Finite Group Databasebeta

← #206014

Subgroups — #206014

Order 512.

1055 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2#20826C211yesmaximal, characteristic, complemented
3C8×#117C211yesmaximal, characteristic, complemented
4#17545C211yesmaximal, characteristic, complemented
5#17544C211yesmaximal, characteristic
6#17385C211yesmaximal, characteristic, complemented
7#17384C211yesmaximal, characteristic
8#9813C218yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3167—21no—
12#3167—21no—
13#3167—21no—
14#3167—21no—
15#3167—22no—
16#3167—22no—
17C2×#498C411yescharacteristic, complemented
18C2×#492C411yescharacteristic, complemented
19C2×#487C411yescharacteristic, complemented
20C2×#486C411yescharacteristic, complemented
21C2×#465C2214yes—
22C2×C4×SD16C414yescomplemented
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C4×#117C2211yescharacteristic
28#1848—21no—
29#1837—21no—
30#1810C2214yes—
31C4×#108—22no—
32C8×C4⋊C4—22no—
33C8×C22⋊C4C2214yes—
34#1744—21no—
35C42×C8—21no—
36#1508C2214yes—
37#1507C2214yes—
38#1506—22no—
39#1505—22no—
40#1503—22no—
41#1502—22no—
42#1472C2214yes—
43#1471C2214yes—
44#1470—22no—
45#1469—22no—
46#1461—22no—
47#1460—22no—
48C22×#41—21no—
49C22×SD16C2×C414yes—
50C23×C8C2312yes—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×#127C2×C411yescharacteristic
71C2×#125—21no—
72C2×#124—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8C2312yes—
75C2×#121—21no—
76C2×#121C2312yes—
77C2×C4×Q8—21no—
78C2×C4×Q8C2×C412yes—
79C2×C4×D4—21no—
80C2×C4×D4C2×C412yes—
81C2×#117—21no—
82C2×#117—21no—
83C22×C42C2311yescharacteristic
84#498—21no—
85#498—22no—
86#498—21no—
87#492—21no—
88#492—22no—
89#492—21no—
90#487—21no—
91#487—21no—
92#487—22no—
93#486—21no—
94#486—21no—
95#486—22no—
96#465—24no—
97#465—24no—
98#465—24no—
99#465—24no—
100C4×SD16—24no—
101C4×SD16—24no—
102C4×SD16—24no—
103C4×SD16—24no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431D412yes—
115#431D412yes—
116C2×#108—24no—
117C2×#106C2×C414yes—
118#423—21no—
119#423—21no—
120C2×C4⋊C8—21no—
121C2×#103C2×C414yes—
122C2×#102C2×C414yes—
123#407C2312yes—
124#407C2312yes—
125C2×C22⋊C8C2312yes—
126C4×M16—21no—
127C4×M16—21no—
128C2×C4×C8—21no—
129C2×C4×C8—24no—
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C411yescharacteristic
132C2×C4×C8C2×C412yes—
133C2×C4×C8D411yescharacteristic
134C2×C4×C8D411yescharacteristic
135C2×C4×C8D411yescharacteristic
136C2×C4×C8D411yescharacteristic
137C2×C4×C8D412yes—
138C2×C4×C8D412yes—
139C4×C4⋊C4—21no—
140C4×C4⋊C4D412yes—
141C4×C22⋊C4—21no—
142C4×C22⋊C4C2312yes—
143#376D412yes—
144C43—21no—
145#341—24no—
146#336—24no—
147#328—28no—
148#327—28no—
149#324—28no—
150#323—28no—
151C2×#41—41no—
152C2×#41—41no—
153C2×#41—41no—
154C2×#41—41no—
155C2×#41—84no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8C22×C412yes—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C22×C412yes—
163C23×C4—22no—
164C23×C4#4111yescharacteristic
165C23×C4#4111yescharacteristic
166C23×C4C2411yescharacteristic
167C2×SD16—24no—
168C2×SD16—24no—
169C2×SD16—48no—
170C2×M16—22no—
171C2×M16—22no—
172C22×C8—22no—
173C22×C8—44no—
174C22×C8C22×C412yes—
175C22×C8C22×C412yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192#127—21no—
193#127—21no—
194#127—21no—
195#127—21no—
196#125—41no—
197#125—41no—
198#125—41no—
199#125—41no—
200#124—41no—
201#124—41no—
202#124—41no—
203#124—41no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212#121—22no—
213#121—22no—
214#121—22no—
215#121—22no—
216#121—41no—
217#121—41no—
218#121—41no—
219#121—41no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236#117—28no—
237#117—41no—
238#117—41no—
239#117—41no—
240#117—41no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
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×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42#4111yescharacteristic
270#108—48no—
271#106—24no—
272#106—24no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279#103—24no—
280#103—24no—
281#102—24no—
282#102—24no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8C2×D414yes—
