Finite Group Databasebeta

← #6386109

Subgroups — #6386109

Order 512.

3230 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2#34178C212yesmaximal, complemented
3#22478C2116yesmaximal, complemented
4#21825C212yesmaximal, complemented
5#21747C214yesmaximal
6#20722C212yesmaximal
7#20688C214yesmaximal, complemented
8C2×#552—24no—
9C2×#552—22no—
10C2×#552—28no—
11C2×#552—22no—
12C2×#552C418yescomplemented
13C2×#552C2218yes—
14C22×#126C412yescomplemented
15C22×#125C411yescharacteristic, complemented
16C22×#125C412yescomplemented
17C22×#123C2212yes—
18C22×C22⋊Q8C411yescharacteristic, complemented
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C411yescharacteristic, complemented
21C22×C4×D4C2211yescharacteristic
22C22×#117C411yescharacteristic, complemented
23C22×#117C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25#2345—28no—
26#2344C22116yescomplemented
27#2316—22no—
28#2313—24no—
29#2280C2214yes—
30#2280C2218yes—
31#2274—22no—
32#1996C22116yescomplemented
33#1899C22132yes—
34#1820C22116yescomplemented
35#1819C2218yes—
36#1819C2214yes—
37#1799C2218yes—
38#1781C2214yes—
39C4×#99C2218yes—
40#1740C22116yescomplemented
41C23×Q8C2×C411yescharacteristic
42C23×D4C2311yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2312yes—
45C24×C4C2311yescharacteristic
46#552—216no—
47#552—216no—
48#552—232no—
49#552—216no—
50#552—232no—
51#552—216no—
52C2×#126—24no—
53C2×#126—24no—
54C2×#126—28no—
55C2×#126C2×C4116yes—
56C2×#125—24no—
57C2×#125—24no—
58C2×#125—28no—
59C2×#125—22no—
60C2×#125—24no—
61C2×#125—22no—
62C2×#125C2×C4116yes—
63C2×#125C2×C418yes—
64C2×#123—24no—
65C2×#123—28no—
66C2×#123—24no—
67C2×#123C2×C418yes—
68C2×#123C2318yes—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—24no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8C2×C418yes—
76C2×C22⋊Q8C2×C414yes—
77C2×C22⋊Q8C2314yes—
78C2×C4×Q8—22no—
79C2×C4×Q8—24no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2×C418yes—
83C2×C4×D4—22no—
84C2×C4×D4—24no—
85C2×C4×D4—22no—
86C2×C4×D4C2×C414yes—
87C2×C4×D4C2314yes—
88C2×#117—24no—
89C2×#117—28no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—22no—
93C2×#117—24no—
94C2×#117—21no—
95C2×#117—21no—
96C2×#117C2×C418yes—
97C2×#117C2×C418yes—
98C2×#117C23116yes—
99C2×#117C2318yes—
100C22×C4⋊C4C2×C411yescharacteristic
101C22×C4⋊C4C2×C412yes—
102C22×C4⋊C4C2×C412yes—
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C4⋊C4C2×C412yes—
105C22×C4⋊C4C2×C412yes—
106C22×C4⋊C4C2×C412yes—
107C22×C4⋊C4C2×C411yescharacteristic
108C22×C4⋊C4C2×C411yescharacteristic
109C22×C4⋊C4C2×C412yes—
110C22×C4⋊C4C2×C414yes—
111C22×C4⋊C4D414yes—
112C22×C4⋊C4D412yes—
113C22×C4⋊C4D418yes—
114C22×C4⋊C4D412yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2×C412yes—
117C22×C22⋊C4C2×C411yescharacteristic
118C22×C22⋊C4C2×C412yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C42C2×C412yes—
122C22×C42C2×C412yes—
123C22×C42C2×C411yescharacteristic
124C2×#99C23116yes—
125#389—216no—
126#387—232no—
127#381—24no—
128#381C23116yes—
129#380—28no—
130#380—28no—
131#380—24no—
132#380C2318yes—
133#379C2318yes—
134C4×C4⋊C4—216no—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4C2318yes—
139#342C23116yes—
140#342C23132yes—
141C25C22×C411yescharacteristic
142C25C2411yescharacteristic
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—44no—
147C22×Q8C22×C414yes—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—44no—
151C22×D4C22×C412yes—
152C22×D4C2412yes—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—44no—
157C23×C4—48no—
158C23×C4—44no—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C414yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C414yes—
170C23×C4C22×C412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D414yes—
173C23×C4C2×D418yes—
174C23×C4C2×D414yes—
175C23×C4C2×D414yes—
176C23×C4C2×D412yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179#126—232no—
180#126—232no—
181#125—232no—
182#125—232no—
183#125—216no—
184#125—216no—
185#123—216no—
186#123—216no—
187#123—216no—
188#123—216no—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C22⋊Q8—216no—
194C22⋊Q8—216no—
195C4×Q8—216no—
196C4×Q8—28no—
197C4×Q8—28no—
198C4×D4—28no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—28no—
202#117—216no—
203#117—216no—
204#117—216no—
205#117—216no—
206#117—216no—
207#117—28no—
208#117—28no—
209#117—416no—
210#117—432no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C412yes—
234C2×C4⋊C4C22⋊C414yes—
235C2×C4⋊C4C22⋊C412yes—
236C2×C4⋊C4C22⋊C414yes—
237C2×C4⋊C4C22⋊C414yes—
238C2×C4⋊C4C22⋊C418yes—
239C2×C4⋊C4C22⋊C418yes—
