Finite Group Databasebeta

← #4388245

Subgroups — #4388245

Order 512.

3124 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic, complemented
2C2×#2598C212yesmaximal, complemented
3C2×#2314C211yesmaximal, characteristic, complemented
4#16512C214yesmaximal, complemented
5#15525C214yesmaximal, complemented
6#15394C214yesmaximal, complemented
7#15338C214yesmaximal, complemented
8#15320C214yesmaximal, complemented
9#14799C214yesmaximal, complemented
10#14676C211yesmaximal, characteristic, complemented
11#14345C212yesmaximal, complemented
12C22×#121C2214yes—
13C22×C4×D4C2212yes—
14C22×#117C2211yescharacteristic
15#2598—216no—
16#2537—28no—
17C2×#400C2212yes—
18C2×#399C2212yes—
19C2×#394C2212yes—
20C2×#394C2214yes—
21C2×#394C2214yes—
22C2×#392C2212yes—
23C2×#392C2214yes—
24#2314—28no—
25C2×#390C2214yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#385C2214yes—
33C2×#384C2211yescharacteristic
34C2×#383C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2214yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2212yes—
39#1445C2214yes—
40#1443C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58C22⋊#95C2214yes—
59C22⋊#95C2214yes—
60C23×D4C2312yes—
61C23×D4C2314yes—
62C23×D4C2314yes—
63C24×C4C2314yes—
64C24×C4C2311yescharacteristic
65C2×#121—216no—
66C2×#121—48no—
67C2×#121—48no—
68C2×C4×D4—28no—
69C2×C4×D4—28no—
70C2×C4×D4—28no—
71C2×#117—24no—
72C2×#117—24no—
73C2×#117D418yescomplemented
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C42C2311yescharacteristic
95C22×C42C2314yes—
96C22×C42C2311yescharacteristic
97#400—28no—
98#399—28no—
99#394—28no—
100#394—216no—
101#394—48no—
102#392—28no—
103#392—48no—
104#390—48no—
105#388—216no—
106#388—216no—
107#388—48no—
108#387—216no—
109#386—216no—
110#386—216no—
111#385—216no—
112#384D418yescomplemented
113#383—28no—
114#382—24no—
115#380—28no—
116#380—216no—
117C4×C4⋊C4—28no—
118C2×#95C2312yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C25C2411yescharacteristic
136C25C2414yes—
137C22×D4—28no—
138C22×D4—28no—
139C22×D4—28no—
140C22×D4—28no—
141C22×D4—28no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—28no—
145C22×D4—24no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—44no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—28no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4C2×D414yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2411yescharacteristic
174C23×C4C2414yes—
175C23×C4C2411yescharacteristic
176C23×C4C2412yes—
177#121—464no—
178C4×D4—432no—
179#117—232no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4C2×D418yes—
199C2×C4⋊C4C2×D4116yes—
200C2×C4⋊C4C2×D418yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4C2×D418yes—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—28no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42C2×D414yes—
246C2×C42C2×D414yes—
247C2×C42C2×D414yes—
248C2×C42#4114yes—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—24no—
256#95—24no—
257#95—28no—
258#95—28no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266C24—21no—
267C24—24no—
268C24—24no—
269C24—21no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24C22×D412yes—
287C24H(5,𝔽2)18yes—
288C24C2511yescharacteristic
289C2×D4—216no—
290C2×D4—416no—
291C2×D4—416no—
292C2×D4—48no—
293C2×D4—416no—
294C2×D4—416no—
295C2×D4—416no—
296C2×D4—416no—
297C2×D4—416no—
298C2×D4—48no—
299C2×D4—88no—
300C2×D4—88no—
301C2×D4—88no—
302C2×D4—88no—
303C2×D4—88no—
304C2×D4—88no—
305C2×D4—88no—
306C2×D4—88no—
307C2×D4—88no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—48no—
359C22×C4—416no—
360C22×C4—416no—
361C22×C4—416no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4C22×D412yes—
367C22×C4C22×D418yes—
368C22×C4C22×D414yes—
369C22×C4C22×D412yes—
370C22×C4C22×D414yes—
371C22×C4C22×D414yes—
372C22×C4C22×D412yes—
373C22×C4C2×#4112yes—
374C22×C4C2×#4114yes—
375C4⋊C4—216no—
376C4⋊C4—48no—
377C4⋊C4—416no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—88no—
381C4⋊C4—88no—
382C22⋊C4—216no—
383C22⋊C4—416no—
384C22⋊C4—416no—
385C22⋊C4—416no—
386C22⋊C4—416no—
387C22⋊C4—416no—
388C22⋊C4—416no—
389C22⋊C4—88no—
390C22⋊C4—88no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C42—28no—
396C42—88no—
397C42#121116yes—
398C23—22no—
399C23—24no—
400C23—28no—
401C23—28no—
402C23—28no—
403C23—28no—
404C23—28no—
405C23—28no—
406C23—28no—
407C23—28no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—42no—
412C23—44no—
413C23—41no—
414C23—44no—
415C23—44no—
416C23—41no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23C23×D411yescharacteristic
447C23C23×D411yescharacteristic
448C23C22×#4111yescharacteristic
449C23C2×H(5,𝔽2)14yes—
450C23#58512yes—
451C23#58512yes—
452C23#58514yes—
453D4—432no—
454D4—832no—
455D4—832no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—416no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—832no—
511C2×C4C2×#12114yes—
512C2×C4C2×#12118yes—
513C22—48no—
514C22—48no—
515C22—48no—
516C22—48no—
517C22—48no—
518C22—42no—
519C22—48no—
520C22—48no—
521C22—48no—
522C22—44no—
523C22—84no—
524C22—84no—
525C22—81no—
526C22—82no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22—81no—
533C22—84no—
534C22—84no—
535C22C22×#12111yescharacteristic
536C22#344912yes—
537C22#348412yes—
538C22#348511yescharacteristic
539C22#348711yescharacteristic
540C22#348712yes—
541C22#348814yes—
542C22#349312yes—
543C22#349414yes—
544C22#353314yes—
545C22#353414yes—
546C22#356014yes—
547C22#356614yes—
548C4—24no—
549C4—28no—
550C4—48no—
551C4—42no—
552C4—44no—
553C4—84no—
554C4—84no—
555C4—84no—
556C4—84no—
557C2—82no—
558C2—88no—
559C2#3515011yescharacteristic
560C2#3556312yes—
561C2#4199514yes—
562C2#4769714yes—
563C2#5200811yescharacteristic
564C2#5201012yes—
565C2#5208411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2537C2semidirect
2. C2×#2598C2semidirect
3. C2×#2314C2semidirect
4. #16512C2semidirect
5. #15525C2semidirect
6. #15394C2semidirect
7. #15338C2semidirect
8. #15320C2semidirect
9. #14799C2semidirect
10. #14676C2semidirect
11. #14345C2semidirect
73. C2×#117D4semidirect
112. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#25982
C2×#23142
#165122
#155252
#153942
#153382
#153202
#147992
#146762
#143452