Finite Group Databasebeta

← #495398

Subgroups — #495398

Order 512.

1074 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3C4×#413C211yesmaximal, characteristic
4#13293C214yesmaximal, complemented
5#13292C214yesmaximal, complemented
6#12575C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic, complemented
8C2×C4×#41C2211yescharacteristic
9C2×#413C2212yes—
10C2×#410C2212yes—
11C2×C4×M16—21no—
12#2857—21no—
13#2857—21no—
14#2628—21no—
15#2628—21no—
16#2568—21no—
17#2567—21no—
18#2563—21no—
19#2542—21no—
20#2487C412yescomplemented
21#2487C2212yes—
22#2409C412yescomplemented
23#2409C2212yes—
24#2317C412yescomplemented
25#2311C2212yes—
26#2285—21no—
27#2285C411yescharacteristic, complemented
28#1799—21no—
29#1799—21no—
30#1799C2212yes—
31#1799C2212yes—
32C4×#101—21no—
33C4×#101—21no—
34C4×#100C2212yes—
35C4×#100C2212yes—
36#1742—21no—
37#1740—21no—
38#1390C22116yes—
39C22×#41C2311yescharacteristic
40C23×Q8C2×C412yes—
41C22×M16—22no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×C4×D4C2311yescharacteristic
64C2×#117—22no—
65C2×#117C2×C412yes—
66C2×#117C2×C411yescharacteristic
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C412yes—
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74#413—24no—
75#413Q818yescomplemented
76C2×#101—22no—
77C2×#100C2314yes—
78#410—28no—
79C2×#99—22no—
80C2×#99C2314yes—
81C4×M16—22no—
82C4×M16—22no—
83C4×M16—41no—
84C4×M16—41no—
85C2×#97—41no—
86C2×C4×C8—41no—
87#400—22no—
88#400—22no—
89#399—22no—
90#399—22no—
91#395—22no—
92#395—22no—
93#393—22no—
94#393—22no—
95#391—22no—
96#391—22no—
97#391—22no—
98#390—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#386C2×C414yes—
104#386C2314yes—
105#385C2×C414yes—
106#385C2×C414yes—
107#384—22no—
108#384—21no—
109#384—41no—
110#382—22no—
111#382—21no—
112#382—41no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—22no—
119#380C2×C412yes—
120#380C2×C412yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—41no—
124C4×C4⋊C4—41no—
125#343—22no—
126#343—22no—
127#342—22no—
128#342—22no—
129C2×#41—41no—
130C2×#41—41no—
131C2×#41—42no—
132C2×#41C2×Q814yes—
133C2×#41#4114yes—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—84no—
139C22×Q8C2×D411yescharacteristic
140C22×D4C22×C411yescharacteristic
141C22×D4C2×D411yescharacteristic
142C22×D4C2411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C414yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×M16—28no—
154C2×M16—24no—
155C2×M16—44no—
156C2×M16—42no—
157C22×C8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C22⋊Q8—84no—
169C22⋊Q8—84no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×Q8—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—41no—
191#117—41no—
192#117—44no—
193#117—44no—
194#117—44no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4C22⋊C411yescharacteristic
221C2×C22⋊C4C22⋊C411yescharacteristic
222C2×C22⋊C4C22×C414yes—
223C2×C22⋊C4C22×C411yescharacteristic
224C2×C22⋊C4C22×C411yescharacteristic
225C2×C22⋊C4C2×D411yescharacteristic
226C2×C22⋊C4C2×D411yescharacteristic
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251#101—28no—
252#100—24no—
253#100C2×Q818yes—
254#99—28no—
255#99—44no—
256#97—42no—
257#97—42no—
258C4×C8—42no—
259C4×C8—42no—
260#95—22no—
261#95—22no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—42no—
267#95—42no—
268C24C2×C22⋊C411yescharacteristic
269C24C23×C411yescharacteristic
270C24C22×D411yescharacteristic
271#41—44no—
272#41—44no—
273#41—44no—
274#41—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—84no—
283C2×Q8—88no—
284C2×Q8—88no—
285C2×Q8C22⋊Q814yes—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4C4×Q814yes—
293C2×D4#12412yes—
294C2×D4C4⋊Q812yes—
295C2×D4C22×Q812yes—
296C2×D4C2×#4112yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—84no—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C4×Q814yes—
341C22×C4C22⋊Q814yes—
342C22×C4#12412yes—
343C22×C4C4⋊Q812yes—
344C22×C4C22×D411yescharacteristic
345M16—28no—
346M16—44no—
347M16—48no—
348C2×C8—44no—
349C2×C8—48no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C4⋊C4—84no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—84no—
380C22⋊C4—84no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—44no—
400C42—84no—
401C42#9911yescharacteristic
402C42#9911yescharacteristic
403C42#9911yescharacteristic
404C42#9911yescharacteristic
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23#39112yes—
418C23C22×C22⋊C411yescharacteristic
419C23C2×C4×Q812yes—
420C23C2×#12411yescharacteristic
421C23C2×C4⋊Q811yescharacteristic
422Q8—44no—
423Q8—84no—
424Q8—88no—
425Q8—84no—
426D4—44no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—42no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
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—41no—
462C2×C4—41no—
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—44no—
472C2×C4—44no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—88no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4#39112yes—
482C2×C4#39112yes—
483C2×C4#39112yes—
484C2×C4C2×#9911yescharacteristic
485C2×C4C2×#9911yescharacteristic
486C2×C4C2×#9911yescharacteristic
487C2×C4C2×#9911yescharacteristic
488C2×C4C2×#9911yescharacteristic
489C2×C4C2×#9911yescharacteristic
490C2×C4#41012yes—
491C2×C4C2×C22⋊Q812yes—
492C8—48no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—21no—
498C22—22no—
499C22—42no—
500C22—41no—
501C22—41no—
502C22—42no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—81no—
511C22—81no—
512C22C2×#39111yescharacteristic
513C22C22×#9911yescharacteristic
514C22C2×#41011yescharacteristic
515C4—21no—
516C4—21no—
517C4—22no—
518C4—22no—
519C4—22no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—84no—
527C4—84no—
528C4—164no—
529C4#199612yes—
530C4#199612yes—
531C4#212212yes—
532C4#212312yes—
533C2—21no—
534C2—21no—
535C2—41no—
536C2—42no—
537C2—41no—
538C2—42no—
539C2C2×#199611yescharacteristic
540C2C2×#212211yescharacteristic
541C2C2×#212311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13293C2semidirect
5. #13292C2semidirect
7. C22×C4⋊Q8C4semidirect
20. #2487C4semidirect
22. #2409C4semidirect
24. #2317C4semidirect
27. #2285C4semidirect
75. #413Q8semidirect

Maximal subgroups

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

subgroupindex
#352792
#206322
C4×#4132
#132932
#132922
#125752