Finite Group Databasebeta

← #116811

Subgroups — #116811

Order 512.

780 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21170C211yesmaximal, characteristic
2#20734C211yesmaximal, characteristic
3C2×#1487C211yesmaximal, characteristic
4#13925C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1949—22no—
17#1948—21no—
18#1947—21no—
19#1824—21no—
20#1803—21no—
21#1803—21no—
22#1787—21no—
23C4×#103—22no—
24#1736—21no—
25#1487—22no—
26#1487C414yescomplemented
27C2×Q8⋊C8—22no—
28#1446—21no—
29#1445—21no—
30#1436—22no—
31C4⋊#95—21no—
32C4⋊#95—21no—
33#1294—22no—
34#1286—22no—
35#1279—22no—
36C23×Q8D411yescharacteristic
37C22×M16—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8Q812yes—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C42—21no—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#428—42no—
68#428—42no—
69C2×#107—22no—
70C2×#106—22no—
71#423—21no—
72#423—21no—
73#423C2×C412yes—
74#423D412yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8D412yes—
79C2×#103—22no—
80C2×#103—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8D412yes—
85#384D411yescharacteristic
86#384D411yescharacteristic
87#382—21no—
88#382—21no—
89#382—21no—
90#382—21no—
91#382—21no—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111Q8⋊C8—28no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8C4⋊C412yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—41no—
133C2×M16—41no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—41no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153#117—41no—
154#117—41no—
155#117—41no—
156#117—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
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—22no—
170C2×C4⋊C4—22no—
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—21no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C4⋊C412yes—
191C2×C4⋊C4SD1612yes—
192C2×C4⋊C4Q1612yes—
193C2×C22⋊C4—41no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—82no—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q811yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216#107—44no—
217#106—44no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8C22⋊C414yes—
226#103—44no—
227#103—44no—
228C22⋊C8—22no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—41no—
236#95—22no—
237#95—22no—
238#95—44no—
239#95—44no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—82no—
247#95—82no—
248C24#12011yescharacteristic
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—44no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—41no—
264C2×Q8—41no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4C2×C4⋊C411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4C22⋊Q811yescharacteristic
314C22×C4C22⋊Q811yescharacteristic
315C22×C4#12311yescharacteristic
316C22×C4#12311yescharacteristic
317C22×C4#12311yescharacteristic
318C22×C4#12411yescharacteristic
319C22×C4#12711yescharacteristic
320C22×C4C2×SD1611yescharacteristic
321C22×C4C2×Q1611yescharacteristic
322C22×C4#13511yescharacteristic
323C22×C4#13511yescharacteristic
324C22×C4AGL(1,ℤ/8)11yescharacteristic
325C22×C4#13711yescharacteristic
326C22×C4#13711yescharacteristic
327C22×C4#13711yescharacteristic
328M16—42no—
329M16—42no—
330M16—42no—
331M16—42no—
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—81no—
340C2×C8—81no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C22⋊C4—81no—
370C22⋊C4—81no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—82no—
376C42—82no—
377C42—81no—
378C42—81no—
379C42C2×C22⋊C411yescharacteristic
380C42#11711yescharacteristic
381C42C4×D411yescharacteristic
382C42C4×D411yescharacteristic
383C42#12111yescharacteristic
384C42#12111yescharacteristic
385C42C22⋊Q811yescharacteristic
386C42C22⋊Q811yescharacteristic
387C42C22⋊Q811yescharacteristic
388C42C22⋊Q811yescharacteristic
389C42#12311yescharacteristic
390C42#12311yescharacteristic
391C42#12411yescharacteristic
392C42#12411yescharacteristic
393C42#12611yescharacteristic
394C42#12611yescharacteristic
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23#38511yescharacteristic
400C23#39211yescharacteristic
401C23#39411yescharacteristic
402C23#44811yescharacteristic
403C23#44911yescharacteristic
404C23#45111yescharacteristic
405C23#45211yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413Q8—42no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
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—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4#38011yescharacteristic
468C2×C4#38511yescharacteristic
469C2×C4#38611yescharacteristic
470C2×C4#38811yescharacteristic
471C2×C4#38811yescharacteristic
472C2×C4#39011yescharacteristic
473C2×C4#39311yescharacteristic
474C2×C4#39411yescharacteristic
475C2×C4#39611yescharacteristic
476C2×C4#39611yescharacteristic
477C2×C4#39711yescharacteristic
478C2×C4#39911yescharacteristic
479C2×C4C4×SD1611yescharacteristic
480C2×C4C4×Q1611yescharacteristic
481C2×C4#44011yescharacteristic
482C2×C4#44111yescharacteristic
483C2×C4#45011yescharacteristic
484C2×C4#45111yescharacteristic
485C2×C4#45211yescharacteristic
486C2×C4#45211yescharacteristic
487C2×C4#46111yescharacteristic
488C2×C4#46211yescharacteristic
489C2×C4#46311yescharacteristic
490C2×C4#46411yescharacteristic
491C2×C4#46511yescharacteristic
492C2×C4#46711yescharacteristic
493C2×C4#46811yescharacteristic
494C2×C4#47011yescharacteristic
495C2×C4#47511yescharacteristic
496C2×C4#47711yescharacteristic
497C2×C4#47911yescharacteristic
498C2×C4#47911yescharacteristic
499C2×C4#48711yescharacteristic
500C2×C4#48811yescharacteristic
501C2×C4#48911yescharacteristic
502C2×C4#49111yescharacteristic
503C8—44no—
504C8—82no—
505C8—82no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22#143911yescharacteristic
519C22#186311yescharacteristic
520C22#186511yescharacteristic
521C22#200011yescharacteristic
522C22#200111yescharacteristic
523C22#200811yescharacteristic
524C22#200911yescharacteristic
525C22#201011yescharacteristic
526C22#201011yescharacteristic
527C22#204211yescharacteristic
528C22#204311yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—44no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—162no—
542C4—161no—
543C4—161no—
544C4#187711yescharacteristic
545C4#187811yescharacteristic
546C4#193311yescharacteristic
547C4#193511yescharacteristic
548C4#202411yescharacteristic
549C4#202611yescharacteristic
550C4#206611yescharacteristic
551C4#206811yescharacteristic
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2#999811yescharacteristic
557C2#999911yescharacteristic
558C2#1000211yescharacteristic
559C2#1000311yescharacteristic
560C2#1106711yescharacteristic
561C2#1110011yescharacteristic
562C2#1111711yescharacteristic
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
26. #1487C4semidirect

Maximal subgroups

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

subgroupindex
#211702
#207342
C2×#14872
#139252
#80262
C2×#12862
#78922