Finite Group Databasebeta

← #1045266

Subgroups — #1045266

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2807C212yesmaximal
2#17221C212yesmaximal, complemented
3#16577C214yesmaximal, complemented
4#16321C212yesmaximal
5#15972C214yesmaximal
6#15363C212yesmaximal, complemented
7#15156C212yesmaximal, complemented
8#15151C212yesmaximal, complemented
9#15135C211yesmaximal, characteristic
10#14130C211yesmaximal, characteristic
11#14005C212yesmaximal, complemented
12#13974C212yesmaximal
13#13910C212yesmaximal, complemented
14#13880C212yesmaximal
15C4×#399C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17#2807—216no—
18C2×#400C2214yes—
19C2×#399C2212yes—
20C2×#399C2211yescharacteristic
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#389C2214yes—
24C2×#389C2214yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#385C2212yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#384C2211yescharacteristic
34C2×#384C2212yes—
35C2×#384C2212yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43#1445C2212yes—
44#1445C2211yescharacteristic
45#1445C2212yes—
46#1445C2214yes—
47#1445C2214yes—
48#1444C2211yescharacteristic
49#1444C2212yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2211yescharacteristic
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C22×C4⋊C4—42no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2314yes—
109#400—216no—
110#399—28no—
111#399D418yes—
112#398—216no—
113#398—48no—
114#389—216no—
115#389—28no—
116#389—216no—
117#389—48no—
118#389—44no—
119#389—44no—
120#388—216no—
121#385—28no—
122#385—216no—
123#385—48no—
124#384—44no—
125#384—42no—
126#384—44no—
127#380—44no—
128#380Q818yes—
129C4×C4⋊C4—28no—
130C4×C4⋊C4—42no—
131C4×C4⋊C4—44no—
132C4×C22⋊C4—28no—
133C4×C22⋊C4—42no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C25C2411yescharacteristic
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—44no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4C2×Q814yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2411yescharacteristic
176C23×C4C2414yes—
177C23×C4C2411yescharacteristic
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2414yes—
181C23×C4C2411yescharacteristic
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
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—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—416no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C2×D418yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×Q818yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4C2×D418yes—
255C2×C22⋊C4C2×D414yes—
256C2×C22⋊C4C2×Q818yes—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—28no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42#4118yes—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—41no—
286#95—41no—
287#95—41no—
288#95—44no—
289#95—42no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95C2×D414yes—
297#95C2×Q818yes—
298C24—22no—
299C24—22no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24C22×D412yes—
307C24C22×Q812yes—
308C242−1+412yes—
309C24C2511yescharacteristic
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—21no—
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—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—48no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—48no—
386C22×C4—84no—
387C22×C4—88no—
388C22×C4—84no—
389C22×C4—88no—
390C22×C4C22×D414yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D414yes—
394C22×C4C22×Q812yes—
395C22×C4C22×Q818yes—
396C22×C4C22×Q812yes—
397C22×C4C2×#4114yes—
398C22×C4C2×#4114yes—
399C22×C4C2×#4114yes—
400C22×C4H(5,𝔽2)14yes—
401C22×C4H(5,𝔽2)14yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C22×C42−1+414yes—
405C22×C42−1+412yes—
406C4⋊C4—216no—
407C4⋊C4—216no—
408C4⋊C4—216no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—416no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—88no—
417C4⋊C4—84no—
418C4⋊C4—88no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C4⋊C4—88no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C22⋊C4—216no—
425C22⋊C4—216no—
426C22⋊C4—216no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—88no—
430C42—216no—
431C42—48no—
432C42—44no—
433C42—44no—
434C42—88no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—42no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23D4×Q814yes—
458C23C23×D411yescharacteristic
459C23C23×Q811yescharacteristic
460C23C22×#4112yes—
461C23C2×H(5,𝔽2)12yes—
462C23C2×H(5,𝔽2)11yescharacteristic
463C23C2×H(5,𝔽2)12yes—
464C23C2×2−1+411yescharacteristic
465C23C2×2−1+412yes—
466C23C2×2−1+411yescharacteristic
467C23C2×2−1+411yescharacteristic
468C23#58511yescharacteristic
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—416no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—816no—
551C2×C4#53918yes—
552C2×C4#54414yes—
553C2×C4#54718yes—
554C2×C4#55418yes—
555C2×C4#55914yes—
556C2×C4#56014yes—
557C22—28no—
558C22—24no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—44no—
568C22C2×#53912yes—
569C22C2×#54411yescharacteristic
570C22C2×#54712yes—
571C22C2×D4×Q811yescharacteristic
572C22C2×#55412yes—
573C22C2×#55911yescharacteristic
574C22#349212yes—
575C22#349212yes—
576C22#349611yescharacteristic
577C22#349812yes—
578C22#349911yescharacteristic
579C22#350211yescharacteristic
580C22#350312yes—
581C22#350712yes—
582C22C2×#56011yescharacteristic
583C22#353711yescharacteristic
584C22#353811yescharacteristic
585C22#354612yes—
586C22#355011yescharacteristic
587C22#355212yes—
588C22#355212yes—
589C22#355212yes—
590C22#356311yescharacteristic
591C4—28no—
592C4—42no—
593C4—44no—
594C4—42no—
595C4—42no—
596C4—48no—
597C4—44no—
598C4—42no—
599C4—44no—
600C4—81no—
601C4—84no—
602C4—84no—
603C4—84no—
604C4#280718yescomplemented
605C2—44no—
606C2#3934812yes—
607C2#3936811yescharacteristic
608C2#4245112yes—
609C2#4293311yescharacteristic
610C2#4492312yes—
611C2#4537511yescharacteristic
612C2#4548711yescharacteristic
613C2C2×#280711yescharacteristic
614C2#5300811yescharacteristic
615C2#5931412yes—
616C2#5982711yescharacteristic
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. #17221C2semidirect
3. #16577C2semidirect
6. #15363C2semidirect
7. #15156C2semidirect
8. #15151C2semidirect
11. #14005C2semidirect
13. #13910C2semidirect
604. C4#2807semidirect

Maximal subgroups

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

subgroupindex
C2×#28072
#172212
#165772
#163212
#159722
#153632
#151562
#151512
#151352
#141302
#140052
#139742
#139102
#138802
C4×#3992