Finite Group Databasebeta

← #829180

Subgroups — #829180

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2607C212yesmaximal
2#16666C212yesmaximal
3#16109C212yesmaximal, complemented
4#16062C214yesmaximal, complemented
5#15980C212yesmaximal, complemented
6#15135C212yesmaximal
7#15125C211yesmaximal, characteristic
8#14772C214yesmaximal
9#14133C212yesmaximal, complemented
10#14053C212yesmaximal, complemented
11#14049C212yesmaximal, complemented
12#13926C212yesmaximal, complemented
13#13904C211yesmaximal, characteristic
14C4×#393C211yesmaximal, characteristic
15C22⋊C4×C4⋊C4C212yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2607—216no—
20C2×#399C2214yes—
21C2×#393C2212yes—
22C2×#393C2211yescharacteristic
23C2×#391C2214yes—
24C2×#391C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#387C2214yes—
28C2×#387C2214yes—
29C2×#387C2214yes—
30C2×#384C2212yes—
31C2×#381C2212yes—
32C2×#381C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40#1446C2212yes—
41#1446C2214yes—
42#1446C2211yescharacteristic
43#1445C2214yes—
44#1445C2214yes—
45#1445C2212yes—
46#1445C2212yes—
47#1445C2214yes—
48#1444C2214yes—
49#1443C2214yes—
50#1442C2212yes—
51#1442C2211yescharacteristic
52#1442C2212yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1439C2212yes—
58#1439C2211yescharacteristic
59#1439C2214yes—
60#1439C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C22⋊#95C2214yes—
74C4×#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×Q8C2311yescharacteristic
77C23×Q8C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—42no—
82C2×C4×Q8—28no—
83C2×C4×Q8—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4Q818yescomplemented
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2314yes—
110C22×C42C2314yes—
111#399—216no—
112#393—28no—
113#393D418yes—
114#391—216no—
115#391—216no—
116#389—28no—
117#389—44no—
118#387—216no—
119#387—216no—
120#387—48no—
121#384—28no—
122#381—28no—
123#381—44no—
124#380—28no—
125#380—42no—
126C4×C4⋊C4—44no—
127C4×C4⋊C4—44no—
128C4×C22⋊C4—216no—
129C4×C22⋊C4—28no—
130C4×C22⋊C4—42no—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C25C2411yescharacteristic
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—28no—
166C22×Q8—24no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—42no—
170C22×Q8—44no—
171C22×Q8—42no—
172C22×Q8C2×D414yes—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C2×Q814yes—
184C23×C4C2×Q818yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C4⋊Q8—416no—
200C4×Q8—416no—
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—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—416no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×Q818yes—
234C2×C4⋊C4C2×Q818yes—
235C2×C4⋊C4#4118yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4C2×D418yes—
253C2×C22⋊C4C2×D414yes—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42#4118yes—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—24no—
275#95—28no—
276#95—28no—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—41no—
289#95—42no—
290#95—42no—
291#95—41no—
292#95—41no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95C2×D414yes—
300#95C2×D418yes—
301C24—22no—
302C24—22no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24C22×D412yes—
310C24C22×Q812yes—
311C242−1+412yes—
312C24C2511yescharacteristic
313C2×Q8—216no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—48no—
317C2×Q8—44no—
318C2×Q8—48no—
319C2×Q8—416no—
320C2×Q8—84no—
321C2×Q8—88no—
322C2×Q8—84no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D414yes—
408C22×C4C22×D414yes—
409C22×C4C22×Q812yes—
410C22×C4C22×Q818yes—
411C22×C4C22×Q812yes—
412C22×C4C2×#4118yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4114yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C42−1+414yes—
421C4⋊C4—216no—
422C4⋊C4—216no—
423C4⋊C4—48no—
424C4⋊C4—416no—
425C4⋊C4—48no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—416no—
429C4⋊C4—416no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—88no—
433C22⋊C4—216no—
434C22⋊C4—44no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—84no—
439C42—216no—
440C42—44no—
441C42—44no—
442C42—48no—
443C42—88no—
444C42—88no—
445C23—24no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23D4×Q814yes—
468C23C23×D411yescharacteristic
469C23C23×Q811yescharacteristic
470C23C22×#4112yes—
471C23C22×#4112yes—
472C23C2×H(5,𝔽2)11yescharacteristic
473C23C2×H(5,𝔽2)11yescharacteristic
474C23C2×H(5,𝔽2)11yescharacteristic
475C23C2×2−1+412yes—
476C23C2×2−1+411yescharacteristic
477C23#58512yes—
478C23#58511yescharacteristic
479Q8—416no—
480Q8—816no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—416no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—816no—
563C2×C4#53318yes—
564C2×C4#54314yes—
565C2×C4#54718yes—
566C2×C4#55114yes—
567C2×C4#55114yes—
568C2×C4#55918yes—
569C22—28no—
570C22—24no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—44no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22C2×#53312yes—
581C22C2×#54311yescharacteristic
582C22C2×#54712yes—
583C22C2×D4×Q811yescharacteristic
584C22D4×#4112yes—
585C22C2×#55111yescharacteristic
586C22C2×#55111yescharacteristic
587C22C2×#55912yes—
588C22#349512yes—
589C22#349512yes—
590C22#349611yescharacteristic
591C22#349811yescharacteristic
592C22#350111yescharacteristic
593C22#350212yes—
594C22#350612yes—
595C22#352112yes—
596C22#352212yes—
597C22#352312yes—
598C22#353611yescharacteristic
599C22#353811yescharacteristic
600C22#355111yescharacteristic
601C22#355412yes—
602C22#356111yescharacteristic
603C4—28no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—44no—
608C4—48no—
609C4—42no—
610C4—48no—
611C4—44no—
612C4—82no—
613C4—81no—
614C4—84no—
615C4—84no—
616C4#260718yescomplemented
617C2—44no—
618C2C2×#260711yescharacteristic
619C2#3578312yes—
620C2#3613312yes—
621C2#3936711yescharacteristic
622C2#4281811yescharacteristic
623C2#4520112yes—
624C2#4525111yescharacteristic
625C2#4541911yescharacteristic
626C2#5919811yescharacteristic
627C2#5964112yes—
628C2#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
3. #16109C2semidirect
4. #16062C2semidirect
5. #15980C2semidirect
9. #14133C2semidirect
10. #14053C2semidirect
11. #14049C2semidirect
12. #13926C2semidirect
86. C22×C4⋊C4Q8semidirect
616. C4#2607semidirect

Maximal subgroups

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

subgroupindex
C2×#26072
#166662
#161092
#160622
#159802
#151352
#151252
#147722
#141332
#140532
#140492
#139262
#139042
C4×#3932
C22⋊C4×C4⋊C42