Finite Group Databasebeta

← #6343656

Subgroups — #6343656

Order 512.

3093 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#2973C211yesmaximal, characteristic
3#34185C211yesmaximal, characteristic
4#17621C2116yesmaximal, complemented
5#17596C214yesmaximal, complemented
6#17578C214yesmaximal, complemented
7C2×#1520C214yesmaximal, complemented
8#3501—24no—
9#3501—24no—
10#3501C418yescomplemented
11#3501C2218yes—
12C2×#554—24no—
13C2×#554C414yescomplemented
14C2×D4×Q8C414yescomplemented
15C2×#544—21no—
16C2×#543—21no—
17C2×#543—21no—
18C2×#533C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C2×#518—21no—
22C22×C4×D4C2211yescharacteristic
23#2973—24no—
24#2973C2218yes—
25C2×#435—24no—
26C2×#432C2212yes—
27C2×#407C2212yes—
28C22×C22⋊C8C2212yes—
29#2353C2218yes—
30#2352—24no—
31#2313—21no—
32#2312—21no—
33#2308—21no—
34#2283C2212yes—
35#2280C2212yes—
36C2×#380C2212yes—
37#2273—21no—
38#1520C22116yes—
39#1520C22116yescomplemented
40#1515C22116yescomplemented
41#1511C22116yescomplemented
42#1508C22116yescomplemented
43#1506C22116yescomplemented
44C2×#328C22116yescomplemented
45C23×Q8C2×C412yes—
46C23×Q8C2×C412yescomplemented
47C23×D4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C23×C8—22no—
52#554—232no—
53#554—216no—
54#554—232no—
55#554C2×C4132yescomplemented
56D4×Q8—216no—
57D4×Q8C2×C4132yescomplemented
58#544—28no—
59#544—28no—
60#543—28no—
61#543—28no—
62#543—28no—
63#543—28no—
64#533—24no—
65#533—24no—
66#533C2×C418yescomplemented
67#533C2318yes—
68C2×C4⋊Q8—24no—
69C2×C4⋊Q8C2×C412yescomplemented
70C2×C4⋊Q8C2×C412yescomplemented
71C2×#125—24no—
72C2×#125—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8C2×C414yes—
87C2×C22⋊Q8C2×C418yes—
88C2×C22⋊Q8C2×C414yescomplemented
89C2×C22⋊Q8C2×C418yescomplemented
90C2×C22⋊Q8C2314yes—
91C2×C22⋊Q8C2318yes—
92C2×C22⋊Q8C2318yes—
93#518—28no—
94#518—28no—
95C2×C4×Q8—22no—
96C2×C4×Q8C2×C414yescomplemented
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4C2×C414yes—
100C2×C4×D4C2×C414yes—
101C2×C4×D4D414yes—
102C2×C4×D4D414yes—
103C2×C4×D4D414yes—
104C2×C4×D4C2314yes—
105C2×#117—22no—
106C2×#117—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2×C412yes—
111C22×C4⋊C4C2×C412yescomplemented
112C22×C4⋊C4D411yescharacteristic
113C22×C22⋊C4—22no—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4D411yescharacteristic
117C22×C22⋊C4D411yescharacteristic
118C22×C42C2311yescharacteristic
119#435—232no—
120#432—24no—
121#432C2318yes—
122C2×C4⋊C8—22no—
123#407—24no—
124#407C2318yes—
125C2×C22⋊C8—22no—
126C2×C22⋊C8—24no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2318yes—
130C2×#97—22no—
131#388—28no—
132#387—28no—
133#386—28no—
134#383C2314yes—
135#382—22no—
136#381—22no—
137#381C2314yes—
138#380—22no—
139#380—22no—
140#380—22no—
141#380C2314yes—
142#380C2314yes—
143#379—22no—
144C4×C4⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4C2314yes—
149#376C2314yes—
150C2×#95C2314yes—
151#328C23164yes—
152C25C2×D411yescharacteristic
153C25C2×D411yescharacteristic
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—84no—
160C22×Q8—84no—
161C22×Q8C22×C414yes—
162C22×Q8C22×C418yes—
163C22×Q8C22×C418yes—
164C22×D4—28no—
165C22×D4C22⋊C414yes—
166C22×D4C22⋊C414yes—
167C22×D4C2×D414yes—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4C22×C412yes—
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2412yes—
192C2×M16—28no—
193C2×M16—84no—
194C22×C8—24no—
195C22×C8—28no—
196C22×C8—84no—
197C4⋊Q8—232no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8C22×C418yes—
201C4⋊Q8C22×C418yes—
202#125—232no—
203#125—28no—
204#125—28no—
205C22⋊Q8—28no—
206C22⋊Q8—28no—
207C22⋊Q8—28no—
208C22⋊Q8—28no—
209C22⋊Q8—28no—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—28no—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8—232no—
220C22⋊Q8—232no—
221C22⋊Q8—232no—
222C22⋊Q8—232no—
223C22⋊Q8C22×C418yes—
224C22⋊Q8C22×C4116yes—
225C22⋊Q8C22×C4116yes—
226C22⋊Q8C2418yes—
227C4×Q8—28no—
228C4×Q8—28no—
229C4×Q8—28no—
230C4×Q8C22×C4116yes—
231C4×D4—232no—
232C4×D4—232no—
233C4×D4C22⋊C4116yes—
234C4×D4C22⋊C4116yes—
235C4×D4C22×C4116yes—
236C4×D4C2×D4116yes—
237#117—28no—
238#117—28no—
239#117—84no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—84no—
258C2×C4⋊C4C22⋊C414yes—
259C2×C4⋊C4C22⋊C414yes—
