Finite Group Databasebeta

← #6379361

Subgroups — #6379361

Order 512.

3196 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×#3503C211yesmaximal, characteristic
2#34216C212yesmaximal, complemented
3#22513C218yesmaximal, complemented
4Q8×#99C218yesmaximal, complemented
5#21334C214yesmaximal, complemented
6#21332C212yesmaximal, complemented
7#20634C212yesmaximal
8#20632C211yesmaximal, characteristic
9#20393C212yesmaximal
10C4×#410C211yesmaximal, characteristic
11#3503—24no—
12#3503—24no—
13#3503—24no—
14#3503—24no—
15C2×Q82C412yescomplemented
16C2×#557C412yescomplemented
17C2×#554C412yescomplemented
18C2×#554C2214yes—
19C2×D4×Q8C412yescomplemented
20C2×D4×Q8C2214yes—
21C2×#544C414yescomplemented
22C2×#544C2212yes—
23C2×#533C412yescomplemented
24C2×#533C2211yescharacteristic
25C22×C4⋊Q8C412yescomplemented
26C22×C4⋊Q8C2211yescharacteristic
27C2×#520C2212yes—
28C2×C4×#41C2211yescharacteristic
29C2×#410C2212yes—
30C2×#410C2211yescharacteristic
31C2×#410C2211yescharacteristic
32#2369—24no—
33#2365—24no—
34#2345C2218yes—
35Q8×C22⋊C4C2218yes—
36#2331—24no—
37#2316C2214yes—
38C4×C4⋊Q8—24no—
39#2285C2212yes—
40#2274C2214yes—
41C4×#117C2212yes—
42#1996C2218yes—
43#1996C2218yes—
44#1996C22132yes—
45#1799C2214yes—
46#1799C2214yes—
47#1799C22116yes—
48C4×#99C2214yes—
49C4×#99C2214yes—
50C4×#99C22116yes—
51#1740C2214yes—
52#1740C2214yes—
53#1740C2212yes—
54#1740C2212yes—
55C22×#41C2311yescharacteristic
56C23×Q8C2×C414yes—
57C23×Q8C2312yes—
58Q82—28no—
59Q82—28no—
60#557—28no—
61#557—28no—
62#554—28no—
63#554—28no—
64#554—216no—
65#554—216no—
66D4×Q8—216no—
67D4×Q8—216no—
68D4×Q8—28no—
69D4×Q8—28no—
70#544—216no—
71#544—216no—
72#544—28no—
73#544—28no—
74#533—24no—
75#533—24no—
76#533—24no—
77#533—24no—
78#533—22no—
79#533—22no—
80#533—22no—
81#533—22no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8C2×C411yescharacteristic
87C2×C4⋊Q8C2×C414yes—
88C2×C4⋊Q8C2×C414yes—
89C2×C4⋊Q8C2×C414yes—
90C2×C4⋊Q8C2×C411yescharacteristic
91C2×C4⋊Q8C2×C418yes—
92C2×C4⋊Q8C2×C414yes—
93C2×C4⋊Q8C2×C412yes—
94C2×C4⋊Q8C2×C412yes—
95C2×C4⋊Q8C2×C412yes—
96C2×C4⋊Q8C2×C412yes—
97C2×C4⋊Q8D414yes—
98C2×C4⋊Q8D414yes—
99C2×C4⋊Q8D414yes—
100C2×C4⋊Q8D414yes—
101C2×#125C2×C414yes—
102C2×#125C2×C412yes—
103C2×#125C2×C418yes—
104C2×#125C2×C414yes—
105C2×C22⋊Q8C2×C414yes—
106C2×C22⋊Q8C2×C414yes—
107C2×C22⋊Q8C2×C418yes—
108C2×C22⋊Q8C2×C418yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2318yes—
112#520—28no—
113#520—28no—
114C4×#41—22no—
115C4×#41—22no—
116C4×#41—22no—
117C4×#41—22no—
118C2×C4×Q8—28no—
119C2×C4×Q8C2×C411yescharacteristic
120C2×C4×Q8C2×C411yescharacteristic
121C2×C4×Q8C2×C414yes—
122C2×C4×Q8C2×C414yes—
123C2×C4×Q8C2×C414yes—
124C2×C4×Q8C2×C412yes—
125C2×C4×Q8C2×C412yes—
126C2×C4×D4C2×C411yescharacteristic
127C2×C4×D4C2×C411yescharacteristic
128C2×C4×D4C2×C414yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2318yes—
132C2×#117C2×C412yes—
133C2×#117C2×C414yes—
134C2×#117C2312yes—
135C2×#117C2314yes—
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C2×#117C2312yes—
139C22×C4⋊C4C2×C414yes—
140C22×C4⋊C4C2×C414yes—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2312yes—
143C22×C42C2×C412yes—
144C22×C42C2311yescharacteristic
145#410—28no—
146#410—24no—
147#410Q818yescomplemented
148C2×#99C2318yes—
149C2×#99C2314yes—
150C2×#99C2314yes—
151#391—28no—
152#391—216no—
153#389—28no—
154#389—28no—
155#389—28no—
156#387—216no—
157#384—22no—
158#382—22no—
159#380C23116yes—
160#380C2314yes—
161#380C2314yes—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C22⋊C4C2314yes—
170C4×C22⋊C4C23116yes—
171C4×C22⋊C4C2314yes—
172#376—22no—
173C43—22no—
174#342C2318yes—
175#342C23116yes—
