Finite Group Databasebeta

← #198399

Subgroups — #198399

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#21804C211yesmaximal, characteristic, complemented
4#21212C211yesmaximal, characteristic
5#17574C211yesmaximal, characteristic
6#9262C218yesmaximal, complemented
7#7981C211yesmaximal, characteristic, complemented
8#7924C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3206—22no—
12#3206—22no—
13#3206—22no—
14#3206—22no—
15C2×#492C412yescomplemented
16C2×#487C412yescomplemented
17C2×#461C414yescomplemented
18C2×#448C2214yes—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#410—21no—
22C2×#410—21no—
23#2369—22no—
24#2365—22no—
25#2345—22no—
26Q8×C22⋊C4C2214yes—
27#2331—22no—
28#2316—21no—
29C4×C4⋊Q8—22no—
30#2285C2211yescharacteristic
31#2285C2211yescharacteristic
32#2274—21no—
33C4×#117C2211yescharacteristic
34#1981—22no—
35#1964—22no—
36#1964C2214yes—
37#1956—22no—
38#1787C2214yes—
39#1519C2214yes—
40#1517—22no—
41#1502—22no—
42#1289C2214yes—
43#1282C2214yes—
44C22×#41—21no—
45C23×Q8C2312yes—
46C22×SD16C2×C418yes—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8D412yes—
66C2×C4⋊Q8D412yes—
67C2×C4⋊Q8D412yes—
68C2×#127C2×C411yescharacteristic
69C2×#124—21no—
70C2×#121—21no—
71C2×#121C2312yes—
72C2×C4×Q8—24no—
73C2×C4×D4—21no—
74C2×C4×D4C2×C412yes—
75C2×#117—21no—
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C42C2311yescharacteristic
82#492—24no—
83#492—24no—
84#487—24no—
85#487—24no—
86#461—28no—
87#461—28no—
88#448—28no—
89#448—28no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2×C412yes—
100C2×#103—28no—
101C2×#103C2×C418yes—
102#410—44no—
103#410—44no—
104C2×#99—24no—
105C2×#99—24no—
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2312yes—
108C2×C4×C8—21no—
109C2×C4×C8C2×C412yes—
110#391—24no—
111#391—28no—
112#389—24no—
113#389—24no—
114#389—24no—
115#387—28no—
116#384—21no—
117#384—21no—
118#382—21no—
119#382D412yes—
120#380—21no—
121#380—24no—
122#380—21no—
123#380C2312yes—
124#380C2312yes—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D412yes—
134C4×C22⋊C4—24no—
135C4×C22⋊C4D412yes—
136C4×C22⋊C4C2312yes—
137#376—21no—
138C43—21no—
139#327—28no—
140C2×#41—41no—
141C2×#41—41no—
142C2×#41—41no—
143C2×#41—41no—
144C2×#41—42no—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—42no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—84no—
153C22×Q8C22×C414yes—
154C22×Q8C2×D414yes—
155C22×Q8C2×D414yes—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22×C412yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2411yescharacteristic
165C2×SD16—216no—
166C2×SD16—88no—
167C22×C8—22no—
168C22×C8C22×C414yes—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8C22⋊C414yes—
186C4⋊Q8C22⋊C414yes—
187C4⋊Q8C22⋊C414yes—
188C4⋊Q8C22⋊C414yes—
189#127—21no—
190#127—21no—
191#127—21no—
192#127—21no—
193#124—41no—
194#124—41no—
195#124—41no—
196#124—41no—
197#121—22no—
198#121—22no—
199#121—22no—
200#121—22no—
201#121—41no—
202#121—41no—
203#121—41no—
204#121—41no—
205C4×Q8—88no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—42no—
218#117—42no—
219#117—42no—
220#117—42no—
221#117SD1614yescomplemented
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4C22×C412yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4C2×D412yes—
250C2×C22⋊C4C2×D412yes—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—28no—
257C2×C42—41no—
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42#4111yescharacteristic
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274#103—216no—
275#103—88no—
276#99—48no—
277#99—48no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—41no—
285C4×C8—41no—
286#95—28no—
287#95—28no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—41no—
293#95C2×D414yes—
294C24—22no—
295C24C22×D411yescharacteristic
