Finite Group Databasebeta

← #2044138

Subgroups — #2044138

Order 512.

2622 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17073C211yesmaximal, characteristic, complemented
2#17068C211yesmaximal, characteristic, complemented
3#16372C214yesmaximal, complemented
4#15709C214yesmaximal, complemented
5#15326C214yesmaximal, complemented
6#15320C214yesmaximal, complemented
7#15307C214yesmaximal, complemented
8#14698C212yesmaximal, complemented
9#14682C211yesmaximal, characteristic, complemented
10#14512C214yesmaximal, complemented
11#14445C212yesmaximal, complemented
12C22×#121C2212yes—
13C23×C42C2211yescharacteristic
14C2×#398C2214yes—
15C2×#394C2214yes—
16C2×#394C2214yes—
17C2×#394C2214yes—
18C2×#392C2214yes—
19C2×#392C2214yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#390C2214yes—
23C2×#388C2214yes—
24C2×#388C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#382C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30#1444C2214yes—
31#1442C2214yes—
32#1441C2214yes—
33#1440C2214yes—
34#1440C2214yes—
35#1439C2214yes—
36#1439C2214yes—
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1438C2214yes—
43#1438C2214yes—
44#1438C2214yes—
45#1437C2214yes—
46#1437C2214yes—
47#1437C2214yes—
48#1436C2214yes—
49C4⋊#95C2211yescharacteristic
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2212yes—
52C4⋊#95C2212yes—
53#1434C2214yes—
54#1434C2214yes—
55#1434C2214yes—
56C22⋊#95C2214yes—
57C23×D4C2314yes—
58C23×D4C2312yes—
59C23×D4C2314yes—
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C2×#121—28no—
63C2×#121—44no—
64C2×#121—44no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C42—24no—
86C22×C42—44no—
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89C22×C42C2314yes—
90C22×C42C2311yescharacteristic
91#398—48no—
92#394—48no—
93#394—48no—
94#394—48no—
95#392—48no—
96#392—48no—
97#390—28no—
98#390—44no—
99#390—48no—
100#388—216no—
101#388—48no—
102#386—48no—
103#386—48no—
104#382—44no—
105#380—44no—
106#380—44no—
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—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C25C2414yes—
124C25C2411yescharacteristic
125C22×D4—28no—
126C22×D4—28no—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—44no—
135C22×D4—42no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—42no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—44no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C2414yes—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2411yescharacteristic
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167#121—432no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
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#4118yes—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
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—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—28no—
233C2×C42—416no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—44no—
237C2×C42—44no—
238C2×C42#4114yes—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255C24—21no—
256C24—22no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24H(5,𝔽2)14yes—
278C242−1+412yes—
279C24C2511yescharacteristic
280C2×D4—416no—
281C2×D4—416no—
282C2×D4—416no—
283C2×D4—48no—
284C2×D4—48no—
285C2×D4—416no—
286C2×D4—48no—
287C2×D4—84no—
288C2×D4—84no—
289C2×D4—84no—
290C2×D4—84no—
291C2×D4—88no—
292C2×D4—88no—
293C2×D4—88no—
294C2×D4—88no—
295C2×D4—88no—
296C2×D4—88no—
297C2×D4—88no—
298C2×D4—88no—
299C2×D4—88no—
300C2×D4—88no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—216no—
309C22×C4—28no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—22no—
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—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—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—48no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4C2×#4118yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4114yes—
367C22×C4C2×#4114yes—
368C22×C4H(5,𝔽2)12yes—
369C22×C4H(5,𝔽2)14yes—
370C22×C4H(5,𝔽2)14yes—
371C22×C4H(5,𝔽2)14yes—
372C4⋊C4—48no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—88no—
378C4⋊C4—88no—
379C22⋊C4—416no—
380C22⋊C4—416no—
381C22⋊C4—416no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C22⋊C4—88no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C22⋊C4—88no—
388C22⋊C4—88no—
389C22⋊C4—88no—
390C22⋊C4—88no—
391C22⋊C4—88no—
392C42—28no—
393C42—44no—
394C42—48no—
395C42—88no—
396C23—24no—
397C23—22no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—22no—
405C23—24no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23#56214yes—
452C23C22×#4111yescharacteristic
453C23C22×#4112yes—
454C23C2×H(5,𝔽2)12yes—
455C23C2×H(5,𝔽2)12yes—
456C23C2×H(5,𝔽2)12yes—
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×H(5,𝔽2)12yes—
459C23C2×2−1+411yescharacteristic
460C23#58512yes—
461D4—816no—
462D4—832no—
463D4—832no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—22no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—432no—
483C2×C4—416no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4#54014yes—
524C2×C4#55518yes—
525C22—28no—
526C22—24no—
527C22—44no—
528C22—44no—
529C22—42no—
530C22—44no—
531C22—42no—
532C22—44no—
533C22—44no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22C2×#54011yescharacteristic
550C22#346012yes—
551C22C2×#55512yes—
552C22#348311yescharacteristic
553C22#349012yes—
554C22#349011yescharacteristic
555C22#349211yescharacteristic
556C22#349412yes—
557C22#351712yes—
558C22#351912yes—
559C22#352012yes—
560C22C2×#56211yescharacteristic
561C22#353412yes—
562C22#353612yes—
563C22#353712yes—
564C22#353712yes—
565C22#354312yes—
566C22#354612yes—
567C22#354911yescharacteristic
568C22#355311yescharacteristic
569C22#355812yes—
570C4—28no—
571C4—24no—
572C4—48no—
573C4—44no—
574C4—42no—
575C4—84no—
576C4—84no—
577C4—84no—
578C4—84no—
579C2—44no—
580C2—164no—
581C2#3672711yescharacteristic
582C2#3721812yes—
583C2#3794912yes—
584C2#3882212yes—
585C2#3894212yes—
586C2#4598811yescharacteristic
587C2#4608812yes—
588C2#5318112yes—
589C2#5923311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17073C2semidirect
2. #17068C2semidirect
3. #16372C2semidirect
4. #15709C2semidirect
5. #15326C2semidirect
6. #15320C2semidirect
7. #15307C2semidirect
8. #14698C2semidirect
9. #14682C2semidirect
10. #14512C2semidirect
11. #14445C2semidirect

Maximal subgroups

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

subgroupindex
#170732
#170682
#163722
#157092
#153262
#153202
#153072
#146982
#146822
#145122
#144452