Finite Group Databasebeta

← #6386223

Subgroups — #6386223

Order 512.

3360 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2#34178C212yesmaximal, complemented
3#22513C2116yesmaximal, complemented
4#21825C214yesmaximal, complemented
5#21825C212yesmaximal, complemented
6#20722C212yesmaximal
7#20722C214yesmaximal
8C2×#554—24no—
9C2×#554—22no—
10C2×#554—28no—
11C2×#554—22no—
12C2×#554C418yescomplemented
13C2×#554C2218yes—
14C22×C4⋊Q8C411yescharacteristic, complemented
15C22×#125C412yescomplemented
16C22×C22⋊Q8C412yescomplemented
17C22×C22⋊Q8C411yescharacteristic, complemented
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C411yescharacteristic, complemented
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2212yes—
23C23×C4⋊C4C411yescharacteristic, complemented
24C23×C4⋊C4C2211yescharacteristic
25#2345C22116yescomplemented
26#2344—28no—
27#2316—22no—
28#2313—24no—
29#2280C2214yes—
30#2280C2218yes—
31#2274—22no—
32#1996C22132yescomplemented
33#1996C22116yescomplemented
34#1819C2218yes—
35#1819C2214yes—
36#1799C22116yes—
37#1799C2218yes—
38#1781C2214yes—
39C4×#99C22116yes—
40C4×#99C2218yes—
41C23×Q8C2×C411yescharacteristic
42C23×Q8C2×C412yes—
43C23×D4C2311yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C412yes—
46C24×C4C2311yescharacteristic
47C24×C4C2312yes—
48#554—216no—
49#554—216no—
50#554—232no—
51#554—216no—
52#554—232no—
53#554—216no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C418yes—
58C2×#125—28no—
59C2×#125—24no—
60C2×#125—24no—
61C2×#125C2×C4116yes—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8C2×C4116yes—
75C2×C22⋊Q8C2×C418yes—
76C2×C22⋊Q8C2×C418yes—
77C2×C22⋊Q8C2×C414yes—
78C2×C22⋊Q8C2318yes—
79C2×C22⋊Q8C2314yes—
80C2×C4×Q8—24no—
81C2×C4×Q8—22no—
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8C2×C418yes—
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—22no—
89C2×C4×D4—24no—
90C2×C4×D4—22no—
91C2×C4×D4C2×C418yes—
92C2×C4×D4C2×C414yes—
93C2×C4×D4C2318yes—
94C2×C4×D4C2314yes—
95C2×#117—24no—
96C2×#117—28no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2×C412yes—
102C22×C4⋊C4C2×C411yescharacteristic
103C22×C4⋊C4C2×C418yes—
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C4⋊C4C2×C412yes—
106C22×C4⋊C4C2×C412yes—
107C22×C4⋊C4C2×C412yes—
108C22×C4⋊C4C2×C412yes—
109C22×C4⋊C4C2×C411yescharacteristic
110C22×C4⋊C4C2×C412yes—
111C22×C4⋊C4C2×C411yescharacteristic
112C22×C4⋊C4C2×C414yes—
113C22×C4⋊C4D414yes—
114C22×C4⋊C4D412yes—
115C22×C4⋊C4D418yes—
116C22×C4⋊C4D412yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2×C412yes—
119C22×C22⋊C4C2×C411yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2×C412yes—
123C22×C42C2×C411yescharacteristic
124C2×#99C23116yes—
125#389—216no—
126#387—232no—
127#381—24no—
128#381—28no—
129#380—28no—
130#380—24no—
131#380C23116yes—
132#380C2318yes—
133#379C2318yes—
134C4×C4⋊C4—216no—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4C2318yes—
138C4×C22⋊C4C23116yes—
139#342C23132yes—
140#342C23116yes—
141C25C22×C411yescharacteristic
142C25C2411yescharacteristic
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—24no—
147C22×Q8—24no—
148C22×Q8—44no—
149C22×Q8—84no—
150C22×Q8C22×C414yes—
151C22×Q8C22×C418yes—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—28no—
155C22×D4C22×C412yes—
156C22×D4C2412yes—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—84no—
166C23×C4—84no—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C22×C414yes—
174C23×C4C22×C418yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C22×C414yes—
180C23×C4C2×D412yes—
181C23×C4C2×D418yes—
182C23×C4C2×D414yes—
183C23×C4C2×D412yes—
184C23×C4C2×D414yes—
185C23×C4C2×D414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C4⋊Q8—216no—
189C4⋊Q8—216no—
190#125—232no—
191#125—232no—
192C22⋊Q8—216no—
193C22⋊Q8—216no—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C22⋊Q8—216no—
197C22⋊Q8—216no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—232no—
201C22⋊Q8—232no—
202C22⋊Q8—216no—
203C22⋊Q8—216no—
204C4×Q8—216no—
205C4×Q8—28no—
206C4×Q8—28no—
207C4×D4—216no—
208C4×D4—216no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—216no—
212C4×D4—216no—
213C4×D4—28no—
214C4×D4—28no—
215#117—416no—
216#117—816no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4C22⋊C414yes—
240C2×C4⋊C4C22⋊C412yes—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C418yes—
244C2×C4⋊C4C22⋊C414yes—
245C2×C4⋊C4C22⋊C414yes—
