Finite Group Databasebeta

← #4312069

Subgroups — #4312069

Order 512.

2634 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2809C211yesmaximal, characteristic, complemented
2C2×#2310C211yesmaximal, characteristic
3#16989C212yesmaximal, complemented
4#16653C212yesmaximal, complemented
5#16300C214yesmaximal
6#15336C214yesmaximal, complemented
7#15333C214yesmaximal, complemented
8#14893C212yesmaximal
9#14893C212yesmaximal
10#14772C214yesmaximal
11#14501C211yesmaximal, characteristic, complemented
12#14467C212yesmaximal, complemented
13#13922C212yesmaximal
14C24×Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#2809—28no—
18C2×#400C2212yes—
19C2×#400C2212yes—
20C2×#399C2212yes—
21C2×#399C2212yes—
22C2×#398C2214yes—
23#2310—28no—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2214yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#387C2214yes—
36C2×#381C2214yes—
37C2×#380C2212yes—
38C2×C4×C22⋊C4C2212yes—
39#1446C2214yes—
40#1446C2214yes—
41#1445C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65#1434C2212yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8—24no—
70C23×Q8—24no—
71C23×Q8—42no—
72C23×Q8—42no—
73C23×Q8—42no—
74C23×Q8—42no—
75C23×Q8Q818yes—
76C23×Q8C2312yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—42no—
84C2×#117—42no—
85C2×#117—42no—
86C2×#117—42no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109#400—28no—
110#400—44no—
111#399—44no—
112#399—44no—
113#398—216no—
114#391—24no—
115#391—28no—
116#391—28no—
117#391—24no—
118#391—24no—
119#391—216no—
120#391—44no—
121#391—44no—
122#391D418yes—
123#389—28no—
124#389—44no—
125#387—216no—
126#381—216no—
127#380—44no—
128C4×C22⋊C4—44no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2411yescharacteristic
156C22×Q8—232no—
157C22×Q8—24no—
158C22×Q8—22no—
159C22×Q8—24no—
160C22×Q8—22no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—416no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—416no—
175C22×Q8—416no—
176C22×Q8—416no—
177C22×Q8—416no—
178C22×Q8—416no—
179C22×Q8C2×D414yes—
180C22×Q8C2×Q818yes—
181C22×Q8C2×Q814yes—
182C22×Q8C2×Q814yes—
183C22×Q8#4114yes—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C2×Q814yes—
197C23×C4C2×Q818yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210#117—416no—
211#117—416no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C2×D414yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42—42no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42C2×D414yes—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—42no—
299#95—42no—
300#95—44no—
301#95C2×D4116yes—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—24no—
307C24—24no—
308C24C22×Q812yes—
309C24H(5,𝔽2)12yes—
310C24H(5,𝔽2)12yes—
311C24C2511yescharacteristic
312C2×Q8—216no—
313C2×Q8—28no—
314C2×Q8—216no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—48no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—4128no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8C22⋊Q8116yes—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—216no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—216no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×Q814yes—
416C22×C4C22×Q812yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q812yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4114yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+418yes—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—416no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C4⋊C4—88no—
437C22⋊C4—416no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—88no—
441C42—44no—
442C42—48no—
443C42—44no—
444C42—84no—
445C42—84no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—28no—
453C23—28no—
454C23—24no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23#54314yes—
463C23C23×D411yescharacteristic
464C23C23×Q811yescharacteristic
465C23C22×#4111yescharacteristic
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×H(5,𝔽2)11yescharacteristic
469C23C2×H(5,𝔽2)11yescharacteristic
470C23C2×2−1+414yes—
471C23#58514yes—
472Q8—416no—
473Q8—432no—
474Q8—416no—
475C2×C4—232no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—432no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4C2×C22⋊Q814yes—
551C2×C4C2×C22⋊Q818yes—
552C2×C4#54314yes—
553C22—28no—
554C22—24no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22C22×C22⋊Q811yescharacteristic
563C22C2×#54311yescharacteristic
564C22C2×#54311yescharacteristic
565C22#348411yescharacteristic
566C22#348411yescharacteristic
567C22#348712yes—
568C22#349011yescharacteristic
569C22#349011yescharacteristic
570C22#349912yes—
571C22#350214yes—
572C22#351614yes—
573C22#353514yes—
574C22#353812yes—
575C22#354412yes—
576C22#354512yes—
577C22#355112yes—
578C22#355414yes—
579C4—24no—
580C4—24no—
581C4—28no—
582C4—44no—
583C4—48no—
584C4—44no—
585C4—44no—
586C4—82no—
587C4—84no—
588C4—82no—
589C4—84no—
590C4—84no—
591C2—44no—
592C2#3498211yescharacteristic
593C2#3584512yes—
594C2#3719412yes—
595C2#4281812yes—
596C2#4636112yes—
597C2#4717112yes—
598C2#5154211yescharacteristic
599C2#5184012yes—
600C2#5208311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2809C2semidirect
3. #16989C2semidirect
4. #16653C2semidirect
6. #15336C2semidirect
7. #15333C2semidirect
11. #14501C2semidirect
12. #14467C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
C2×#23102
#169892
#166532
#163002
#153362
#153332
#148932
#147722
#145012
#144672
#139222