Finite Group Databasebeta

← #4414922

Subgroups — #4414922

Order 512.

2006 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17076C211yesmaximal, characteristic, complemented
2#17071C212yesmaximal
3#17070C211yesmaximal, characteristic, complemented
4#16463C214yesmaximal, complemented
5#16317C214yesmaximal, complemented
6#15098C212yesmaximal, complemented
7#14831C214yesmaximal, complemented
8#14825C212yesmaximal
9#14771C214yesmaximal, complemented
10#14699C212yesmaximal, complemented
11#14667C212yesmaximal
12#14639C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic
14C22×C4×D4C2211yescharacteristic
15C2×#400C2212yes—
16C2×#399C2214yes—
17C2×#399C2212yes—
18C2×#398C2214yes—
19C2×#396C2214yes—
20C2×#396C2214yes—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387C2214yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2214yes—
30C2×C43C2211yescharacteristic
31#1444C2214yes—
32#1444C2214yes—
33#1444C2212yes—
34#1444C2214yes—
35#1444C2212yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1441C2212yes—
40#1441C2214yes—
41#1441C2212yes—
42#1441C2212yes—
43#1441C2214yes—
44#1441C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1437C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63#1434C2214yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C23×D4C2311yescharacteristic
68C24×C4C2312yes—
69C2×C4×D4—24no—
70C2×C4×D4—42no—
71C2×C4×D4—42no—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C42C2311yescharacteristic
95C22×C42C2312yes—
96C22×C42C2312yes—
97C22×C42C2312yes—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#400—28no—
101#399—48no—
102#399—44no—
103#398—48no—
104#396—48no—
105#396—48no—
106#390—24no—
107#390Q818yes—
108#389—28no—
109#389—44no—
110#389—44no—
111#387—48no—
112#380—44no—
113C4×C4⋊C4—42no—
114C4×C22⋊C4—48no—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2312yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C43—42no—
139C25C2412yes—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—41no—
143C22×D4—41no—
144C22×D4—41no—
145C22×D4—41no—
146C22×D4C2×Q814yes—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—28no—
150C23×C4—42no—
151C23×C4—44no—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2412yes—
155C23×C4C2414yes—
156C23×C4C2414yes—
157C23×C4C2412yes—
158C23×C4C2412yes—
159C23×C4C2412yes—
160C23×C4C2411yescharacteristic
161C23×C4C2411yescharacteristic
162C23×C4C2412yes—
163C23×C4C2411yescharacteristic
164C4×D4—416no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C2×Q814yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4C2×Q8116yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—42no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—42no—
242C2×C42—44no—
243C2×C42C2×Q814yes—
244C2×C42#4114yes—
245#95—28no—
246#95—24no—
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—42no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—42no—
268C24—22no—
269C24—22no—
270C24—24no—
271C24—24no—
272C24—22no—
273C24—22no—
274C24—24no—
275C24—24no—
276C24—22no—
277C24—22no—
278C24C22×Q814yes—
279C24C2511yescharacteristic
280C2×D4—28no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—48no—
350C22×C4—416no—
351C22×C4—48no—
352C22×C4—88no—
353C22×C4—84no—
354C22×C4C22×Q814yes—
355C22×C4C22×Q814yes—
356C22×C4C22×Q812yes—
357C22×C4C2×#4114yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4114yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)14yes—
363C22×C4H(5,𝔽2)14yes—
364C22×C4H(5,𝔽2)12yes—
365C22×C42−1+414yes—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—416no—
372C4⋊C4—48no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—88no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—88no—
380C4⋊C4—88no—
381C22⋊C4—416no—
382C22⋊C4—48no—
383C22⋊C4—416no—
384C22⋊C4—88no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C22⋊C4—88no—
388C42—28no—
389C42—44no—
390C42—44no—
391C42—48no—
392C42—48no—
393C42—84no—
394C23—24no—
395C23—28no—
396C23—24no—
397C23—24no—
398C23—28no—
399C23—44no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—42no—
413C23—44no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23C23×Q811yescharacteristic
418C23C22×#4111yescharacteristic
419C23C22×#4111yescharacteristic
420C23C2×H(5,𝔽2)12yes—
421C23C2×H(5,𝔽2)12yes—
422C23C2×H(5,𝔽2)11yescharacteristic
423C23C2×H(5,𝔽2)11yescharacteristic
424C23C2×2−1+412yes—
425C23#58511yescharacteristic
426C23#58511yescharacteristic
427C23#58512yes—
428D4—416no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—816no—
501C2×C4#53314yes—
502C2×C4#54018yes—
503C2×C4#56314yes—
504C22—44no—
505C22—48no—
506C22—44no—
507C22—44no—
508C22—48no—
509C22—84no—
510C22—82no—
511C22—82no—
512C22—84no—
513C22—82no—
514C22—82no—
515C22C2×#53311yescharacteristic
516C22C2×#54012yes—
517C22#345911yescharacteristic
518C22#346011yescharacteristic
519C22#346112yes—
520C22#349011yescharacteristic
521C22#349311yescharacteristic
522C22#350111yescharacteristic
523C22#350112yes—
524C22#350111yescharacteristic
525C22#350312yes—
526C22#351712yes—
527C22#352012yes—
528C22C2×#56311yescharacteristic
529C22#352911yescharacteristic
530C22#353112yes—
531C22#353412yes—
532C22#353612yes—
533C22#354311yescharacteristic
534C22#354612yes—
535C22#354911yescharacteristic
536C22#355912yes—
537C22#356012yes—
538C4—28no—
539C4—24no—
540C4—24no—
541C4—44no—
542C4—44no—
543C4—44no—
544C4—42no—
545C4—84no—
546C4—84no—
547C4—84no—
548C4—82no—
549C4—84no—
550C2—84no—
551C2#3501211yescharacteristic
552C2#3517612yes—
553C2#3669811yescharacteristic
554C2#3746512yes—
555C2#3762611yescharacteristic
556C2#4579312yes—
557C2#4595712yes—
558C2#4615211yescharacteristic
559C2#4619911yescharacteristic
560C2#4628211yescharacteristic
561C2#4631811yescharacteristic
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
1. #17076C2semidirect
3. #17070C2semidirect
4. #16463C2semidirect
5. #16317C2semidirect
6. #15098C2semidirect
7. #14831C2semidirect
9. #14771C2semidirect
10. #14699C2semidirect
12. #14639C2semidirect

Maximal subgroups

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

subgroupindex
#170762
#170712
#170702
#164632
#163172
#150982
#148312
#148252
#147712
#146992
#146672
#146392
C4×#3902