Finite Group Databasebeta

← #116292

Subgroups — #116292

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21385C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4#14150C211yesmaximal, characteristic
5C2×#1286C211yesmaximal, characteristic
6#7901C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#389—21no—
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1989—22no—
17#1988—21no—
18#1987—21no—
19#1987—21no—
20#1981—21no—
21#1872—21no—
22#1872—21no—
23C4×#103—22no—
24#1736—21no—
25#1736—21no—
26#1518—22no—
27#1518C414yescomplemented
28C2×#328—22no—
29#1444—21no—
30#1442—21no—
31#1436—22no—
32C4⋊#95—21no—
33#1286—22no—
34#1279—22no—
35#1279—22no—
36C22×M16—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533Q812yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C42—21no—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2×C412yes—
70#423—21no—
71#423—21no—
72#423—21no—
73#423—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8D412yes—
78C2×#103—22no—
79C2×#103—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8D412yes—
84#389—42no—
85#389—42no—
86#382—21no—
87#382—21no—
88#382C2×C412yes—
89#382D412yes—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95D412yes—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#328—28no—
112C22×Q8—21no—
113C22×Q8—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—41no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8C4⋊C412yescomplemented
138C4⋊Q8C4⋊C412yescomplemented
139#125—22no—
140#125—22no—
141#125—41no—
142#125—41no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159#117—22no—
160#117—22no—
161#117—41no—
162#117—41no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4SD1612yes—
195C2×C4⋊C4Q1612yes—
196C2×C22⋊C4—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—82no—
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×Q811yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226C4⋊C8—41no—
227#103—44no—
228#103—44no—
229C22⋊C8—22no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—41no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—44no—
241#95—44no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—82no—
247#95C22⋊C414yes—
248C24#12011yescharacteristic
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—81no—
256C2×Q8—81no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4C22⋊Q811yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12411yescharacteristic
309C22×C4#12411yescharacteristic
310C22×C4#12411yescharacteristic
311C22×C4#12711yescharacteristic
312C22×C4C2×SD1611yescharacteristic
313C22×C4C2×Q1611yescharacteristic
314C22×C4#13511yescharacteristic
315C22×C4#13511yescharacteristic
316C22×C4AGL(1,ℤ/8)11yescharacteristic
317C22×C4#13711yescharacteristic
318C22×C4#13711yescharacteristic
319C22×C4#13711yescharacteristic
320M16—81no—
321M16—81no—
322M16—81no—
323M16—81no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—81no—
332C2×C8—81no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—88no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C42—42no—
369C42—42no—
370C42—82no—
371C42—82no—
372C42—81no—
373C42—81no—
374C42—81no—
375C42—81no—
376C42C2×C4⋊C411yescharacteristic
377C42#11711yescharacteristic
378C42C4×D411yescharacteristic
379C42C4×D411yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42#12111yescharacteristic
383C42C22⋊Q811yescharacteristic
384C42C22⋊Q811yescharacteristic
385C42C22⋊Q811yescharacteristic
386C42#12311yescharacteristic
387C42#12311yescharacteristic
388C42#12311yescharacteristic
389C42#12311yescharacteristic
390C42#12611yescharacteristic
391C42#12611yescharacteristic
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23#38611yescharacteristic
397C23#39211yescharacteristic
398C23#39311yescharacteristic
399C23#45011yescharacteristic
400C23#45111yescharacteristic
401C23#45211yescharacteristic
402C23#45211yescharacteristic
403Q8—44no—
404Q8—82no—
405Q8—82no—
406Q8—82no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
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—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4#38011yescharacteristic
461C2×C4#38511yescharacteristic
462C2×C4#38511yescharacteristic
463C2×C4#38811yescharacteristic
464C2×C4#38811yescharacteristic
465C2×C4#39011yescharacteristic
466C2×C4#39411yescharacteristic
467C2×C4#39411yescharacteristic
468C2×C4#39611yescharacteristic
469C2×C4#39611yescharacteristic
470C2×C4#39711yescharacteristic
471C2×C4#39911yescharacteristic
472C2×C4C4×SD1611yescharacteristic
473C2×C4C4×Q1611yescharacteristic
474C2×C4#44011yescharacteristic
475C2×C4#44111yescharacteristic
476C2×C4#44811yescharacteristic
477C2×C4#44911yescharacteristic
478C2×C4#45111yescharacteristic
479C2×C4#45211yescharacteristic
480C2×C4#46111yescharacteristic
481C2×C4#46211yescharacteristic
482C2×C4#46311yescharacteristic
483C2×C4#46411yescharacteristic
484C2×C4#46511yescharacteristic
485C2×C4#46711yescharacteristic
486C2×C4#46811yescharacteristic
487C2×C4#47011yescharacteristic
488C2×C4#47511yescharacteristic
489C2×C4#47711yescharacteristic
490C2×C4#47911yescharacteristic
491C2×C4#47911yescharacteristic
492C2×C4#48711yescharacteristic
493C2×C4#48811yescharacteristic
494C2×C4#48911yescharacteristic
495C2×C4#49111yescharacteristic
496C8—82no—
497C8—82no—
498C8—82no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22#143911yescharacteristic
512C22#187711yescharacteristic
513C22#187811yescharacteristic
514C22#200011yescharacteristic
515C22#200111yescharacteristic
516C22#200811yescharacteristic
517C22#200911yescharacteristic
518C22#201011yescharacteristic
519C22#201011yescharacteristic
520C22#202411yescharacteristic
521C22#202611yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—84no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4#186311yescharacteristic
538C4#186511yescharacteristic
539C4#193311yescharacteristic
540C4#193511yescharacteristic
541C4#204211yescharacteristic
542C4#204311yescharacteristic
543C4#206611yescharacteristic
544C4#206811yescharacteristic
545C2—21no—
546C2—21no—
547C2—21no—
548C2—21no—
549C2#1000011yescharacteristic
550C2#1000111yescharacteristic
551C2#1000211yescharacteristic
552C2#1000311yescharacteristic
553C2#1106711yescharacteristic
554C2#1108311yescharacteristic
555C2#1109311yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1518C4semidirect
137. C4⋊Q8C4⋊C4semidirect
138. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#213852
#213122
C2×#15182
#141502
C2×#12862
#79012
#78922