Finite Group Databasebeta

← #803459

Subgroups — #803459

Order 512.

2882 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊C4C212yesmaximal
2#17059C212yesmaximal
3#16110C212yesmaximal, complemented
4#16109C212yesmaximal, complemented
5#16108C214yesmaximal, complemented
6#15150C211yesmaximal, characteristic
7#15132C212yesmaximal
8#15061C214yesmaximal
9#14248C212yesmaximal
10#14017C212yesmaximal, complemented
11#13976C212yesmaximal, complemented
12#13931C212yesmaximal, complemented
13#13931C212yesmaximal, complemented
14#13922C211yesmaximal, characteristic
15C4×#397C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4×Q8C2212yes—
18C22×C4×Q8C2212yes—
19C2×#397C2211yescharacteristic
20C2×#397C2212yes—
21C2×#397C2214yes—
22C2×#397C2212yes—
23Q8×C22⋊C4—216no—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2214yes—
27C2×#391C2214yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#384C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2214yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2214yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39#1446C2212yes—
40#1446C2212yes—
41#1445C2211yescharacteristic
42#1445C2212yes—
43#1445C2212yes—
44#1445C2214yes—
45#1445C2214yes—
46#1445C2214yes—
47#1445C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1443C2212yes—
53#1442C2214yes—
54#1442C2211yescharacteristic
55#1442C2214yes—
56#1440C2214yes—
57#1440C2212yes—
58#1439C2211yescharacteristic
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×Q8—28no—
77C23×Q8—28no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8D418yes—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C2×C4×Q8—28no—
86C2×C4×Q8—28no—
87C2×C4×Q8—28no—
88C2×C4×Q8—28no—
89C2×C4×Q8—44no—
90C2×C4×Q8—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113#397—24no—
114#397—28no—
115#397—44no—
116#397—48no—
117#391—216no—
118#391—216no—
119#391—28no—
120#391—28no—
121#391—28no—
122#391—28no—
123#389—44no—
124#384—42no—
125#380—28no—
126#380—216no—
127#380—42no—
128C4×C4⋊C4—42no—
129C4×C22⋊C4—28no—
130C4×C22⋊C4—216no—
131C4×C22⋊C4—42no—
132C2×#95C2311yescharacteristic
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C25C2411yescharacteristic
166C22×Q8—264no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—28no—
170C22×Q8—28no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—416no—
176C22×Q8—416no—
177C22×Q8—416no—
178C22×Q8—432no—
179C22×Q8C2×D418yes—
180C22×Q8C2×D418yes—
181C22×Q8#4118yes—
182C22×Q8#4118yes—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4C2×D414yes—
192C23×C4C2×D418yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C4×Q8—432no—
208C4×Q8—432no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4#4114yes—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—44no—
271C2×C42—44no—
272#95—24no—
273#95—24no—
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—28no—
283#95—22no—
284#95—24no—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—41no—
294#95—44no—
295#95—41no—
296#95—41no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—42no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—22no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24C22×D412yes—
314C24C2×#4112yes—
315C24H(5,𝔽2)12yes—
316C24C2511yescharacteristic
317C2×Q8—216no—
318C2×Q8—232no—
319C2×Q8—216no—
320C2×Q8—216no—
321C2×Q8—216no—
322C2×Q8—216no—
323C2×Q8—216no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—416no—
328C2×Q8—4128no—
329C2×Q8#123132yes—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
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—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—232no—
363C22×C4—216no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—416no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4C22×D412yes—
413C22×C4C22×D418yes—
414C22×C4C22×D412yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4118yes—
418C22×C4C2×#4118yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4114yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+414yes—
424C22×C42−1+414yes—
425C22×C42−1+412yes—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—416no—
429C4⋊C4—416no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—88no—
436C22⋊C4—216no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—416no—
441C22⋊C4—84no—
442C42—416no—
443C42—416no—
444C42—48no—
445C42—48no—
446C42—44no—
447C42—44no—
448C23—24no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23#55014yes—
471C23C23×D411yescharacteristic
472C23C22×#4112yes—
473C23C22×#4112yes—
474C23C22×#4111yescharacteristic
475C23C2×H(5,𝔽2)11yescharacteristic
476C23C2×H(5,𝔽2)11yescharacteristic
477C23C2×H(5,𝔽2)11yescharacteristic
478C23C2×2−1+412yes—
479C23C2×2−1+412yes—
480C23C2×2−1+411yescharacteristic
481C23#58511yescharacteristic
482Q8—264no—
483Q8—432no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—432no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—416no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4C2×#123116yes—
567C2×C4C2×#12318yes—
568C2×C4#53414yes—
569C2×C4#55114yes—
570C2×C4#55114yes—
571C2×C4#55118yes—
572C22—24no—
573C22—28no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—44no—
582C22—42no—
583C22C22×#12312yes—
584C22C2×#53411yescharacteristic
585C22C2×#55011yescharacteristic
586C22C2×#55111yescharacteristic
587C22C2×#55111yescharacteristic
588C22C2×#55112yes—
589C22#348212yes—
590C22#348212yes—
591C22#348612yes—
592C22#349711yescharacteristic
593C22#350511yescharacteristic
594C22#351111yescharacteristic
595C22#351612yes—
596C22#351812yes—
597C22#352212yes—
598C22#352212yes—
599C22#352312yes—
600C22#352312yes—
601C22#353611yescharacteristic
602C22#353811yescharacteristic
603C22#355011yescharacteristic
604C22#355212yes—
605C22#355811yescharacteristic
606C4—216no—
607C4—42no—
608C4—44no—
609C4—42no—
610C4—44no—
611C4—42no—
612C4—42no—
613C4—81no—
614C4—82no—
615C4—84no—
616C4—84no—
617C4—82no—
618C4—84no—
619C4#248318yes—
620C2—44no—
621C2C2×#248311yescharacteristic
622C2#3572312yes—
623C2#3648312yes—
624C2#3974311yescharacteristic
625C2#4280711yescharacteristic
626C2#4529212yes—
627C2#4535111yescharacteristic
628C2#4541911yescharacteristic
629C2#5937911yescharacteristic
630C2#5948112yes—
631C2#5994611yescharacteristic
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. #16110C2semidirect
4. #16109C2semidirect
5. #16108C2semidirect
10. #14017C2semidirect
11. #13976C2semidirect
12. #13931C2semidirect
13. #13931C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C42
#170592
#161102
#161092
#161082
#151502
#151322
#150612
#142482
#140172
#139762
#139312
#139222
C4×#3972