Finite Group Databasebeta

← #772057

Subgroups — #772057

Order 512.

2622 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C212yesmaximal
2#16002C212yesmaximal, complemented
3#15976C212yesmaximal, complemented
4#15973C214yesmaximal, complemented
5#15966C211yesmaximal, characteristic
6#15949C211yesmaximal, characteristic
7#15717C214yesmaximal, complemented
8#15463C212yesmaximal
9#15206C212yesmaximal, complemented
10#15122C212yesmaximal, complemented
11#14202C214yesmaximal, complemented
12#14043C214yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14C22×#123C2212yes—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2212yes—
17C2×#397C2214yes—
18#2322—216no—
19C2×#389C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#388C2214yes—
23C2×#387C2214yes—
24C2×#387C2214yes—
25C2×#386C2212yes—
26C2×#385C2212yes—
27C2×#385C2214yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#381C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2214yes—
34C2×#376C2211yescharacteristic
35#1445C2212yes—
36#1445C2212yes—
37#1445C2211yescharacteristic
38#1445C2211yescharacteristic
39#1444C2212yes—
40#1444C2212yes—
41#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1442C2212yes—
45#1442C2214yes—
46#1442C2212yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2211yescharacteristic
58#1438C2211yescharacteristic
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64#1434C2212yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C2×#123—28no—
73C2×#123—216no—
74C2×C4×D4—42no—
75C2×C4×D4—42no—
76C2×C4×D4D418yes—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4C2314yes—
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⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2314yes—
101C22×C42C2312yes—
102#397—216no—
103#389—216no—
104#389—44no—
105#389—44no—
106#388—216no—
107#387—216no—
108#387—48no—
109#386—28no—
110#385—216no—
111#385—28no—
112#385—216no—
113#385—48no—
114#381—28no—
115C4×C4⋊C4—42no—
116C4×C22⋊C4—28no—
117C4×C22⋊C4—48no—
118#376—42no—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2311yescharacteristic
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C25C2412yes—
147C22×D4—28no—
148C22×D4—41no—
149C22×D4—41no—
150C22×D4C2×D414yes—
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—28no—
156C23×C4—24no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4C2×D418yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2411yescharacteristic
178#123—264no—
179C4×D4—416no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—432no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D414yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4C2×D418yes—
229C2×C22⋊C4#41116yes—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—28no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42C2×D414yes—
241#95—28no—
242#95—28no—
243#95—28no—
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—24no—
254#95—24no—
255#95—24no—
256#95—44no—
257#95—42no—
258#95—42no—
259#95—41no—
260#95—41no—
261#95—44no—
262#95—41no—
263#95—41no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268C24—24no—
269C24—24no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24C22×D414yes—
276C24C2×#4118yes—
277C24C2511yescharacteristic
278C2×D4—216no—
279C2×D4—44no—
280C2×D4—48no—
281C2×D4—48no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—216no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—216no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—416no—
341C22×C4—416no—
342C22×C4—416no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—88no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D414yes—
356C22×C4C22×D412yes—
357C22×C4C2×#4118yes—
358C22×C4C2×#4118yes—
359C22×C4H(5,𝔽2)14yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+414yes—
363C22×C42−1+414yes—
364C4⋊C4—232no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—416no—
369C4⋊C4—48no—
370C4⋊C4—416no—
371C4⋊C4—416no—
372C4⋊C4—88no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C22⋊C4—232no—
376C22⋊C4—216no—
377C22⋊C4—232no—
378C22⋊C4—48no—
379C22⋊C4—416no—
380C22⋊C4—48no—
381C22⋊C4—88no—
382C42—28no—
383C42—48no—
384C42—84no—
385C42—88no—
386C23—24no—
387C23—24no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—28no—
392C23—28no—
393C23—28no—
394C23—28no—
395C23—28no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23#52518yes—
407C23C23×D411yescharacteristic
408C23C22×#4114yes—
409C23C2×H(5,𝔽2)11yescharacteristic
410C23C2×H(5,𝔽2)11yescharacteristic
411C23C2×H(5,𝔽2)12yes—
412C23C2×2−1+412yes—
413C23C2×2−1+412yes—
414C23#58512yes—
415D4—416no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—432no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—44no—
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—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—816no—
484C2×C4—816no—
485C2×C4#53414yes—
486C22—216no—
487C22—28no—
488C22—44no—
489C22—44no—
490C22—48no—
491C22—48no—
492C22—48no—
493C22—48no—
494C22—48no—
495C22C2×#52512yes—
496C22C2×#53411yescharacteristic
497C22#348112yes—
498C22#348112yes—
499C22#349214yes—
500C22#349612yes—
501C22#349912yes—
502C22#350714yes—
503C22#351514yes—
504C22#352014yes—
505C22#353412yes—
506C22#353812yes—
507C22#354311yescharacteristic
508C22#354511yescharacteristic
509C22#355011yescharacteristic
510C22#355011yescharacteristic
511C4—24no—
512C4—44no—
513C4—42no—
514C4—48no—
515C4—44no—
516C4—44no—
517C4—48no—
518C4—42no—
519C4—44no—
520C4—84no—
521C4—84no—
522C4—82no—
523C2—48no—
524C2#3495611yescharacteristic
525C2#3536314yes—
526C2#3712911yescharacteristic
527C2#3738611yescharacteristic
528C2#4340312yes—
529C2#4345212yes—
530C2#4346212yes—
531C2#4380312yes—
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
2. #16002C2semidirect
3. #15976C2semidirect
4. #15973C2semidirect
7. #15717C2semidirect
9. #15206C2semidirect
10. #15122C2semidirect
11. #14202C2semidirect
12. #14043C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
#160022
#159762
#159732
#159662
#159492
#157172
#154632
#152062
#151222
#142022
#140432
#138472