Finite Group Databasebeta

← #4474580

Subgroups — #4474580

Order 512.

2686 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17282C214yesmaximal, complemented
2#16903C211yesmaximal, characteristic, complemented
3#16706C214yesmaximal, complemented
4#16654C214yesmaximal, complemented
5#16252C211yesmaximal, characteristic
6#15369C214yesmaximal, complemented
7#15368C214yesmaximal, complemented
8#15270C214yesmaximal, complemented
9#14719C212yesmaximal, complemented
10#14417C212yesmaximal, complemented
11#14339C211yesmaximal, characteristic, complemented
12C22×#123C2214yes—
13C22×#121C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C2×#394C2212yes—
16C2×#394C2212yes—
17C2×#392C2214yes—
18C2×#392C2214yes—
19C2×#386C2214yes—
20C2×#386C2214yes—
21C2×#386C2214yes—
22C2×#386C2214yes—
23C2×#386C2214yes—
24C2×#381C2212yes—
25C2×#380C2212yes—
26#1445C2214yes—
27#1445C2214yes—
28#1443C2214yes—
29#1443C2214yes—
30#1442C2214yes—
31#1442C2214yes—
32#1442C2214yes—
33#1441C2214yes—
34#1441C2214yes—
35#1441C2212yes—
36#1441C2212yes—
37#1440C2214yes—
38#1440C2214yes—
39#1440C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1438C2214yes—
49#1438C2214yes—
50#1438C2214yes—
51C22⋊#95C2211yescharacteristic
52C22⋊#95C2211yescharacteristic
53C22⋊#95C2212yes—
54C22⋊#95C2214yes—
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C23×D4C2314yes—
58C23×D4C2312yes—
59C23×D4C2314yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2314yes—
63C24×C4C2311yescharacteristic
64C2×#123—48no—
65C2×#123—48no—
66C2×#123—48no—
67C2×#121—28no—
68C2×#121—48no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89#394—44no—
90#394—44no—
91#392—48no—
92#392—48no—
93#386—48no—
94#386—48no—
95#386—48no—
96#386—48no—
97#386—48no—
98#381—44no—
99#380—44no—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2312yes—
104C2×#95C2312yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C25C2414yes—
121C25C2411yescharacteristic
122C22×D4—28no—
123C22×D4—28no—
124C22×D4—28no—
125C22×D4—28no—
126C22×D4—28no—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—24no—
130C22×D4—44no—
131C22×D4—44no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—44no—
139C22×D4—42no—
140C22×D4—42no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4#4114yes—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C23×C4C2411yescharacteristic
168#123—464no—
169#121—432no—
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—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—416no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
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—
224#95—28no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—42no—
243#95—42no—
244C24—21no—
245C24—22no—
246C24—24no—
247C24—21no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—24no—
255C24—24no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24C2×#4112yes—
264C24H(5,𝔽2)18yes—
265C24H(5,𝔽2)14yes—
266C24C2511yescharacteristic
267C2×D4—216no—
268C2×D4—416no—
269C2×D4—416no—
270C2×D4—416no—
271C2×D4—416no—
272C2×D4—416no—
273C2×D4—416no—
274C2×D4—84no—
275C2×D4—84no—
276C2×D4—88no—
277C2×D4—88no—
278C2×D4—88no—
279C2×D4—88no—
280C2×D4—88no—
281C2×D4—88no—
282C2×D4—88no—
283C2×D4—88no—
284C2×D4—88no—
285C2×D4—88no—
286C22×C4—28no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
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—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—44no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—416no—
346C22×C4—88no—
347C22×C4—84no—
348C22×C4—88no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4—88no—
355C22×C4C2×#4114yes—
356C22×C4H(5,𝔽2)12yes—
357C22×C4H(5,𝔽2)14yes—
358C22×C42−1+412yes—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—416no—
363C4⋊C4—416no—
364C22⋊C4—416no—
365C22⋊C4—416no—
366C22⋊C4—416no—
367C22⋊C4—416no—
368C22⋊C4—88no—
369C22⋊C4—88no—
370C22⋊C4—88no—
371C22⋊C4—84no—
372C22⋊C4—84no—
373C22⋊C4—88no—
374C22⋊C4—88no—
375C22⋊C4—88no—
376C22⋊C4—88no—
377C23—22no—
378C23—24no—
379C23—22no—
380C23—24no—
381C23—28no—
382C23—28no—
383C23—24no—
384C23—24no—
385C23—28no—
386C23—28no—
387C23—28no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—28no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23#54014yes—
425C23C22×#4111yescharacteristic
426C23C2×H(5,𝔽2)12yes—
427C23C2×H(5,𝔽2)14yes—
428C23C2×H(5,𝔽2)12yes—
429C23C2×H(5,𝔽2)11yescharacteristic
430C23C2×2−1+411yescharacteristic
431C23#58514yes—
432D4—816no—
433D4—832no—
434D4—832no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—416no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
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—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—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—88no—
492C2×C4—88no—
493C2×C4—832no—
494C22—28no—
495C22—24no—
496C22—48no—
497C22—48no—
498C22—42no—
499C22—42no—
500C22—44no—
501C22—48no—
502C22—48no—
503C22—48no—
504C22—48no—
505C22—48no—
506C22—48no—
507C22—48no—
508C22—84no—
509C22—84no—
510C22—84no—
511C22—84no—
512C22—84no—
513C22—84no—
514C22—84no—
515C22—84no—
516C22C2×#54011yescharacteristic
517C22#349012yes—
518C22#349011yescharacteristic
519C22#349211yescharacteristic
520C22#351012yes—
521C22#353214yes—
522C22#353414yes—
523C22#353414yes—
524C22#353612yes—
525C22#353712yes—
526C22#354214yes—
527C22#355914yes—
528C22#356214yes—
529C4—42no—
530C4—42no—
531C4—44no—
532C4—44no—
533C4—84no—
534C4—84no—
535C4—84no—
536C4—84no—
537C4—84no—
538C2—44no—
539C2—88no—
540C2#3672711yescharacteristic
541C2#4054612yes—
542C2#4645611yescharacteristic
543C2#4699611yescharacteristic
544C2#4959312yes—
545C2#5509414yes—
546C2#5638114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#16252C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17282C2semidirect
2. #16903C2semidirect
3. #16706C2semidirect
4. #16654C2semidirect
6. #15369C2semidirect
7. #15368C2semidirect
8. #15270C2semidirect
9. #14719C2semidirect
10. #14417C2semidirect
11. #14339C2semidirect

Maximal subgroups

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

subgroupindex
#172822
#169032
#167062
#166542
#162522
#153692
#153682
#152702
#147192
#144172
#143392