Finite Group Databasebeta

← #6502276

Subgroups — #6502276

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2446C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#29241C212yesmaximal
4#27586C211yesmaximal, characteristic
5#27586C211yesmaximal, characteristic
6#27196C214yesmaximal, complemented
7#26694C214yesmaximal, complemented
8#26590C214yesmaximal, complemented
9#26171C214yesmaximal, complemented
10#24212C212yesmaximal
11#23934C211yesmaximal, characteristic
12#23934C211yesmaximal, characteristic
13#22044C212yesmaximal
14#21912C211yesmaximal, characteristic
15C4×#451C212yesmaximal
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#462C2214yes—
20C2×#451C2212yes—
21C2×#451C2212yes—
22C2×C4×Q16C2214yes—
23#2446—22no—
24#2446—22no—
25#2446C2214yes—
26#2446C2214yes—
27C2×#400—21no—
28C2×#398—21no—
29#2370—22no—
30#2369C2212yes—
31#2365C2212yes—
32#2355—22no—
33#2355C2212yes—
34Q8×C4⋊C4C2212yes—
35#2329—21no—
36#2329—21no—
37#2317—21no—
38C4×C22⋊Q8C2212yes—
39C4×C22⋊Q8C2212yes—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#387—21no—
46#2285C2211yescharacteristic
47#2274—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2095C2212yes—
50#2095C2212yes—
51#2087C2214yes—
52#2072C2212yes—
53#2072C2212yes—
54#2068C2214yes—
55#2043C2214yes—
56#2024C2212yes—
57#2024C2212yes—
58#2016C2214yes—
59#2016C2212yes—
60#2016C2212yes—
61#1981C2214yes—
62#1969C2214yes—
63#1933C2214yes—
64#1928C2214yes—
65#1928C2214yes—
66#1928C2214yes—
67#1921C2214yes—
68#1916C2214yes—
69#1916C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1878C2214yes—
74#1872C2212yes—
75#1872C2212yes—
76#1863C2214yes—
77#1863C2214yes—
78#1852C2212yes—
79#1852C2212yes—
80#1817C2211yescharacteristic
81#1817C2211yescharacteristic
82#1797C2212yes—
83#1787C2212yes—
84#1787C2212yes—
85C4×#103C2214yes—
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90C22×Q16C2314yes—
91#533—28no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2314yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—22no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×#117—21no—
108C2×#117—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#462—48no—
122#451—28no—
123#451—28no—
124C4×Q16—48no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C4×C8C2314yes—
139#400—24no—
140#400—24no—
141#398—24no—
142#398—24no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—21no—
147#391—21no—
148#391—22no—
149#391—22no—
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#391C2314yes—
155#391C2312yes—
156#391C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2314yes—
163#389C2312yes—
164#389C2312yes—
165#387—24no—
166#387—24no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#387—24no—
175#385—22no—
176#385—22no—
177#384—22no—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#382—22no—
181#382—21no—
182#380—21no—
183#380C2312yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—24no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C22×Q8C2412yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×Q16—28no—
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—44no—
246C2×Q16—44no—
247C2×Q16—44no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—28no—
253C4⋊Q8—44no—
254C4⋊Q8—48no—
255#125—28no—
256C22⋊Q8—28no—
257C22⋊Q8—24no—
258C22⋊Q8#4118yes—
259C4×Q8—28no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—28no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8—48no—
267#117—28no—
268#117—44no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4Q1618yescomplemented
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2412yes—
325C2×C42C2411yescharacteristic
326#107—44no—
327#107—44no—
328C4⋊C8—44no—
329#103—28no—
330#103—28no—
331#103—28no—
332#103—44no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337C22⋊C8—24no—
338C22⋊C8—24no—
339C4×C8—44no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365C24C22×D411yescharacteristic
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—416no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—81no—
385C2×Q8—81no—
386C2×Q8—88no—
387C2×Q8C2×#4114yes—
388C2×Q8H(5,𝔽2)14yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4C2×Q1614yes—
427C22×C4C2×Q1614yes—
428C22×C4C2×Q1614yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4114yes—
439C22×C4C2511yescharacteristic
440Q16—416no—
441Q16—88no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—28no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4C2×Q1618yes—
475C4⋊C4C2×Q1618yes—
476C4⋊C4C2×#4114yes—
477C22⋊C4—28no—
478C22⋊C4—44no—
479C42—28no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—42no—
487C42—42no—
488C42—48no—
489C42—44no—
490C42—82no—
491C42—82no—
492C42—82no—
493C42—82no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23#52512yes—
499C23C22×Q1612yes—
500C23#57512yes—
501C23C23×D411yescharacteristic
502Q8—28no—
503Q8—28no—
504Q8—48no—
505Q8—44no—
506Q8—44no—
507Q8—44no—
508Q8—44no—
509Q8—84no—
510Q8—84no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4#52512yes—
569C2×C4C22×Q1612yes—
570C2×C4C22×Q1612yes—
571C2×C4C22×Q1614yes—
572C2×C4C22×Q1614yes—
573C2×C4C22×#4112yes—
574C2×C4C2×H(5,𝔽2)12yes—
575C2×C4#58512yes—
576C8—44no—
577C8—44no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22#310414yes—
585C22C2×#52511yescharacteristic
586C22#348411yescharacteristic
587C22#348711yescharacteristic
588C22C23×Q1611yescharacteristic
589C22C2×#57511yescharacteristic
590C22#358611yescharacteristic
591C22#358611yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—44no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4#351314yes—
610C2—24no—
611C2#3535611yescharacteristic
612C2C2×#310411yescharacteristic
613C2#6056911yescharacteristic
614C2#6156311yescharacteristic
615C2#6157811yescharacteristic
616C2#6188411yescharacteristic
617C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
6. #27196C2semidirect
7. #26694C2semidirect
8. #26590C2semidirect
9. #26171C2semidirect
295. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#24462
#342132
#292412
#275862
#271962
#266942
#265902
#261712
#242122
#239342
#220442
#219122
C4×#4512