Finite Group Databasebeta

← #359819

Subgroups — #359819

Order 512.

3146 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2280C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3C2×#1824C212yesmaximal, complemented
4C2×#1779C211yesmaximal, characteristic
5#11199C214yesmaximal, complemented
6#10731C214yesmaximal, complemented
7#10730C212yesmaximal, complemented
8C25×C4C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C22⋊C4—21no—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16#2280—28no—
17C2×#381—21no—
18C2×#380—24no—
19C2×#380—21no—
20C2×#380—21no—
21C2×#380C418yescomplemented
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#379—21no—
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4—21no—
29C22×#95—21no—
30#1824—28no—
31#1779—24no—
32C2×#340C2218yescomplemented
33C2×#340C2218yescomplemented
34C26D411yescharacteristic
35C24×C4—24no—
36C24×C4—28no—
37C24×C4—24no—
38C24×C4—21no—
39C24×C4—21no—
40C24×C4—44no—
41C24×C4C2×C414yes—
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D414yes—
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2314yes—
50C2×#117—28no—
51C2×#117—28no—
52C2×#117—44no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4C2×C414yes—
61C22×C4⋊C4C2×C414yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2314yes—
65C22×C22⋊C4—28no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—48no—
72C22×C42—22no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—28no—
79C2×C22⋊C8—44no—
80#381—28no—
81#380—28no—
82#380—28no—
83#380—24no—
84#380—24no—
85#380—24no—
86#380—28no—
87#380C2×C418yes—
88#380C2×C418yes—
89#380D418yes—
90#379—28no—
91C4×C22⋊C4—28no—
92C4×C22⋊C4—28no—
93C2×#95—28no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—216no—
99#340—48no—
100C25—24no—
101C25—24no—
102C25—24no—
103C25—22no—
104C25—22no—
105C25—24no—
106C25—42no—
107C25—42no—
108C25C22⋊C414yes—
109C25C2×D411yescharacteristic
110C25C2×D411yescharacteristic
111C25C2×D411yescharacteristic
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—216no—
121C23×C4—216no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—216no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—216no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—48no—
140C23×C4—48no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—432no—
145C23×C4—432no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—44no—
153C23×C4C22⋊C414yes—
154C23×C4C22⋊C414yes—
155C23×C4C22⋊C414yes—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D414yes—
166C23×C4#4118yes—
167C23×C4#4114yes—
168C23×C4C2411yescharacteristic
169C22×C8—24no—
170C22×C8—24no—
171C22×C8—24no—
172C22×C8—28no—
173C22×C8—28no—
174#117—432no—
175C2×C4⋊C4—216no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C22×C414yes—
193C2×C4⋊C4C22×C414yes—
194C2×C4⋊C4C2×D418yes—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—432no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—48no—
208C2×C22⋊C4—44no—
209C2×C42—28no—
210C2×C42—44no—
211C2×C42—44no—
212C22⋊C8—48no—
213C22⋊C8—48no—
214C22⋊C8—48no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—28no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—232no—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—28no—
239C24—28no—
240C24—28no—
241C24—28no—
242C24—28no—
243C24—44no—
244C24—44no—
245C24—48no—
246C24—48no—
247C24—416no—
248C24—48no—
249C24—416no—
250C24—48no—
251C24—416no—
252C24—48no—
253C24—48no—
254C24C2×C22⋊C412yes—
255C24C2×C22⋊C412yes—
256C24C2×C22⋊C412yes—
257C24#12012yes—
258C24#12314yes—
259C24#12312yes—
260C24#13514yes—
261C24AGL(1,ℤ/8)14yes—
262C24#13714yes—
263C24C22×D411yescharacteristic
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—216no—
278C22×C4—216no—
279C22×C4—216no—
