Finite Group Databasebeta

← #220098

Subgroups — #220098

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36121C211yesmaximal, characteristic
2#22751C212yesmaximal, complemented
3C2×#1560C211yesmaximal, characteristic, complemented
4C2×#1555C211yesmaximal, characteristic, complemented
5#9699C212yesmaximal, complemented
6#9141C212yesmaximal, complemented
7#9136C212yesmaximal, complemented
8#8592C211yesmaximal, characteristic, complemented
9#8344C211yesmaximal, characteristic, complemented
10#8339C211yesmaximal, characteristic, complemented
11C4×#331C211yesmaximal, characteristic, complemented
12C2×#546C412yes—
13C2×#532C2211yescharacteristic
14C22×#127C2211yescharacteristic
15C2×#432C2212yes—
16C2×#423C2212yescomplemented
17#2858C2211yescharacteristic
18#2857C2211yescharacteristic
19#2856C2211yescharacteristic
20#2544C412yes—
21#2445—22no—
22#2435C412yes—
23#2431C412yes—
24#2428—22no—
25#2408—22no—
26D4×C22⋊C4—22no—
27C4×#127C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1923—22no—
30#1917—22no—
31#1870—22no—
32#1831C2212yescomplemented
33#1797C2212yes—
34#1797C2212yescomplemented
35C4×C4⋊C8C2212yescomplemented
36#1753—22no—
37#1560—24no—
38#1555—24no—
39C2×#333C2212yescomplemented
40C2×#333C2212yescomplemented
41C2×#331C2211yescharacteristic
42C2×#331C2211yescharacteristic
43C2×#331C2212yes—
44C2×#331C2211yescharacteristic
45C2×#331C2212yes—
46C2×#331C2211yescharacteristic
47#1296C2212yescomplemented
48#1296C2212yescomplemented
49C22×#41C2×C412yes—
50C23×D4—22no—
51C23×D4C2×C412yes—
52C22×M16—22no—
53#546—48no—
54#532—22no—
55#532—22no—
56#532—42no—
57C2×C4⋊Q8C2311yescharacteristic
58C2×#127—22no—
59C2×#127—22no—
60C2×#127—22no—
61C2×#127—42no—
62C2×#127C2×C412yes—
63C2×#127C2311yescharacteristic
64C2×#127C2311yescharacteristic
65C2×#124C2×C412yes—
66C2×#124C2×C412yes—
67C2×#124C2×C412yes—
68C2×#123—22no—
69C2×C22⋊Q8C2×C412yes—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121C2×C412yes—
74C2×#120—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D412yes—
80C22×C22⋊C4—22no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#432—24no—
84#432—42no—
85#423—24no—
86#423—42no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×#97—22no—
96C2×C4×C8—22no—
97#394—24no—
98#394—24no—
99#392—24no—
100#391C2×C412yes—
101#390—22no—
102#390—22no—
103#390—22no—
104#390—22no—
105#390C2×C412yes—
106#390C2×C412yes—
107#389—22no—
108#386—24no—
109#386—24no—
110#384C2311yescharacteristic
111#380—22no—
112#379—24no—
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4D412yes—
115C4×C22⋊C4C2×C411yescharacteristic
116C4×C22⋊C4C2×C411yescharacteristic
117C4×C22⋊C4D412yes—
118C4×C22⋊C4D412yes—
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#333—24no—
126#333—42no—
127#331—22no—
128#331—22no—
129#331—22no—
130#331—24no—
131#331—42no—
132#331D414yescomplemented
133C25—22no—
134C2×#41—24no—
135C2×#41—42no—
136C2×#41—44no—
137C2×#41—82no—
138C22×Q8C22×C412yes—
139C22×D4—24no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—42no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—82no—
153C22×D4—82no—
154C22×D4—82no—
155C22×D4C22⋊C411yescharacteristic
156C22×D4C22×C412yes—
157C22×D4C22×C412yes—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—82no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—22no—
174#127—24no—
175#127—21no—
176#127—21no—
177#127—22no—
178#127—22no—
179#127—42no—
180#127—44no—
181#127—44no—
182#127C2×D412yes—
183#127#4112yes—
184#124—24no—
185#124—42no—
186#124—44no—
187#124—82no—
188#123—48no—
189C22⋊Q8—44no—
190#121—48no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—82no—
195#121—82no—
196#120—48no—
197C4×D4—24no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—82no—
204C4×D4—82no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4C22⋊C411yescharacteristic
