Finite Group Databasebeta

← #109706

Subgroups — #109706

Order 512.

910 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#21940C211yesmaximal, characteristic
2#13802C211yesmaximal, characteristic
3#10854C211yesmaximal, characteristic
4#8023C211yesmaximal, characteristic
5C2×#1294C211yesmaximal, characteristic
6C2×#1294C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊C8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C2×#380C2211yescharacteristic
11C2×#380C2211yescharacteristic
12C2×#380C2211yescharacteristic
13C2×C4×C22⋊C4—21no—
14C2×C4×C22⋊C4C2211yescharacteristic
15C22×#95C2211yescharacteristic
16#1852—22no—
17#1852C414yescomplemented
18#1808—22no—
19C2×#342—21no—
20C2×#342—21no—
21#1436—22no—
22#1434—21no—
23C22⋊#95—21no—
24C4×#95—21no—
25#1294—22no—
26#1294—21no—
27#1294—21no—
28#1294C414yescomplemented
29#1282C414yescomplemented
30C24×C4—21no—
31C24×C4C2311yescharacteristic
32C23×C8—21no—
33C23×C8—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4D412yes—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C22⋊C4—21no—
40C22×C22⋊C4D411yescharacteristic
41C22×C22⋊C4D411yescharacteristic
42C22×C42—21no—
43C22×C42—22no—
44C2×#107—24no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8C2×C412yes—
54C2×C22⋊C8C2×C412yes—
55C2×C22⋊C8C2×C412yes—
56C2×C22⋊C8C2×C412yes—
57#380—21no—
58#380—21no—
59#380—21no—
60#380—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380—21no—
65#380—21no—
66#380C2×C412yes—
67#380C2×C412yes—
68#380D412yes—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—42no—
72C4×C22⋊C4—42no—
73C4×C22⋊C4C2×C412yes—
74C4×C22⋊C4Q812yes—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95C2×C412yes—
86C2×#95D411yescharacteristic
87C2×#95D411yescharacteristic
88C2×#95Q812yes—
89#342—42no—
90#342—42no—
91#340—24no—
92C25C2×D411yescharacteristic
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—44no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—82no—
108C23×C4—82no—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×Q811yescharacteristic
116C23×C4C2×Q811yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—82no—
130C22×C8—82no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—82no—
137C2×C4⋊C4—82no—
138C2×C4⋊C4—82no—
139C2×C4⋊C4C22⋊C414yes—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4C4⋊C412yes—
167C2×C22⋊C4C4⋊C412yes—
168C2×C22⋊C4D812yes—
169C2×C22⋊C4Q1612yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—44no—
173C2×C42—44no—
174C2×C42—82no—
175C2×C42—82no—
176#107—84no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8C4214yescomplemented
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—21no—
192#95—21no—
193#95—22no—
194#95—44no—
195#95—44no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—82no—
200#95C4⋊C412yes—
201#95C4⋊C412yes—
202C24—42no—
203C24—42no—
204C24C2×C4⋊C411yescharacteristic
205C24C4×D411yescharacteristic
206C24C4×D411yescharacteristic
207C24#12011yescharacteristic
208C24#12111yescharacteristic
209C24C22⋊Q811yescharacteristic
210C24#12311yescharacteristic
211C24C2×D811yescharacteristic
212C24C2×Q1611yescharacteristic
213C24#13511yescharacteristic
214C24#13511yescharacteristic
215C24AGL(1,ℤ/8)11yescharacteristic
216C24AGL(1,ℤ/8)11yescharacteristic
217C24#13711yescharacteristic
218C24#13711yescharacteristic
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C2×C4211yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C4⋊C411yescharacteristic
285C22×C4#11711yescharacteristic
286C22×C4#11711yescharacteristic
287C22×C4#11711yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C4×Q811yescharacteristic
295C22×C4C4×Q811yescharacteristic
296C22×C4C4×Q811yescharacteristic
297C22×C4C4×Q811yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4C22⋊Q811yescharacteristic
300C22×C4C22⋊Q811yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4#12311yescharacteristic
303C22×C4#12311yescharacteristic
304C22×C4#12411yescharacteristic
305C22×C4#12511yescharacteristic
306C22×C4#12511yescharacteristic
307C22×C4#12611yescharacteristic
308C22×C4#12611yescharacteristic
309C22×C4C4⋊Q811yescharacteristic
310C2×C8—24no—
311C2×C8—44no—
312C2×C8—42no—
313C2×C8—82no—
314C2×C8—82no—
315C2×C8—82no—
316C2×C8—82no—
317C2×C8—82no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—81no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C22⋊C4—81no—
342C22⋊C4—81no—
343C22⋊C4#10714yescomplemented
344C42—82no—
345C42—82no—
346C42—88no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23C4×C4⋊C411yescharacteristic
370C23#38511yescharacteristic
371C23#38511yescharacteristic
372C23#38611yescharacteristic
373C23#38711yescharacteristic
374C23#38811yescharacteristic
375C23#38911yescharacteristic
376C23C2×#10711yescharacteristic
377C23#42711yescharacteristic
378C23#42811yescharacteristic
379C23#42811yescharacteristic
380C23C4×D811yescharacteristic
381C23C4×D811yescharacteristic
382C23C4×Q1611yescharacteristic
383C23C4×Q1611yescharacteristic
384C23#44011yescharacteristic
385C23#44011yescharacteristic
386C23#44111yescharacteristic
387C23#44211yescharacteristic
388C23#44711yescharacteristic
389C23#44911yescharacteristic
390C23#45111yescharacteristic
391C23#45211yescharacteristic
392C23#45911yescharacteristic
393C23#46211yescharacteristic
394C23#46311yescharacteristic
395C23#46411yescharacteristic
396C23#47411yescharacteristic
397C23#47711yescharacteristic
398C23#47811yescharacteristic
399C23#47911yescharacteristic
400C23#48011yescharacteristic
401C23#48211yescharacteristic
402C23#48411yescharacteristic
403C23#48511yescharacteristic
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
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—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—162no—
470C2×C4—162no—
471C2×C4—162no—
472C2×C4—162no—
473C2×C4C4×C22⋊C411yescharacteristic
474C2×C4C4×C4⋊C411yescharacteristic
475C2×C4#38711yescharacteristic
476C2×C4#38711yescharacteristic
477C2×C4#38711yescharacteristic
478C2×C4#38811yescharacteristic
479C2×C4#38911yescharacteristic
480C2×C4#39111yescharacteristic
481C8—44no—
482C8—82no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22#143611yescharacteristic
524C22C4×#10711yescharacteristic
525C22#177611yescharacteristic
526C22#185211yescharacteristic
527C22#185411yescharacteristic
528C22#186211yescharacteristic
529C22#186311yescharacteristic
530C22#187811yescharacteristic
531C22#187911yescharacteristic
532C22#190612yes—
533C22#191312yes—
534C22#191811yescharacteristic
535C22#192111yescharacteristic
536C22#192511yescharacteristic
537C22#192811yescharacteristic
538C22#194711yescharacteristic
539C22#194911yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—44no—
549C4—44no—
550C4—82no—
551C4—82no—
552C4—84no—
553C4—162no—
554C4—162no—
555C2—21no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2#927612yes—
568C2#927712yes—
569C2#1089511yescharacteristic
570C2#1090711yescharacteristic
571C2#1091511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1852C4semidirect
28. #1294C4semidirect
29. #1282C4semidirect
184. C22⋊C8C42semidirect
343. C22⋊C4#107semidirect

Maximal subgroups

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

subgroupindex
#219402
#138022
#108542
#80232
C2×#12942
C2×#12822