280C22×C4—216no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—28no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—464no—
306C22×C4—464no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—88no—
333C22×C4—816no—
334C22×C4—88no—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4#11718yes—
339C22×C4C4×D418yes—
340C22×C4#12012yes—
341C22×C4#12118yes—
342C22×C4#12318yes—
343C22×C4#12314yes—
344C22×C4#12312yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D412yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4114yes—
349C22×C4C2×#4112yes—
350C22×C4H(5,𝔽2)14yes—
351C2×C8—44no—
352C2×C8—44no—
353C2×C8—44no—
354C2×C8—48no—
355C2×C8—48no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—216no—
359C4⋊C4—48no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C22⋊C4—416no—
365C22⋊C4—416no—
366C22⋊C4—416no—
367C22⋊C4—416no—
368C22⋊C4—416no—
369C42—416no—
370C23—22no—
371C23—28no—
372C23—28no—
373C23—24no—
374C23—22no—
375C23—22no—
376C23—24no—
377C23—28no—
378C23—28no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—232no—
383C23—22no—
384C23—22no—
385C23—28no—
386C23—28no—
387C23—28no—
388C23—24no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—24no—
397C23—28no—
398C23—24no—
399C23—24no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—416no—
404C23—416no—
405C23—416no—
406C23—416no—
407C23—416no—
408C23—464no—
409C23—464no—
410C23—42no—
411C23—42no—
412C23—44no—
413C23—42no—
414C23—416no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23#38114yes—
424C23#38614yes—
425C23#41614yes—
426C23#41814yes—
427C23#41914yes—
428C23#45218yes—
429C23#48518yes—
430C23C22×C22⋊C411yescharacteristic
431C23C2×#12011yescharacteristic
432C23C2×#12311yescharacteristic
433C23C2×#12312yes—
434C23#53412yes—
435C23C2×#13512yes—
436C23C2×AGL(1,ℤ/8)12yes—
437C23C2×#13712yes—
438C23#57512yes—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—216no—
448C2×C4—216no—
449C2×C4—216no—
450C2×C4—216no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—416no—
455C2×C4—416no—
456C2×C4—416no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—816no—
471C2×C4—816no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—88no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4#38114yes—
480C2×C4#38614yes—
481C2×C4C2×#11712yes—
482C2×C4C2×C4×D412yes—
483C2×C4#51812yes—
484C2×C4C2×#12112yes—
485C2×C4C2×#12312yes—
486C2×C4#53412yes—
487C2×C4#54618yes—
488C2×C4#55118yes—
489C8—84no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—216no—
495C22—216no—
496C22—24no—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22—24no—
504C22—464no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—48no—
513C22—48no—
514C22—48no—
515C22—48no—
516C22—42no—
517C22#187318yes—
518C22C2×#38111yescharacteristic
519C22C2×#38611yescharacteristic
520C22#230811yescharacteristic
521C22#240512yes—
522C22#248312yes—
523C22C2×#41611yescharacteristic
524C22C2×#41811yescharacteristic
525C22C2×#41911yescharacteristic
526C22#289611yescharacteristic
527C22C2×#45212yes—
528C22#300612yes—
529C22C2×#48512yes—
530C22#309114yes—
531C22#309214yes—
532C22#309412yes—
533C22#318814yes—
534C22#319614yes—
535C4—24no—
536C4—24no—
537C4—42no—
538C4—84no—
539C4—84no—
540C4—88no—
541C4#234214yes—
542C4#241214yes—
543C2—28no—
544C2—44no—
545C2—44no—
546C2#1386911yescharacteristic
547C2#2069911yescharacteristic
548C2#2070011yescharacteristic
549C2C2×#187311yescharacteristic
550C2#2091211yescharacteristic
551C2#2176111yescharacteristic
552C2#2176811yescharacteristic
553C2#2400714yes—
554C2#2760414yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1824C2semidirect
5. #11199C2semidirect
6. #10731C2semidirect
7. #10730C2semidirect
21. C2×#380C4semidirect
32. C2×#340C22semidirect
33. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22802
C22×#3802
C2×#18242
C2×#17792
#111992
#107312
#107302