Finite Group Databasebeta

← #188521

Subgroups — #188521

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34205C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic, complemented
3C2×#1519C211yesmaximal, characteristic
4C22×#327C211yesmaximal, characteristic
5#13966C211yesmaximal, characteristic, complemented
6#8575C214yesmaximal, complemented
7#8573C214yesmaximal, complemented
8#8023C211yesmaximal, characteristic, complemented
9C2×#1294C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C2×#518—21no—
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2364—22no—
15#2361C414yescomplemented
16#2344C412yescomplemented
17#2342—22no—
18D4×C22⋊C4C412yescomplemented
19#2325—22no—
20#2308—21no—
21C4×#121C2214yes—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C22×#95C2211yescharacteristic
26#1850—22no—
27#1850—22no—
28#1797C2214yescomplemented
29#1519—24no—
30C2×#327—24no—
31C2×#327C2214yes—
32C2×#327C2212yes—
33C2×#327C2212yes—
34#1442—22no—
35#1434C2212yescomplemented
36#1434C2212yescomplemented
37C22⋊#95—21no—
38C22⋊#95—21no—
39#1294C2214yescomplemented
40#1294C2212yescomplemented
41#1294C2212yescomplemented
42C23×D4—21no—
43C23×D4C2×C411yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8—21no—
47C23×C8—21no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121C2×C412yes—
53C2×#121D414yes—
54C2×#121Q814yes—
55C2×#121C2312yes—
56#518—44no—
57#518—44no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4C2×C414yescomplemented
61C2×#117—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42C2312yes—
72C2×C4⋊C8—24no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81#390C2×C414yescomplemented
82#389C2×C414yescomplemented
83#388—24no—
84#388—24no—
85#387—24no—
86#386—24no—
87#386—24no—
88#386—24no—
89#385—24no—
90#381—22no—
91#380—21no—
92#380—21no—
93#380—22no—
94#380—22no—
95#380C2×C412yes—
96#380C2312yes—
97#379—22no—
98C4×C4⋊C4C2×C414yescomplemented
99C4×C22⋊C4—24no—
100C4×C22⋊C4C2×C412yes—
101C4×C22⋊C4C2×C412yes—
102C4×C22⋊C4D412yes—
103C4×C22⋊C4D412yes—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113C2×#95C2312yes—
114C2×#95C2311yescharacteristic
115C2×#95C2311yescharacteristic
116#336—24no—
117#327—28no—
118#327—24no—
119#327—24no—
120C25—21no—
121C25C2×D411yescharacteristic
122C22×D4—22no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—82no—
133C22×D4C4⋊C414yes—
134C22×D4C22×C412yes—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×Q812yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—42no—
162C22×C8—42no—
163C22×C8—82no—
164C22×C8—82no—
165#121—24no—
166#121—24no—
167#121—24no—
168#121—24no—
169#121—24no—
170#121—22no—
171#121—22no—
172#121C4⋊C418yes—
173#121C2×D414yes—
174#121C2×Q814yes—
175C4×D4—416no—
176C4×D4—48no—
177C4×D4—48no—
178C4×D4—84no—
179#117—48no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C4⋊C414yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C414yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C42—24no—
214C2×C42—82no—
215C2×C42C22×C414yes—
216C2×C42C2×D414yes—
217C4⋊C8—84no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224#95—22no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—44no—
235#95—44no—
236#95—42no—
237#95—82no—
238#95—82no—
239#95C22⋊C414yes—
240#95C22⋊C414yes—
241C24—22no—
242C24—24no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—82no—
248C24—82no—
249C24C2×C22⋊C411yescharacteristic
250C24C2×C22⋊C411yescharacteristic
251C24#12112yes—
252C24C22⋊Q812yes—
253C24#13512yes—
254C24#13512yes—
255C24AGL(1,ℤ/8)12yes—
256C24AGL(1,ℤ/8)12yes—
257C24C22×D411yescharacteristic
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—28no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—44no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—48no—
