Finite Group Databasebeta

← #878109

Subgroups — #878109

Order 512.

1950 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16894C214yesmaximal
2#16235C214yesmaximal, complemented
3#16172C214yesmaximal, complemented
4#15232C212yesmaximal, complemented
5#15223C211yesmaximal, characteristic
6#15223C211yesmaximal, characteristic
7#15221C212yesmaximal, complemented
8#14697C212yesmaximal
9#14457C214yesmaximal
10#14150C212yesmaximal, complemented
11#14006C212yesmaximal, complemented
12#13875C212yesmaximal
13C4×#385C211yesmaximal, characteristic
14C23×C42C2211yescharacteristic
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#385C2214yes—
20C2×#385C2212yes—
21C2×#382C2212yes—
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1445C2214yes—
31#1445C2214yes—
32#1445C2214yes—
33#1445C2214yes—
34#1444C2214yes—
35#1444C2214yes—
36#1443C2214yes—
37#1443C2214yes—
38#1443C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1437C2214yes—
49#1436C2214yes—
50#1436C2212yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2212yes—
55#1436C2214yes—
56C4⋊#95C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C4×#95C2212yes—
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2311yescharacteristic
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2311yescharacteristic
84C22×C42—44no—
85C22×C42—42no—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2314yes—
93#389—28no—
94#389—44no—
95#389—44no—
96#389—44no—
97#385—28no—
98#385—216no—
99#382—28no—
100#382—42no—
101#382—42no—
102#380—28no—
103#380—44no—
104C4×C4⋊C4—44no—
105C4×C4⋊C4—44no—
106C4×C22⋊C4—42no—
107C4×C22⋊C4Q818yes—
108C2×#95C2312yes—
109C2×#95C2312yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2312yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C25C2411yescharacteristic
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C2×Q814yes—
149C23×C4#4118yes—
150C23×C4C2412yes—
151C23×C4C2414yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2412yes—
155C23×C4C2412yes—
156C23×C4C2412yes—
157C23×C4C2411yescharacteristic
158C23×C4C2414yes—
159C23×C4C2412yes—
160C23×C4C2412yes—
161C23×C4C2411yescharacteristic
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4#4118yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4C2×Q814yes—
204C2×C22⋊C4C2×Q814yes—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—41no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—41no—
218C2×C42—416no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—44no—
222C2×C42C2×Q818yes—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—24no—
227#95—24no—
228#95—28no—
229#95—28no—
230#95—28no—
231#95—42no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—44no—
240#95—44no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95C2×Q818yes—
252C24—22no—
253C24—21no—
254C24—22no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24C22×Q812yes—
260C24C2×#4114yes—
261C24C2511yescharacteristic
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—28no—
297C22×C4—216no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4C22×Q812yes—
344C22×C4C22×Q814yes—
345C22×C4C22×Q814yes—
346C22×C4C22×Q812yes—
347C22×C4C2×#4118yes—
348C22×C4C2×#4114yes—
349C22×C4C2×#4114yes—
350C22×C4C2×#4114yes—
351C22×C4H(5,𝔽2)14yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+412yes—
354C4⋊C4—216no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—88no—
361C4⋊C4—88no—
362C4⋊C4—88no—
363C4⋊C4—88no—
364C22⋊C4—28no—
365C22⋊C4—44no—
366C22⋊C4—416no—
367C22⋊C4—48no—
368C22⋊C4—88no—
369C42—216no—
370C42—44no—
371C42—44no—
372C42—44no—
373C42—44no—
374C42—48no—
375C42—88no—
376C23—22no—
377C23—22no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23—42no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—41no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—41no—
395C23—41no—
396C23#53314yes—
397C23C23×Q811yescharacteristic
398C23C22×#4112yes—
399C23C22×#4112yes—
400C23C2×H(5,𝔽2)12yes—
401C23C2×H(5,𝔽2)11yescharacteristic
402C23C2×2−1+411yescharacteristic
403C23#58512yes—
404C23#58512yes—
405C23#58512yes—
406C2×C4—24no—
407C2×C4—28no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—28no—
423C2×C4—432no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—416no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4#53918yes—
484C2×C4#54414yes—
485C2×C4#55418yes—
486C2×C4#55914yes—
487C2×C4#56014yes—
488C22—24no—
489C22—28no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22C2×#53311yescharacteristic
498C22C2×#53912yes—
499C22C2×#54411yescharacteristic
500C22#345811yescharacteristic
501C22C2×#55412yes—
502C22#347512yes—
503C22C2×#55911yescharacteristic
504C22#348012yes—
505C22#350412yes—
506C22#350512yes—
507C22#350512yes—
508C22#350512yes—
509C22#350712yes—
510C22#352112yes—
511C22#352412yes—
512C22C2×#56011yescharacteristic
513C22#353912yes—
514C22#354012yes—
515C22#354012yes—
516C22#355711yescharacteristic
517C22#355711yescharacteristic
518C4—44no—
519C4—42no—
520C4—48no—
521C4—44no—
522C4—44no—
523C4—44no—
524C4—84no—
525C4—82no—
526C4—84no—
527C4—84no—
528C4—82no—
529C4#280718yes—
530C2—44no—
531C2#3510411yescharacteristic
532C2#3638712yes—
533C2#3742711yescharacteristic
534C2#3810612yes—
535C2#3953612yes—
536C2#3979412yes—
537C2#4550812yes—
538C2C2×#280711yescharacteristic
539C2#5962712yes—
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. #16235C2semidirect
3. #16172C2semidirect
4. #15232C2semidirect
7. #15221C2semidirect
10. #14150C2semidirect
11. #14006C2semidirect

Maximal subgroups

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

subgroupindex
#168942
#162352
#161722
#152322
#152232
#152212
#146972
#144572
#141502
#140062
#138752
C4×#3852