176#342C23116yes—
177#342C2318yes—
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—24no—
181C2×#41C2×Q814yes—
182C2×#41#4114yes—
183C22×Q8—28no—
184C22×Q8—28no—
185C22×Q8—84no—
186C22×Q8—88no—
187C22×Q8C22×C411yescharacteristic
188C22×Q8C22×C412yes—
189C22×Q8C22×C412yes—
190C22×Q8C22×C418yes—
191C22×Q8C22×C414yes—
192C22×Q8C22×C414yes—
193C22×Q8C2×D418yes—
194C22×Q8C2×D418yes—
195C22×D4C22×C411yescharacteristic
196C22×D4C2412yes—
197C23×C4C22×C412yes—
198C23×C4C22×C412yes—
199C23×C4C22×C412yes—
200C23×C4C22×C418yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C4⋊Q8—28no—
206C4⋊Q8—28no—
207C4⋊Q8—232no—
208C4⋊Q8—28no—
209C4⋊Q8—28no—
210C4⋊Q8—28no—
211C4⋊Q8—28no—
212C4⋊Q8—216no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8C22⋊C414yes—
222C4⋊Q8C22⋊C414yes—
223C4⋊Q8C22⋊C414yes—
224C4⋊Q8C22⋊C414yes—
225C4⋊Q8C22⋊C418yes—
226C4⋊Q8C22⋊C418yes—
227C4⋊Q8C22⋊C418yes—
228C4⋊Q8C22⋊C418yes—
229C4⋊Q8C22⋊C418yes—
230C4⋊Q8C22⋊C418yes—
231#125—28no—
232#125—28no—
233#125—232no—
234#125—216no—
235#125—24no—
236#125—24no—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—232no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—24no—
243C22⋊Q8—24no—
244C22⋊Q8—232no—
245C22⋊Q8—28no—
246C22⋊Q8—28no—
247C22⋊Q8—232no—
248C4×Q8—28no—
249C4×Q8—28no—
250C4×Q8—216no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—816no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—232no—
267C4×D4—216no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272#117—22no—
273#117—22no—
274#117—24no—
275#117—24no—
276#117—24no—
277#117—28no—
278#117—28no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—28no—
283#117—24no—
284#117—28no—
285#117C2×Q818yescomplemented
286C2×C4⋊C4—22no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—216no—
296C2×C4⋊C4C22×C412yes—
297C2×C4⋊C4C22×C412yes—
298C2×C4⋊C4C22×C418yes—
299C2×C4⋊C4C22×C418yes—
300C2×C4⋊C4C22×C412yes—
301C2×C4⋊C4C22×C412yes—
302C2×C4⋊C4C22×C412yes—
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C4⋊C4C22×C412yes—
306C2×C4⋊C4C22×C418yes—
307C2×C4⋊C4C22×C414yes—
308C2×C4⋊C4C22×C418yes—
309C2×C4⋊C4C22×C414yes—
310C2×C4⋊C4C22×C414yes—
311C2×C4⋊C4C22×C414yes—
312C2×C4⋊C4C22×C414yes—
313C2×C4⋊C4C22×C414yes—
314C2×C4⋊C4C2×D418yes—
315C2×C4⋊C4C2×D418yes—
316C2×C4⋊C4C2×D418yes—
317C2×C4⋊C4C2×D418yes—
318C2×C22⋊C4C22×C418yes—
319C2×C22⋊C4C22×C412yes—
320C2×C22⋊C4C22×C412yes—
321C2×C22⋊C4C2414yes—
322C2×C22⋊C4C2418yes—
323C2×C22⋊C4C2414yes—
324C2×C22⋊C4C2412yes—
325C2×C22⋊C4C2414yes—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—216no—
336C2×C42C22×C412yes—
337C2×C42C22×C418yes—
338C2×C42C22×C414yes—
339C2×C42C22×C412yes—
340C2×C42C22×C412yes—
341C2×C42C22×C411yescharacteristic
342C2×C42C22×C411yescharacteristic
343C2×C42C22×C411yescharacteristic
344C2×C42C22×C411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349#99—216no—
350#99—28no—
351#99C2×Q8116yescomplemented
352#95—24no—
353#95—24no—
354#95—216no—
355#95—24no—
356#95—24no—
357#95—216no—
358C24C23×C412yes—
359C24C2511yescharacteristic
360#41—28no—
361#41—28no—
362#41—28no—
363#41—28no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—216no—
370C2×Q8—22no—
371C2×Q8—416no—
372C2×Q8—48no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—84no—
376C2×Q8—88no—
377C2×Q8—88no—
378C2×Q8—816no—
379C2×Q8—816no—
380C2×Q8C2×C22⋊C418yes—
381C2×Q8C2×C22⋊C418yes—
382C2×Q8C2×C22⋊C4116yes—
383C2×Q8C4×Q814yescomplemented
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4C4×Q814yescomplemented
389C2×D4C22×Q814yes—
390C2×D4C2×#4114yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—216no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—84no—
407C22×C4—88no—
