Finite Group Databasebeta

← #4386934

Subgroups — #4386934

Order 512.

2240 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3#16392C214yesmaximal, complemented
4#16069C214yesmaximal, complemented
5#15704C214yesmaximal, complemented
6#15641C214yesmaximal, complemented
7#15456C214yesmaximal, complemented
8#14974C214yesmaximal, complemented
9#14659C211yesmaximal, characteristic
10#14297C211yesmaximal, characteristic, complemented
11#14290C211yesmaximal, characteristic
12#13953C211yesmaximal, characteristic, complemented
13#13904C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16#2504—28no—
17C2×#399C2214yes—
18C2×#397C2214yes—
19#2313—28no—
20C2×#390C2212yes—
21C2×#388C2214yes—
22C2×#387C2214yes—
23C2×#387C2214yes—
24C2×#386C2212yes—
25C2×#386C2212yes—
26C2×#382C2212yes—
27C2×#381C2211yescharacteristic
28C2×#381C2211yescharacteristic
29C2×#381C2212yes—
30C2×#381C2212yes—
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1445C2214yes—
37#1445C2214yes—
38#1445C2212yes—
39#1445C2214yes—
40#1445C2212yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2212yes—
57#1438C2212yes—
58#1437C2212yes—
59#1437C2212yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2212yes—
68#1434C2214yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C2×#121—24no—
75C2×#121—24no—
76C2×#121—24no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4D418yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2311yescharacteristic
100C22×C42C2314yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103#399—48no—
104#397—216no—
105#390—28no—
106#388—48no—
107#387—216no—
108#387—216no—
109#386—28no—
110#386—28no—
111#382—28no—
112#381—24no—
113#381—24no—
114#381—44no—
115#381—44no—
116#380—28no—
117#380—24no—
118C4×C22⋊C4—28no—
119C4×C22⋊C4—24no—
120C4×C22⋊C4—24no—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
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—
146C25C2412yes—
147C25C2411yescharacteristic
148C22×D4—24no—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—44no—
157C22×D4—44no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C2×D414yes—
169C23×C4C2×D414yes—
170C23×C4C2×D414yes—
171C23×C4#4114yes—
172C23×C4#4114yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184#121—416no—
185C2×C4⋊C4—232no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C2×D418yes—
206C2×C4⋊C4C2×D418yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C42—24no—
243C2×C42—28no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—44no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—44no—
252#95—28no—
253#95—24no—
254#95—28no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—24no—
263#95—28no—
264#95—28no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24C22×D412yes—
292C24C2×#4112yes—
293C24C2×#4112yes—
294C24C2511yescharacteristic
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—82no—
301C2×D4—84no—
302C2×D4—82no—
303C2×D4—88no—
304C2×D4—88no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—48no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4#123116yes—
374C22×C4C22×D414yes—
375C22×C4C22×D414yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C4H(5,𝔽2)14yes—
383C4⋊C4—216no—
384C4⋊C4—216no—
385C4⋊C4—44no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C4⋊C4—88no—
389C4⋊C4—88no—
390C22⋊C4—48no—
391C22⋊C4—416no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C22⋊C4—84no—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C42—48no—
403C42—82no—
404C42—82no—
405C42—88no—
406C23—24no—
407C23—24no—
408C23—22no—
409C23—24no—
410C23—22no—
411C23—24no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—42no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23C2×#12314yes—
443C23C23×D411yescharacteristic
444C23C22×#4111yescharacteristic
445C23C22×#4111yescharacteristic
446C23C2×H(5,𝔽2)12yes—
447C23#58512yes—
448C23#58512yes—
449C23#58514yes—
450C23#58512yes—
451D4—88no—
452D4—816no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—416no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
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—88no—
516C2×C4—88no—
517C2×C4C2×#12314yes—
518C2×C4C2×#12314yes—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—48no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—82no—
528C22—84no—
529C22—84no—
530C22—82no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—82no—
535C22—84no—
536C22C22×#12311yescharacteristic
537C22#344912yes—
538C22#348712yes—
539C22#349111yescharacteristic
540C22#349112yes—
541C22#349111yescharacteristic
542C22#349412yes—
543C22#349612yes—
544C22#350811yescharacteristic
545C22#351012yes—
546C22#351012yes—
547C22#351011yescharacteristic
548C22#353314yes—
549C22#356214yes—
550C22#356614yes—
551C22#356714yes—
552C4—24no—
553C4—24no—
554C4—44no—
555C4—44no—
556C4—44no—
557C4—84no—
558C4—84no—
559C4—84no—
560C4—82no—
561C4—84no—
562C4—82no—
563C2—44no—
564C2—162no—
565C2#3509711yescharacteristic
566C2#3554111yescharacteristic
567C2#3567911yescharacteristic
568C2#4052214yes—
569C2#4722312yes—
570C2#4722612yes—
571C2#5066312yes—
572C2#5104212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2313C2semidirect
3. #16392C2semidirect
4. #16069C2semidirect
5. #15704C2semidirect
6. #15641C2semidirect
7. #15456C2semidirect
8. #14974C2semidirect
10. #14297C2semidirect
12. #13953C2semidirect
13. #13904C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#23132
#163922
#160692
#157042
#156412
#154562
#149742
#146592
#142972
#142902
#139532
#139042