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C22×C418yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4C22⋊C414yes—
277C2×C22⋊C4C22⋊C414yes—
278C2×C22⋊C4C22⋊C414yes—
279C2×C22⋊C4C22⋊C414yes—
280C2×C22⋊C4C2×D414yes—
281C2×C22⋊C4C2×D414yes—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42C22×C412yes—
286C2×C42C22×C412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2412yes—
291C4⋊C8—28no—
292C22⋊C8—24no—
293C22⋊C8—28no—
294C22⋊C8—28no—
295C22⋊C8C2418yes—
296#97—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95C2418yes—
303C24—24no—
304C24—81no—
305C24—81no—
306C24—81no—
307C24—81no—
308C24C2×C22⋊C412yes—
309C24C2×C22⋊C412yes—
310C24C2×C22⋊C412yes—
311C24C2×C22⋊C412yes—
312C24#13718yes—
313C24C22×D412yes—
314C24C22×D411yescharacteristic
315C24C22×D412yes—
316C2×Q8—24no—
317C2×Q8—28no—
318C2×Q8—28no—
319C2×Q8—28no—
320C2×Q8—48no—
321C2×Q8—88no—
322C2×Q8—88no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—88no—
327C2×Q8—88no—
328C2×Q8C23×C418yes—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—216no—
332C2×D4—216no—
333C2×D4C2×C22⋊C4116yes—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—48no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—81no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C414yes—
368C22×C4C2×C22⋊C414yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C414yes—
376C22×C4C2×C22⋊C414yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C23×C411yescharacteristic
379C22×C4C23×C414yes—
380C22×C4C23×C414yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D414yes—
389C22×C4C22×D414yes—
390C22×C4C2511yescharacteristic
391M16—88no—
392C2×C8—28no—
393C2×C8—48no—
394C2×C8—88no—
395C4⋊C4—28no—
396C4⋊C4—24no—
397C4⋊C4—216no—
398C4⋊C4—216no—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—28no—
403C4⋊C4—28no—
404C4⋊C4—216no—
405C4⋊C4—28no—
406C4⋊C4—48no—
407C4⋊C4—84no—
408C4⋊C4C2×C22⋊C4116yes—
409C4⋊C4C23×C418yes—
410C22⋊C4—28no—
411C22⋊C4—216no—
412C22⋊C4—216no—
413C22⋊C4—28no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C22⋊C4C2×C22⋊C4116yes—
419C22⋊C4C2×C22⋊C4116yes—
420C42—216no—
421C42—84no—
422C42—84no—
423C42C2×C22⋊C414yes—
424C42C2×C22⋊C414yes—
425C42C23×C414yes—
426C42C22×D414yes—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—88no—
445C23—88no—
446C23#41918yes—
447C23#41918yes—
448C23C22×C22⋊C414yes—
449C23C22×C22⋊C411yescharacteristic
450C23C22×C22⋊C414yes—
451C23C22×C22⋊C411yescharacteristic
452C23C2×#13714yes—
453C23C2×#13714yes—
454C23C2×#13714yes—
455C23C23×D411yescharacteristic
456Q8—48no—
457Q8—48no—
458Q8—88no—
459Q8—88no—
460Q8—88no—
461D4—232no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—88no—
496C2×C4—88no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4—168no—
500C2×C4C22×C22⋊C414yes—
501C2×C4C22×C22⋊C411yescharacteristic
502C2×C4C22×C22⋊C414yes—
503C2×C4C22×C22⋊C414yes—
504C2×C4C22×C22⋊C414yes—
505C2×C4C22×C22⋊C411yescharacteristic
506C2×C4C24×C411yescharacteristic
507C2×C4C23×D411yescharacteristic
508C8—48no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—24no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22—24no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—48no—
524C22—48no—
525C22—48no—
526C22—84no—
527C22—84no—
528C22C2×#41912yes—
529C22C2×#41912yes—
530C22C2×#41918yes—
531C22C23×C22⋊C411yescharacteristic
532C22C22×#13712yes—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—24no—
537C4—24no—
538C4—24no—
539C4—24no—
540C4—48no—
541C4—48no—
542C4—88no—
543C4#288312yes—
544C2—22no—
545C2—22no—
546C2—22no—
547C2—28no—
548C2—44no—
549C2C2×#288311yescharacteristic
550C2C22×#41911yescharacteristic
551C2#6009311yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17621C2semidirect
5. #17596C2semidirect
6. #17578C2semidirect
7. C2×#1520C2semidirect
10. #3501C4semidirect
13. C2×#554C4semidirect
14. C2×D4×Q8C4semidirect
39. #1520C22semidirect
40. #1515C22semidirect
41. #1511C22semidirect
42. #1508C22semidirect
43. #1506C22semidirect
44. C2×#328C22semidirect
46. C23×Q8C2×C4semidirect
55. #554C2×C4semidirect
57. D4×Q8C2×C4semidirect
66. #533C2×C4semidirect
69. C2×C4⋊Q8C2×C4semidirect
70. C2×C4⋊Q8C2×C4semidirect
88. C2×C22⋊Q8C2×C4semidirect
89. C2×C22⋊Q8C2×C4semidirect
96. C2×C4×Q8C2×C4semidirect
111. C22×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#29732
#341852
#176212
#175962
#175782
C2×#15202