408C22×C4C4×Q818yescomplemented
409C22×C4C23×C411yescharacteristic
410C22×C4C23×C411yescharacteristic
411C22×C4C23×C412yes—
412C22×C4C23×C414yes—
413C22×C4C23×C412yes—
414C22×C4C23×C412yes—
415C22×C4C23×C414yes—
416C22×C4C23×C411yescharacteristic
417C22×C4C23×C411yescharacteristic
418C22×C4C23×C418yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D4116yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×Q812yes—
429C22×C4C2×#4112yes—
430C22×C42−1+414yes—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—216no—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—816no—
455C4⋊C4C2×C22⋊C418yes—
456C4⋊C4C2×C22⋊C418yes—
457C4⋊C4C2×C22⋊C418yes—
458C4⋊C4C2×C22⋊C418yes—
459C4⋊C4C2×C22⋊C4116yes—
460C4⋊C4C2×C22⋊C4116yes—
461C22⋊C4—28no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—216no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—28no—
469C22⋊C4—24no—
470C22⋊C4—28no—
471C22⋊C4C22×Q818yes—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—216no—
479C42—28no—
480C42—28no—
481C42—48no—
482C42—48no—
483C42—48no—
484C42—44no—
485C42—816no—
486C42C2×C22⋊C411yescharacteristic
487C42C2×C22⋊C411yescharacteristic
488C42C2×C22⋊C411yescharacteristic
489C42C2×C22⋊C411yescharacteristic
490C42C2×C22⋊C412yes—
491C42C2×C22⋊C412yes—
492C42C2×C22⋊C412yes—
493C42C2×C22⋊C412yes—
494C42C2×C22⋊C412yes—
495C42C2×C22⋊C412yes—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23C2×C4×Q814yes—
503C23C24×C411yescharacteristic
504C23C23×D412yes—
505C23C23×Q811yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×2−1+412yes—
508Q8—28no—
509Q8—216no—
510Q8—48no—
511Q8—416no—
512Q8—44no—
513Q8—44no—
514Q8—44no—
515Q8—44no—
516Q8—816no—
517D4—28no—
518D4—28no—
519D4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—416no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—816no—
556C2×C4C22×C22⋊C411yescharacteristic
557C2×C4C22×C22⋊C411yescharacteristic
558C2×C4C22×C22⋊C4116yes—
559C2×C4C22×C22⋊C412yes—
560C2×C4C22×C22⋊C412yes—
561C2×C4C22×C22⋊C412yes—
562C2×C4C22×C22⋊C411yescharacteristic
563C2×C4C22×C22⋊C411yescharacteristic
564C2×C4C22×C22⋊C411yescharacteristic
565C2×C4C22×C22⋊C411yescharacteristic
566C2×C4C2×C4×Q814yes—
567C2×C4C2×C4×Q812yes—
568C2×C4C2×C4×Q812yes—
569C2×C4#51912yes—
570C2×C4#51912yes—
571C2×C4#54814yes—
572C2×C4#54814yes—
573C2×C4D4×Q814yes—
574C2×C4D4×Q814yes—
575C22—22no—
576C22—21no—
577C22—22no—
578C22—21no—
579C22—24no—
580C22—41no—
581C22—41no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22C23×C22⋊C411yescharacteristic
588C22C22×C4×Q811yescharacteristic
589C22C2×#51911yescharacteristic
590C22C2×#54812yes—
591C22C2×D4×Q812yes—
592C4—24no—
593C4—21no—
594C4—22no—
595C4—216no—
596C4—21no—
597C4—22no—
598C4—48no—
599C4—42no—
600C4—44no—
601C4—42no—
602C4—44no—
603C4—84no—
604C4—88no—
605C4Q8×C22⋊C412yes—
606C4Q8×C22⋊C412yes—
607C4Q8×C22⋊C414yes—
608C4#288312yes—
609C4#288412yes—
610C2—22no—
611C2—22no—
612C2—21no—
613C2—21no—
614C2—44no—
615C2C2×Q8×C22⋊C411yescharacteristic
616C2C2×#288311yescharacteristic
617C2C2×#288411yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC22×#41C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #22513C2semidirect
4. Q8×#99C2semidirect
5. #21334C2semidirect
6. #21332C2semidirect
15. C2×Q82C4semidirect
16. C2×#557C4semidirect
17. C2×#554C4semidirect
19. C2×D4×Q8C4semidirect
21. C2×#544C4semidirect
23. C2×#533C4semidirect
25. C22×C4⋊Q8C4semidirect
147. #410Q8semidirect
285. #117C2×Q8semidirect
351. #99C2×Q8semidirect
383. C2×Q8C4×Q8semidirect
388. C2×D4C4×Q8semidirect
408. C22×C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
#342162
#225132
Q8×#992
#213342
#213322
#206342
#206322
#203932
C4×#4102