Finite Group Databasebeta

← #4453314

Subgroups — #4453314

Order 512.

2422 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic
3#16354C214yesmaximal, complemented
4#16337C214yesmaximal, complemented
5#16328C214yesmaximal, complemented
6#16279C214yesmaximal, complemented
7#15015C211yesmaximal, characteristic
8#14955C212yesmaximal
9#14914C211yesmaximal, characteristic
10#14843C212yesmaximal
11#14822C211yesmaximal, characteristic
12#14720C211yesmaximal, characteristic
13#14349C211yesmaximal, characteristic
14#14039C212yesmaximal
15#14039C212yesmaximal
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#400C2214yes—
22C2×#400C2212yes—
23C2×#398—28no—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#395C2212yes—
27#2329—28no—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#380C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42#1446C2212yes—
43#1445C2214yes—
44#1445C2212yes—
45#1444C2212yes—
46#1444C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2212yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114#400—48no—
115#400—44no—
116#398—216no—
117#398—216no—
118#395—44no—
119#391—28no—
120#389—28no—
121#387—28no—
122#387—28no—
123#387—48no—
124#387—48no—
125#385—28no—
126#385—28no—
127#385—28no—
128#380—24no—
129C4×C22⋊C4—28no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95D418yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C25C2411yescharacteristic
166C22×Q8—22no—
167C22×Q8—24no—
168C22×Q8—42no—
169C22×Q8—42no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C22⋊Q8—232no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4#4118yes—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—44no—
268C2×C42—44no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—44no—
284#95—44no—
285#95—42no—
286#95—42no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95—44no—
298#95—44no—
299#95C2×D418yes—
300#95C2×D418yes—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24C22×D412yes—
306C24C2×#4114yes—
307C242−1+414yes—
308C242−1+412yes—
309C242−1+412yes—
310C24C2511yescharacteristic
311C2×Q8—28no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—88no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
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—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—416no—
390C22×C4—416no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—416no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—88no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4C22×D414yes—
409C22×C4C22×D412yes—
410C22×C4C22×D414yes—
411C22×C4C22×D412yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4114yes—
414C22×C4H(5,𝔽2)14yes—
415C22×C4H(5,𝔽2)14yes—
416C22×C42−1+414yes—
417C4⋊C4—28no—
418C4⋊C4—216no—
419C4⋊C4—416no—
420C4⋊C4—416no—
421C4⋊C4—416no—
422C4⋊C4—416no—
423C4⋊C4—48no—
424C4⋊C4—416no—
425C4⋊C4—48no—
426C4⋊C4—84no—
427C4⋊C4—82no—
428C4⋊C4—84no—
429C4⋊C4—82no—
430C22⋊C4—216no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C42—84no—
435C42—84no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—28no—
441C23—28no—
442C23—24no—
443C23—24no—
444C23—28no—
445C23—28no—
446C23—24no—
447C23—24no—
448C23#52514yes—
449C23#53614yes—
450C23#53614yes—
451C23#54118yes—
452C23#54118yes—
453C23C23×D411yescharacteristic
454C23C22×#4112yes—
455C23C2×H(5,𝔽2)12yes—
456C23C2×H(5,𝔽2)12yes—
457C23C2×2−1+411yescharacteristic
458C23C2×2−1+411yescharacteristic
459C23C2×2−1+412yes—
460C23C2×2−1+412yes—
461C23#58512yes—
462Q8—88no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—416no—
487C2×C4—416no—
488C2×C4—416no—
489C2×C4—416no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—88no—
539C2×C4—88no—
540C2×C4—88no—
541C22—24no—
542C22—24no—
543C22—216no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—216no—
549C22#263418yes—
550C22C2×#52511yescharacteristic
551C22C2×#53611yescharacteristic
552C22C2×#53611yescharacteristic
553C22C2×#54112yes—
554C22C2×#54112yes—
555C22#348411yescharacteristic
556C22#348411yescharacteristic
557C22#348511yescharacteristic
558C22#348611yescharacteristic
559C22#351314yes—
560C22#352014yes—
561C22#353612yes—
562C22#353612yes—
563C22#354114yes—
564C22#355214yes—
565C22#356312yes—
566C22#356312yes—
567C4—44no—
568C4—42no—
569C4—42no—
570C4—44no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—44no—
576C4—84no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—84no—
581C2—28no—
582C2C2×#263411yescharacteristic
583C2#3535411yescharacteristic
584C2#3535711yescharacteristic
585C2#3671611yescharacteristic
586C2#3671811yescharacteristic
587C2#3695911yescharacteristic
588C2#3723311yescharacteristic
589C2#4069514yes—
590C2#4075714yes—
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
3. #16354C2semidirect
4. #16337C2semidirect
5. #16328C2semidirect
6. #16279C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3982
C2×#23292
#163542
#163372
#163282
#162792
#150152
#149552
#149142
#148432
#148222
#147202
#143492
#140392