Finite Group Databasebeta

← #826772

Subgroups — #826772

Order 512.

2206 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2508C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#16694C212yesmaximal
4#15609C214yesmaximal
5#15601C214yesmaximal
6#15230C211yesmaximal, characteristic
7#15194C211yesmaximal, characteristic
8#15128C211yesmaximal, characteristic
9#14879C212yesmaximal
10#14872C212yesmaximal
11#14150C212yesmaximal
12#14045C212yesmaximal
13#14006C212yesmaximal
14#14006C212yesmaximal
15#13925C212yesmaximal
16C4×#398C211yesmaximal, characteristic
17#13821C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20#2508—28no—
21C2×#400C2212yes—
22C2×#398C2211yescharacteristic
23C2×#398C2212yes—
24C2×#395C2212yes—
25#2365—28no—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#382C2212yes—
38C2×#382C2212yes—
39C2×#382C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47#1446C2212yes—
48#1446C2212yes—
49#1446C2212yes—
50#1445C2212yes—
51#1445C2211yescharacteristic
52#1444C2212yes—
53#1444C2212yes—
54#1444C2214yes—
55#1444C2211yescharacteristic
56#1444C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2211yescharacteristic
62#1441C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—24no—
88C2×C4×Q8—28no—
89C2×C4×Q8—28no—
90C2×C4×Q8—44no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2314yes—
120#400—28no—
121#398—24no—
122#398—28no—
123#395—44no—
124#391—28no—
125#389—28no—
126#389—24no—
127#389—24no—
128#389—24no—
129#389—28no—
130#389—24no—
131#389—44no—
132#389Q8116yescomplemented
133#387—28no—
134#387—216no—
135#382—24no—
136#382—28no—
137#382—28no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—44no—
142C4×C4⋊C4—44no—
143C4×C4⋊C4—44no—
144C4×C4⋊C4D418yes—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C22×Q8—24no—
181C22×Q8—28no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—44no—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C4⋊Q8—232no—
203C4×Q8—432no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×Q818yes—
256C2×C4⋊C4C2×Q8116yes—
257C2×C4⋊C4C2×Q818yes—
258C2×C4⋊C4#4114yes—
259C2×C4⋊C4#4118yes—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C2×D418yes—
280C2×C42C2×D414yes—
281C2×C42C2×Q818yes—
282#95—22no—
283#95—24no—
284#95—28no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—22no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95C2×D418yes—
316#95C2×Q8116yes—
317C24C2511yescharacteristic
318C2×Q8—216no—
319C2×Q8—416no—
320C2×Q8—416no—
321C2×Q8—48no—
322C2×Q8—48no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C22×D414yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×Q814yes—
412C22×C4C22×Q818yes—
413C22×C4C22×Q814yes—
414C22×C4C22×Q818yes—
415C22×C4C22×Q814yes—
416C22×C4C2×#4118yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C42−1+412yes—
423C22×C42−1+412yes—
424C4⋊C4—216no—
425C4⋊C4—216no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—216no—
429C4⋊C4—416no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—416no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C4⋊C4C4⋊Q8116yes—
445C42—28no—
446C42—216no—
447C42—216no—
448C42—416no—
449C42—44no—
450C42—48no—
451C42—44no—
452C42—88no—
453C23C23×D411yescharacteristic
454C23C23×Q812yes—
455C23C22×#4112yes—
456C23C22×#4111yescharacteristic
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×H(5,𝔽2)11yescharacteristic
459C23C2×2−1+411yescharacteristic
460C23C2×2−1+411yescharacteristic
461C23#58512yes—
462C23#58512yes—
463C23#58511yescharacteristic
464Q8—432no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4#52514yes—
560C2×C4C2×C4⋊Q818yes—
561C2×C4C2×C4⋊Q814yes—
562C2×C4#53714yes—
563C2×C4#54614yes—
564C2×C4#55418yes—
565C2×C4#55718yes—
566C22C2×#52511yescharacteristic
567C22C22×C4⋊Q811yescharacteristic
568C22C2×#53711yescharacteristic
569C22C2×#54611yescharacteristic
570C22C2×#55412yes—
571C22#347512yes—
572C22C2×#55712yes—
573C22Q8×#4112yes—
574C22#348711yescharacteristic
575C22#348911yescharacteristic
576C22#350011yescharacteristic
577C22#350112yes—
578C22#350212yes—
579C22#350412yes—
580C22#351312yes—
581C22#351511yescharacteristic
582C22#351812yes—
583C22#352011yescharacteristic
584C22#352412yes—
585C22#354012yes—
586C22#354611yescharacteristic
587C22#355211yescharacteristic
588C22#356512yes—
589C4—24no—
590C4—28no—
591C4—44no—
592C4—48no—
593C4—48no—
594C4—44no—
595C4—42no—
596C4—44no—
597C4—44no—
598C4—42no—
599C4—81no—
600C4—82no—
601C4—81no—
602C4—82no—
603C4—81no—
604C4—82no—
605C4#258418yes—
606C2C2×#258411yescharacteristic
607C2#3595711yescharacteristic
608C2#3607711yescharacteristic
609C2#3616211yescharacteristic
610C2#3627611yescharacteristic
611C2#3890811yescharacteristic
612C2#4288211yescharacteristic
613C2#4528112yes—
614C2#4536112yes—
615C2#5955112yes—
616C2#5958712yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
132. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25082
C2×#23652
#166942
#156092
#156012
#152302
#151942
#151282
#148792
#148722
#141502
#140452
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C4×#3982
#138212