Finite Group Databasebeta

← #186621

Subgroups — #186621

Order 512.

2438 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q82C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3#20734C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic
7C2×#1479C211yesmaximal, characteristic
8#8451C218yesmaximal, complemented
9C2×Q82—24no—
10C2×Q82—22no—
11C2×Q82—24no—
12C2×Q82—24no—
13C2×Q82—22no—
14C2×Q82—22no—
15C2×Q82C418yescomplemented
16C2×Q82C2214yes—
17C22×C4⋊Q8—22no—
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8—21no—
22C22×C4×Q8—21no—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419—21no—
29#2285C2211yescharacteristic
30#1981—22no—
31#1956C2214yes—
32#1824—21no—
33#1805—22no—
34#1803C2214yes—
35#1803C2214yes—
36#1787—22no—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1548—24no—
41C2×#333C2214yes—
42#1487—24no—
43#1479—24no—
44C2×Q8⋊C8C2214yes—
45C2×Q8⋊C8C2214yes—
46C23×Q8—22no—
47C23×Q8D411yescharacteristic
48C23×Q8D411yescharacteristic
49C23×Q8D412yes—
50C22×M16—21no—
51Q82—216no—
52Q82—216no—
53Q82—216no—
54Q82—216no—
55Q82—28no—
56Q82—216no—
57Q82—216no—
58Q82—28no—
59Q82—28no—
60Q82C2×C4116yes—
61C2×C4⋊Q8—28no—
62C2×C4⋊Q8—216no—
63C2×C4⋊Q8—28no—
64C2×C4⋊Q8—28no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—24no—
67C2×C4⋊Q8—216no—
68C2×C4⋊Q8—28no—
69C2×C4⋊Q8—28no—
70C2×C4⋊Q8—28no—
71C2×C4⋊Q8—24no—
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8C2×C414yes—
78C2×C4⋊Q8D412yes—
79C2×C4⋊Q8D414yes—
80C2×C4⋊Q8C2312yes—
81C2×C4×Q8—24no—
82C2×C4×Q8—24no—
83C2×C4×Q8—28no—
84C2×C4×Q8—28no—
85C2×C4×Q8—28no—
86C2×C4×Q8—28no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8C2×C414yes—
98C2×C4×Q8C2×C414yes—
99C2×C4×Q8D414yes—
100C2×C4×Q8D414yes—
101C2×C4×Q8D412yes—
102C2×C4×Q8D412yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×#117—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C42—22no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118#431—22no—
119#431—22no—
120#431—22no—
121#431—22no—
122#423—22no—
123#423—22no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128#419—44no—
129C2×#103—24no—
130C2×#103—24no—
131C2×#103—28no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8C2312yes—
136#384C2312yes—
137#382—21no—
138#380—21no—
139#380—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142#340—24no—
143#340—24no—
144#333—28no—
145Q8⋊C8—28no—
146Q8⋊C8—28no—
147C22×Q8—28no—
148C22×Q8—28no—
149C22×Q8—216no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—84no—
159C22×Q8—88no—
160C22×Q8C22⋊C414yes—
161C22×Q8C22⋊C414yes—
162C22×Q8C22⋊C418yes—
163C22×Q8SD1614yes—
164C22×Q8Q1614yes—
165C22×Q8C2×D412yes—
166C22×Q8C2×D412yes—
167C22×Q8C2×D414yes—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—42no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C4⋊Q8—232no—
183C4⋊Q8—264no—
184C4⋊Q8—216no—
185C4⋊Q8—216no—
186C4⋊Q8—232no—
187C4⋊Q8—232no—
188C4⋊Q8—232no—
189C4⋊Q8—216no—
190C4⋊Q8—216no—
191C4⋊Q8—24no—
192C4⋊Q8—24no—
193C4⋊Q8—24no—
194C4⋊Q8—24no—
195C4⋊Q8—24no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198C4⋊Q8C22⋊C418yes—
199C4⋊Q8C22×C414yes—
200C4⋊Q8C2×D414yes—
201C4×Q8—216no—
202C4×Q8—216no—
203C4×Q8—232no—
204C4×Q8—232no—
205C4×Q8—216no—
206C4×Q8—216no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8C22⋊C418yes—
222C4×Q8C22⋊C418yes—
223C4×Q8C22×C414yes—
224C4×Q8C22×C414yes—
225C4×Q8C2×D414yes—
226C4×Q8C2×D414yes—
227#117—42no—
228#117—42no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—216no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C22⋊C414yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C22⋊C4—41no—
261C2×C42—28no—
262C2×C42—216no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—41no—
