Finite Group Databasebeta

← #222070

Subgroups — #222070

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36121C211yesmaximal, characteristic
2#22797C212yesmaximal, complemented
3C2×#1555C211yesmaximal, characteristic, complemented
4C2×#1543C211yesmaximal, characteristic, complemented
5#9707C212yesmaximal, complemented
6#9136C212yesmaximal, complemented
7#9134C212yesmaximal, complemented
8#8585C211yesmaximal, characteristic, complemented
9#8344C211yesmaximal, characteristic, complemented
10#8339C211yesmaximal, characteristic, complemented
11#8231C211yesmaximal, characteristic, complemented
12C2×#546C412yes—
13C2×#532C2211yescharacteristic
14C22×#127C2211yescharacteristic
15C2×#432C2212yes—
16C2×#431C2212yescomplemented
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#1955—22no—
30#1924—22no—
31#1923—22no—
32#1842C2212yescomplemented
33#1797C2212yes—
34#1797C2212yescomplemented
35#1771—22no—
36#1767C2212yescomplemented
37#1555—24no—
38#1543—24no—
39C2×#333C2212yescomplemented
40C2×#333C2212yescomplemented
41C2×#331C2211yescharacteristic
42C2×#331C2211yescharacteristic
43C2×#331C2211yescharacteristic
44C2×#331C2212yes—
45C2×#331C2212yes—
46C2×#331C2211yescharacteristic
47#1296C2212yescomplemented
48#1296C2212yescomplemented
49C22×#41C2×C412yes—
50C23×D4—22no—
51C23×D4C2×C412yes—
52#546—48no—
53#532—22no—
54#532—22no—
55#532—42no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×#127—22no—
58C2×#127—22no—
59C2×#127—22no—
60C2×#127—42no—
61C2×#127C2×C412yes—
62C2×#127C2311yescharacteristic
63C2×#127C2311yescharacteristic
64C2×#124C2×C412yes—
65C2×#124C2×C412yes—
66C2×#124C2×C412yes—
67C2×#123—22no—
68C2×C22⋊Q8C2×C412yes—
69C2×#121—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121C2×C412yes—
73C2×#120—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4C2×C412yes—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D412yes—
79C22×C22⋊C4—22no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#432—24no—
83#432—42no—
84#431—24no—
85#431—42no—
86C2×C4⋊C8—22no—
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×C22⋊C8—22no—
96C2×#97—22no—
97C2×#97—22no—
98C2×C4×C8—22no—
99#394—24no—
100#394—24no—
101#392—24no—
102#391C2×C412yes—
103#390—22no—
104#390—22no—
105#390—22no—
106#390—22no—
107#390C2×C412yes—
108#390C2×C412yes—
109#389—22no—
110#386—24no—
111#386—24no—
112#384C2311yescharacteristic
113#380—22no—
114#379—24no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4D412yes—
117C4×C22⋊C4C2×C411yescharacteristic
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
121C43C2311yescharacteristic
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—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—22no—
171#127—24no—
172#127—21no—
173#127—21no—
174#127—22no—
175#127—22no—
176#127—42no—
177#127—44no—
178#127—44no—
179#127C2×D412yes—
180#127#4112yes—
181#124—24no—
182#124—42no—
183#124—44no—
184#124—82no—
185#123—48no—
186C22⋊Q8—44no—
187#121—48no—
188#121—44no—
189#121—44no—
190#121—44no—
191#121—82no—
192#121—82no—
193#120—48no—
194C4×D4—24no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—82no—
201C4×D4—82no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—48no—
218C2×C22⋊C4C22⋊C411yescharacteristic
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C411yescharacteristic
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2411yescharacteristic
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—44no—
245C4⋊C8—82no—
246C4⋊C8C2×D414yes—
247C22⋊C8—44no—
248C22⋊C8—44no—
249C22⋊C8—82no—
250C22⋊C8—82no—
251#97—44no—
252#97—44no—
253C4×C8—44no—
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—
348C2×C8—24no—
349C2×C8—42no—
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—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C22⋊C4—22no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—48no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—84no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C22⋊C4#10014yes—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—41no—
395C42C22×D411yescharacteristic
396C42C2×#4111yescharacteristic
397C42H(5,𝔽2)12yes—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—24no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—22no—
407C23—22no—
408C23—42no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—88no—
418C23—88no—
419C23—816no—
420C23—164no—
421C23—168no—
422C23C2×#10012yes—
423C23C22×C22⋊C411yescharacteristic
424Q8—82no—
425Q8—82no—
426D4—44no—
427D4—44no—
428D4—44no—
429D4—44no—
430D4—44no—
431D4—44no—
432D4—44no—
433D4—82no—
434D4—82no—
435D4—88no—
436D4—816no—
437D4—88no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4—164no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
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—44no—
479C2×C4—88no—
480C2×C4—164no—
481C2×C4C2×#10012yes—
482C2×C4C2×#10012yes—
483C2×C4C2×C4×D411yescharacteristic
484C2×C4#51811yescharacteristic
485C2×C4D4212yes—
486C2×C4#54612yes—
487C2×C4#57512yes—
488C2×C4#57512yes—
489C2×C4#57612yes—
490C2×C4#57712yes—
491C8—44no—
492C8—82no—
493C8—82no—
494C8—82no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—44no—
505C22—42no—
506C22—48no—
507C22—82no—
508C22—84no—
509C22—82no—
510C22—82no—
511C22—84no—
512C22—82no—
513C22—82no—
514C22—84no—
515C22—1616no—
516C22D4×C22⋊C411yescharacteristic
517C22C22×#10011yescharacteristic
518C22#288811yescharacteristic
519C22#289612yes—
520C22#289712yes—
521C4—22no—
522C4—21no—
523C4—21no—
524C4—22no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—82no—
531C4—82no—
532C4#326712yes—
533C4#326812yes—
534C4#331112yes—
535C4#331912yes—
536C2—42no—
537C2—81no—
538C2—84no—
539C2—162no—
540C2#1773811yescharacteristic
541C2#1776612yes—
542C2#1777011yescharacteristic
543C2D4×#10011yescharacteristic
544C2#2242412yes—
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. #22797C2semidirect
3. C2×#1555C2semidirect
4. C2×#1543C2semidirect
5. #9707C2semidirect
6. #9136C2semidirect
7. #9134C2semidirect
8. #8585C2semidirect
9. #8344C2semidirect
10. #8339C2semidirect
11. #8231C2semidirect
16. C2×#431C22semidirect
32. #1842C22semidirect
34. #1797C22semidirect
36. #1767C22semidirect
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
#227972
C2×#15552
C2×#15432
#97072
#91362
#91342
#85852
#83442
#83392
#82312