Finite Group Databasebeta

← #188533

Subgroups — #188533

Order 512.

1040 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34224C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3C2×#1512C211yesmaximal, characteristic
4C2×#1509C211yesmaximal, characteristic
5#13955C211yesmaximal, characteristic, complemented
6#8576C212yesmaximal, complemented
7#8575C212yesmaximal
8#8574C212yesmaximal, complemented
9#8573C212yesmaximal
10#8021C211yesmaximal, characteristic, complemented
11#8017C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15#2373—22no—
16#2370C412yescomplemented
17#2367C412yescomplemented
18#2357C411yescharacteristic, complemented
19#2356C411yescharacteristic, complemented
20#2351C411yescharacteristic, complemented
21#2349C411yescharacteristic, complemented
22#2347—21no—
23#2343—21no—
24#2322—22no—
25#2318—21no—
26C4×C22⋊Q8C2212yes—
27C4×#121C2212yes—
28#2283C2211yescharacteristic
29C2×#382C2211yescharacteristic
30#2275—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1940—21no—
34#1940—21no—
35#1940—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1512—24no—
39#1509—24no—
40C2×#328C2211yescharacteristic
41C2×#328C2212yes—
42C2×#328C2211yescharacteristic
43C2×#327C2211yescharacteristic
44C2×#327C2212yes—
45C2×#327C2211yescharacteristic
46#1442—21no—
47#1442—21no—
48C4⋊#95—22no—
49#1434C2212yes—
50#1434C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55#529—22no—
56#529—22no—
57#529D414yes—
58#529Q814yes—
59C2×#126—21no—
60C2×#125C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#123—21no—
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121C2311yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C412yes—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#432—22no—
79#432—22no—
80#432—22no—
81#432—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C22⋊C8C2311yescharacteristic
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2311yescharacteristic
89C2×#97—21no—
90C2×#97—21no—
91#391C2×C412yes—
92#390C2×C412yes—
93#389—22no—
94#389—22no—
95#389C2×C412yes—
96#389C2×C412yes—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#386—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#383—21no—
110#383—21no—
111#382—21no—
112#382—22no—
113#382—22no—
114#382C2×C411yescharacteristic
115#382C2×C411yescharacteristic
116#381—21no—
117#381—21no—
118#380—21no—
119#380C2312yes—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D412yes—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—22no—
139#376—21no—
140#376—21no—
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152#336—22no—
153#336—22no—
154#328—24no—
155#328—22no—
156#328—22no—
157#327—24no—
158#327—22no—
159#327—22no—
160C22×Q8C22×C411yescharacteristic
161C22×D4—21no—
162C22×D4C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—21no—
176#126—24no—
177#126—24no—
178#125—22no—
179#125C4⋊C414yes—
180#124—22no—
181#124C4⋊C414yes—
182#123—24no—
183#123—24no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8C2×D412yes—
187C22⋊Q8C2×Q812yes—
188#121—21no—
189#121—21no—
190#121C2×D412yes—
191#121C2×Q812yes—
192C4×Q8—22no—
193C4×Q8—82no—
194C4×Q8C4⋊C414yes—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—22no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—82no—
201C4×D4C4⋊C414yes—
202#117—24no—
203#117—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—82no—
242C2×C42C22×C412yes—
243C2×C42C22×C412yes—
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×Q812yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—82no—
257C4⋊C8—82no—
258C22⋊C8—22no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—22no—
263C22⋊C8—21no—
264#97—42no—
265#97—42no—
266#95—21no—
267#95—22no—
268#95—21no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95—21no—
283#95—21no—
284#95—44no—
285#95—44no—
286#95—42no—
287#95—82no—
288#95—82no—
289#95C22⋊C411yescharacteristic
290#95C22⋊C412yes—
291#95C22⋊C411yescharacteristic
292#95C22⋊C411yescharacteristic
293#95C22⋊C411yescharacteristic
294#95C22⋊C412yes—
295C24—21no—
296C24C22×D411yescharacteristic
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8C2×C4⋊C412yes—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4C2×C4⋊C412yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4C22⋊Q812yes—
360C22×C4C22⋊Q812yes—
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—84no—
394C4⋊C4C2×C4⋊C412yes—
395C4⋊C4C2×C4⋊C412yes—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C42—24no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—44no—
414C42—82no—
415C42—82no—
416C42—82no—
417C42—84no—
418C42C2×C4⋊C412yes—
419C42C2×C4⋊C412yes—
420C42#11712yes—
421C42#11712yes—
422C42#12112yes—
423C42#12112yes—
424C42C22⋊Q812yes—
425C42C22⋊Q812yes—
426C23—21no—
427C23—21no—
428C23—22no—
429C23—22no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23C22×C22⋊C411yescharacteristic
435C23C2×#12111yescharacteristic
436C23C2×C22⋊Q811yescharacteristic
437C23#57512yes—
438C23#57512yes—
439Q8—42no—
440Q8—42no—
441D4—42no—
442D4—42no—
443D4—44no—
444D4—42no—
445D4—42no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—84no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—164no—
511C2×C4#38012yes—
512C2×C4#38012yes—
513C2×C4#38012yes—
514C2×C4#38012yes—
515C2×C4C22×C4⋊C411yescharacteristic
516C2×C4C2×#11711yescharacteristic
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×C22⋊Q811yescharacteristic
519C8—81no—
520C8—81no—
521C8—82no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—82no—
534C22—82no—
535C22C2×#38011yescharacteristic
536C22#288311yescharacteristic
537C22#288411yescharacteristic
538C22#289612yes—
539C22#290112yes—
540C22#304011yescharacteristic
541C22#304011yescharacteristic
542C22#308012yes—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—84no—
559C4—82no—
560C4—82no—
561C4—161no—
562C4—161no—
563C2—21no—
564C2—21no—
565C2—22no—
566C2—82no—
567C2#1762412yes—
568C2#1762512yes—
569C2#2063411yescharacteristic
570C2#2064911yescharacteristic
571C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21955C2semidirect
5. #13955C2semidirect
6. #8576C2semidirect
8. #8574C2semidirect
10. #8021C2semidirect
11. #8017C2semidirect
16. #2370C4semidirect
17. #2367C4semidirect
18. #2357C4semidirect
19. #2356C4semidirect
20. #2351C4semidirect
21. #2349C4semidirect

Maximal subgroups

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

subgroupindex
#342242
#219552
C2×#15122
C2×#15092
#139552
#85762
#85752
#85742
#85732
#80212
#80172