Finite Group Databasebeta

← #521707

Subgroups — #521707

Order 512.

2912 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2290C211yesmaximal, characteristic
2#14686C211yesmaximal, characteristic, complemented
3#14472C214yesmaximal, complemented
4#14447C214yesmaximal, complemented
5#14350C211yesmaximal, characteristic, complemented
6#14343C211yesmaximal, characteristic, complemented
7#13999C214yesmaximal
8#13980C214yesmaximal, complemented
9#13884C212yesmaximal, complemented
10#13876C212yesmaximal, complemented
11#13872C212yesmaximal
12C22⋊C4×C4⋊C4C212yesmaximal, complemented
13#13797C211yesmaximal, characteristic, complemented
14#13777C211yesmaximal, characteristic, complemented
15C2×C4⋊#95C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C42C2211yescharacteristic
20#2290C4116yescomplemented
21C2×#388C2214yes—
22C2×#388C2212yes—
23C2×#387C2214yes—
24C2×#387C2212yes—
25C2×#386C2214yes—
26C2×#386C2212yes—
27C2×#385C2212yes—
28C2×#384C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2214yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2214yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×#376C2211yescharacteristic
40C22×#95C2214yes—
41#1445C2214yes—
42#1443C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2212yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2211yescharacteristic
68#1434C2214yes—
69#1434C2214yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C23×D4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2314yes—
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4C2×C418yes—
80C2×#117—44no—
81C2×#117C2×C418yescomplemented
82C22×C4⋊C4—28no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—44no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C42—44no—
104C22×C42C2×C418yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111C22×C42C2314yes—
112C22×C42C2312yes—
113#388—48no—
114#388C2×C4116yescomplemented
115#387—48no—
116#387C2×C4116yescomplemented
117#386—48no—
118#386C2×C4116yes—
119#385C2×C4116yes—
120#384—44no—
121#380—28no—
122#380—28no—
123#380—48no—
124#380—44no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—44no—
127C4×C4⋊C4C2×C418yescomplemented
128C4×C22⋊C4—48no—
129C4×C22⋊C4C2×C418yes—
130C4×C22⋊C4C2×C418yescomplemented
131#376C2×C418yescomplemented
132C2×#95—48no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
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×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C25C2411yescharacteristic
158C25C2411yescharacteristic
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4C22×C414yes—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4C22×C414yes—
184C23×C4C22×C418yes—
185C23×C4C22×C414yes—
186C23×C4#4114yes—
187C23×C4#4114yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C4×D4—232no—
202#117—416no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—432no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C418yes—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4#4118yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4C22×C414yes—
254C2×C22⋊C4C22×C414yes—
255C2×C22⋊C4C22×C418yes—
256C2×C22⋊C4C22×C418yes—
257C2×C22⋊C4C22×C418yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—416no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42C22×C414yes—
274C2×C42C22×C414yes—
275C2×C42C22×C414yes—
276C2×C42C22×C414yes—
277C2×C42C22×C414yes—
278C2×C42C22×C418yes—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
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—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95C22×C418yes—
299#95C22×C418yes—
300#95C22×C418yes—
301#95C22×C418yes—
302#95C22×C418yes—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24C23×C412yes—
316C24C23×C412yes—
317C24C2×#4112yes—
318C24C2×#4112yes—
319C24H(5,𝔽2)14yes—
320C242−1+414yes—
321C24C2511yescharacteristic
322C2×D4—28no—
323C2×D4—216no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—48no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—216no—
336C22×C4—216no—
337C22×C4—216no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
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—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—416no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4#117116yes—
414C22×C4C23×C414yes—
415C22×C4C23×C412yes—
416C22×C4C23×C414yes—
417C22×C4C23×C412yes—
418C22×C4C23×C414yes—
419C22×C4C23×C412yes—
420C22×C4C23×C412yes—
421C22×C4C23×C414yes—
422C22×C4C23×C412yes—
423C22×C4C2×#4118yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4114yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C42−1+414yes—
432C4⋊C4—28no—
433C4⋊C4—216no—
434C4⋊C4—216no—
435C4⋊C4—48no—
436C4⋊C4—416no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C22⋊C4—28no—
443C22⋊C4—216no—
444C22⋊C4—28no—
445C22⋊C4—216no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—28no—
451C42—28no—
452C42—28no—
453C42—28no—
454C42—28no—
455C42—48no—
456C42—88no—
457C42—88no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23C2×#11714yes—
496C23#52018yes—
497C23#54014yes—
498C23#54114yes—
499C23#54218yes—
500C23C24×C411yescharacteristic
501C23C22×#4111yescharacteristic
502C23C22×#4111yescharacteristic
503C23C22×#4112yes—
504C23C2×H(5,𝔽2)12yes—
505C23C2×H(5,𝔽2)12yes—
506C23C2×H(5,𝔽2)12yes—
507C23C2×2−1+412yes—
508C23C2×2−1+412yes—
509D4—416no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—232no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—432no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—416no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—832no—
588C2×C4C2×#11714yes—
589C2×C4C2×#11714yes—
590C2×C4C4×#4114yes—
591C2×C4#51814yes—
592C2×C4#52018yes—
593C2×C4#55118yes—
594C2×C4#55318yes—
595C22—28no—
596C22—24no—
597C22—24no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—42no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22#231418yes—
613C22C22×#11711yescharacteristic
614C22C2×C4×#4111yescharacteristic
615C22C2×#51811yescharacteristic
616C22C2×#52012yes—
617C22C2×#52012yes—
618C22C2×#54011yescharacteristic
619C22C2×#54111yescharacteristic
620C22C2×#54212yes—
621C22C2×#55112yes—
622C22C2×#55312yes—
623C22#350811yescharacteristic
624C22#350911yescharacteristic
625C22#351112yes—
626C22#351912yes—
627C22#352012yes—
628C22#352212yes—
629C22#352312yes—
630C22#353612yes—
631C22#353612yes—
632C22#353712yes—
633C22#353812yes—
634C4—28no—
635C4—24no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—48no—
642C4—42no—
643C4—84no—
644C4—84no—
645C4—84no—
646C4#235318yes—
647C2—28no—
648C2—82no—
649C2C2×#231411yescharacteristic
650C2C2×#235311yescharacteristic
651C2#3418811yescharacteristic
652C2#3424312yes—
653C2#3436012yes—
654C2#3811012yes—
655C2#3811112yes—
656C2#3812212yes—
657C2#3812312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14686C2semidirect
3. #14472C2semidirect
4. #14447C2semidirect
5. #14350C2semidirect
6. #14343C2semidirect
8. #13980C2semidirect
9. #13884C2semidirect
10. #13876C2semidirect
12. C22⋊C4×C4⋊C4C2semidirect
13. #13797C2semidirect
14. #13777C2semidirect
15. C2×C4⋊#95C2semidirect
20. #2290C4semidirect
81. C2×#117C2×C4semidirect
114. #388C2×C4semidirect
116. #387C2×C4semidirect
127. C4×C4⋊C4C2×C4semidirect
130. C4×C22⋊C4C2×C4semidirect
131. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22902
#146862
#144722
#144472
#143502
#143432
#139992
#139802
#138842
#138762
#138722
C22⋊C4×C4⋊C42
#137972
#137772
C2×C4⋊#952