Finite Group Databasebeta

← #948618

Subgroups — #948618

Order 512.

2332 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C212yesmaximal, complemented
2C2×#2628C212yesmaximal
3#17099C214yesmaximal, complemented
4#16531C214yesmaximal
5#16375C214yesmaximal
6#15699C214yesmaximal, complemented
7#15230C212yesmaximal, complemented
8#15223C211yesmaximal, characteristic
9#15221C212yesmaximal, complemented
10#15216C211yesmaximal, characteristic
11#13982C212yesmaximal, complemented
12#13875C212yesmaximal
13#13844C211yesmaximal, characteristic
14C22×C4⋊Q8C2212yes—
15C22×#125C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#2628—216no—
18#2550—216no—
19C2×#400C2214yes—
20C2×#399C2214yes—
21C2×#398C2214yes—
22C2×#395C2214yes—
23C2×#395C2214yes—
24C2×#393C2214yes—
25C2×#391C2214yes—
26C2×#389C2212yes—
27C2×#389C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#382C2212yes—
31C2×#382C2212yes—
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39#1446C2214yes—
40#1446C2214yes—
41#1445C2214yes—
42#1445C2214yes—
43#1445C2214yes—
44#1444C2212yes—
45#1444C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2212yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1439C2212yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66#1434C2214yes—
67#1434C2214yes—
68C4×#95C2212yes—
69C23×Q8C2314yes—
70C24×C4C2311yescharacteristic
71C2×C4⋊Q8—28no—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—44no—
74C2×#125—28no—
75C2×#125—42no—
76C2×#117—28no—
77C2×#117—42no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4C2311yescharacteristic
91C22×C22⋊C4C2311yescharacteristic
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2312yes—
99C22×C42C2312yes—
100C22×C42C2314yes—
101#400—216no—
102#399—216no—
103#398—216no—
104#395—216no—
105#395—216no—
106#393—216no—
107#391—216no—
108#389—216no—
109#389—28no—
110#389—28no—
111#389D4116yes—
112#382—28no—
113#382—28no—
114#382—42no—
115#380—28no—
116#380—28no—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—28no—
119C4×C22⋊C4—42no—
120C4×C22⋊C4Q818yescomplemented
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C25C2411yescharacteristic
150C22×Q8—28no—
151C22×Q8—28no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8—44no—
155C22×Q8—44no—
156C22×Q8—44no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4C2×Q814yes—
162C23×C4C2412yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C4⋊Q8—432no—
175#125—416no—
176#117—416no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
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×C4⋊C4C2×D418yes—
212C2×C4⋊C4C2×D4116yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4C2×Q818yes—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—28no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42C2×D418yes—
241C2×C42C2×Q818yes—
242#95—24no—
243#95—24no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—44no—
254#95—42no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—44no—
260#95—42no—
261#95—42no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—44no—
267#95C2×D4116yes—
268#95C2×Q814yes—
269#95C2×Q814yes—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24C22×Q812yes—
279C24C2511yescharacteristic
280C2×Q8—416no—
281C2×Q8—416no—
282C2×Q8—88no—
283C2×Q8—88no—
284C2×Q8—88no—
285C2×Q8—88no—
286C2×Q8—88no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—216no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—48no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4C22×D418yes—
361C22×C4C22×D414yes—
362C22×C4C22×D418yes—
363C22×C4C22×D414yes—
364C22×C4C22×D414yes—
365C22×C4C22×Q812yes—
366C22×C4C22×Q814yes—
367C22×C4C22×Q814yes—
368C22×C4C22×Q812yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C42−1+412yes—
371C22×C42−1+414yes—
372C4⋊C4—216no—
373C4⋊C4—216no—
374C4⋊C4—416no—
375C4⋊C4—48no—
376C4⋊C4—416no—
377C4⋊C4—416no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C22⋊C4—48no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—48no—
388C22⋊C4—88no—
389C22⋊C4—88no—
390C42—216no—
391C42—216no—
392C42—44no—
393C42—44no—
394C42—88no—
395C23—22no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—42no—
401C23—41no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23C23×D412yes—
417C23C23×Q811yescharacteristic
418C23C2×H(5,𝔽2)11yescharacteristic
419C23C2×2−1+411yescharacteristic
420C23C2×2−1+412yes—
421C23#58512yes—
422C23#58512yes—
423C23#58512yes—
424C23#58512yes—
425Q8—832no—
426C2×C4—28no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—416no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4#53618yes—
503C2×C4#54414yes—
504C2×C4D4214yes—
505C2×C4D4×Q818yes—
506C2×C4#56314yes—
507C22—42no—
508C22—44no—
509C22—42no—
510C22—42no—
511C22—44no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22C2×#53612yes—
517C22C2×#54411yescharacteristic
518C22#345812yes—
519C22C2×D4211yescharacteristic
520C22C2×D4×Q812yes—
521C22#348512yes—
522C22#348712yes—
523C22#350012yes—
524C22#350012yes—
525C22#350411yescharacteristic
526C22#350411yescharacteristic
527C22C2×#56311yescharacteristic
528C22#354012yes—
529C22#355112yes—
530C22#355712yes—
531C22#355711yescharacteristic
532C22#355711yescharacteristic
533C22#356512yes—
534C22#356512yes—
535C22#356512yes—
536C22#356512yes—
537C4—44no—
538C4—48no—
539C4—44no—
540C4—44no—
541C4—42no—
542C4—48no—
543C4—44no—
544C4—82no—
545C4—84no—
546C4—82no—
547C4—84no—
548C4#267718yes—
549C2—82no—
550C2#3742811yescharacteristic
551C2#3756511yescharacteristic
552C2#3769112yes—
553C2#3936612yes—
554C2#4264712yes—
555C2#4287512yes—
556C2#4509312yes—
557C2C2×#267711yescharacteristic
558C2#5307212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2550C2semidirect
3. #17099C2semidirect
6. #15699C2semidirect
7. #15230C2semidirect
9. #15221C2semidirect
11. #13982C2semidirect
120. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
C2×#26282
#170992
#165312
#163752
#156992
#152302
#152232
#152212
#152162
#139822
#138752
#138442