Finite Group Databasebeta

← #4929363

Subgroups — #4929363

Order 512.

3254 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×#2406C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3#14964C214yesmaximal, complemented
4#14905C214yesmaximal, complemented
5#14787C214yesmaximal, complemented
6#14778C214yesmaximal, complemented
7#14297C211yesmaximal, characteristic, complemented
8#14178C214yesmaximal, complemented
9#14177C214yesmaximal, complemented
10#13953C211yesmaximal, characteristic, complemented
11#13937C211yesmaximal, characteristic
12#13923C211yesmaximal, characteristic, complemented
13#13851C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×#123C2212yes—
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18#2406—28no—
19C2×#399C2214yes—
20C2×#397C2214yes—
21C2×#395C2212yes—
22C2×#394C2212yes—
23C2×#391C2212yes—
24C2×#390C2212yes—
25C2×#388C2214yes—
26C2×#387C2214yes—
27C2×#386C2212yes—
28C2×#385C2214yes—
29C2×#381C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#2274—28no—
32C2×C4×C4⋊C4C2214yes—
33C2×C4×C4⋊C4C2214yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40#1446C2212yes—
41#1445C2214yes—
42#1445C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1438C2214yes—
52#1438C2212yes—
53#1438C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65#1434C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41—24no—
72C22×#41—44no—
73C22×#41—44no—
74C23×Q8C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×#123—28no—
82C2×#123—44no—
83C2×#123—44no—
84C2×#117—28no—
85C2×#117—216no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4D418yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2314yes—
113#399—216no—
114#397—216no—
115#395—28no—
116#394—28no—
117#391—44no—
118#390—44no—
119#388—48no—
120#387—48no—
121#386—28no—
122#385—216no—
123#381—24no—
124#380—24no—
125C4×C4⋊C4—216no—
126C4×C4⋊C4—216no—
127C4×C22⋊C4—28no—
128C4×C22⋊C4—24no—
129C4×C22⋊C4—24no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—44no—
132C4×C22⋊C4—44no—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2412yes—
156C25C2411yescharacteristic
157C2×#41—432no—
158C2×#41—432no—
159C2×#41—416no—
160C2×#41—416no—
161C2×#41—416no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—24no—
166C22×Q8—42no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—24no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4#4114yes—
199C23×C4#4114yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211#123—432no—
212#117—264no—
213C2×C4⋊C4—232no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D418yes—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4#41116yescomplemented
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42—42no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304C24—24no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24C22×D412yes—
320C24C2×#4112yes—
321C24C2×#4112yes—
322C24H(5,𝔽2)14yes—
323C24C2511yescharacteristic
324#41—4128no—
325C2×Q8—28no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×D4—216no—
331C2×D4—28no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—44no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—44no—
342C22×C4—216no—
343C22×C4—216no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
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—24no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
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—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—82no—
412C22×C4—84no—
413C22×C4—82no—
414C22×C4#123116yes—
415C22×C4C22×D414yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D414yes—
419C22×C4C2×#4118yes—
420C22×C4C2×#4118yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4118yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C42−1+414yes—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—232no—
431C4⋊C4—416no—
432C4⋊C4—416no—
433C4⋊C4—84no—
434C4⋊C4—88no—
435C22⋊C4—216no—
436C22⋊C4—216no—
437C22⋊C4—416no—
438C22⋊C4—48no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—416no—
442C22⋊C4—48no—
443C22⋊C4—416no—
444C22⋊C4—88no—
445C22⋊C4—84no—
446C42—232no—
447C42—84no—
448C42—84no—
449C42—82no—
450C42—82no—
451C42—88no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23C2×#12314yes—
491C23C23×D411yescharacteristic
492C23C22×#4111yescharacteristic
493C23C22×#4111yescharacteristic
494C23C22×#4114yes—
495C23C2×H(5,𝔽2)12yes—
496C23C2×H(5,𝔽2)12yes—
497C23C2×2−1+412yes—
498C23#58512yes—
499Q8—416no—
500D4—416no—
501D4—432no—
502C2×C4—232no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—416no—
529C2×C4—432no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
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—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
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—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—88no—
568C2×C4C2×#12314yes—
569C2×C4C2×#12314yes—
570C2×C4#52518yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—48no—
579C22—44no—
580C22—48no—
581C22—44no—
582C22—44no—
583C22—48no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—84no—
588C22—82no—
589C22—84no—
590C22C22×#12311yescharacteristic
591C22C2×#52512yes—
592C22#344412yes—
593C22#344412yes—
594C22#348412yes—
595C22#349011yescharacteristic
596C22#349012yes—
597C22#349011yescharacteristic
598C22#349512yes—
599C22#349712yes—
600C22#350911yescharacteristic
601C22#351011yescharacteristic
602C22#351314yes—
603C22#351714yes—
604C22#352014yes—
605C22#352214yes—
606C4—28no—
607C4—24no—
608C4—24no—
609C4—44no—
610C4—44no—
611C4—44no—
612C4—48no—
613C4—48no—
614C4—82no—
615C4—84no—
616C4—84no—
617C2—44no—
618C2—84no—
619C2#3535011yescharacteristic
620C2#3543012yes—
621C2#3543512yes—
622C2#3567311yescharacteristic
623C2#3571514yes—
624C2#3591312yes—
625C2#3594312yes—
626C2#3662611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2274C2semidirect
3. #14964C2semidirect
4. #14905C2semidirect
5. #14787C2semidirect
6. #14778C2semidirect
7. #14297C2semidirect
8. #14178C2semidirect
9. #14177C2semidirect
10. #13953C2semidirect
12. #13923C2semidirect
234. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#24062
C2×#22742
#149642
#149052
#147872
#147782
#142972
#141782
#141772
#139532
#139372
#139232
#138512