Finite Group Databasebeta

← #539373

Subgroups — #539373

Order 512.

2478 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C211yesmaximal, characteristic
2#14434C214yesmaximal, complemented
3#14368C214yesmaximal, complemented
4#14077C214yesmaximal, complemented
5#14022C212yesmaximal
6#14001C212yesmaximal
7#13936C212yesmaximal, complemented
8#13910C211yesmaximal, characteristic, complemented
9#13906C211yesmaximal, characteristic
10#13891C214yesmaximal, complemented
11#13848C211yesmaximal, characteristic, complemented
12#13804C211yesmaximal, characteristic, complemented
13C4⋊C42C212yesmaximal, complemented
14#13785C211yesmaximal, characteristic
15#13779C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17#2357C4116yescomplemented
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#389C2214yes—
23C2×#388C2214yes—
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#388C2214yes—
27C2×#384C2211yescharacteristic
28C2×#384C2212yes—
29C2×#382C2211yescharacteristic
30C2×#381C2212yes—
31C2×#380C2214yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2214yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×#376C2211yescharacteristic
42#1444C2212yes—
43#1444C2212yes—
44#1444C2212yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2212yes—
48#1441C2212yes—
49#1441C2212yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70#1434C2214yes—
71#1434C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4C2×C418yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2314yes—
108C22×C42C2314yes—
109C22×C42C2311yescharacteristic
110#390—24no—
111#390—24no—
112#390—44no—
113#390C2×C4116yes—
114#389—48no—
115#388—48no—
116#388—48no—
117#388—48no—
118#388C2×C4132yes—
119#384—28no—
120#384—24no—
121#382C2×C418yes—
122#381C2×C4116yes—
123#380—48no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—48no—
128C4×C4⋊C4—44no—
129C4×C4⋊C4—44no—
130C4×C4⋊C4C2×C418yes—
131C4×C4⋊C4C2×C4116yes—
132C4×C22⋊C4C2×C4116yes—
133#376—24no—
134C2×#95C2311yescharacteristic
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2311yescharacteristic
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C25C2412yes—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4C22×C414yes—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4C22×C418yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C4×D4—232no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
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—22no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C22×C418yes—
225C2×C4⋊C4C22×C418yes—
226C2×C4⋊C4#4114yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4C22×C418yes—
246C2×C22⋊C4C22×C4116yes—
247C2×C22⋊C4C22×C4116yes—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42C22×C418yes—
269C2×C42C22×C4116yes—
270C2×C42C22×C414yes—
271C2×C42#4114yes—
272C2×C42#4114yes—
273C2×C42#4114yes—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—22no—
279#95—24no—
280#95—22no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—24no—
286#95—44no—
287#95—44no—
288#95—42no—
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#95C22×C414yes—
298#95C22×C418yes—
299#95C22×C414yes—
300#95C22×C4116yes—
301#95C22×C418yes—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—24no—
308C24—22no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24C23×C414yes—
313C24C2511yescharacteristic
314C2×D4—28no—
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
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—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—216no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—416no—
389C22×C4—88no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4C23×C414yes—
393C22×C4C23×C418yes—
394C22×C4C23×C414yes—
395C22×C4C23×C412yes—
396C22×C4C23×C412yes—
397C22×C4C23×C414yes—
398C22×C4C23×C412yes—
399C22×C4C2×#4114yes—
400C22×C4C2×#4114yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C42−1+414yes—
413C22×C42−1+412yes—
414C4⋊C4—216no—
415C4⋊C4—28no—
416C4⋊C4—216no—
417C4⋊C4—216no—
418C4⋊C4—28no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—416no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4—88no—
427C22⋊C4—216no—
428C22⋊C4—232no—
429C22⋊C4—416no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C42—216no—
435C42—28no—
436C42—416no—
437C42—48no—
438C42—44no—
439C42—84no—
440C42—88no—
441C42—88no—
442C42#117116yes—
443C23—24no—
444C23—24no—
445C23—28no—
446C23—24no—
447C23—28no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23C24×C411yescharacteristic
467C23C22×#4111yescharacteristic
468C23C22×#4111yescharacteristic
469C23C22×#4111yescharacteristic
470C23C22×#4111yescharacteristic
471C23C2×H(5,𝔽2)11yescharacteristic
472C23C2×H(5,𝔽2)12yes—
473C23C2×H(5,𝔽2)12yes—
474C23C2×2−1+411yescharacteristic
475C23C2×2−1+412yes—
476C23#58512yes—
477D4—416no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—816no—
557C2×C4C2×#11714yes—
558C2×C4C2×#11714yes—
559C2×C4C2×#11714yes—
560C2×C4C4×#4114yes—
561C2×C4#52018yes—
562C2×C4#52014yes—
563C2×C4#55314yes—
564C2×C4#55314yes—
565C2×C4#55614yes—
566C2×C4#55614yes—
567C22—48no—
568C22—44no—
569C22—44no—
570C22—48no—
571C22—44no—
572C22—82no—
573C22—82no—
574C22—84no—
575C22—82no—
576C22—84no—
577C22—82no—
578C22C22×#11711yescharacteristic
579C22C2×C4×#4111yescharacteristic
580C22C2×#52012yes—
581C22C2×#52011yescharacteristic
582C22C4×H(5,𝔽2)12yes—
583C22C2×#55311yescharacteristic
584C22C2×#55311yescharacteristic
585C22C2×#55611yescharacteristic
586C22C2×#55611yescharacteristic
587C22#349212yes—
588C22#349412yes—
589C22#349412yes—
590C22#350811yescharacteristic
591C22#350811yescharacteristic
592C22#350911yescharacteristic
593C22#351011yescharacteristic
594C22#351112yes—
595C22#351612yes—
596C22#351912yes—
597C22#354312yes—
598C22#355012yes—
599C22#355412yes—
600C22#355412yes—
601C4—28no—
602C4—24no—
603C4—24no—
604C4—44no—
605C4—44no—
606C4—44no—
607C4—48no—
608C4—44no—
609C4—44no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—84no—
614C4#234718yes—
615C2—84no—
616C2C2×#234711yescharacteristic
617C2#3418811yescharacteristic
618C2#3425511yescharacteristic
619C2#3430512yes—
620C2#3449812yes—
621C2#4497412yes—
622C2#4497911yescharacteristic
623C2#4503712yes—
624C2#4504111yescharacteristic
625C2#4534911yescharacteristic
626C2#4543011yescharacteristic
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. #14434C2semidirect
3. #14368C2semidirect
4. #14077C2semidirect
7. #13936C2semidirect
8. #13910C2semidirect
10. #13891C2semidirect
11. #13848C2semidirect
12. #13804C2semidirect
13. C4⋊C42C2semidirect
15. #13779C2semidirect
17. #2357C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
#144342
#143682
#140772
#140222
#140012
#139362
#139102
#139062
#138912
#138482
#138042
C4⋊C422
#137852
#137792