240C2×C4⋊C4C22⋊C418yes—
241C2×C4⋊C4C22⋊C4116yes—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C414yes—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C418yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D4116yes—
254C2×C4⋊C4C2×D418yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C414yes—
270C2×C22⋊C4C22×C418yes—
271C2×C22⋊C4C22×C412yes—
272C2×C22⋊C4C22×C414yes—
273C2×C22⋊C4C24116yes—
274C2×C22⋊C4C2412yes—
275C2×C22⋊C4C2414yes—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—28no—
286C2×C42—216no—
287C2×C42C22×C418yes—
288C2×C42C22×C418yes—
289C2×C42C22×C414yes—
290#99—416no—
291#95—216no—
292#95—216no—
293#95—28no—
294#95—28no—
295C24—24no—
296C24—21no—
297C24—21no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24C23×C411yescharacteristic
304C24C23×C412yes—
305C24C23×C411yescharacteristic
306C24C22×D412yes—
307C24C22×D412yes—
308C24C2511yescharacteristic
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—48no—
312C2×Q8—44no—
313C2×Q8—48no—
314C2×Q8—44no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—28no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—48no—
321C22×C4—216no—
322C22×C4—28no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—216no—
337C22×C4—44no—
338C22×C4—48no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—48no—
344C22×C4—44no—
345C22×C4—48no—
346C22×C4—416no—
347C22×C4—48no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C414yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C414yes—
364C22×C4C2×C22⋊C414yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C414yes—
367C22×C4C2×C22⋊C418yes—
368C22×C4C2×C22⋊C418yes—
369C22×C4C2×C22⋊C414yes—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C23×C412yes—
372C22×C4C23×C414yes—
373C22×C4C23×C412yes—
374C22×C4C23×C414yes—
375C22×C4C23×C412yes—
376C22×C4C23×C412yes—
377C22×C4C23×C414yes—
378C22×C4C23×C414yes—
379C22×C4C23×C412yes—
380C22×C4C22×D418yes—
381C22×C4C22×D418yes—
382C22×C4C22×D414yes—
383C22×C4C22×D414yes—
384C4⋊C4—216no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—216no—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—216no—
392C4⋊C4—216no—
393C4⋊C4—416no—
394C4⋊C4—816no—
395C4⋊C4C2×C22⋊C418yes—
396C4⋊C4C2×C22⋊C414yes—
397C4⋊C4C2×C22⋊C414yes—
398C4⋊C4C2×C22⋊C418yes—
399C4⋊C4C2×C22⋊C418yes—
400C4⋊C4C2×C22⋊C4116yes—
401C4⋊C4C2×C22⋊C4116yes—
402C22⋊C4—24no—
403C22⋊C4—28no—
404C22⋊C4—24no—
405C22⋊C4—28no—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—216no—
410C22⋊C4—232no—
411C22⋊C4—416no—
412C22⋊C4—416no—
413C42—28no—
414C42—28no—
415C42—28no—
416C42—28no—
417C42—24no—
418C42—24no—
419C42—416no—
420C42—816no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—22no—
430C23—24no—
431C23—41no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—41no—
437C23—416no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—88no—
447C23C22×C22⋊C412yes—
448C23C22×C22⋊C411yescharacteristic
449C23C22×C22⋊C411yescharacteristic
450C23C24×C411yescharacteristic
451C23C23×D412yes—
452C23#58514yes—
453Q8—48no—
454Q8—48no—
455D4—48no—
456D4—48no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—216no—
462C2×C4—216no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—416no—
501C2×C4—416no—
502C2×C4—416no—
503C2×C4#41014yes—
504C2×C4C22×C22⋊C414yes—
505C2×C4C22×C22⋊C414yes—
506C2×C4C22×C22⋊C412yes—
507C2×C4C22×C22⋊C418yes—
508C2×C4C22×C22⋊C414yes—
509C2×C4C22×C22⋊C412yes—
510C22—28no—
511C22—21no—
512C22—21no—
513C22—22no—
514C22—21no—
515C22—21no—
516C22—22no—
517C22—21no—
518C22—21no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—44no—
523C22—48no—
524C22—48no—
525C22—44no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22C2×#41012yes—
533C22#288412yes—
534C22C23×C22⋊C411yescharacteristic
535C22#342812yes—
536C22#348514yes—
537C4—24no—
538C4—28no—
539C4—24no—
540C4—22no—
541C4—24no—
542C4—42no—
543C4—42no—
544C4—44no—
545C4—42no—
546C4—44no—
547C4—816no—
548C4C2×#41014yes—
549C2—24no—
550C2—22no—
551C2—22no—
552C2—22no—
553C2—42no—
554C2—82no—
555C2#3417211yescharacteristic
556C2C22×#41011yescharacteristic
557C2C2×#288411yescharacteristic
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. #22478C2semidirect
4. #21825C2semidirect
7. #20688C2semidirect
12. C2×#552C4semidirect
14. C22×#126C4semidirect
15. C22×#125C4semidirect
16. C22×#125C4semidirect
18. C22×C22⋊Q8C4semidirect
20. C22×C4×Q8C4semidirect
22. C22×#117C4semidirect
26. #2344C22semidirect
32. #1996C22semidirect
34. #1820C22semidirect
40. #1740C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
#341782
#224782
#218252
#217472
#207222
#206882