Finite Group Databasebeta

← #2037255

Subgroups — #2037255

Order 512.

2238 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic
2C2×#2861C211yesmaximal, characteristic
3#17080C212yesmaximal, complemented
4#16278C214yesmaximal, complemented
5#16032C212yesmaximal
6#15884C212yesmaximal, complemented
7#15850C212yesmaximal, complemented
8#15832C214yesmaximal, complemented
9#15342C212yesmaximal
10#15298C214yesmaximal, complemented
11#14703C212yesmaximal, complemented
12#14445C211yesmaximal, characteristic
13#14405C214yesmaximal, complemented
14C22×#125C2211yescharacteristic
15C22×#124C2211yescharacteristic
16#2861—28no—
17#2804—28no—
18C2×#400C2212yes—
19C2×#398C2212yes—
20C2×#397C2214yes—
21C2×#393C2214yes—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#386C2212yes—
34C2×#385C2214yes—
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#384C2212yes—
38C2×#381C2212yes—
39C2×C43C2211yescharacteristic
40#1446C2214yes—
41#1446C2214yes—
42#1445C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2212yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67#1434C2214yes—
68#1434C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×#125—24no—
74C2×#125—42no—
75C2×#125—42no—
76C2×#124—24no—
77C2×#124—42no—
78C2×#124—42no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105#400—28no—
106#398—44no—
107#397—48no—
108#393—216no—
109#391—24no—
110#391—28no—
111#391—24no—
112#391—28no—
113#391—44no—
114#390—24no—
115#390D418yes—
116#389—44no—
117#388—216no—
118#388—216no—
119#388—216no—
120#386—28no—
121#385—216no—
122#385—48no—
123#385—44no—
124#384—44no—
125#381—44no—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C43—24no—
150C25C2412yes—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×D4—24no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—42no—
167C22×D4C2×D414yes—
168C23×C4—24no—
169C23×C4—44no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2414yes—
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185#125—416no—
186#124—416no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4#4118yes—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4C2×D4116yes—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—28no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C2×D414yes—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
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—28no—
273#95—28no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—42no—
282C24—24no—
283C24—24no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24C22×D414yes—
291C24C2511yescharacteristic
292C2×Q8—44no—
293C2×Q8—48no—
294C2×Q8—44no—
295C2×Q8—48no—
296C2×Q8—44no—
297C2×Q8—48no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×D4—28no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—84no—
308C2×D4—84no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
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—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—48no—
372C22×C4—84no—
373C22×C4—88no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4C22×D414yes—
377C22×C4C22×D412yes—
378C22×C4C22×D414yes—
379C22×C4C2×#4118yes—
380C22×C4C2×#4114yes—
381C22×C4H(5,𝔽2)14yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C42−1+414yes—
385C22×C42−1+414yes—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—44no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C22⋊C4—232no—
398C22⋊C4—48no—
399C22⋊C4—416no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—416no—
403C22⋊C4—84no—
404C22⋊C4—88no—
405C22⋊C4—84no—
406C42—28no—
407C42—44no—
408C42—48no—
409C42—48no—
410C42—44no—
411C42—48no—
412C23—24no—
413C23—28no—
414C23—24no—
415C23—24no—
416C23—28no—
417C23—44no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23C23×D411yescharacteristic
434C23C22×#4112yes—
435C23C2×H(5,𝔽2)12yes—
436C23C2×H(5,𝔽2)11yescharacteristic
437C23C2×H(5,𝔽2)11yescharacteristic
438C23C2×2−1+412yes—
439C23C2×2−1+412yes—
440C23#58512yes—
441C23#58512yes—
442Q8—88no—
443Q8—816no—
444D4—88no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
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—816no—
516C2×C4#53614yes—
517C2×C4#54718yes—
518C22—48no—
519C22—44no—
520C22—48no—
521C22—44no—
522C22—44no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—84no—
527C22C2×#53611yescharacteristic
528C22#345912yes—
529C22C2×#54712yes—
530C22#348111yescharacteristic
531C22#348411yescharacteristic
532C22#348511yescharacteristic
533C22#348812yes—
534C22#349312yes—
535C22#351512yes—
536C22#352212yes—
537C22#352212yes—
538C22#352911yescharacteristic
539C22#354412yes—
540C22#354911yescharacteristic
541C22#355011yescharacteristic
542C22#355512yes—
543C22#355812yes—
544C22#356112yes—
545C22#356112yes—
546C22#356112yes—
547C22#356412yes—
548C4—24no—
549C4—28no—
550C4—44no—
551C4—48no—
552C4—42no—
553C4—44no—
554C4—42no—
555C4—82no—
556C4—84no—
557C4—82no—
558C4—84no—
559C4—84no—
560C2—84no—
561C2#3696611yescharacteristic
562C2#3726712yes—
563C2#3776212yes—
564C2#3876912yes—
565C2#3892112yes—
566C2#4596512yes—
567C2#4622511yescharacteristic
568C2#5336612yes—
569C2#5907011yescharacteristic
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
3. #17080C2semidirect
4. #16278C2semidirect
6. #15884C2semidirect
7. #15850C2semidirect
8. #15832C2semidirect
10. #15298C2semidirect
11. #14703C2semidirect
13. #14405C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#28612
#170802
#162782
#160322
#158842
#158502
#158322
#153422
#152982
#147032
#144452
#144052