Finite Group Databasebeta

← #4387207

Subgroups — #4387207

Order 512.

2456 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3#16360C214yesmaximal, complemented
4#15650C214yesmaximal, complemented
5#15649C214yesmaximal, complemented
6#15456C214yesmaximal, complemented
7#15434C214yesmaximal, complemented
8#14964C214yesmaximal, complemented
9#14659C211yesmaximal, characteristic
10#14297C211yesmaximal, characteristic, complemented
11#14290C211yesmaximal, characteristic
12#13953C211yesmaximal, characteristic, complemented
13#13924C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#123C2212yes—
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#2501—28no—
19C2×#400C2214yes—
20C2×#399C2214yes—
21C2×#399C2214yes—
22#2313—28no—
23C2×#390C2212yes—
24C2×#388C2214yes—
25C2×#387C2214yes—
26C2×#387C2214yes—
27C2×#387C2214yes—
28C2×#384C2212yes—
29C2×#381C2211yescharacteristic
30C2×#381C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1445C2212yes—
37#1445C2212yes—
38#1445C2214yes—
39#1442C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2212yes—
54#1438C2214yes—
55#1438C2212yes—
56#1438C2214yes—
57#1437C2212yes—
58#1437C2214yes—
59#1437C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C2×#123—28no—
75C2×#123—28no—
76C2×#123—48no—
77C2×#123—48no—
78C2×#121—24no—
79C2×#121—24no—
80C2×#121—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4D418yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2314yes—
109#400—48no—
110#399—216no—
111#399—48no—
112#390—28no—
113#388—48no—
114#387—216no—
115#387—216no—
116#387—48no—
117#384—28no—
118#381—24no—
119#381—24no—
120#380—28no—
121#380—24no—
122C4×C22⋊C4—28no—
123C4×C22⋊C4—24no—
124C4×C22⋊C4—24no—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C25C2412yes—
149C25C2411yescharacteristic
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—42no—
159C22×D4—42no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4C2×D414yes—
169C23×C4C2×D414yes—
170C23×C4C2×D414yes—
171C23×C4#4114yes—
172C23×C4#4114yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184#123—432no—
185#123—432no—
186#121—416no—
187C2×C4⋊C4—232no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C2×D418yes—
214C2×C4⋊C4C2×D418yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C42—24no—
251C2×C42—28no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—42no—
259C2×C42—42no—
260#95—28no—
261#95—24no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283C24—24no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24C22×D412yes—
298C24C2×#4112yes—
299C24C2×#4112yes—
300C24C2511yescharacteristic
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—44no—
306C2×D4—84no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—88no—
310C2×D4—88no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
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—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—48no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4#123116yes—
377C22×C4C22×D414yes—
378C22×C4C22×D414yes—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4H(5,𝔽2)18yes—
386C22×C42−1+414yes—
387C4⋊C4—216no—
388C4⋊C4—216no—
389C4⋊C4—416no—
390C4⋊C4—416no—
391C4⋊C4—416no—
392C4⋊C4—44no—
393C4⋊C4—84no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C22⋊C4—216no—
397C22⋊C4—416no—
398C22⋊C4—416no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—416no—
403C22⋊C4—48no—
404C22⋊C4—88no—
405C22⋊C4—84no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C42—48no—
409C42—82no—
410C42—82no—
411C42—88no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23C2×#12314yes—
449C23C23×D411yescharacteristic
450C23C22×#4111yescharacteristic
451C23C22×#4111yescharacteristic
452C23C2×H(5,𝔽2)14yes—
453C23C2×2−1+412yes—
454C23#58512yes—
455C23#58512yes—
456C23#58512yes—
457D4—88no—
458D4—816no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—416no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—88no—
522C2×C4—88no—
523C2×C4C2×#12314yes—
524C2×C4C2×#12314yes—
525C22—24no—
526C22—24no—
527C22—24no—
528C22—44no—
529C22—44no—
530C22—48no—
531C22—44no—
532C22—44no—
533C22—84no—
534C22—84no—
535C22—82no—
536C22—84no—
537C22—84no—
538C22—82no—
539C22—84no—
540C22—82no—
541C22—84no—
542C22C22×#12311yescharacteristic
543C22#344912yes—
544C22#348412yes—
545C22#349112yes—
546C22#349111yescharacteristic
547C22#349111yescharacteristic
548C22#349512yes—
549C22#349712yes—
550C22#350812yes—
551C22#350812yes—
552C22#350911yescharacteristic
553C22#351011yescharacteristic
554C22#353514yes—
555C22#353914yes—
556C22#355414yes—
557C22#356014yes—
558C4—24no—
559C4—24no—
560C4—48no—
561C4—44no—
562C4—44no—
563C4—44no—
564C4—84no—
565C4—84no—
566C4—84no—
567C4—82no—
568C4—82no—
569C2—44no—
570C2—162no—
571C2#3509711yescharacteristic
572C2#3555411yescharacteristic
573C2#3567311yescharacteristic
574C2#4049914yes—
575C2#4685412yes—
576C2#4685612yes—
577C2#5067612yes—
578C2#5105512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2313C2semidirect
3. #16360C2semidirect
4. #15650C2semidirect
5. #15649C2semidirect
6. #15456C2semidirect
7. #15434C2semidirect
8. #14964C2semidirect
10. #14297C2semidirect
12. #13953C2semidirect
13. #13924C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#23132
#163602
#156502
#156492
#154562
#154342
#149642
#146592
#142972
#142902
#139532
#139242