Finite Group Databasebeta

← #4458119

Subgroups — #4458119

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16998C214yesmaximal, complemented
2#16933C212yesmaximal, complemented
3#16877C212yesmaximal, complemented
4#16706C212yesmaximal
5#16405C214yesmaximal
6#16358C212yesmaximal
7#15242C212yesmaximal, complemented
8#15235C212yesmaximal, complemented
9#15146C212yesmaximal, complemented
10#15133C212yesmaximal, complemented
11#14340C211yesmaximal, characteristic
12#14130C212yesmaximal
13#13879C212yesmaximal
14#13852C211yesmaximal, characteristic
15#13795C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#400C2214yes—
20C2×#398C2214yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#385C2214yes—
24C2×#384C2211yescharacteristic
25C2×#384C2212yes—
26C2×#382C2211yescharacteristic
27C2×#382C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32#1445C2214yes—
33#1445C2214yes—
34#1444C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1441C2212yes—
43#1441C2212yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2211yescharacteristic
48#1441C2212yes—
49#1441C2211yescharacteristic
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1439C2212yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2211yescharacteristic
59#1439C2211yescharacteristic
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C2×#117—24no—
78C2×#117—42no—
79C2×#117—42no—
80C2×#117—42no—
81C2×#117—42no—
82C2×#117Q818yes—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113#400—48no—
114#398—48no—
115#389—44no—
116#389—44no—
117#385—48no—
118#384—42no—
119#384—44no—
120#382—42no—
121#382—44no—
122#380—28no—
123C4×C4⋊C4—42no—
124C4×C4⋊C4—42no—
125C4×C22⋊C4—28no—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
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×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C25C2411yescharacteristic
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4C2×Q814yes—
167C23×C4#4114yes—
168C23×C4#4114yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183#117—416no—
184#117—416no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
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—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—416no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C2×Q814yes—
229C2×C4⋊C4C2×Q814yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4C2×Q814yes—
247C2×C22⋊C4C2×Q814yes—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C2×Q814yes—
262C2×C42C2×Q814yes—
263#95—24no—
264#95—24no—
265#95—28no—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—41no—
279#95—41no—
280#95—42no—
281#95—42no—
282#95—44no—
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—44no—
293C24—22no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24C22×Q812yes—
302C24C2×#4112yes—
303C24C2×#4112yes—
304C24C2511yescharacteristic
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
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—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—48no—
386C22×C4—88no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—88no—
390C22×C4#125116yes—
391C22×C4C22×Q812yes—
392C22×C4C22×Q812yes—
393C22×C4C22×Q812yes—
394C22×C4C22×Q812yes—
395C22×C4C22×Q812yes—
396C22×C4C22×Q812yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4H(5,𝔽2)14yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)12yes—
405C22×C42−1+414yes—
406C22×C42−1+412yes—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—88no—
418C4⋊C4—88no—
419C4⋊C4—88no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4—88no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—416no—
428C22⋊C4—48no—
429C42—28no—
430C42—28no—
431C42—48no—
432C42—48no—
433C42—48no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C2×#12514yes—
457C23C23×Q811yescharacteristic
458C23C22×#4111yescharacteristic
459C23C22×#4111yescharacteristic
460C23C2×H(5,𝔽2)11yescharacteristic
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×H(5,𝔽2)11yescharacteristic
463C23C2×H(5,𝔽2)12yes—
464C23C2×2−1+411yescharacteristic
465C23C2×2−1+412yes—
466C23#58512yes—
467C23#58512yes—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—232no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—816no—
552C2×C4C2×#12514yes—
553C2×C4C2×#12514yes—
554C2×C4#53814yes—
555C2×C4#53814yes—
556C2×C4#53914yes—
557C2×C4#53914yes—
558C2×C4#54314yes—
559C2×C4#54414yes—
560C22—24no—
561C22—28no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—44no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22C22×#12511yescharacteristic
572C22C2×#53811yescharacteristic
573C22C2×#53811yescharacteristic
574C22C2×#53911yescharacteristic
575C22C2×#53911yescharacteristic
576C22C2×#54311yescharacteristic
577C22C2×#54411yescharacteristic
578C22#349412yes—
579C22#349512yes—
580C22#350312yes—
581C22#350412yes—
582C22#350612yes—
583C22#350712yes—
584C22#353711yescharacteristic
585C22#353811yescharacteristic
586C22#353912yes—
587C22#354712yes—
588C22#354812yes—
589C22#355111yescharacteristic
590C22#355111yescharacteristic
591C22#355512yes—
592C22#355812yes—
593C22#356112yes—
594C4—24no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—44no—
600C4—44no—
601C4—84no—
602C4—84no—
603C4—84no—
604C4—82no—
605C4—82no—
606C4—84no—
607C4#264618yes—
608C2—44no—
609C2C2×#264611yescharacteristic
610C2#3554312yes—
611C2#3730711yescharacteristic
612C2#3731412yes—
613C2#3742511yescharacteristic
614C2#4251912yes—
615C2#4252211yescharacteristic
616C2#4269511yescharacteristic
617C2#4293512yes—
618C2#4295811yescharacteristic
619C2#4298211yescharacteristic
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
1. #16998C2semidirect
2. #16933C2semidirect
3. #16877C2semidirect
7. #15242C2semidirect
8. #15235C2semidirect
9. #15146C2semidirect
10. #15133C2semidirect

Maximal subgroups

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

subgroupindex
#169982
#169332
#168772
#167062
#164052
#163582
#152422
#152352
#151462
#151332
#143402
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