Finite Group Databasebeta

← #1015672

Subgroups — #1015672

Order 512.

2202 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2742C212yesmaximal
2#17219C212yesmaximal, complemented
3#16473C212yesmaximal
4#16410C214yesmaximal, complemented
5#16081C212yesmaximal, complemented
6#15992C214yesmaximal
7#15158C212yesmaximal, complemented
8#15139C212yesmaximal, complemented
9#15135C211yesmaximal, characteristic
10#14133C212yesmaximal, complemented
11#14130C211yesmaximal, characteristic
12#14005C212yesmaximal, complemented
13#13994C212yesmaximal
14#13907C212yesmaximal
15C4×#393C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#2742—216no—
19C2×#400C2214yes—
20C2×#399C2212yes—
21C2×#395C2214yes—
22C2×#393C2211yescharacteristic
23C2×#389C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#385C2212yes—
32C2×#384C2211yescharacteristic
33C2×#380C2214yes—
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39#1446C2211yescharacteristic
40#1445C2212yes—
41#1444C2212yes—
42#1444C2212yes—
43#1444C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2212yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2211yescharacteristic
50#1442C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1439C2212yes—
61#1439C2211yescharacteristic
62#1439C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×Q8C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4×Q8—44no—
80C2×C4×Q8—42no—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2314yes—
108#400—216no—
109#399—28no—
110#395—216no—
111#393D418yes—
112#389—28no—
113#389—28no—
114#389—216no—
115#389—44no—
116#389—44no—
117#389—44no—
118#388—216no—
119#387—216no—
120#385—28no—
121#384—42no—
122#380—216no—
123#380—44no—
124#380Q818yescomplemented
125C4×C4⋊C4—44no—
126C4×C22⋊C4—44no—
127C4×C22⋊C4—42no—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C25C2411yescharacteristic
160C22×Q8—42no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—42no—
164C22×Q8C2×D414yes—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—28no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4C2×Q814yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C4×Q8—416no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—416no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C2×Q818yes—
229C2×C4⋊C4#4118yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4C2×D418yes—
246C2×C22⋊C4C2×D414yes—
247C2×C22⋊C4C2×Q818yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—28no—
259C2×C42—44no—
260C2×C42—44no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—41no—
277#95—42no—
278#95—42no—
279#95—44no—
280#95—42no—
281#95—42no—
282#95—41no—
283#95—41no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95C2×D414yes—
290#95C2×D418yes—
291#95C2×Q818yes—
292C24—22no—
293C24—22no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24C22×D412yes—
301C24C22×Q812yes—
302C242−1+412yes—
303C24C2511yescharacteristic
304C2×Q8—48no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—48no—
308C2×Q8—44no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—216no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—48no—
385C22×C4—416no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—88no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4C22×D414yes—
393C22×C4C22×D414yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×Q812yes—
397C22×C4C22×Q818yes—
398C22×C4C22×Q812yes—
399C22×C4C2×#4118yes—
400C22×C4C2×#4114yes—
401C22×C4H(5,𝔽2)14yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C22×C42−1+412yes—
405C4⋊C4—216no—
406C4⋊C4—216no—
407C4⋊C4—416no—
408C4⋊C4—416no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—84no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C4⋊C4—84no—
419C22⋊C4—216no—
420C22⋊C4—48no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—48no—
424C22⋊C4—88no—
425C42—48no—
426C42—44no—
427C42—48no—
428C42—44no—
429C42—88no—
430C23—22no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—42no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—42no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23D4×Q814yes—
453C23C23×D411yescharacteristic
454C23C23×Q811yescharacteristic
455C23C22×#4112yes—
456C23C2×H(5,𝔽2)12yes—
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×2−1+411yescharacteristic
459C23C2×2−1+411yescharacteristic
460C23C2×2−1+411yescharacteristic
461C23#58512yes—
462C23#58511yescharacteristic
463C23#58512yes—
464Q8—416no—
465C2×C4—22no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—416no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—816no—
548C2×C4#54218yes—
549C2×C4#54414yes—
550C2×C4#55114yes—
551C2×C4#55418yes—
552C2×C4#56014yes—
553C22—28no—
554C22—24no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—42no—
563C22—44no—
564C22C2×#54212yes—
565C22C2×#54411yescharacteristic
566C22C2×D4×Q811yescharacteristic
567C22D4×#4112yes—
568C22C2×#55111yescharacteristic
569C22C2×#55412yes—
570C22#349612yes—
571C22#349611yescharacteristic
572C22#349911yescharacteristic
573C22#350211yescharacteristic
574C22#350412yes—
575C22#351612yes—
576C22#351612yes—
577C22#352112yes—
578C22C2×#56011yescharacteristic
579C22#353811yescharacteristic
580C22#353912yes—
581C22#354011yescharacteristic
582C22#354112yes—
583C22#354712yes—
584C22#355111yescharacteristic
585C22#356311yescharacteristic
586C22#356512yes—
587C4—42no—
588C4—44no—
589C4—42no—
590C4—48no—
591C4—42no—
592C4—48no—
593C4—42no—
594C4—44no—
595C4—81no—
596C4—82no—
597C4—84no—
598C4—82no—
599C4—84no—
600C4#274218yescomplemented
601C2—44no—
602C2#3936811yescharacteristic
603C2#3939812yes—
604C2#4272212yes—
605C2#4298111yescharacteristic
606C2#4540512yes—
607C2#4542011yescharacteristic
608C2#4547811yescharacteristic
609C2C2×#274211yescharacteristic
610C2#5300611yescharacteristic
611C2#5983112yes—
612C2#5984311yescharacteristic
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. #17219C2semidirect
4. #16410C2semidirect
5. #16081C2semidirect
7. #15158C2semidirect
8. #15139C2semidirect
10. #14133C2semidirect
12. #14005C2semidirect
124. #380Q8semidirect
600. C4#2742semidirect

Maximal subgroups

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

subgroupindex
C2×#27422
#172192
#164732
#164102
#160812
#159922
#151582
#151392
#151352
#141332
#141302
#140052
#139942
#139072
C4×#3932