Finite Group Databasebeta

← #2811769

Subgroups — #2811769

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3#16689C212yesmaximal, complemented
4#16685C214yesmaximal, complemented
5#15850C212yesmaximal, complemented
6#15820C212yesmaximal
7#15765C214yesmaximal, complemented
8#15294C214yesmaximal, complemented
9#15084C212yesmaximal
10#14825C212yesmaximal
11#14454C211yesmaximal, characteristic, complemented
12#14393C214yesmaximal, complemented
13#14223C212yesmaximal, complemented
14C22×#125C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16#2571—28no—
17#2566—28no—
18C2×#400C2212yes—
19C2×#400C2212yes—
20C2×#399C2212yes—
21C2×#399C2214yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#396C2214yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2214yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#385C2214yes—
32C2×#384C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2214yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39#1445C2212yes—
40#1445C2214yes—
41#1445C2212yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1442C2212yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2212yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1437C2214yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2214yes—
64#1434C2214yes—
65#1434C2214yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×D4C2311yescharacteristic
70C24×C4C2312yes—
71C2×#125—24no—
72C2×#125—42no—
73C2×#125—42no—
74C2×C4×D4—24no—
75C2×C4×D4—42no—
76C2×C4×D4—42no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2314yes—
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104#400—28no—
105#400—28no—
106#399—48no—
107#399—44no—
108#398—28no—
109#398—44no—
110#396—216no—
111#390—24no—
112#390—24no—
113#389—216no—
114#388—216no—
115#388—48no—
116#387—48no—
117#385—216no—
118#384—44no—
119#384D418yes—
120#382—24no—
121#382—216no—
122#380—44no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—28no—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C25C2412yes—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—24no—
153C22×D4—41no—
154C22×D4—41no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—42no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4C2412yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172#125—416no—
173C4×D4—416no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D4116yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
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⋊C4—44no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—28no—
243C2×C42—22no—
244C2×C42—28no—
245C2×C42—28no—
246C2×C42—24no—
247C2×C42—42no—
248C2×C42—44no—
249C2×C42C2×D414yes—
250#95—24no—
251#95—28no—
252#95—24no—
253#95—24no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—28no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—42no—
272#95—42no—
273#95#4118yes—
274C24—22no—
275C24—24no—
276C24—24no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24—24no—
284C242−1+414yes—
285C24C2511yescharacteristic
286C2×D4—28no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—44no—
290C2×D4—44no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
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—48no—
349C22×C4—48no—
350C22×C4—88no—
351C22×C4—84no—
352C22×C4—88no—
353C22×C4C22×D414yes—
354C22×C4C22×D414yes—
355C22×C4C22×D412yes—
356C22×C4C22×D414yes—
357C22×C4C2×#4118yes—
358C22×C4C2×#4114yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)14yes—
362C22×C42−1+414yes—
363C4⋊C4—416no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—416no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4—88no—
376C4⋊C4—84no—
377C4⋊C4—88no—
378C22⋊C4—48no—
379C22⋊C4—416no—
380C22⋊C4—416no—
381C22⋊C4—416no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C22⋊C4—88no—
385C42—28no—
386C42—28no—
387C42—48no—
388C42—44no—
389C42—416no—
390C23—24no—
391C23—28no—
392C23—24no—
393C23—24no—
394C23—28no—
395C23—42no—
396C23—44no—
397C23—42no—
398C23—42no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23C23×D411yescharacteristic
414C23C22×#4112yes—
415C23C2×H(5,𝔽2)12yes—
416C23C2×H(5,𝔽2)11yescharacteristic
417C23C2×H(5,𝔽2)11yescharacteristic
418C23C2×2−1+412yes—
419C23C2×2−1+412yes—
420C23#58514yes—
421D4—416no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—816no—
488C2×C4#53614yes—
489C2×C4#55018yes—
490C22—48no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—48no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—84no—
500C22—84no—
501C22C2×#53611yescharacteristic
502C22C2×#55012yes—
503C22#348311yescharacteristic
504C22#348411yescharacteristic
505C22#348512yes—
506C22#348611yescharacteristic
507C22#349514yes—
508C22#351514yes—
509C22#351912yes—
510C22#353312yes—
511C22#353812yes—
512C22#353812yes—
513C22#354012yes—
514C22#354114yes—
515C22#354911yescharacteristic
516C22#355914yes—
517C4—24no—
518C4—28no—
519C4—44no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—48no—
524C4—84no—
525C4—84no—
526C4—82no—
527C4—84no—
528C4—82no—
529C2—84no—
530C2#3670811yescharacteristic
531C2#3913412yes—
532C2#4471612yes—
533C2#4513812yes—
534C2#4809714yes—
535C2#5205411yescharacteristic
536C2#5549212yes—
537C2#5905611yescharacteristic
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
1. C2×#2571C2semidirect
3. #16689C2semidirect
4. #16685C2semidirect
5. #15850C2semidirect
7. #15765C2semidirect
8. #15294C2semidirect
11. #14454C2semidirect
12. #14393C2semidirect
13. #14223C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#25662
#166892
#166852
#158502
#158202
#157652
#152942
#150842
#148252
#144542
#143932
#142232