272C2×C42C22×C412yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2411yescharacteristic
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—42no—
285C4⋊C8—42no—
286#103—48no—
287#103—48no—
288#103—88no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—42no—
294#95—41no—
295#95—41no—
296C24#12011yescharacteristic
297C2×Q8—216no—
298C2×Q8—28no—
299C2×Q8—216no—
300C2×Q8—232no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—28no—
310C2×Q8—28no—
311C2×Q8—24no—
312C2×Q8—28no—
313C2×Q8—24no—
314C2×Q8—28no—
315C2×Q8—48no—
316C2×Q8—84no—
317C2×Q8—88no—
318C2×Q8—88no—
319C2×Q8—832no—
320C2×Q8—816no—
321C2×Q8#103116yescomplemented
322C2×Q8C2×C22⋊C414yes—
323C2×Q8C2×C22⋊C414yes—
324C2×Q8C2×C22⋊C418yes—
325C2×Q8C2×SD1614yes—
326C2×Q8C2×Q1614yes—
327C2×Q8#13718yes—
328C22×C4—28no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—42no—
343C22×C4—48no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4#12012yes—
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4#12011yescharacteristic
356C22×C4C2×SD1612yes—
357C22×C4C2×Q1612yes—
358C22×C4#13712yes—
359C22×C4#13712yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363M16—84no—
364C2×C8—44no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—44no—
370C2×C8—42no—
371C4⋊C4—232no—
372C4⋊C4—232no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4C2×C22⋊C414yes—
394C4⋊C4C2×C22⋊C414yes—
395C22⋊C4—82no—
396C42—232no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—28no—
402C42—28no—
403C42—82no—
404C42C2×C22⋊C412yes—
405C42C2×C22⋊C412yes—
406C42C2×C22⋊C412yes—
407C42#12012yes—
408C42#12012yes—
409C42#12012yes—
410C42C23×C411yescharacteristic
411C42C22×D411yescharacteristic
412C42C22×D411yescharacteristic
413C42C22×D411yescharacteristic
414C23—22no—
415C23—22no—
416C23#37912yes—
417C23#44812yes—
418C23#45112yes—
419C23C2×#12011yescharacteristic
420Q8—216no—
421Q8—24no—
422Q8—24no—
423Q8—24no—
424Q8—24no—
425Q8—28no—
426Q8—216no—
427Q8—48no—
428Q8—84no—
429Q8—88no—
430Q8—832no—
431Q8C2×#10318yes—
432Q8#41918yescomplemented
433C2×C4—216no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—216no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—416no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4#37912yes—
460C2×C4#37912yes—
461C2×C4#37912yes—
462C2×C4C2×#10314yes—
463C2×C4#41914yes—
464C2×C4#44814yes—
465C2×C4#44812yes—
466C2×C4#45114yes—
467C2×C4#45112yes—
468C2×C4C22×C22⋊C411yescharacteristic
469C2×C4C22×C22⋊C411yescharacteristic
470C2×C4C22×C22⋊C411yescharacteristic
471C2×C4C2×#12011yescharacteristic
472C2×C4C2×#12011yescharacteristic
473C2×C4C2×#12011yescharacteristic
474C2×C4C22×SD1611yescharacteristic
475C2×C4C22×Q1611yescharacteristic
476C2×C4C2×#13711yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C2×C4C2×#13714yes—
479C8—82no—
480C8—82no—
481C8—82no—
482C22—24no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22#178714yes—
487C22C2×#37911yescharacteristic
488C22C2×#44811yescharacteristic
489C22C2×#45111yescharacteristic
490C22#300412yes—
491C22#301912yes—
492C4—28no—
493C4—28no—
494C4—22no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—48no—
499C4—44no—
500C4—44no—
501C4—162no—
502C4#178714yes—
503C4C22×#10311yescharacteristic
504C4C2×#41911yescharacteristic
505C4C2×#41912yes—
506C4C2×#44811yescharacteristic
507C4C2×#45111yescharacteristic
508C4#300412yes—
509C2—24no—
510C2#1876812yes—
511C2#1876912yes—
512C2C2×#178711yescharacteristic
513C2#2059911yescharacteristic
514C2#2061611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #8451C2semidirect
15. C2×Q82C4semidirect
321. C2×Q8#103semidirect
432. Q8#419semidirect

Maximal subgroups

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

subgroupindex
C22×Q822
#218042
#207342
#206472
C2×#15482
C2×#14872
C2×#14792
#84512