Finite Group Databasebeta

← #951620

Subgroups — #951620

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C212yesmaximal, complemented
2#17182C214yesmaximal, complemented
3#16548C214yesmaximal
4#16499C214yesmaximal, complemented
5#15365C214yesmaximal, complemented
6#15181C212yesmaximal
7#15135C212yesmaximal, complemented
8#15131C212yesmaximal, complemented
9#14277C211yesmaximal, characteristic, complemented
10#14276C211yesmaximal, characteristic, complemented
11#14046C212yesmaximal, complemented
12#13875C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14C22×C4×D4C2211yescharacteristic
15#2494—216no—
16C2×#401C2214yes—
17C2×#399C2214yes—
18C2×#397C2214yes—
19C2×#396C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2214yes—
24C2×#389C2212yes—
25C2×#388C2214yes—
26C2×#387C2214yes—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×#376C2211yescharacteristic
40#1445C2212yes—
41#1445C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C2×C4×D4—24no—
71C2×C4×D4—42no—
72C2×C4×D4—42no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2314yes—
100#401—216no—
101#399—48no—
102#397—216no—
103#396—48no—
104#389—28no—
105#389—28no—
106#389—28no—
107#389—48no—
108#389—44no—
109#388—216no—
110#387—216no—
111#384—24no—
112#384—42no—
113#382—42no—
114#382D418yescomplemented
115#380—28no—
116#380—28no—
117#380—44no—
118#380—44no—
119C4×C4⋊C4—28no—
120C4×C4⋊C4—42no—
121C4×C22⋊C4—28no—
122C4×C22⋊C4—28no—
123#376—42no—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C25C2412yes—
149C22×D4—24no—
150C22×D4—22no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4—42no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2411yescharacteristic
169C23×C4C2412yes—
170C23×C4C2411yescharacteristic
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C4×D4—416no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4C2×D418yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
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—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
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—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—28no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42C2×D414yes—
251C2×C42#4114yes—
252C2×C42#4114yes—
253#95—24no—
254#95—24no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95C2×D4116yes—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C242−1+414yes—
292C24C2511yescharacteristic
293C2×D4—48no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—48no—
297C2×D4—44no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4C22×D418yes—
373C22×C4C22×D414yes—
374C22×C4C22×D412yes—
375C22×C4C2×#4112yes—
376C22×C4C2×#4114yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)14yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+412yes—
384C22×C42−1+412yes—
385C4⋊C4—216no—
386C4⋊C4—416no—
387C4⋊C4—44no—
388C4⋊C4—416no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C22⋊C4—48no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—416no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—88no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C42—28no—
408C42—28no—
409C42—28no—
410C42—48no—
411C42—88no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23C23×D411yescharacteristic
448C23C22×#4111yescharacteristic
449C23C22×#4111yescharacteristic
450C23C2×H(5,𝔽2)11yescharacteristic
451C23C2×H(5,𝔽2)11yescharacteristic
452C23C2×H(5,𝔽2)12yes—
453C23C2×2−1+411yescharacteristic
454C23C2×2−1+411yescharacteristic
455C23C2×2−1+412yes—
456C23#58512yes—
457C23#58512yes—
458D4—416no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—816no—
533C2×C4#53714yes—
534C2×C4#53814yes—
535C2×C4#53914yes—
536C2×C4#54714yes—
537C2×C4#55014yes—
538C2×C4#55614yes—
539C2×C4#56314yes—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22C2×#53711yescharacteristic
556C22C2×#53811yescharacteristic
557C22C2×#53911yescharacteristic
558C22#346012yes—
559C22C2×#54711yescharacteristic
560C22C2×#55011yescharacteristic
561C22C2×#55611yescharacteristic
562C22#348812yes—
563C22#348912yes—
564C22#349412yes—
565C22#349512yes—
566C22#349512yes—
567C22C2×#56311yescharacteristic
568C22#353711yescharacteristic
569C22#353811yescharacteristic
570C22#353911yescharacteristic
571C22#353911yescharacteristic
572C22#354112yes—
573C22#354612yes—
574C22#355212yes—
575C22#355212yes—
576C22#356512yes—
577C22#356512yes—
578C4—42no—
579C4—44no—
580C4—48no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—84no—
585C4—82no—
586C4—84no—
587C4—84no—
588C4—82no—
589C4#272718yes—
590C2—84no—
591C2#3738011yescharacteristic
592C2#3751511yescharacteristic
593C2#3776712yes—
594C2#3796212yes—
595C2#4292812yes—
596C2#4293412yes—
597C2#4542311yescharacteristic
598C2#4543211yescharacteristic
599C2C2×#272711yescharacteristic
600C2#5298711yescharacteristic
601C2#5307511yescharacteristic
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×#2494C2semidirect
2. #17182C2semidirect
4. #16499C2semidirect
5. #15365C2semidirect
7. #15135C2semidirect
8. #15131C2semidirect
9. #14277C2semidirect
10. #14276C2semidirect
11. #14046C2semidirect
12. #13875C2semidirect
114. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
#171822
#165482
#164992
#153652
#151812
#151352
#151312
#142772
#142762
#140462
#138752
#138472