Finite Group Databasebeta

← #418673

Subgroups — #418673

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic
4C2×#2024C211yesmaximal, characteristic
5C2×#1865C212yesmaximal
6#11380C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11208C212yesmaximal
9#11067C212yesmaximal
10#10932C211yesmaximal, characteristic
11#10884C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18#2581—24no—
19#2581—24no—
20#2567—28no—
21C2×#400—21no—
22C2×#395—21no—
23C2×#395—21no—
24C2×#395—21no—
25C2×#391—21no—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#387—21no—
33C2×#387—21no—
34C2×#384—21no—
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#2058—24no—
38#2024—24no—
39#1865—28no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2211yescharacteristic
46C2×#336C2212yes—
47C2×#336C2211yescharacteristic
48C2×#336C2212yes—
49C23×Q8—21no—
50C23×Q8D411yescharacteristic
51C23×Q8C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—24no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8—42no—
65C2×C4×Q8—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—42no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#103Q818yescomplemented
94#400—28no—
95#395—28no—
96#395—28no—
97#395—28no—
98#391—28no—
99#391—22no—
100#391—22no—
101#391—22no—
102#391D414yes—
103#389—24no—
104#389—24no—
105#389—22no—
106#389—22no—
107#389—24no—
108#389—22no—
109#389D418yes—
110#389D414yes—
111#387—28no—
112#387—28no—
113#384—28no—
114#382—22no—
115#382—22no—
116C4×C4⋊C4—24no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#336—22no—
134#336—24no—
135#336—22no—
136#336—42no—
137C22×Q8—24no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—82no—
154C22×Q8—82no—
155C22×Q8C2×D412yes—
156C22×Q8C2×Q814yes—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×Q814yes—
184C4⋊Q8—48no—
185C4⋊Q8—48no—
186C4×Q8—48no—
187C4×Q8—48no—
188C4×Q8—48no—
189C4×Q8—48no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×Q814yes—
240C2×C4⋊C4#4114yes—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C2×D414yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259#106—48no—
260#103—216no—
261#103—48no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—28no—
273C24C22×D411yescharacteristic
274C2×Q8—28no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—48no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—44no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—88no—
300C2×Q8—88no—
301C2×Q8C4⋊Q814yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4#12012yes—
368C22×C4#12012yes—
369C22×C4#12012yes—
370C22×C4#12012yes—
371C22×C4#13512yes—
372C22×C4#13712yes—
373C22×C4#13712yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×Q812yes—
378C22×C4C2×#4112yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C42−1+412yes—
381C22×C42−1+412yes—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—48no—
418C4⋊C4—88no—
419C4⋊C4—88no—
420C4⋊C4C22⋊Q818yes—
421C42—24no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—48no—
426C42—48no—
427C42—42no—
428C42—42no—
429C23C2×#12011yescharacteristic
430C23C2×#12011yescharacteristic
431C23#53411yescharacteristic
432C23#54812yes—
433C23D4×Q812yes—
434C23C2×#13511yescharacteristic
435C23C2×#13711yescharacteristic
436C23C2×#13711yescharacteristic
437C23#57511yescharacteristic
438C23#57712yes—
439C23#57712yes—
440Q8—28no—
441Q8—416no—
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—816no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—28no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#45214yes—
518C2×C4C2×C22⋊Q812yes—
519C2×C4C2×C4⋊Q811yescharacteristic
520C2×C4#53211yescharacteristic
521C2×C4#53611yescharacteristic
522C2×C4#53612yes—
523C2×C4#53611yescharacteristic
524C2×C4#54212yes—
525C2×C4#54412yes—
526C2×C4D4212yes—
527C2×C4D4212yes—
528C2×C4#54712yes—
529C2×C4#54812yes—
530C2×C4D4×Q812yes—
531C2×C4D4×Q812yes—
532C2×C4#55412yes—
533C2×C4#56312yes—
534C8—44no—
535C8—82no—
536C22#240311yescharacteristic
537C22#240912yes—
538C22#243211yescharacteristic
539C22#243411yescharacteristic
540C22#272912yes—
541C22C2×#45211yescharacteristic
542C22#300211yescharacteristic
543C22#300411yescharacteristic
544C22#300711yescharacteristic
545C22#300912yes—
546C22#301011yescharacteristic
547C22#301011yescharacteristic
548C22#319011yescharacteristic
549C22#319611yescharacteristic
550C22#320012yes—
551C22#336012yes—
552C22#336912yes—
553C22#336912yes—
554C22#337612yes—
555C22#338112yes—
556C22#338712yes—
557C22#338712yes—
558C22#338812yes—
559C4—24no—
560C4—48no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—81no—
570C4—84no—
571C4#244012yes—
572C4#245212yes—
573C4#267412yes—
574C4#267712yes—
575C2#1526511yescharacteristic
576C2#2392211yescharacteristic
577C2#2392611yescharacteristic
578C2#2433712yes—
579C2#2434512yes—
580C2#2564611yescharacteristic
581C2#2566211yescharacteristic
582C2#2583611yescharacteristic
583C2#2584811yescharacteristic
584C2#3257512yes—
585C2#3258212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
93. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25812
C2×#20582
C2×#20242
C2×#18652
#113802
#113272
#112082
#110672
#109322
#108842