—
473C2×C4C22×C22⋊C411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4C2×C4×Q812yes—
476C2×C4#51912yes—
477C2×C4C2×#12111yescharacteristic
478C2×C4C2×#12311yescharacteristic
479C2×C4C2×C4⋊Q811yescharacteristic
480C2×C4#53211yescharacteristic
481C2×C4#53312yes—
482C2×C4#53611yescharacteristic
483C2×C4#53611yescharacteristic
484C2×C4#54112yes—
485C2×C4#54812yes—
486C2×C4#54812yes—
487C2×C4D4×Q812yes—
488C2×C4D4×Q812yes—
489C2×C4#55612yes—
490C2×C4#55712yes—
491C8—28no—
492C8—82no—
493C22—22no—
494C22—22no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22C2×#38611yescharacteristic
502C22#236512yes—
503C22#242912yes—
504C22#243211yescharacteristic
505C22#243411yescharacteristic
506C22C2×#45311yescharacteristic
507C22C2×#45511yescharacteristic
508C22C2×#45411yescharacteristic
509C22C2×#45611yescharacteristic
510C4—22no—
511C4—22no—
512C4—48no—
513C4—44no—
514C4—44no—
515C4—42no—
516C4—42no—
517C4—81no—
518C4—84no—
519C4—81no—
520C4—84no—
521C4#188214yes—
522C4#188314yes—
523C4C4×C4⋊Q811yescharacteristic
524C4#232711yescharacteristic
525C4Q8×C22⋊C412yes—
526C4#244612yes—
527C4#254511yescharacteristic
528C4#254611yescharacteristic
529C2—22no—
530C2—22no—
531C2#1402411yescharacteristic
532C2C2×#188211yescharacteristic
533C2C2×#188311yescharacteristic
534C2#2568711yescharacteristic
535C2#2568811yescharacteristic
536C2#2582911yescharacteristic
537C2#2583011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3165C4semidirect
142. #339Q8semidirect
344. M16C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#342142
C2×#19942
#213132
#147142
#112172
#108992
#108572
C4×#3392