Finite Group Databasebeta

← #6487669

Subgroups — #6487669

Order 512.

2132 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic, complemented
2C2×#3103C211yesmaximal, characteristic
3#35180C211yesmaximal, characteristic
4C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#32909C214yesmaximal, complemented
6#29761C214yesmaximal, complemented
7#24818C214yesmaximal, complemented
8#24814C214yesmaximal, complemented
9#24733C214yesmaximal, complemented
10#24729C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21860C211yesmaximal, characteristic, complemented
13#21859C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2212yes—
18#3104—28no—
19#3103—28no—
20C2×#484C2212yes—
21C2×#481C2212yes—
22C2×#477C2212yes—
23C2×#475C2212yes—
24C2×#464C2218yes—
25C2×#462C2212yes—
26C2×#460C2212yes—
27C2×#451C2212yes—
28C2×#448C2212yes—
29C2×C4×Q16C2214yes—
30C2×C4×SD16C2214yes—
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#2850C2214yes—
34#2639—22no—
35#2592—22no—
36#2568C2212yes—
37#2563C2212yes—
38#2562—22no—
39#2548C2212yes—
40#2547C2212yes—
41Q8×C4⋊C4—22no—
42Q8×C4⋊C4—22no—
43Q8×C4⋊C4C2214yes—
44Q8×C4⋊C4C2214yes—
45#2311—21no—
46#2311—21no—
47C2×#389—22no—
48C2×#389C2212yes—
49C2×#384—21no—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2085C2218yes—
54#2067C2218yes—
55#2066C2218yes—
56#2042C2218yes—
57#1990C2212yes—
58#1990C2212yes—
59#1957C2212yes—
60#1956C2212yes—
61#1939C2214yes—
62#1939C2214yes—
63#1880C2218yes—
64#1878C2218yes—
65#1874C2214yes—
66#1872C2214yes—
67#1805C2218yes—
68#1803C2214yes—
69#1802C2214yes—
70#1797C2214yes—
71C22×#41—21no—
72C23×Q8C2311yescharacteristic
73C22×Q16C2314yes—
74C22×SD16C2314yes—
75#533—24no—
76#533—24no—
77#532—24no—
78#532—24no—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#127C2311yescharacteristic
83C2×#125—21no—
84C2×#124—21no—
85C2×#124C2314yes—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8C2312yes—
88C2×#121—21no—
89C2×#121C2312yes—
90C2×C4×Q8—21no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—48no—
96C2×C4×Q8D418yes—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2318yes—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×D4—21no—
102C2×C4×D4C2312yes—
103C2×#117—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C42—21no—
114C22×C42—22no—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#484—28no—
118#481—28no—
119#477—28no—
120#475—28no—
121#464—416no—
122#462—28no—
123#460—28no—
124#451—28no—
125#448—28no—
126C4×Q16—216no—
127C4×SD16—216no—
128#431—24no—
129#431—24no—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2318yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2318yes—
139C2×#102C2314yes—
140C2×#102C2318yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#397—24no—
147#394—24no—
148#393—24no—
149#391—22no—
150#391—22no—
151#391C2314yes—
152#390—22no—
153#390—22no—
154#390C2314yes—
155#389—24no—
156#389—28no—
157#389—28no—
158#389C2318yes—
159#388—24no—
160#388—24no—
161#386—24no—
162#386—24no—
163#384—24no—
164#384—24no—
165#384—21no—
166#384—21no—
167#384C2312yes—
168#384C2312yes—
169#382C2314yes—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—22no—
174C4×C4⋊C4—24no—
175C4×C4⋊C4—24no—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C2×#95—22no—
181C2×#95D412yes—
182#340C2318yes—
183#340C2318yes—
184#336C2318yes—
185C2×#41—44no—
186C2×#41—44no—
187C2×#41—44no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×Q8—22no—
194C22×Q8C2×D414yes—
195C22×Q8C2412yes—
196C22×Q8C2412yes—
197C22×D4—22no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4C2412yes—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
210C2×Q16—28no—
211C2×Q16—28no—
212C2×Q16—28no—
213C2×SD16—28no—
214C2×SD16—28no—
215C2×SD16—28no—
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224#127—22no—
225#127—22no—
226#125—44no—
227#124—48no—
228#124—44no—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—44no—
232#121—24no—
233#121—24no—