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22⋊C411yescharacteristic
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2411yescharacteristic
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—44no—
248C4⋊C8C2×D414yes—
249C22⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8—82no—
252#97—44no—
253C4×C8—82no—
254#95—24no—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258C24—22no—
259C24—22no—
260C24—44no—
261C24—48no—
262C24—44no—
263C24—84no—
264C24—82no—
265C24C2×C22⋊C411yescharacteristic
266C24C2×C22⋊C411yescharacteristic
267#41—88no—
268#41—88no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—48no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—82no—
302C2×D4—84no—
303C2×D4—88no—
304C2×D4—88no—
305C2×D4—88no—
306C2×D4—88no—
307C2×D4—88no—
308C2×D4—88no—
309C2×D4—82no—
310C2×D4—84no—
311C2×D4C4×D414yes—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—82no—
338C22×C4—84no—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D412yes—
348M16—88no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—22no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C22⋊C4—22no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—48no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C22⋊C4—84no—
381C22⋊C4#10014yes—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—41no—
396C42C22×D411yescharacteristic
397C42C2×#4111yescharacteristic
398C42H(5,𝔽2)12yes—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—22no—
403C23—24no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—88no—
419C23—88no—
420C23—816no—
421C23—164no—
422C23—168no—
423C23C2×#10012yes—
424C23C22×C22⋊C411yescharacteristic
425Q8—82no—
426Q8—82no—
427D4—44no—
428D4—44no—
429D4—44no—
430D4—44no—
431D4—44no—
432D4—44no—
433D4—44no—
434D4—82no—
435D4—82no—
436D4—88no—
437D4—816no—
438D4—88no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4—82no—
443D4—164no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—88no—
481C2×C4—164no—
482C2×C4C2×#10012yes—
483C2×C4C2×#10012yes—
484C2×C4C2×C4×D411yescharacteristic
485C2×C4#51811yescharacteristic
486C2×C4D4212yes—
487C2×C4#54612yes—
488C2×C4#57512yes—
489C2×C4#57512yes—
490C2×C4#57612yes—
491C2×C4#57712yes—
492C8—44no—
493C8—82no—
494C8—82no—
495C8—82no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—42no—
503C22—41no—
504C22—41no—
505C22—44no—
506C22—42no—
507C22—48no—
508C22—82no—
509C22—84no—
510C22—82no—
511C22—82no—
512C22—84no—
513C22—82no—
514C22—82no—
515C22—84no—
516C22—1616no—
517C22D4×C22⋊C411yescharacteristic
518C22C22×#10011yescharacteristic
519C22#288811yescharacteristic
520C22#289612yes—
521C22#289712yes—
522C4—22no—
523C4—21no—
524C4—21no—
525C4—22no—
526C4—41no—
527C4—42no—
528C4—41no—
529C4—42no—
530C4—42no—
531C4—82no—
532C4—82no—
533C4#327312yes—
534C4#327512yes—
535C4#332212yes—
536C4#332312yes—
537C2—42no—
538C2—81no—
539C2—84no—
540C2—162no—
541C2#1773811yescharacteristic
542C2#1776612yes—
543C2#1777011yescharacteristic
544C2D4×#10011yescharacteristic
545C2#2242712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36121C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22751C2semidirect
3. C2×#1560C2semidirect
4. C2×#1555C2semidirect
5. #9699C2semidirect
6. #9141C2semidirect
7. #9136C2semidirect
8. #8592C2semidirect
9. #8344C2semidirect
10. #8339C2semidirect
11. C4×#331C2semidirect
16. C2×#423C22semidirect
32. #1831C22semidirect
34. #1797C22semidirect
35. C4×C4⋊C8C22semidirect
39. C2×#333C22semidirect
40. C2×#333C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect
132. #331D4semidirect

Maximal subgroups

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

subgroupindex
#361212
#227512
C2×#15602
C2×#15552
#96992
#91412
#91362
#85922
#83442
#83392
C4×#3312