Finite Group Databasebeta

← #207250

Subgroups — #207250

Order 512.

886 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic
3#13806C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9912C211yesmaximal, characteristic
6#9911C211yesmaximal, characteristic
7#9910C212yesmaximal
8#9909C211yesmaximal, characteristic
9#9908C211yesmaximal, characteristic
10#8025C212yesmaximal
11#8013C212yesmaximal
12C2×#431C2212yes—
13#2857C2211yescharacteristic
14#2857C2211yescharacteristic
15#2851C2212yes—
16#2627—21no—
17#2627—21no—
18#2606—22no—
19#2605—22no—
20#2570C412yescomplemented
21#2570C412yescomplemented
22#2550C412yescomplemented
23#2550C412yescomplemented
24#2329—21no—
25#2329—21no—
26#2283C2212yes—
27C2×#383C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1939—22no—
31#1939—22no—
32#1797C2211yescharacteristic
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1436—21no—
36#1436—21no—
37#1434—21no—
38#1434C2211yescharacteristic
39#1434C2211yescharacteristic
40C4×#95—21no—
41C4×#95C2211yescharacteristic
42C4×#95C2211yescharacteristic
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125C2×C412yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C42—21no—
59C22×C42—21no—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#431—24no—
65#431—24no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2312yes—
72C2×C4×C8—22no—
73#400—22no—
74#400—22no—
75#398—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#395—22no—
80#395—22no—
81#391—21no—
82#391—21no—
83#391—22no—
84#391—22no—
85#389—21no—
86#389—21no—
87#389—22no—
88#389—22no—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#385—22no—
100#385—22no—
101#384C2×C411yescharacteristic
102#384C2×C411yescharacteristic
103#383—22no—
104#383—22no—
105#383Q814yescomplemented
106#382C2×C412yes—
107#382C2×C412yes—
108#382C2×C412yes—
109#381—22no—
110#380—21no—
111#380—21no—
112#380C2312yes—
113#380C2312yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4D412yes—
120C4×C22⋊C4—22no—
121#376—22no—
122#376—22no—
123#376Q814yescomplemented
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139#336—22no—
140#336—21no—
141#336—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—81no—
158C4⋊Q8—81no—
159#125—82no—
160C22⋊Q8—42no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—81no—
200C2×C42—81no—
201C2×C42—81no—
202C2×C42—81no—
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×Q812yes—
212C2×C42C2×Q812yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215C4⋊C8—44no—
216C4⋊C8—81no—
217C4⋊C8—81no—
218C4⋊C8—82no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C4×C8—44no—
224#95—22no—
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—24no—
237#95—24no—
238#95—24no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—81no—
250#95—81no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C2×Q812yes—
256#95C2×Q812yes—
257#95C2×Q812yes—
258#95C2×Q812yes—
259C24C22×D411yescharacteristic
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—81no—
263C2×Q8—81no—
264C2×Q8—81no—
265C2×Q8—81no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
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×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C22⋊Q812yes—
329C22×C4C22⋊Q812yes—
330C22×C4C22⋊Q812yes—
331C22×C4C22⋊Q812yes—
332C22×C4#13512yes—
333C22×C4#13512yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×Q811yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—81no—
365C4⋊C4—81no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C42—41no—
371C42—41no—
372C42—42no—
373C42—42no—
374C42—41no—
375C42—41no—
376C42—48no—
377C42—82no—
378C42—82no—
379C42—84no—
380C42—84no—
381C42C4×Q814yes—
382C42C4×Q814yes—
383C42#12414yes—
384C42C4⋊Q814yes—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23C22×C22⋊C411yescharacteristic
390C23C2×C22⋊Q811yescharacteristic
391C23C2×C22⋊Q811yescharacteristic
392C23C2×#13511yescharacteristic
393C23C2×#13511yescharacteristic
394C23#57511yescharacteristic
395C23#57511yescharacteristic
396Q8—162no—
397Q8—162no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
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—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
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—42no—
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×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4#39114yes—
481C2×C4#39114yes—
482C2×C4#41612yes—
483C2×C4#41612yes—
484C2×C4C2×C4×Q811yescharacteristic
485C2×C4C2×C4×Q811yescharacteristic
486C2×C4C2×#12411yescharacteristic
487C2×C4C2×C4⋊Q811yescharacteristic
488C8—82no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22C2×#39111yescharacteristic
503C22#288311yescharacteristic
504C22#288411yescharacteristic
505C22C2×#41611yescharacteristic
506C22#289611yescharacteristic
507C22#290111yescharacteristic
508C22#290111yescharacteristic
509C22#307611yescharacteristic
510C22#307611yescharacteristic
511C22#307611yescharacteristic
512C22#307611yescharacteristic
513C4—24no—
514C4—24no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—22no—
520C4—22no—
521C4—44no—
522C4—42no—
523C4—42no—
524C4—41no—
525C4—41no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—82no—
533C2—21no—
534C2—21no—
535C2—21no—
536C2—21no—
537C2#1761511yescharacteristic
538C2#1761611yescharacteristic
539C2#1762311yescharacteristic
540C2#1762411yescharacteristic
541C2#2131111yescharacteristic
542C2#2133011yescharacteristic
543C2#2133411yescharacteristic
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. #2570C4semidirect
21. #2570C4semidirect
22. #2550C4semidirect
23. #2550C4semidirect
105. #383Q8semidirect
123. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
#219512
#138062
#99132
#99122
#99112
#99102
#99092
#99082
#80252
#80132