271C2×D4—48no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—82no—
275C2×D4—84no—
276C2×D4—84no—
277C2×D4C2×C4⋊C414yes—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—48no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—48no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4C2×C22⋊C414yes—
330C22×C4C2×C22⋊C414yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C4⋊C412yes—
334C22×C4C2×C4⋊C412yes—
335C22×C4#11712yes—
336C22×C4#11712yes—
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4C22⋊Q812yes—
341C22×C4C22⋊Q812yes—
342C22×C4C22⋊Q812yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C2×C8—44no—
350C2×C8—44no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—84no—
354C2×C8—84no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—82no—
360C4⋊C4—84no—
361C4⋊C4C2×C4⋊C414yes—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—28no—
369C22⋊C4—48no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—44no—
373C22⋊C4—84no—
374C22⋊C4—84no—
375C22⋊C4—84no—
376C42—44no—
377C42—44no—
378C42—48no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—24no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—42no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—48no—
399C23—84no—
400C23—84no—
401C23—164no—
402C23—164no—
403C23#38012yes—
404C23#38012yes—
405C23#41612yes—
406C23#41612yes—
407C23#41812yes—
408C23#41812yes—
409C23#46314yes—
410C23#46314yes—
411C23#47814yes—
412C23#47814yes—
413C23C22×C22⋊C411yescharacteristic
414C23C2×#12111yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23C2×#13511yescharacteristic
417C23C2×#13511yescharacteristic
418C23C2×AGL(1,ℤ/8)11yescharacteristic
419C23C2×AGL(1,ℤ/8)11yescharacteristic
420D4—44no—
421D4—44no—
422D4—44no—
423D4—44no—
424D4—416no—
425D4—44no—
426D4—44no—
427D4—84no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
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—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—816no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—168no—
482C2×C4—164no—
483C2×C4#38012yes—
484C2×C4#38012yes—
485C2×C4C22×C4⋊C411yescharacteristic
486C2×C4C2×#11711yescharacteristic
487C2×C4C2×#12111yescharacteristic
488C2×C4C2×C22⋊Q811yescharacteristic
489C8—82no—
490C8—82no—
491C22—21no—
492C22—22no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—22no—
497C22—21no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—24no—
502C22—24no—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—24no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—84no—
515C22—84no—
516C22—168no—
517C22#180414yes—
518C22#180414yes—
519C22C2×#38011yescharacteristic
520C22#288312yes—
521C22C2×#41611yescharacteristic
522C22C2×#41811yescharacteristic
523C22#290112yes—
524C22C2×#46311yescharacteristic
525C22C2×#46311yescharacteristic
526C22C2×#47811yescharacteristic
527C22C2×#47811yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—22no—
533C4—22no—
534C4—44no—
535C4—44no—
536C4—44no—
537C4—84no—
538C4—84no—
539C4—161no—
540C4—161no—
541C2—21no—
542C2—21no—
543C2—24no—
544C2—21no—
545C2—21no—
546C2—24no—
547C2—84no—
548C2#1761512yes—
549C2#1762012yes—
550C2#2063311yescharacteristic
551C2C2×#180411yescharacteristic
552C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21930C2semidirect
5. #13966C2semidirect
6. #8575C2semidirect
7. #8573C2semidirect
8. #8023C2semidirect
9. C2×#1294C2semidirect
15. #2361C4semidirect
16. #2344C4semidirect
18. D4×C22⋊C4C4semidirect
28. #1797C22semidirect
35. #1434C22semidirect
36. #1434C22semidirect
39. #1294C22semidirect
40. #1294C22semidirect
41. #1294C22semidirect
60. C2×C4×D4C2×C4semidirect
81. #390C2×C4semidirect
82. #389C2×C4semidirect
98. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342052
#219302
C2×#15192
C22×#3272
#139662
#85752
#85732
#80232
C2×#12942