Finite Group Databasebeta

← #4386819

Subgroups — #4386819

Order 512.

2714 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic
3#16697C214yesmaximal
4#16109C214yesmaximal, complemented
5#15783C214yesmaximal, complemented
6#15503C214yesmaximal, complemented
7#15500C214yesmaximal, complemented
8#14906C214yesmaximal
9#14660C211yesmaximal, characteristic
10#14297C211yesmaximal, characteristic
11#14292C211yesmaximal, characteristic
12#13953C211yesmaximal, characteristic
13#13922C211yesmaximal, characteristic
14C24×Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2212yes—
18#2434—28no—
19C2×#400C2214yes—
20C2×#398C2214yes—
21C2×#395C2214yes—
22#2310—28no—
23C2×#391C2212yes—
24C2×#391C2214yes—
25C2×#391C2214yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#381C2212yes—
31C2×#381C2211yescharacteristic
32C2×#381C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#1446C2212yes—
37#1446C2214yes—
38#1446C2212yes—
39#1445C2214yes—
40#1445C2214yes—
41#1445C2214yes—
42#1445C2212yes—
43#1445C2212yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2212yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C23×Q8—24no—
70C23×Q8—24no—
71C23×Q8—44no—
72C23×Q8—44no—
73C23×Q8D418yes—
74C23×Q8C2312yes—
75C23×Q8C2312yes—
76C23×Q8C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—44no—
82C2×C4⋊Q8—44no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×#117—28no—
87C2×#117—44no—
88C2×#117—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C42C2314yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#400—48no—
108#398—48no—
109#395—216no—
110#391—216no—
111#391—216no—
112#391—28no—
113#391—44no—
114#391—44no—
115#385—28no—
116#385—28no—
117#381—28no—
118#381—24no—
119#381—24no—
120#380—24no—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—24no—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2411yescharacteristic
153C22×Q8—232no—
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—432no—
172C22×Q8—416no—
173C22×Q8—416no—
174C22×Q8—432no—
175C22×Q8C2×D418yes—
176C22×Q8C2×D418yes—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2×D414yes—
190C23×C4#4114yes—
191C23×C4#4114yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C4⋊Q8—432no—
205C22⋊Q8—416no—
206#117—432no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C42—28no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—28no—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287C24—22no—
288C24—22no—
289C24—24no—
290C24—22no—
291C24—22no—
292C24C22×D412yes—
293C24C2×#4112yes—
294C24C2×#4112yes—
295C24C2511yescharacteristic
296C2×Q8—216no—
297C2×Q8—216no—
298C2×Q8—216no—
299C2×Q8—216no—
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—48no—
308C2×Q8—48no—
309C2×Q8—48no—
310C2×Q8—4128no—
311C2×Q8—48no—
312C2×Q8—44no—
313C2×Q8—84no—
314C2×Q8—82no—
315C2×Q8—82no—
316C22×C4—216no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—216no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
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—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4#123116yes—
391C22×C4C22×D414yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D414yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4H(5,𝔽2)14yes—
400C22×C4H(5,𝔽2)14yes—
401C22×C42−1+414yes—
402C22×C42−1+418yes—
403C4⋊C4—416no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—416no—
407C4⋊C4—416no—
408C4⋊C4—84no—
409C4⋊C4—88no—
410C4⋊C4—88no—
411C22⋊C4—216no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—84no—
417C42—416no—
418C42—48no—
419C42—82no—
420C42—82no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23C2×#12314yes—
437C23C23×D411yescharacteristic
438C23C22×#4111yescharacteristic
439C23C22×#4111yescharacteristic
440C23C2×H(5,𝔽2)12yes—
441C23C2×H(5,𝔽2)12yes—
442C23C2×2−1+414yes—
443C23C2×2−1+412yes—
444C23#58512yes—
445Q8—432no—
446Q8—432no—
447Q8—88no—
448C2×C4—232no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—416no—
472C2×C4—416no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
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×C4C2×#12314yes—
521C2×C4C2×#12314yes—
522C2×C4#53418yes—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—48no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22C22×#12311yescharacteristic
532C22C2×#53412yes—
533C22#348612yes—
534C22#349012yes—
535C22#349011yescharacteristic
536C22#349011yescharacteristic
537C22#349512yes—
538C22#349712yes—
539C22#350911yescharacteristic
540C22#350912yes—
541C22#350912yes—
542C22#351011yescharacteristic
543C22#355214yes—
544C22#355214yes—
545C22#355414yes—
546C22#356114yes—
547C4—28no—
548C4—24no—
549C4—24no—
550C4—48no—
551C4—44no—
552C4—44no—
553C4—84no—
554C4—84no—
555C4—82no—
556C4—84no—
557C4—82no—
558C4—82no—
559C2—44no—
560C2#3499211yescharacteristic
561C2#3567311yescharacteristic
562C2#3567411yescharacteristic
563C2#4278714yes—
564C2#4645312yes—
565C2#4645512yes—
566C2#5133612yes—
567C2#5134812yes—
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
4. #16109C2semidirect
5. #15783C2semidirect
6. #15503C2semidirect
7. #15500C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#23102
#166972
#161092
#157832
#155032
#155002
#149062
#146602
#142972
#142922
#139532
#139222