Finite Group Databasebeta

← #2044434

Subgroups — #2044434

Order 512.

2754 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17081C211yesmaximal, characteristic, complemented
2#17068C211yesmaximal, characteristic, complemented
3#16723C214yesmaximal, complemented
4#15958C214yesmaximal, complemented
5#15391C214yesmaximal, complemented
6#15376C214yesmaximal, complemented
7#15273C214yesmaximal, complemented
8#14698C212yesmaximal, complemented
9#14682C211yesmaximal, characteristic, complemented
10#14519C214yesmaximal, complemented
11#14440C212yesmaximal, complemented
12C22×#121C2212yes—
13C23×C42C2211yescharacteristic
14C2×#394C2214yes—
15C2×#392C2214yes—
16C2×#392C2214yes—
17C2×#390C2212yes—
18C2×#390C2212yes—
19C2×#386C2214yes—
20C2×#386C2214yes—
21C2×#386C2214yes—
22C2×#386C2214yes—
23C2×#386C2214yes—
24C2×#386C2214yes—
25C2×#385C2214yes—
26C2×#385C2214yes—
27C2×#382C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30#1445C2214yes—
31#1443C2214yes—
32#1443C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1441C2214yes—
37#1440C2214yes—
38#1440C2214yes—
39#1440C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1438C2214yes—
47#1438C2214yes—
48#1437C2214yes—
49#1437C2214yes—
50C4⋊#95C2212yes—
51C4⋊#95C2211yescharacteristic
52C4⋊#95C2211yescharacteristic
53C4⋊#95C2212yes—
54#1434C2214yes—
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C23×D4C2314yes—
58C23×D4C2314yes—
59C23×D4C2312yes—
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C24×C4C2314yes—
63C2×#121—28no—
64C2×#121—44no—
65C2×#121—44no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2314yes—
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×C42—24no—
85C22×C42—44no—
86C22×C42C2311yescharacteristic
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89#394—48no—
90#392—216no—
91#392—48no—
92#390—28no—
93#390—44no—
94#386—48no—
95#386—48no—
96#386—48no—
97#386—48no—
98#386—48no—
99#386—48no—
100#385—48no—
101#385—48no—
102#382—44no—
103#380—44no—
104#380—44no—
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—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C25C2414yes—
124C25C2411yescharacteristic
125C22×D4—28no—
126C22×D4—28no—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—28no—
130C22×D4—28no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—44no—
136C22×D4—42no—
137C22×D4—44no—
138C22×D4—44no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—42no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4#4118yes—
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×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2412yes—
173C23×C4C2411yescharacteristic
174#121—432no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
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×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—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
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×C22⋊C4—44no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—416no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42#4114yes—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—44no—
244#95—44no—
245#95—44no—
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—
276C24C2×#4118yes—
277C24H(5,𝔽2)14yes—
278C242−1+412yes—
279C24C2511yescharacteristic
280C2×D4—216no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—416no—
284C2×D4—416no—
285C2×D4—416no—
286C2×D4—416no—
287C2×D4—416no—
288C2×D4—416no—
289C2×D4—84no—
290C2×D4—84no—
291C2×D4—84no—
292C2×D4—84no—
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—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
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—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—24no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—44no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4C2×#4112yes—
373C22×C4C2×#4114yes—
374C22×C4C2×#4114yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)14yes—
377C22×C4H(5,𝔽2)14yes—
378C22×C4H(5,𝔽2)14yes—
379C4⋊C4—48no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—88no—
385C22⋊C4—416no—
386C22⋊C4—416no—
387C22⋊C4—416no—
388C22⋊C4—88no—
389C22⋊C4—88no—
390C22⋊C4—88no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C42—28no—
398C42—44no—
399C42—48no—
400C23—24no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—22no—
409C23—24no—
410C23—28no—
411C23—28no—
412C23—28no—
413C23—28no—
414C23—28no—
415C23—28no—
416C23—28no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—41no—
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—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23#56214yes—
456C23C22×#4111yescharacteristic
457C23C22×#4112yes—
458C23C2×H(5,𝔽2)12yes—
459C23C2×H(5,𝔽2)12yes—
460C23C2×H(5,𝔽2)12yes—
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×H(5,𝔽2)12yes—
463C23C2×2−1+411yescharacteristic
464C23#58512yes—
465D4—816no—
466D4—832no—
467D4—832no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—432no—
484C2×C4—416no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
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—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—832no—
528C2×C4#54014yes—
529C2×C4#55318yes—
530C22—28no—
531C22—24no—
532C22—44no—
533C22—44no—
534C22—42no—
535C22—44no—
536C22—42no—
537C22—44no—
538C22—44no—
539C22—48no—
540C22—48no—
541C22—48no—
542C22—48no—
543C22—48no—
544C22—48no—
545C22—48no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22C2×#54011yescharacteristic
555C22#345912yes—
556C22C2×#55312yes—
557C22#348111yescharacteristic
558C22#349012yes—
559C22#349011yescharacteristic
560C22#349211yescharacteristic
561C22#349412yes—
562C22#351412yes—
563C22#351912yes—
564C22#352012yes—
565C22C2×#56211yescharacteristic
566C22#353412yes—
567C22#353612yes—
568C22#353712yes—
569C22#353712yes—
570C22#354312yes—
571C22#354612yes—
572C22#354911yescharacteristic
573C22#355311yescharacteristic
574C22#355812yes—
575C4—28no—
576C4—24no—
577C4—44no—
578C4—42no—
579C4—84no—
580C4—84no—
581C4—84no—
582C4—84no—
583C4—84no—
584C2—44no—
585C2—88no—
586C2#3672711yescharacteristic
587C2#3715512yes—
588C2#3794012yes—
589C2#3876012yes—
590C2#3914812yes—
591C2#4598811yescharacteristic
592C2#4608812yes—
593C2#5318112yes—
594C2#5922011yescharacteristic
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. #17081C2semidirect
2. #17068C2semidirect
3. #16723C2semidirect
4. #15958C2semidirect
5. #15391C2semidirect
6. #15376C2semidirect
7. #15273C2semidirect
8. #14698C2semidirect
9. #14682C2semidirect
10. #14519C2semidirect
11. #14440C2semidirect

Maximal subgroups

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

subgroupindex
#170812
#170682
#167232
#159582
#153912
#153762
#152732
#146982
#146822
#145192
#144402