Finite Group Databasebeta

← #892314

Subgroups — #892314

Order 512.

2406 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2727C212yesmaximal
2#17075C211yesmaximal, characteristic, complemented
3#17070C211yesmaximal, characteristic, complemented
4#16596C212yesmaximal
5#15927C214yesmaximal, complemented
6#15706C214yesmaximal, complemented
7#14824C212yesmaximal
8#14771C214yesmaximal, complemented
9#14707C212yesmaximal, complemented
10#14587C212yesmaximal, complemented
11#14226C212yesmaximal, complemented
12#14102C214yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic
14C22×#126C2214yes—
15C22×C4×D4C2211yescharacteristic
16#2727—216no—
17C2×#399C2214yes—
18C2×#398C2212yes—
19C2×#398C2212yes—
20C2×#398C2214yes—
21C2×#397C2212yes—
22C2×#396C2214yes—
23C2×#394C2214yes—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#382C2212yes—
32C2×#382C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2214yes—
36C2×C43C2211yescharacteristic
37#1445C2214yes—
38#1445C2212yes—
39#1444C2212yes—
40#1444C2214yes—
41#1444C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2212yes—
45#1441C2212yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1437C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64#1434C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C4×#95C2212yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C2×#126—216no—
71C2×#126—216no—
72C2×#126—48no—
73C2×C4×D4—24no—
74C2×C4×D4—42no—
75C2×C4×D4—42no—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2314yes—
101C22×C42C2312yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#399—216no—
106#398—28no—
107#398—28no—
108#398—48no—
109#397—44no—
110#396—216no—
111#394—216no—
112#390—24no—
113#390D418yes—
114#389—216no—
115#389—28no—
116#389—28no—
117#389—44no—
118#388—216no—
119#382—28no—
120#382—44no—
121#380—44no—
122C4×C4⋊C4—42no—
123C4×C22⋊C4—48no—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C43—42no—
147C25C2412yes—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—41no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4C2×D414yes—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—28no—
158C23×C4—42no—
159C23×C4—44no—
160C23×C4C2414yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2411yescharacteristic
168C23×C4C2412yes—
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172#126—464no—
173C4×D4—416no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4#4118yes—
215C2×C4⋊C4#4114yes—
216C2×C4⋊C4#4114yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—28no—
245C2×C42—28no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—44no—
251C2×C42—41no—
252C2×C42—42no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—44no—
257C2×C42C2×D414yes—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95—44no—
277#95—42no—
278#95—44no—
279#95—44no—
280C24—24no—
281C24—22no—
282C24—22no—
283C24—24no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24C22×D414yes—
291C24C2511yescharacteristic
292C2×D4—28no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
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—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—48no—
360C22×C4—416no—
361C22×C4—48no—
362C22×C4—88no—
363C22×C4—84no—
364C22×C4C22×D414yes—
365C22×C4C22×D412yes—
366C22×C4C22×D414yes—
367C22×C4C2×#4114yes—
368C22×C4C2×#4114yes—
369C22×C4C2×#4118yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4114yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)14yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+414yes—
377C22×C42−1+414yes—
378C4⋊C4—216no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—416no—
383C4⋊C4—48no—
384C4⋊C4—416no—
385C4⋊C4—48no—
386C4⋊C4—416no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—88no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C22⋊C4—232no—
394C22⋊C4—416no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—416no—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C42—216no—
401C42—28no—
402C42—44no—
403C42—48no—
404C42—416no—
405C42—44no—
406C42—84no—
407C23—24no—
408C23—28no—
409C23—24no—
410C23—24no—
411C23—28no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23C23×D411yescharacteristic
431C23C22×#4111yescharacteristic
432C23C22×#4111yescharacteristic
433C23C22×#4112yes—
434C23C2×H(5,𝔽2)11yescharacteristic
435C23C2×H(5,𝔽2)11yescharacteristic
436C23C2×H(5,𝔽2)12yes—
437C23C2×2−1+412yes—
438C23C2×2−1+412yes—
439C23#58511yescharacteristic
440C23#58511yescharacteristic
441D4—416no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
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—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—816no—
514C2×C4#53214yes—
515C2×C4#54618yes—
516C2×C4#56314yes—
517C22—44no—
518C22—48no—
519C22—44no—
520C22—44no—
521C22—48no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—84no—
526C22—82no—
527C22—84no—
528C22C2×#53211yescharacteristic
529C22#345911yescharacteristic
530C22#346011yescharacteristic
531C22#346112yes—
532C22C2×#54612yes—
533C22#348212yes—
534C22#348212yes—
535C22#348311yescharacteristic
536C22#349011yescharacteristic
537C22#349212yes—
538C22#349311yescharacteristic
539C22#349611yescharacteristic
540C22#349611yescharacteristic
541C22#349912yes—
542C22#351712yes—
543C22#352012yes—
544C22#352012yes—
545C22C2×#56311yescharacteristic
546C22#353612yes—
547C22#354311yescharacteristic
548C22#355011yescharacteristic
549C22#355912yes—
550C22#356312yes—
551C4—24no—
552C4—24no—
553C4—28no—
554C4—44no—
555C4—48no—
556C4—44no—
557C4—44no—
558C4—42no—
559C4—44no—
560C4—84no—
561C4—84no—
562C4—84no—
563C2—84no—
564C2#3500311yescharacteristic
565C2#3640612yes—
566C2#3725411yescharacteristic
567C2#3879112yes—
568C2#3880312yes—
569C2#3941312yes—
570C2#4614711yescharacteristic
571C2#4619911yescharacteristic
572C2#5230611yescharacteristic
573C2#5941911yescharacteristic
574C2#6004711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17075C2semidirect
3. #17070C2semidirect
5. #15927C2semidirect
6. #15706C2semidirect
8. #14771C2semidirect
9. #14707C2semidirect
10. #14587C2semidirect
11. #14226C2semidirect
12. #14102C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27272
#170752
#170702
#165962
#159272
#157062
#148242
#147712
#147072
#145872
#142262
#141022
C4×#3902