246C2×C4⋊C4C22⋊C414yes—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C22⋊C418yes—
250C2×C4⋊C4C22⋊C418yes—
251C2×C4⋊C4C22⋊C4116yes—
252C2×C4⋊C4C22×C418yes—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C418yes—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D4116yes—
264C2×C4⋊C4C2×D418yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C22×C414yes—
277C2×C22⋊C4C22×C414yes—
278C2×C22⋊C4C22×C412yes—
279C2×C22⋊C4C24116yes—
280C2×C22⋊C4C2414yes—
281C2×C22⋊C4C2412yes—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—28no—
289C2×C42—216no—
290C2×C42C22×C418yes—
291C2×C42C22×C414yes—
292#99—232no—
293#95—216no—
294#95—28no—
295#95—28no—
296#95—216no—
297C24—21no—
298C24—21no—
299C24—24no—
300C24—24no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24C23×C411yescharacteristic
306C24C23×C411yescharacteristic
307C24C23×C412yes—
308C24C22×D412yes—
309C24C22×D412yes—
310C24C2511yescharacteristic
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—216no—
314C2×Q8—48no—
315C2×Q8—44no—
316C2×Q8—48no—
317C2×Q8—44no—
318C2×Q8—88no—
319C2×Q8—88no—
320C2×Q8—88no—
321C2×D4—216no—
322C2×D4—216no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D42−1+418yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—216no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4—44no—
350C22×C4—416no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—88no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—88no—
371C22×C4C2×C22⋊C414yes—
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C414yes—
376C22×C4C2×C22⋊C414yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C414yes—
381C22×C4C2×C22⋊C414yes—
382C22×C4C2×C22⋊C418yes—
383C22×C4C2×C22⋊C418yes—
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C23×C412yes—
386C22×C4C23×C412yes—
387C22×C4C23×C414yes—
388C22×C4C23×C412yes—
389C22×C4C23×C412yes—
390C22×C4C23×C414yes—
391C22×C4C23×C414yes—
392C22×C4C23×C412yes—
393C22×C4C23×C414yes—
394C22×C4C22×D414yes—
395C22×C4C22×D414yes—
396C22×C4C22×D418yes—
397C22×C4C22×D418yes—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—216no—
404C4⋊C4—28no—
405C4⋊C4—28no—
406C4⋊C4—216no—
407C4⋊C4—216no—
408C4⋊C4—416no—
409C4⋊C4—816no—
410C4⋊C4C2×C22⋊C418yes—
411C4⋊C4C2×C22⋊C414yes—
412C4⋊C4C2×C22⋊C414yes—
413C4⋊C4C2×C22⋊C4116yes—
414C4⋊C4C2×C22⋊C4116yes—
415C4⋊C4C2×C22⋊C418yes—
416C4⋊C4C2×C22⋊C418yes—
417C22⋊C4—28no—
418C22⋊C4—24no—
419C22⋊C4—28no—
420C22⋊C4—24no—
421C22⋊C4—216no—
422C22⋊C4—28no—
423C22⋊C4—232no—
424C22⋊C4—416no—
425C22⋊C4—416no—
426C42—28no—
427C42—28no—
428C42—24no—
429C42—24no—
430C42—416no—
431C42—816no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—41no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—416no—
455C23—416no—
456C23—42no—
457C23—42no—
458C23C22×C22⋊C412yes—
459C23C22×C22⋊C411yescharacteristic
460C23C22×C22⋊C411yescharacteristic
461C23#51914yes—
462C23C24×C411yescharacteristic
463C23C23×D412yes—
464C23C2×2−1+414yes—
465Q8—48no—
466Q8—48no—
467Q8—816no—
468D4—216no—
469D4—216no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—416no—
514C2×C4—416no—
515C2×C4—416no—
516C2×C4—88no—
517C2×C4—88no—
518C2×C4#41014yes—
519C2×C4C22×C22⋊C414yes—
520C2×C4C22×C22⋊C414yes—
521C2×C4C22×C22⋊C412yes—
522C2×C4C22×C22⋊C414yes—
523C2×C4C22×C22⋊C418yes—
524C2×C4C22×C22⋊C412yes—
525C2×C4#51914yes—
526C22—28no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—24no—
537C22—22no—
538C22—22no—
539C22—44no—
540C22—48no—
541C22—48no—
542C22—44no—
543C22—48no—
544C22—48no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—42no—
549C22C2×#41012yes—
550C22#288312yes—
551C22C23×C22⋊C411yescharacteristic
552C22C2×#51912yes—
553C22#348614yes—
554C4—24no—
555C4—22no—
556C4—24no—
557C4—24no—
558C4—24no—
559C4—28no—
560C4—42no—
561C4—44no—
562C4—44no—
563C4—42no—
564C4—816no—
565C4C2×#41014yes—
566C2—24no—
567C2—22no—
568C2—22no—
569C2—22no—
570C2—42no—
571C2—44no—
572C2#3417311yescharacteristic
573C2C22×#41011yescharacteristic
574C2C2×#288311yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34178C2semidirect
3. #22513C2semidirect
4. #21825C2semidirect
5. #21825C2semidirect
12. C2×#554C4semidirect
14. C22×C4⋊Q8C4semidirect
15. C22×#125C4semidirect
16. C22×C22⋊Q8C4semidirect
17. C22×C22⋊Q8C4semidirect
20. C22×C4×Q8C4semidirect
23. C23×C4⋊C4C4semidirect
25. #2345C22semidirect
32. #1996C22semidirect
33. #1996C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
#341782
#225132
#218252
#207222