Finite Group Databasebeta

← #207936

Subgroups — #207936

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2#20533C211yesmaximal, characteristic
3#13949C211yesmaximal, characteristic
4#9885C211yesmaximal, characteristic
5#9884C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic
7#9679C211yesmaximal, characteristic
8#9112C212yesmaximal
9#9112C212yesmaximal
10#8221C212yesmaximal
11C4×#328C212yesmaximal
12C2×#431C2212yes—
13#2857C2211yescharacteristic
14#2857C2211yescharacteristic
15#2855C2212yes—
16#2646—21no—
17#2627—21no—
18#2590—22no—
19#2588—22no—
20#2511C412yescomplemented
21#2507C412yescomplemented
22#2451C412yescomplemented
23#2446C412yescomplemented
24#2329—21no—
25#2313—21no—
26C4×C22⋊Q8C2212yes—
27C2×#387—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C8×C4⋊C4—22no—
30C8×C4⋊C4—22no—
31C4×C22⋊C8C2212yes—
32C4×C22⋊C8C2211yescharacteristic
33C4×C22⋊C8C2211yescharacteristic
34C2×#328C2212yes—
35C2×#328C2212yes—
36C2×#328C2212yes—
37C2×#328C2212yes—
38#1442—21no—
39#1441—21no—
40#1436—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434C2211yescharacteristic
44#1434C2211yescharacteristic
45#1434C2211yescharacteristic
46C4×#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8C2×C412yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#431—24no—
65#431—24no—
66C2×C4⋊C8—22no—
67C2×C22⋊C8C2312yes—
68C2×C22⋊C8C2312yes—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73#400—22no—
74#400—22no—
75#400—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#395—22no—
81#391—21no—
82#391—22no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2×C412yes—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—44no—
105#385—22no—
106#385—22no—
107#384C2×C411yescharacteristic
108#384C2×C411yescharacteristic
109#381—22no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380C2312yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4D412yes—
120C4×C22⋊C4—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132C2×#95C2311yescharacteristic
133C2×#95C2311yescharacteristic
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136#336—22no—
137#336—21no—
138#336—21no—
139#328—24no—
140#328—24no—
141#328—24no—
142#328—24no—
143C22×Q8—21no—
144C22×Q8C22×C412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—81no—
159C4⋊Q8—81no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C4×Q8—82no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4—81no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4SD1614yes—
193C2×C4⋊C4Q1614yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—81no—
206C2×C42—81no—
207C2×C42—81no—
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C4⋊C8—28no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8#4114yes—
220C22⋊C8#4114yes—
221C4×C8—28no—
222C4×C8—82no—
223C4×C8—81no—
224C4×C8—81no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—42no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—81no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24C22×D411yescharacteristic
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—81no—
263C2×Q8—81no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×SD1612yes—
325C22×C4C2×SD1612yes—
326C22×C4C2×SD1612yes—
327C22×C4C2×Q1612yes—
328C22×C4C2×Q1612yes—
329C22×C4C2×Q1612yes—
330C22×C4#13512yes—
331C22×C4#13512yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4112yes—
335C22×C4C2×#4112yes—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4#10318yes—
365C4⋊C4#10318yes—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C42—24no—
371C42—22no—
372C42—22no—
373C42—42no—
374C42—41no—
375C42—41no—
376C42—42no—
377C42—41no—
378C42—41no—
379C42—82no—
380C42—82no—
381C42—84no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23C22×C22⋊C411yescharacteristic
387C23#53211yescharacteristic
388C23#53211yescharacteristic
389C23C22×SD1611yescharacteristic
390C23C22×Q1611yescharacteristic
391C23C2×#13511yescharacteristic
392C23C2×#13511yescharacteristic
393Q8—82no—
394Q8—82no—
395Q8—162no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4C2×#10314yes—
475C2×C4C2×#10314yes—
476C2×C4C2×#10312yes—
477C2×C4C2×#10312yes—
478C2×C4#41612yes—
479C2×C4#41612yes—
480C2×C4C4×#4112yes—
481C2×C4#52512yes—
482C8—44no—
483C8—44no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22#229711yescharacteristic
497C22#288311yescharacteristic
498C22#288411yescharacteristic
499C22C22×#10311yescharacteristic
500C22C2×#41611yescharacteristic
501C22#290111yescharacteristic
502C22#290111yescharacteristic
503C22#316711yescharacteristic
504C22#316711yescharacteristic
505C22#317011yescharacteristic
506C22#317011yescharacteristic
507C4—24no—
508C4—22no—
509C4—22no—
510C4—24no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—161no—
524C4—161no—
525C2—21no—
526C2—21no—
527C2—21no—
528C2—21no—
529C2#1761411yescharacteristic
530C2#1761511yescharacteristic
531C2#1761611yescharacteristic
532C2#1761911yescharacteristic
533C2#2138411yescharacteristic
534C2#2140011yescharacteristic
535C2#2142311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #2511C4semidirect
21. #2507C4semidirect
22. #2451C4semidirect
23. #2446C4semidirect

Maximal subgroups

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

subgroupindex
#350152
#205332
#139492
#98852
#98842
#96792
#91122
#82212
C4×#3282