296#41—82no—
297#41—82no—
298#41—82no—
299#41—82no—
300#41—82no—
301#41—82no—
302#41—82no—
303#41—82no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—48no—
308C2×Q8—48no—
309C2×Q8—48no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—84no—
315C2×Q8—84no—
316C2×Q8—84no—
317C2×Q8C2×C22⋊C418yes—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—44no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—82no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—84no—
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4#12011yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4C2×SD1612yes—
362C22×C4#13512yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4#13712yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371SD16—816no—
372C2×C8—24no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—42no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—82no—
395C4⋊C4—88no—
396C4⋊C4—82no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4C2×SD1614yes—
409C42—24no—
410C42—44no—
411C42—44no—
412C42—42no—
413C42—44no—
414C42—88no—
415C42—82no—
416C42C2×C22⋊C411yescharacteristic
417C42C2×C22⋊C411yescharacteristic
418C42C2×C22⋊C411yescharacteristic
419C42C2×C22⋊C411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42C4×D411yescharacteristic
423C42C4×D411yescharacteristic
424C42#12111yescharacteristic
425C42#12111yescharacteristic
426C42#12111yescharacteristic
427C42#12111yescharacteristic
428C42#12311yescharacteristic
429C42#12311yescharacteristic
430C42#12311yescharacteristic
431C42#12311yescharacteristic
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—82no—
438C23—82no—
439C23C2×C4×D411yescharacteristic
440C23C2×#12011yescharacteristic
441C23C2×#12111yescharacteristic
442C23C22×SD1611yescharacteristic
443C23C2×#13511yescharacteristic
444C23C2×AGL(1,ℤ/8)11yescharacteristic
445C23C2×#13711yescharacteristic
446Q8—216no—
447Q8—48no—
448Q8—84no—
449Q8—84no—
450Q8—82no—
451Q8—82no—
452D4—44no—
453D4—82no—
454D4—82no—
455D4—82no—
456D4—82no—
457D4—82no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—82no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—48no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4#38611yescharacteristic
496C2×C4#38611yescharacteristic
497C2×C4#38611yescharacteristic
498C2×C4#38611yescharacteristic
499C2×C4#38611yescharacteristic
500C2×C4#38611yescharacteristic
501C2×C4#38611yescharacteristic
502C2×C4#38611yescharacteristic
503C2×C4C4×SD1612yes—
504C2×C4C4×SD1612yes—
505C2×C4#44012yes—
506C2×C4#44012yes—
507C2×C4#44812yes—
508C2×C4#44812yes—
509C2×C4#44912yes—
510C2×C4#44912yes—
511C2×C4#46012yes—
512C2×C4#46012yes—
513C2×C4#46412yes—
514C2×C4#46412yes—
515C2×C4C22×C22⋊C411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×#12311yescharacteristic
519C8—44no—
520C8—82no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—162no—
535C22—162no—
536C22C2×#38611yescharacteristic
537C22C2×C4×SD1611yescharacteristic
538C22C2×#44011yescharacteristic
539C22C2×#44811yescharacteristic
540C22C2×#44911yescharacteristic
541C22C2×#46011yescharacteristic
542C22C2×#46411yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—44no—
550C4—48no—
551C4—82no—
552C4—84no—
553C4—82no—
554C4—162no—
555C4#188012yes—
556C4#188012yes—
557C4#188012yes—
558C4#188012yes—
559C4#190512yes—
560C4#288312yes—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—162no—
568C2C2×#188011yescharacteristic
569C2#2098511yescharacteristic
570C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #21804C2semidirect
6. #9262C2semidirect
7. #7981C2semidirect
8. #7924C2semidirect
15. C2×#492C4semidirect
16. C2×#487C4semidirect
17. C2×#461C4semidirect
221. #117SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
#342162
#218042
#212122
#175742
#92622
#79812
#79242