Finite Group Databasebeta

← #205032

Subgroups — #205032

Order 512.

1311 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35085C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic, complemented
3C2×#1504C211yesmaximal, characteristic
4C2×#1508C211yesmaximal, characteristic, complemented
5C2×#1501C211yesmaximal, characteristic
6C22×#328C211yesmaximal, characteristic, complemented
7#13796C211yesmaximal, characteristic, complemented
8#9716C214yesmaximal, complemented
9#9714C214yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2857C2212yes—
15#2850C2212yes—
16#2663—22no—
17#2564—22no—
18#2503C411yescharacteristic, complemented
19#2501C411yescharacteristic, complemented
20#2499C412yescomplemented
21#2436C411yescharacteristic, complemented
22#2434C411yescharacteristic, complemented
23#2432C412yescomplemented
24#2412—22no—
25#2322—22no—
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C22×#95C2211yescharacteristic
29#1851—22no—
30#1850—22no—
31#1797C2214yes—
32#1508—24no—
33#1504—24no—
34#1501—24no—
35C2×#328—24no—
36C2×#328C2214yes—
37C2×#328C2212yes—
38C2×#328C2214yes—
39C2×#328C2212yes—
40C2×#327C2212yes—
41C2×#327C2212yes—
42#1434—22no—
43#1434C2214yes—
44C22⋊#95—22no—
45C22×#41C2×C411yescharacteristic
46C23×Q8C2×C411yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×M16—21no—
50C23×C8—21no—
51C2×C4⋊Q8C2×C412yescomplemented
52C2×#124C2×C412yescomplemented
53#525—22no—
54#525—22no—
55#525—22no—
56#525—22no—
57C2×#123—21no—
58C2×#123—22no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8C2×C412yes—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2312yes—
68C2×C22⋊Q8C2311yescharacteristic
69C2×#121C2×C411yescharacteristic
70C2×#121C2311yescharacteristic
71C2×#120—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×#117C2×C411yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42C2312yes—
82C2×C4⋊C8—24no—
83#407—22no—
84#407—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91#400—22no—
92#399—22no—
93#397—24no—
94#395—22no—
95#394—24no—
96#393—22no—
97#391C2×C412yescomplemented
98#391C2×C412yescomplemented
99#391C2×C412yescomplemented
100#390C2×C412yescomplemented
101#389—24no—
102#388—24no—
103#387—24no—
104#386—24no—
105#386—22no—
106#385—22no—
107#385—24no—
108#384C2×C412yescomplemented
109#382C2×C412yescomplemented
110#381—22no—
111#380—24no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4C2×C411yescharacteristic
115C4×C22⋊C4C2×C411yescharacteristic
116C4×C22⋊C4D412yes—
117C4×C22⋊C4D412yes—
118C4×C22⋊C4D414yes—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95D414yes—
124C2×#95D412yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127#336—24no—
128#328—28no—
129#328—24no—
130#328—24no—
131#328—28no—
132#327—24no—
133#327—24no—
134C25C2×D411yescharacteristic
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—82no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—82no—
142C22×Q8C22×C412yes—
143C22×Q8C22×C411yescharacteristic
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—21no—
154C23×C4—44no—
155C23×C4—82no—
156C23×C4—82no—
157C23×C4—82no—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D414yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C2×M16—42no—
170C2×M16—82no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—42no—
174C22×C8—82no—
175C4⋊Q8—82no—
176#124—82no—
177#123—48no—
178#123—42no—
179#123—42no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—21no—
187C22⋊Q8—21no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196#121—22no—
197#121—22no—
198#121—21no—
199#121—21no—
200#121—21no—
201#121—21no—
202#120—42no—
203#120—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208#117—22no—
209#117—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C414yes—
241C2×C42—21no—
242C2×C42—24no—
243C2×C42—82no—
244C2×C42—82no—
245C2×C42C22×C412yes—
246C2×C42C22×C412yes—
247C2×C42C2×D414yes—
248C4⋊C8—84no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—42no—
254C22⋊C8—42no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—22no—
262#95—84no—
263#95—84no—
264#95C22⋊C414yes—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267#95C2×D414yes—
268#95C2×D414yes—
269C24—22no—
270C24—22no—
271C24—24no—
272C24—21no—
273C24—42no—
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C24#12112yes—
277C24#12112yes—
278C24C22×D411yescharacteristic
279#41—84no—
280#41—84no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—82no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—82no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—48no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—84no—
338C22×C4—82no—
339C22×C4—84no—
340C22×C4—816no—
341C22×C4C2×C22⋊C414yes—
342C22×C4C2×C22⋊C414yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12412yes—
352C22×C4#12412yes—
353C22×C4#12712yes—
354C22×C4#12712yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361M16—84no—
362C2×C8—44no—
363C2×C8—44no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—84no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—48no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—84no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—44no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—44no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—42no—
397C22⋊C4—84no—
398C42—44no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—84no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—42no—
416C23—42no—
417C23—48no—
418C23—48no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23#39012yes—
424C23#39012yes—
425C23#42114yes—
426C23C22×C22⋊C411yescharacteristic
427C23C2×#12111yescharacteristic
428C23C2×#12111yescharacteristic
429C23#57712yes—
430C23#57812yes—
431Q8—82no—
432Q8—82no—
433Q8—81no—
434Q8—81no—
435Q8—84no—
436D4—81no—
437D4—81no—
438D4—84no—
439D4—84no—
440D4—81no—
441D4—81no—
442D4—82no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—88no—
490C2×C4—84no—
491C2×C4—816no—
492C2×C4—84no—
493C2×C4#39012yes—
494C2×C4#39012yes—
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×#12411yescharacteristic
498C2×C4C2×#12711yescharacteristic
499C8—82no—
500C8—82no—
501C22—22no—
502C22—24no—
503C22—24no—
504C22—28no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—44no—
514C22—48no—
515C22—44no—
516C22—44no—
517C22—81no—
518C22—84no—
519C22—81no—
520C22—81no—
521C22#196314yes—
522C22C2×#39011yescharacteristic
523C22#288312yes—
524C22#289812yes—
525C22C2×#42112yes—
526C22#304211yescharacteristic
527C22#304411yescharacteristic
528C22#304612yes—
529C4—21no—
530C4—22no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—22no—
535C4—44no—
536C4—44no—
537C4—84no—
538C4—84no—
539C4—81no—
540C4—82no—
541C4—81no—
542C4—81no—
543C4—82no—
544C2—21no—
545C2—22no—
546C2—24no—
547C2—21no—
548C2—42no—
549C2—161no—
550C2#1759814yes—
551C2C2×#196311yescharacteristic
552C2#2121311yescharacteristic
553C2#2126211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21932C2semidirect
4. C2×#1508C2semidirect
6. C22×#328C2semidirect
7. #13796C2semidirect
8. #9716C2semidirect
9. #9714C2semidirect
18. #2503C4semidirect
19. #2501C4semidirect
20. #2499C4semidirect
21. #2436C4semidirect
22. #2434C4semidirect
23. #2432C4semidirect
51. C2×C4⋊Q8C2×C4semidirect
52. C2×#124C2×C4semidirect
97. #391C2×C4semidirect
98. #391C2×C4semidirect
99. #391C2×C4semidirect
100. #390C2×C4semidirect
108. #384C2×C4semidirect
109. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350852
#219322
C2×#15042
C2×#15082
C2×#15012
C22×#3282
#137962
#97162
#97142