Finite Group Databasebeta

← #900775

Subgroups — #900775

Order 512.

2614 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic
2C2×#2683C211yesmaximal, characteristic
3#17074C212yesmaximal, complemented
4#17004C212yesmaximal, complemented
5#15670C214yesmaximal, complemented
6#15385C214yesmaximal, complemented
7#15097C212yesmaximal, complemented
8#14833C214yesmaximal, complemented
9#14707C212yesmaximal, complemented
10#14505C212yesmaximal
11#14052C214yesmaximal, complemented
12#13968C212yesmaximal
13C4×#390C211yesmaximal, characteristic
14C22×#123C2212yes—
15C22×C22⋊Q8C2214yes—
16C22×C4×D4C2211yescharacteristic
17#2724—28no—
18#2683—28no—
19C2×#400C2214yes—
20C2×#398C2212yes—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#386C2212yes—
30C2×#385C2214yes—
31C2×#382C2212yes—
32C2×#382C2212yes—
33C2×#381C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2214yes—
36C22×#95C2212yes—
37C2×C43C2211yescharacteristic
38#1446C2212yes—
39#1446C2214yes—
40#1445C2212yes—
41#1445C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1437C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63#1434C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C2×#123—28no—
75C2×#123—28no—
76C2×#123—28no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—48no—
80C2×C4×D4—24no—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#400—48no—
108#398—28no—
109#397—216no—
110#396—216no—
111#395—28no—
112#391—44no—
113#390—24no—
114#390D418yes—
115#389—28no—
116#389—28no—
117#386—28no—
118#385—216no—
119#382—28no—
120#382—44no—
121#381—44no—
122C4×C4⋊C4—42no—
123C4×C22⋊C4—48no—
124C2×#95—44no—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C43—42no—
150C25C2412yes—
151C22×Q8—24no—
152C22×Q8—28no—
153C22×Q8—44no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×D4—41no—
157C22×D4—41no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4C2×D414yes—
161C22×D4#4114yes—
162C22×D4#4114yes—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—28no—
169C23×C4—24no—
170C23×C4—28no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192#123—432no—
193C22⋊Q8—464no—
194C4×D4—416no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4#4118yes—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4C2×D4116yes—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—41no—
260C2×C42—42no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42—44no—
265C2×C42C2×D414yes—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290C24—24no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24—24no—
295C24—24no—
296C24—22no—
297C24—22no—
298C24C22×D414yes—
299C24C2×#4114yes—
300C24C2×#4114yes—
301C24C2511yescharacteristic
302C2×Q8—416no—
303C2×Q8—48no—
304C2×Q8—84no—
305C2×Q8—88no—
306C2×Q8—84no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4#124116yes—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—416no—
379C22×C4—416no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—88no—
385C22×C4—84no—
386C22×C4—88no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4C22×D414yes—
394C22×C4C22×D412yes—
395C22×C4C22×D414yes—
396C22×C4C2×#4118yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4114yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)14yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+414yes—
404C22×C42−1+414yes—
405C4⋊C4—216no—
406C4⋊C4—44no—
407C4⋊C4—416no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—416no—
411C4⋊C4—48no—
412C4⋊C4—416no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C22⋊C4—232no—
417C22⋊C4—416no—
418C22⋊C4—48no—
419C22⋊C4—416no—
420C22⋊C4—48no—
421C22⋊C4—416no—
422C22⋊C4—84no—
423C42—216no—
424C42—28no—
425C42—44no—
426C42—48no—
427C42—44no—
428C42—84no—
429C23—24no—
430C23—24no—
431C23—28no—
432C23—28no—
433C23—28no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—28no—
439C23—28no—
440C23—28no—
441C23—44no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23C2×#12418yes—
452C23C23×D411yescharacteristic
453C23C22×#4111yescharacteristic
454C23C22×#4111yescharacteristic
455C23C22×#4112yes—
456C23C2×H(5,𝔽2)11yescharacteristic
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×H(5,𝔽2)12yes—
459C23C2×2−1+412yes—
460C23C2×2−1+412yes—
461C23#58511yescharacteristic
462C23#58511yescharacteristic
463Q8—816no—
464D4—416no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
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—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—816no—
537C2×C4—816no—
538C2×C4—816no—
539C2×C4C2×#12414yes—
540C2×C4#54618yes—
541C2×C4#56314yes—
542C22—28no—
543C22—216no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—48no—
549C22—44no—
550C22—44no—
551C22—48no—
552C22—48no—
553C22C22×#12411yescharacteristic
554C22#345911yescharacteristic
555C22#345911yescharacteristic
556C22#346112yes—
557C22C2×#54612yes—
558C22#348111yescharacteristic
559C22#348212yes—
560C22#348212yes—
561C22#349011yescharacteristic
562C22#349111yescharacteristic
563C22#349212yes—
564C22#349611yescharacteristic
565C22#349611yescharacteristic
566C22#349912yes—
567C22#351412yes—
568C22#352012yes—
569C22#352012yes—
570C22C2×#56311yescharacteristic
571C22#353612yes—
572C22#354311yescharacteristic
573C22#355011yescharacteristic
574C22#355912yes—
575C22#356312yes—
576C4—24no—
577C4—24no—
578C4—28no—
579C4—48no—
580C4—44no—
581C4—44no—
582C4—42no—
583C4—82no—
584C4—82no—
585C4—84no—
586C4—84no—
587C4—84no—
588C2—48no—
589C2#3499311yescharacteristic
590C2#3630312yes—
591C2#3725411yescharacteristic
592C2#3878112yes—
593C2#3880812yes—
594C2#3941312yes—
595C2#4614711yescharacteristic
596C2#4619911yescharacteristic
597C2#5230411yescharacteristic
598C2#5942011yescharacteristic
599C2#6004611yescharacteristic
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
3. #17074C2semidirect
4. #17004C2semidirect
5. #15670C2semidirect
6. #15385C2semidirect
7. #15097C2semidirect
8. #14833C2semidirect
9. #14707C2semidirect
11. #14052C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#26832
#170742
#170042
#156702
#153852
#150972
#148332
#147072
#145052
#140522
#139682
C4×#3902