Finite Group Databasebeta

← #630045

Subgroups — #630045

Order 512.

2816 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3#15810C214yesmaximal, complemented
4#15806C214yesmaximal, complemented
5#15768C214yesmaximal, complemented
6#15701C214yesmaximal, complemented
7#14735C214yesmaximal, complemented
8#14698C211yesmaximal, characteristic
9#14682C211yesmaximal, characteristic, complemented
10#14347C211yesmaximal, characteristic, complemented
11#14271C211yesmaximal, characteristic
12#14135C214yesmaximal, complemented
13C2×#1436C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#121C2211yescharacteristic
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2618—28no—
20C2×#398C2212yes—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#394C2212yes—
24#2316—28no—
25C2×#390C2212yes—
26C2×#389C2214yes—
27C2×#389C2214yes—
28C2×#386C2212yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#382C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C22×#95C2211yescharacteristic
37C22×#95C2212yes—
38C22×#95C2212yes—
39#1445C2214yes—
40#1445C2212yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2212yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2212yes—
58#1437C2212yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×#123—28no—
77C2×#123—216no—
78C2×#121—24no—
79C2×#121—24no—
80C2×#121—24no—
81C2×#117—28no—
82C2×#117—48no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4D418yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42—24no—
106C22×C42—44no—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111#398—28no—
112#397—48no—
113#396—216no—
114#394—28no—
115#390—44no—
116#389—216no—
117#389—216no—
118#386—28no—
119#386—48no—
120#386—48no—
121#382—44no—
122#380—28no—
123#380—24no—
124#380—24no—
125#380—24no—
126C2×#95—28no—
127C2×#95—24no—
128C2×#95—44no—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
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×#95C2314yes—
151C2×#95C2314yes—
152C25C2412yes—
153C25C2411yescharacteristic
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—42no—
163C22×D4—42no—
164C23×C4—22no—
165C23×C4—22no—
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—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2×D414yes—
190C23×C4#4118yes—
191C23×C4#4114yes—
192C23×C4#4114yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204#123—432no—
205#121—416no—
206#117—432no—
207C2×C4⋊C4—232no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4C2×D418yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—416no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—42no—
272C2×C42—42no—
273#95—28no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
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—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—24no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24C22×D412yes—
310C24C2×#4112yes—
311C24C2×#4112yes—
312C24C2×#4114yes—
313C24H(5,𝔽2)14yes—
314C24C2511yescharacteristic
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—416no—
320C2×D4—416no—
321C2×D4—44no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—84no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—216no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
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—44no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4#123116yes—
409C22×C4#124116yes—
410C22×C4C22×D414yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D414yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4118yes—
419C22×C4H(5,𝔽2)14yes—
420C22×C42−1+414yes—
421C4⋊C4—216no—
422C4⋊C4—216no—
423C4⋊C4—416no—
424C4⋊C4—44no—
425C4⋊C4—416no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—88no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—416no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C42—28no—
441C42—48no—
442C42—44no—
443C42—416no—
444C23—24no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23C2×#12314yes—
482C23C2×#12414yes—
483C23#53514yes—
484C23#55318yes—
485C23#55318yes—
486C23C23×D411yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4112yes—
490C23C2×H(5,𝔽2)12yes—
491C23C2×H(5,𝔽2)12yes—
492C23C2×2−1+412yes—
493C23#58512yes—
494C23#58512yes—
495D4—88no—
496D4—816no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—232no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—416no—
519C2×C4—432no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—88no—
565C2×C4—816no—
566C2×C4C2×#12314yes—
567C2×C4C2×#12314yes—
568C2×C4C2×#12418yes—
569C22—24no—
570C22—28no—
571C22—28no—
572C22—28no—
573C22—24no—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—82no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22—82no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—82no—
586C22#250418yes—
587C22C22×#12311yescharacteristic
588C22C22×#12411yescharacteristic
589C22C2×#53511yescharacteristic
590C22#346012yes—
591C22#346112yes—
592C22C2×#55312yes—
593C22C2×#55312yes—
594C22#349011yescharacteristic
595C22#349111yescharacteristic
596C22#349212yes—
597C22#349312yes—
598C22#349512yes—
599C22#349612yes—
600C22#349712yes—
601C22#350512yes—
602C22#350911yescharacteristic
603C22#351011yescharacteristic
604C22#353412yes—
605C22#353612yes—
606C22#355912yes—
607C22#356012yes—
608C4—24no—
609C4—24no—
610C4—28no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—84no—
615C4—84no—
616C4—84no—
617C4—84no—
618C4—82no—
619C2—28no—
620C2—162no—
621C2C2×#250411yescharacteristic
622C2#3515812yes—
623C2#3536011yescharacteristic
624C2#3567311yescharacteristic
625C2#3674112yes—
626C2#3794112yes—
627C2#3794312yes—
628C2#3954912yes—
629C2#3955112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2316C2semidirect
3. #15810C2semidirect
4. #15806C2semidirect
5. #15768C2semidirect
6. #15701C2semidirect
7. #14735C2semidirect
9. #14682C2semidirect
10. #14347C2semidirect
12. #14135C2semidirect
13. C2×#1436C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C2×#23162
#158102
#158062
#157682
#157012
#147352
#146982
#146822
#143472
#142712
#141352
C2×#14362