Finite Group Databasebeta

← #2807718

Subgroups — #2807718

Order 512.

3190 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2677C214yesmaximal, complemented
2C2×#2542C211yesmaximal, characteristic
3C2×#2551C211yesmaximal, characteristic, complemented
4#16522C212yesmaximal, complemented
5#16511C214yesmaximal, complemented
6#15762C212yesmaximal, complemented
7#15662C212yesmaximal
8#15408C214yesmaximal, complemented
9#15354C214yesmaximal, complemented
10#15084C212yesmaximal
11#14723C212yesmaximal
12#14424C211yesmaximal, characteristic, complemented
13#14223C212yesmaximal, complemented
14C22×C4⋊Q8C2214yes—
15C22×C4⋊Q8C2212yes—
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2677—232no—
21#2551—28no—
22#2542—28no—
23C2×#399C2214yes—
24C2×#399C2214yes—
25C2×#398C2214yes—
26C2×#398C2212yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#395C2212yes—
32C2×#393C2214yes—
33C2×#393C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2214yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#387C2214yes—
41C2×#387C2214yes—
42C2×#385C2214yes—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47#1446C2212yes—
48#1445C2214yes—
49#1445C2212yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1436C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66#1434C2214yes—
67#1434C2214yes—
68#1434C2212yes—
69C22⋊#95C2212yes—
70C23×Q8C2312yes—
71C23×Q8C2312yes—
72C23×Q8C2312yes—
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C2×C4⋊Q8—216no—
76C2×C4⋊Q8—216no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—216no—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—48no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—44no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—48no—
87C2×C22⋊Q8—44no—
88C2×C4×D4—24no—
89C2×C4×D4—42no—
90C2×C4×D4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118#399—216no—
119#399—216no—
120#398—216no—
121#398D4116yescomplemented
122#395—216no—
123#395—28no—
124#395—28no—
125#395—216no—
126#395—44no—
127#393—216no—
128#393—44no—
129#390—24no—
130#390—24no—
131#389—216no—
132#388—216no—
133#388—216no—
134#388—48no—
135#387—216no—
136#387—216no—
137#385—216no—
138#384—24no—
139#384D418yes—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—28no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C25C2412yes—
164C22×Q8—24no—
165C22×Q8—28no—
166C22×Q8—28no—
167C22×Q8—28no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—44no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—41no—
183C22×D4—41no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C4⋊Q8—416no—
202C4⋊Q8—432no—
203C4⋊Q8—464no—
204C22⋊Q8—432no—
205C22⋊Q8—464no—
206C4×D4—416no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D418yes—
245C2×C4⋊C4C2×D4116yes—
246C2×C4⋊C4C2×D4116yes—
247C2×C4⋊C4C2×D418yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—22no—
279C2×C42—28no—
280C2×C42—24no—
281C2×C42—42no—
282C2×C42C2×D414yes—
283#95—28no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—44no—
300#95—42no—
301#95—42no—
302#95C2×D418yes—
303#95C2×D4116yes—
304C24—24no—
305C24—22no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24—24no—
311C24—22no—
312C24—22no—
313C24—22no—
314C242−1+414yes—
315C24C2511yescharacteristic
316C2×Q8—416no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—48no—
320C2×Q8—416no—
321C2×Q8—416no—
322C2×Q8—48no—
323C2×Q8—88no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×Q8—88no—
328C2×Q8—88no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×D4—28no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—44no—
335C2×D4—44no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—48no—
390C22×C4—416no—
391C22×C4—48no—
392C22×C4—88no—
393C22×C4—84no—
394C22×C4C22×D414yes—
395C22×C4C22×D418yes—
396C22×C4C22×D414yes—
397C22×C4C22×D414yes—
398C22×C4C22×D418yes—
399C22×C4C22×D412yes—
400C22×C4C22×D414yes—
401C22×C4C22×D414yes—
402C22×C4C22×D414yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)14yes—
405C22×C4H(5,𝔽2)12yes—
406C22×C42−1+418yes—
407C22×C42−1+414yes—
408C4⋊C4—216no—
409C4⋊C4—216no—
410C4⋊C4—232no—
411C4⋊C4—232no—
412C4⋊C4—216no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—416no—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—416no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—88no—
423C22⋊C4—232no—
424C22⋊C4—48no—
425C22⋊C4—416no—
426C22⋊C4—416no—
427C22⋊C4—416no—
428C22⋊C4—416no—
429C22⋊C4—88no—
430C42—28no—
431C42—28no—
432C42—48no—
433C42—44no—
434C42—416no—
435C23—28no—
436C23—24no—
437C23—28no—
438C23—24no—
439C23—24no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—42no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23C23×D411yescharacteristic
459C23C23×D412yes—
460C23C2×H(5,𝔽2)11yescharacteristic
461C23C2×H(5,𝔽2)12yes—
462C23C2×H(5,𝔽2)11yescharacteristic
463C23C2×2−1+412yes—
464C23C2×2−1+412yes—
465C23C2×2−1+414yes—
466Q8—816no—
467Q8—816no—
468Q8—816no—
469D4—416no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—816no—
535C2×C4#53614yes—
536C2×C4D4218yes—
537C2×C4D4214yes—
538C22—44no—
539C22—44no—
540C22—48no—
541C22—44no—
542C22—48no—
543C22—82no—
544C22—82no—
545C22—84no—
546C22—82no—
547C22—84no—
548C22—82no—
549C22C2×#53611yescharacteristic
550C22C2×D4211yescharacteristic
551C22C2×D4212yes—
552C22#348411yescharacteristic
553C22#348614yes—
554C22#348611yescharacteristic
555C22#348612yes—
556C22#349812yes—
557C22#349914yes—
558C22#353812yes—
559C22#353814yes—
560C22#353812yes—
561C22#354311yescharacteristic
562C22#354414yes—
563C22#354412yes—
564C22#355012yes—
565C4—24no—
566C4—28no—
567C4—24no—
568C4—48no—
569C4—42no—
570C4—48no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4—48no—
575C4—84no—
576C4—82no—
577C2—84no—
578C2#3670811yescharacteristic
579C2#3933712yes—
580C2#4532112yes—
581C2#4541812yes—
582C2#5154314yes—
583C2#5207211yescharacteristic
584C2#5485412yes—
585C2#5912611yescharacteristic
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. C2×#2677C2semidirect
3. C2×#2551C2semidirect
4. #16522C2semidirect
5. #16511C2semidirect
6. #15762C2semidirect
8. #15408C2semidirect
9. #15354C2semidirect
12. #14424C2semidirect
13. #14223C2semidirect
121. #398D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26772
C2×#25422
C2×#25512
#165222
#165112
#157622
#156622
#154082
#153542
#150842
#147232
#144242
#142232