234#121—44no—
235C4×Q8—28no—
236C4×Q8—28no—
237C4×Q8—24no—
238C4×Q8—24no—
239C4×Q8—216no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—48no—
243C4×Q8—416no—
244C4×Q8—44no—
245C4×Q8C2×D418yes—
246C4×Q8C2×D418yes—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—44no—
250#117—44no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—84no—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C42—28no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—28no—
297C2×C42—44no—
298C2×C42—48no—
299C2×C42C2×D414yes—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D414yes—
303C2×C42#4114yes—
304C2×C42C2414yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#107—28no—
308#106—28no—
309C4⋊C8—24no—
310C4⋊C8—24no—
311C4⋊C8—48no—
312#103—28no—
313#103—28no—
314#103—28no—
315#103—48no—
316#102—28no—
317#102—48no—
318C22⋊C8—24no—
319C22⋊C8—24no—
320C4×C8—24no—
321C4×C8—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95C2×D414yes—
328#95C2×D414yes—
329C24—22no—
330C24C22×D411yescharacteristic
331#41—816no—
332C2×Q8—24no—
333C2×Q8—216no—
334C2×Q8—24no—
335C2×Q8—216no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×Q8C22×D414yes—
351C2×Q8H(5,𝔽2)14yes—
352C2×Q82−1+414yes—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—81no—
365C2×D4—81no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4#12114yes—
395C22×C4#12114yes—
396C22×C4#13514yes—
397C22×C4#13714yes—
398C22×C4C22×D414yes—
399C22×C4C22×D414yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4C2511yescharacteristic
408Q16—416no—
409SD16—416no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—28no—
429C4⋊C4—416no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—88no—
442C4⋊C4C22×D414yes—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C42—28no—
451C42—28no—
452C42—28no—
453C42—44no—
454C42—416no—
455C42—44no—
456C42#12114yes—
457C42#12114yes—
458C42C22×D412yes—
459C42C22×D412yes—
460C42C2×#4112yes—
461C42C2×#4112yes—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C2×#12112yes—
474C23C2×#13512yes—
475C23C2×#13712yes—
476C23C23×D411yescharacteristic
477Q8—216no—
478Q8—24no—
479Q8—24no—
480Q8—24no—
481Q8—44no—
482Q8—44no—
483Q8—84no—
484Q8—84no—
485Q8#53718yes—
486D4—44no—
487D4—44no—
488D4—84no—
489D4—88no—
490D4—84no—
491C2×C4—216no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—84no—
528C2×C4—88no—
529C2×C4—88no—
530C2×C4#464116yescomplemented
531C2×C4C2×#12112yes—
532C2×C4C2×#12112yes—
533C2×C4C2×#12112yes—
534C2×C4C2×#12112yes—
535C2×C4C2×#12112yes—
536C2×C4C2×#13512yes—
537C2×C4C2×#13512yes—
538C2×C4C2×#13712yes—
539C2×C4C2×#13712yes—
540C2×C4C23×D411yescharacteristic
541C2×C4C22×#4111yescharacteristic
542C2×C4C2×H(5,𝔽2)12yes—
543C2×C4C2×2−1+412yes—
544C8—44no—
545C8—44no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22C2×#46414yes—
560C22C22×#12111yescharacteristic
561C22#348411yescharacteristic
562C22#348611yescharacteristic
563C22C22×#13511yescharacteristic
564C22C22×#13711yescharacteristic
565C22#358511yescharacteristic
566C22#358611yescharacteristic
567C4—22no—
568C4—22no—
569C4—28no—
570C4—22no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4—82no—
575C4—84no—
576C4—82no—
577C4—82no—
578C4C2×#46414yes—
579C4C2×#46414yes—
580C4C2×#53712yes—
581C4#349212yes—
582C2—24no—
583C2—162no—
584C2#3522311yescharacteristic
585C2C22×#46411yescharacteristic
586C2#6042511yescharacteristic
587C2#6156611yescharacteristic
588C2#6158411yescharacteristic
589C2#6182211yescharacteristic
590C2#6183011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3104C2semidirect
4. C2×Q8×C4⋊C4C2semidirect
5. #32909C2semidirect
6. #29761C2semidirect
7. #24818C2semidirect
8. #24814C2semidirect
9. #24733C2semidirect
10. #24729C2semidirect
11. #21951C2semidirect
12. #21860C2semidirect
530. C2×C4#464semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C2×#31032
#351802
C2×Q8×C4⋊C42
#329092
#297612
#248182
#248142
#247332
#247292
#219512
#218602
#218592