Finite Group Databasebeta

← #878114

Subgroups — #878114

Order 512.

2070 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16404C214yesmaximal
2#16239C214yesmaximal, complemented
3#15519C214yesmaximal, complemented
4#15223C211yesmaximal, characteristic
5#15223C211yesmaximal, characteristic
6#15221C212yesmaximal, complemented
7#15146C212yesmaximal, complemented
8#14697C212yesmaximal
9#14442C214yesmaximal
10#14151C212yesmaximal, complemented
11#14006C212yesmaximal, complemented
12#13876C212yesmaximal
13C4×#385C211yesmaximal, characteristic
14C23×C42C2211yescharacteristic
15C2×#400C2214yes—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#389C2214yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#388C2214yes—
22C2×#385C2212yes—
23C2×#385C2214yes—
24C2×#384C2212yes—
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1445C2214yes—
34#1445C2214yes—
35#1445C2214yes—
36#1444C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2212yes—
55#1436C2212yes—
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C4×#95C2212yes—
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2311yescharacteristic
85C22×C22⋊C4C2311yescharacteristic
86C22×C42—44no—
87C22×C42—42no—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2312yes—
92C22×C42C2314yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#400—48no—
96#389—28no—
97#389—48no—
98#389—44no—
99#389—44no—
100#389—44no—
101#388—216no—
102#385—28no—
103#385—48no—
104#384—28no—
105#382—42no—
106#382—42no—
107#380—28no—
108#380—44no—
109C4×C4⋊C4—44no—
110C4×C4⋊C4—44no—
111C4×C22⋊C4—42no—
112C4×C22⋊C4Q818yes—
113C2×#95C2312yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2312yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C25C2411yescharacteristic
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C2×Q814yes—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2412yes—
156C23×C4C2414yes—
157C23×C4C2412yes—
158C23×C4C2412yes—
159C23×C4C2411yescharacteristic
160C23×C4C2414yes—
161C23×C4C2412yes—
162C23×C4C2412yes—
163C23×C4C2412yes—
164C23×C4C2411yescharacteristic
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4#4118yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4C2×Q814yes—
212C2×C22⋊C4C2×Q814yes—
213C2×C22⋊C4#4118yes—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—28no—
224C2×C42—41no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—41no—
228C2×C42—416no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42C2×Q818yes—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—24no—
241#95—24no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95C2×Q818yes—
259C24—22no—
260C24—22no—
261C24—21no—
262C24—22no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24C22×Q812yes—
267C24C2×#4114yes—
268C24C2511yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—28no—
302C22×C4—216no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
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—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4C22×Q812yes—
349C22×C4C22×Q814yes—
350C22×C4C22×Q814yes—
351C22×C4C22×Q812yes—
352C22×C4C2×#4118yes—
353C22×C4C2×#4114yes—
354C22×C4C2×#4114yes—
355C22×C4C2×#4114yes—
356C22×C4H(5,𝔽2)14yes—
357C22×C4H(5,𝔽2)14yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+414yes—
360C22×C42−1+412yes—
361C4⋊C4—216no—
362C4⋊C4—416no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—88no—
369C4⋊C4—88no—
370C4⋊C4—88no—
371C4⋊C4—88no—
372C4⋊C4—88no—
373C22⋊C4—44no—
374C22⋊C4—416no—
375C22⋊C4—48no—
376C22⋊C4—88no—
377C22⋊C4#125116yes—
378C42—216no—
379C42—44no—
380C42—44no—
381C42—44no—
382C42—44no—
383C42—48no—
384C42—88no—
385C23—22no—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—42no—
396C23—41no—
397C23—42no—
398C23—42no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23C2×#12514yes—
406C23C23×Q811yescharacteristic
407C23C22×#4112yes—
408C23C22×#4112yes—
409C23C2×H(5,𝔽2)12yes—
410C23C2×H(5,𝔽2)11yescharacteristic
411C23C2×H(5,𝔽2)12yes—
412C23C2×2−1+411yescharacteristic
413C23C2×2−1+412yes—
414C23#58512yes—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—432no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—416no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4C2×#12518yes—
492C2×C4#53918yes—
493C2×C4#54414yes—
494C2×C4#55418yes—
495C2×C4#55914yes—
496C2×C4#56014yes—
497C22—24no—
498C22—28no—
499C22—44no—
500C22—42no—
501C22—44no—
502C22—42no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22C22×#12511yescharacteristic
507C22C2×#53912yes—
508C22C2×#54411yescharacteristic
509C22#345811yescharacteristic
510C22C2×#55412yes—
511C22C2×#55911yescharacteristic
512C22#348112yes—
513C22#348212yes—
514C22#350312yes—
515C22#350612yes—
516C22#350712yes—
517C22#351412yes—
518C22#351812yes—
519C22#351912yes—
520C22#352012yes—
521C22C2×#56011yescharacteristic
522C22#353612yes—
523C22#354112yes—
524C22#354212yes—
525C22#354811yescharacteristic
526C22#355611yescharacteristic
527C4—28no—
528C4—42no—
529C4—44no—
530C4—44no—
531C4—48no—
532C4—44no—
533C4—82no—
534C4—84no—
535C4—84no—
536C4—82no—
537C4—84no—
538C4#280718yes—
539C2—44no—
540C2#3518111yescharacteristic
541C2#3629012yes—
542C2#3730911yescharacteristic
543C2#3810512yes—
544C2#4014912yes—
545C2#4022012yes—
546C2#4564612yes—
547C2C2×#280711yescharacteristic
548C2#5963012yes—
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
2. #16239C2semidirect
3. #15519C2semidirect
6. #15221C2semidirect
7. #15146C2semidirect
10. #14151C2semidirect
11. #14006C2semidirect

Maximal subgroups

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

subgroupindex
#164042
#162392
#155192
#152232
#152212
#151462
#146972
#144422
#141512
#140062
#138762
C4×#3852