Finite Group Databasebeta

← #192431

Subgroups — #192431

Order 512.

1526 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34205C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic, complemented
3#21758C211yesmaximal, characteristic
4C22×#327C211yesmaximal, characteristic
5#13908C211yesmaximal, characteristic, complemented
6#8869C214yesmaximal, complemented
7#8866C214yesmaximal, complemented
8C2×#1294C211yesmaximal, characteristic, complemented
9#7907C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C2×#518—21no—
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2364—22no—
15#2361—22no—
16#2344—21no—
17#2342—22no—
18D4×C22⋊C4—21no—
19#2325C2214yes—
20#2308—21no—
21C4×#121—22no—
22C2×#380—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1893—22no—
28#1874C2214yes—
29#1852—22no—
30#1852C2214yescomplemented
31#1823—21no—
32#1808—22no—
33#1786—21no—
34C2×#327C418yescomplemented
35C2×#327C2212yes—
36C2×#327C2212yes—
37#1444C2212yescomplemented
38#1442C2212yescomplemented
39#1436—22no—
40C22⋊#95—21no—
41#1294C2212yescomplemented
42#1294C2212yescomplemented
43#1280C2214yescomplemented
44C23×D4—21no—
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8—21no—
49C23×C8—21no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121C2×C414yes—
55C2×#121D412yes—
56C2×#121Q814yes—
57C2×#121C2312yes—
58#518—44no—
59#518—44no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—22no—
63C2×#117—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42D412yes—
75C2×#107—24no—
76C2×#102—24no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8C2×C414yescomplemented
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85#390—22no—
86#389—22no—
87#388—24no—
88#388—24no—
89#387—24no—
90#386—24no—
91#386—24no—
92#386—24no—
93#385—24no—
94#381—22no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—44no—
102#380D412yes—
103#380C2312yes—
104#379—22no—
105C4×C4⋊C4—22no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95C2×C414yescomplemented
119C2×#95C2312yes—
120C2×#95C2311yescharacteristic
121C2×#95C2311yescharacteristic
122#340—24no—
123#340—24no—
124#327C2×C418yescomplemented
125#327C2×C418yescomplemented
126C25—21no—
127C25C2×D411yescharacteristic
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—82no—
139C22×D4C4⋊C414yescomplemented
140C22×D4C2×D412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C22×C412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×Q812yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—24no—
167C22×C8—22no—
168C22×C8—42no—
169C22×C8—82no—
170C22×C8—82no—
171#121—24no—
172#121—24no—
173#121—24no—
174#121—24no—
175#121—24no—
176#121—22no—
177#121—22no—
178#121C4⋊C418yescomplemented
179#121C22×C414yes—
180#121C2×Q814yes—
181C4×D4—416no—
182C4×D4—48no—
183C4×D4—48no—
184C4×D4—84no—
185#117—48no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C4⋊C414yescomplemented
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D414yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—48no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4D814yes—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—82no—
228#107—84no—
229#102—84no—
230C22⋊C8—24no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—42no—
235C22⋊C8C22×C414yes—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—44no—
246#95—41no—
247#95—41no—
248#95—44no—
249#95—82no—
250#95C22×C412yes—
251#95C22×C412yes—
252C24—22no—
253C24—24no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—82no—
259C24—82no—
260C24C4×D412yes—
261C24#12011yescharacteristic
262C24#12011yescharacteristic
263C24C22⋊Q812yes—
264C24C2×D812yes—
265C24#13512yes—
266C24AGL(1,ℤ/8)12yes—
267C24#13712yes—
268C24C22×D411yescharacteristic
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—28no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—48no—
281C2×D4—48no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—84no—
287C2×D4—84no—
288C2×D4C2×C4⋊C414yes—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—48no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—84no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—88no—
340C22×C4—88no—
341C22×C4C2×C4⋊C412yes—
342C22×C4C2×C4⋊C412yes—
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4C4×D412yes—
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4C22⋊Q812yes—
349C22×C4C22⋊Q812yes—
350C22×C4C22⋊Q812yes—
351C22×C4#12312yes—
352C22×C4#12312yes—
353C22×C4C2×D814yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C2×C8—24no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—42no—
364C2×C8—84no—
365C2×C8—84no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—88no—
374C4⋊C4C2×C4⋊C414yes—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—28no—
380C22⋊C4—48no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C42—44no—
391C42—48no—
392C42—82no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—44no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—42no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—48no—
413C23—84no—
414C23—84no—
415C23—164no—
416C23—164no—
417C23#38512yes—
418C23#38512yes—
419C23C4×D814yes—
420C23#44014yes—
421C23#44712yes—
422C23#44712yes—
423C23#45212yes—
424C23#45212yes—
425C23#47414yes—
426C23#47814yes—
427C23C2×C4×D411yescharacteristic
428C23C2×#12011yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430C23C22×D811yescharacteristic
431C23C2×#13511yescharacteristic
432C23C2×AGL(1,ℤ/8)11yescharacteristic
433C23C2×#13711yescharacteristic
434D4—44no—
435D4—48no—
436D4—44no—
437D4—416no—
438D4—44no—
439D4—44no—
440D4—44no—
441D4—82no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
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—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—82no—
484C2×C4—816no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—168no—
496C2×C4—164no—
497C2×C4#38512yes—
498C2×C4#38512yes—
499C2×C4C22×C4⋊C411yescharacteristic
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×#12311yescharacteristic
503C8—44no—
504C8—82no—
505C22—21no—
506C22—22no—
507C22—21no—
508C22—22no—
509C22—21no—
510C22—22no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—24no—
516C22—24no—
517C22—22no—
518C22—22no—
519C22—24no—
520C22—24no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—84no—
529C22—84no—
530C22—168no—
531C22#186214yes—
532C22#186214yes—
533C22C2×#38511yescharacteristic
534C22C2×C4×D811yescharacteristic
535C22C2×#44011yescharacteristic
536C22C2×#44711yescharacteristic
537C22C2×#45211yescharacteristic
538C22#301812yes—
539C22#302612yes—
540C22C2×#47411yescharacteristic
541C22C2×#47811yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—44no—
549C4—44no—
550C4—44no—
551C4—82no—
552C4—84no—
553C4—82no—
554C4—162no—
555C2—21no—
556C2—24no—
557C2—21no—
558C2—24no—
559C2—21no—
560C2—21no—
561C2—84no—
562C2#1838012yes—
563C2#1838512yes—
564C2#2086311yescharacteristic
565C2C2×#186211yescharacteristic
566C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21940C2semidirect
5. #13908C2semidirect
6. #8869C2semidirect
7. #8866C2semidirect
8. C2×#1294C2semidirect
9. #7907C2semidirect
30. #1852C22semidirect
34. C2×#327C4semidirect
37. #1444C22semidirect
38. #1442C22semidirect
41. #1294C22semidirect
42. #1294C22semidirect
43. #1280C22semidirect
82. C2×C22⋊C8C2×C4semidirect
118. C2×#95C2×C4semidirect
124. #327C2×C4semidirect
125. #327C2×C4semidirect
139. C22×D4C4⋊C4semidirect
178. #121C4⋊C4semidirect
197. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342052
#219402
#217582
C22×#3272
#139082
#88692
#88662
C2×#12942
#79072