Finite Group Databasebeta

← #1421295

Subgroups — #1421295

Order 512.

2798 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#16093C212yesmaximal, complemented
4#15698C214yesmaximal, complemented
5#15466C212yesmaximal
6#15292C214yesmaximal
7#15104C212yesmaximal, complemented
8#14787C214yesmaximal, complemented
9#14467C212yesmaximal, complemented
10#14450C211yesmaximal, characteristic, complemented
11#14240C212yesmaximal
12#14178C214yesmaximal
13#13922C212yesmaximal
14C24×Q8C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18#2434—28no—
19C2×#399C2212yes—
20C2×#399C2212yes—
21C2×#398C2212yes—
22C2×#395C2214yes—
23#2369—28no—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2214yes—
32C2×#387C2214yes—
33C2×#385C2214yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2214yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39#1446C2214yes—
40#1446C2214yes—
41#1446C2212yes—
42#1445C2214yes—
43#1445C2212yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1437C2212yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8—24no—
70C23×Q8—28no—
71C23×Q8—42no—
72C23×Q8—42no—
73C23×Q8—44no—
74C23×Q8C2312yes—
75C23×Q8C2311yescharacteristic
76C23×Q8C2311yescharacteristic
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—44no—
83C2×C4⋊Q8D418yes—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—42no—
86C2×C22⋊Q8—42no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109#399—44no—
110#399Q8116yescomplemented
111#398—44no—
112#395—216no—
113#391—28no—
114#391—28no—
115#391—24no—
116#391—28no—
117#391—24no—
118#391—44no—
119#391—44no—
120#389—216no—
121#387—48no—
122#385—216no—
123#380—44no—
124C4×C4⋊C4—216no—
125C4×C4⋊C4—28no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—44no—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C25C2411yescharacteristic
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—42no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—42no—
173C22×Q8—416no—
174C22×Q8—432no—
175C22×Q8—416no—
176C22×Q8—432no—
177C22×Q8—416no—
178C22×Q8C2×D414yes—
179C22×Q8C2×D414yes—
180C22×Q8#4118yes—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C4⋊Q8—232no—
206C4⋊Q8—416no—
207C22⋊Q8—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
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—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C2×D4116yes—
241C2×C4⋊C4C2×Q818yescomplemented
242C2×C4⋊C4C2×Q8116yescomplemented
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4C2×Q818yes—
254C2×C22⋊C4C2×Q8116yes—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—28no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—44no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C2×D414yes—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—42no—
294#95C2×Q818yescomplemented
295C24—21no—
296C24—22no—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—24no—
301C24C22×Q814yes—
302C24H(5,𝔽2)12yes—
303C24C2511yescharacteristic
304C2×Q8—28no—
305C2×Q8—216no—
306C2×Q8—28no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—44no—
320C2×Q8—48no—
321C2×Q8—4128no—
322C2×Q8—44no—
323C2×Q8—48no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8#123116yes—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—216no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
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—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—48no—
397C22×C4C22×D414yes—
398C22×C4C22×D412yes—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×Q818yes—
402C22×C4C22×Q814yes—
403C22×C4C22×Q818yes—
404C22×C4C22×Q814yes—
405C22×C4C2×#4114yes—
406C22×C4C2×#4114yes—
407C22×C4C2×#4114yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C42−1+414yes—
411C4⋊C4—232no—
412C4⋊C4—216no—
413C4⋊C4—416no—
414C4⋊C4—48no—
415C4⋊C4—416no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C22⋊C4—216no—
424C22⋊C4—416no—
425C22⋊C4—48no—
426C22⋊C4—88no—
427C42—28no—
428C42—416no—
429C42—44no—
430C42—84no—
431C42—84no—
432C42—84no—
433C23—24no—
434C23—28no—
435C23—22no—
436C23—24no—
437C23—28no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—42no—
447C23—44no—
448C23Q8214yes—
449C23C23×D411yescharacteristic
450C23C23×Q812yes—
451C23C22×#4112yes—
452C23C2×H(5,𝔽2)11yescharacteristic
453C23C2×H(5,𝔽2)12yes—
454C23C2×H(5,𝔽2)11yescharacteristic
455C23C2×2−1+412yes—
456C23#58514yes—
457Q8—416no—
458Q8—432no—
459Q8—416no—
460Q8—88no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—432no—
492C2×C4—416no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
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—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
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×C4C2×#12314yes—
533C2×C4C2×#12318yes—
534C2×C4D4×Q818yes—
535C22—24no—
536C22—28no—
537C22—42no—
538C22—44no—
539C22—42no—
540C22—48no—
541C22—44no—
542C22—44no—
543C22C22×#12311yescharacteristic
544C22C2×D4×Q812yes—
545C22C2×Q8211yescharacteristic
546C22Q8×#4114yes—
547C22#348411yescharacteristic
548C22#348411yescharacteristic
549C22#348712yes—
550C22#349012yes—
551C22#349012yes—
552C22#350114yes—
553C22#350312yes—
554C22#351614yes—
555C22#354411yescharacteristic
556C22#354512yes—
557C22#355112yes—
558C22#355414yes—
559C4—28no—
560C4—24no—
561C4—28no—
562C4—42no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—48no—
567C4—48no—
568C4—82no—
569C4—84no—
570C4—84no—
571C2—44no—
572C2#3535011yescharacteristic
573C2Q8×#12312yes—
574C2#4313514yes—
575C2#4535712yes—
576C2#5184012yes—
577C2#5208311yescharacteristic
578C2#5915111yescharacteristic
579C2#5992312yes—
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
2. C2×#2369C2semidirect
3. #16093C2semidirect
4. #15698C2semidirect
7. #15104C2semidirect
8. #14787C2semidirect
9. #14467C2semidirect
10. #14450C2semidirect
110. #399Q8semidirect
241. C2×C4⋊C4C2×Q8semidirect
242. C2×C4⋊C4C2×Q8semidirect
294. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#23692
#160932
#156982
#154662
#152922
#151042
#147872
#144672
#144502
#142402
#141782
#139222