Finite Group Databasebeta

← #948170

Subgroups — #948170

Order 512.

3096 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2406C212yesmaximal
2#17219C212yesmaximal, complemented
3#17218C212yesmaximal, complemented
4#16657C214yesmaximal, complemented
5#16309C214yesmaximal, complemented
6#15966C211yesmaximal, characteristic, complemented
7#15949C211yesmaximal, characteristic, complemented
8#15463C212yesmaximal, complemented
9#15204C212yesmaximal, complemented
10#14837C214yesmaximal, complemented
11#14313C212yesmaximal, complemented
12#14043C214yesmaximal, complemented
13#13851C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×#123C2212yes—
16C22×#123C2212yes—
17C23×C4⋊C4C2212yes—
18#2406—216no—
19C2×#397C2212yes—
20C2×#397C2214yes—
21C2×#395C2214yes—
22C2×#394C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#386C2212yes—
26C2×#386C2212yes—
27C2×#385C2214yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#381C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2214yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1446C2211yescharacteristic
36#1446C2211yescharacteristic
37#1445C2212yes—
38#1445C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1444C2212yes—
42#1444C2212yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2212yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1438C2214yes—
60#1438C2211yescharacteristic
61#1438C2211yescharacteristic
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22×#41—42no—
69C22×#41—42no—
70C22×#41—48no—
71C22×#41D418yes—
72C23×Q8C2311yescharacteristic
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C2×#123—28no—
80C2×#123—28no—
81C2×#123—48no—
82C2×#123—48no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—44no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107#397—28no—
108#397—48no—
109#395—216no—
110#394—216no—
111#389—44no—
112#389—44no—
113#386—28no—
114#386—28no—
115#385—216no—
116#385—216no—
117#385—216no—
118#385—48no—
119#381—28no—
120C4×C22⋊C4—42no—
121C4×C22⋊C4—42no—
122C4×C22⋊C4—48no—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2311yescharacteristic
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C25C2411yescharacteristic
149C25C2412yes—
150C2×#41—416no—
151C2×#41—432no—
152C2×#41—432no—
153C2×#41—432no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—44no—
157C22×Q8C2×D414yes—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—44no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4C2×D414yes—
166C22×D4C2×D418yes—
167C23×C4—28no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—28no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—44no—
184C23×C4C2×D418yes—
185C23×C4C2×D414yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197#123—432no—
198#123—432no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—432no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4#4118yescomplemented
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—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—44no—
255C2×C42—44no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—24no—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—44no—
267#95—42no—
268#95—41no—
269#95—44no—
270#95—44no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281C24—24no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24C22×D412yes—
296C24C22×D414yes—
297C24H(5,𝔽2)18yes—
298C24C2511yescharacteristic
299#41—4128no—
300C2×Q8—44no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×D4—216no—
305C2×D4—216no—
306C2×D4—48no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—416no—
312C2×D4—416no—
313C22×C4—24no—
314C22×C4—216no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—216no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
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—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—416no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—416no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4C22×D414yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C2×#4114yes—
388C22×C4C2×#4118yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+414yes—
392C22×C42−1+414yes—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—416no—
396C4⋊C4—416no—
397C4⋊C4—416no—
398C4⋊C4—416no—
399C4⋊C4—88no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C22⋊C4—232no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C22⋊C4—416no—
406C22⋊C4—416no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—416no—
410C22⋊C4—48no—
411C22⋊C4—416no—
412C42—84no—
413C42—84no—
414C23—24no—
415C23—24no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—28no—
422C23—28no—
423C23—28no—
424C23—24no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—44no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—44no—
438C23—42no—
439C23—41no—
440C23—44no—
441C23—44no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—44no—
450C23#53418yes—
451C23C23×D411yescharacteristic
452C23C22×#4112yes—
453C23C2×H(5,𝔽2)11yescharacteristic
454C23C2×H(5,𝔽2)11yescharacteristic
455C23C2×H(5,𝔽2)14yes—
456C23C2×2−1+412yes—
457C23C2×2−1+412yes—
458C23#58512yes—
459Q8—416no—
460D4—416no—
461D4—432no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—432no—
480C2×C4—416no—
481C2×C4—432no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
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—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—816no—
526C2×C4#53414yes—
527C22—216no—
528C22—28no—
529C22—44no—
530C22—48no—
531C22—48no—
532C22—48no—
533C22—48no—
534C22—44no—
535C22—48no—
536C22—42no—
537C22—48no—
538C22—42no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22C2×#53411yescharacteristic
545C22C2×#53412yes—
546C22#345912yes—
547C22D4×#4112yes—
548C22#348111yescharacteristic
549C22#349214yes—
550C22#349912yes—
551C22#352214yes—
552C22#352911yescharacteristic
553C22#353212yes—
554C22#353212yes—
555C22#353514yes—
556C22#353812yes—
557C22#355011yescharacteristic
558C22#355011yescharacteristic
559C22#355914yes—
560C4—24no—
561C4—44no—
562C4—44no—
563C4—44no—
564C4—42no—
565C4—48no—
566C4—44no—
567C4—84no—
568C4—84no—
569C4—84no—
570C4—82no—
571C2—48no—
572C2—82no—
573C2#3738611yescharacteristic
574C2#3743611yescharacteristic
575C2#3842514yes—
576C2#4097412yes—
577C2#4212212yes—
578C2#4430612yes—
579C2#5216311yescharacteristic
580C2#5242312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17219C2semidirect
3. #17218C2semidirect
4. #16657C2semidirect
5. #16309C2semidirect
6. #15966C2semidirect
7. #15949C2semidirect
8. #15463C2semidirect
9. #15204C2semidirect
10. #14837C2semidirect
11. #14313C2semidirect
12. #14043C2semidirect
223. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#24062
#172192
#172182
#166572
#163092
#159662
#159492
#154632
#152042
#148372
#143132
#140432
#138512