289#97—41no—
290#97—41no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—48no—
294C4×C8—41no—
295C4×C8—41no—
296C4×C8C22⋊C412yes—
297C4×C8C22⋊C412yes—
298C4×C8C22⋊C412yes—
299C4×C8C22⋊C412yes—
300C4×C8C22⋊C411yescharacteristic
301C4×C8C22⋊C411yescharacteristic
302C4×C8C22⋊C412yes—
303C4×C8C22⋊C411yescharacteristic
304C4×C8C22⋊C412yes—
305C4×C8C22⋊C411yescharacteristic
306#95—22no—
307#95C2×D414yes—
308C24—22no—
309C24C2×#4111yescharacteristic
310#41—84no—
311#41—84no—
312#41—84no—
313#41—84no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—44no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—44no—
322C2×Q8—82no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—44no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—44no—
333C2×D4—82no—
334C2×D4—84no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—84no—
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4C4×D411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12411yescharacteristic
354C22×C4#12411yescharacteristic
355C22×C4#12411yescharacteristic
356C22×C4#12411yescharacteristic
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363SD16—48no—
364SD16—48no—
365M16—44no—
366M16—44no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—44no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8C2×C22⋊C412yes—
378C2×C8C2×C22⋊C412yes—
379C2×C8C2×C22⋊C412yes—
380C2×C8C2×C22⋊C412yes—
381C4⋊C4—28no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C22⋊C4—28no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—28no—
406C42—44no—
407C42—42no—
408C42—42no—
409C42—44no—
410C42C2×C22⋊C411yescharacteristic
411C42C2×C22⋊C411yescharacteristic
412C42C2×C22⋊C411yescharacteristic
413C42C2×C22⋊C411yescharacteristic
414C42C4×D411yescharacteristic
415C42C4×D411yescharacteristic
416C42C4×D411yescharacteristic
417C42C4×D411yescharacteristic
418C42#12111yescharacteristic
419C42#12111yescharacteristic
420C42#12111yescharacteristic
421C42#12111yescharacteristic
422C42#12711yescharacteristic
423C42#12711yescharacteristic
424C42#12711yescharacteristic
425C42#12711yescharacteristic
426C23—22no—
427C23—22no—
428C23—41no—
429C23—42no—
430C23—44no—
431C23—82no—
432C23—82no—
433C23C2×C4×D411yescharacteristic
434C23C2×#12111yescharacteristic
435C23C2×#12411yescharacteristic
436Q8—44no—
437Q8—82no—
438Q8—82no—
439Q8—82no—
440Q8—82no—
441Q8—82no—
442D4—44no—
443D4—82no—
444D4—82no—
445D4—84no—
446D4—84no—
447D4—82no—
448D4—82no—
449D4—82no—
450C2×C4—22no—
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—28no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4#39011yescharacteristic
478C2×C4#39011yescharacteristic
479C2×C4#39011yescharacteristic
480C2×C4#39011yescharacteristic
481C2×C4#39011yescharacteristic
482C2×C4#39011yescharacteristic
483C2×C4#39011yescharacteristic
484C2×C4#39011yescharacteristic
485C2×C4#44314yes—
486C2×C4#47114yes—
487C2×C4C22×C22⋊C411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12711yescharacteristic
491C8—22no—
492C8—22no—
493C8—22no—
494C8—44no—
495C8—82no—
496C8—82no—
497C8#41014yes—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—44no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—164no—
510C22C2×#39011yescharacteristic
511C22C2×#44312yes—
512C22C2×#47112yes—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—48no—
521C4—82no—
522C4—82no—
523C4—82no—
524C4#197412yes—
525C4#197412yes—
526C4#197712yes—
527C4#197712yes—
528C4C2×#41012yes—
529C2—22no—
530C2—22no—
531C2—41no—
532C2—162no—
533C2#2121211yescharacteristic
534C2C2×#197411yescharacteristic
535C2C2×#197711yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#390
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20826C2semidirect
3. C8×#117C2semidirect
4. #17545C2semidirect
6. #17385C2semidirect
8. #9813C2semidirect
17. C2×#498C4semidirect
18. C2×#492C4semidirect
19. C2×#487C4semidirect
20. C2×#486C4semidirect
22. C2×C4×SD16C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
#208262
C8×#1172
#175452
#175442
#173852
#173842
#98132