Finite Group Databasebeta

← #201537

Subgroups — #201537

Order 512.

1467 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic, complemented
2#34935C211yesmaximal, characteristic
3#20637C211yesmaximal, characteristic, complemented
4C2×#1486C212yesmaximal, complemented
5#9490C214yesmaximal, complemented
6#9489C214yesmaximal, complemented
7#7957C212yesmaximal, complemented
8C2×#525—21no—
9C2×C4×#41C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2989—24no—
12#2983C2212yescomplemented
13#2975C2212yescomplemented
14#2973—22no—
15#2965—24no—
16#2920—22no—
17C2×#431C2212yes—
18C2×#423C2212yes—
19C2×#413—21no—
20#2538—21no—
21#2537—21no—
22#2536C2212yes—
23#2535C2212yes—
24#2534—22no—
25#2486C412yes—
26#2408C412yes—
27#2392C414yes—
28C4×#121—22no—
29#2285C2211yescharacteristic
30#2280—21no—
31#1801—21no—
32#1801C2212yescomplemented
33#1800—21no—
34#1800C2212yescomplemented
35#1742—21no—
36#1486—24no—
37#1486C2218yes—
38C2×D4⋊C8—24no—
39#1286C2218yescomplemented
40C22×#41D411yescharacteristic
41C23×D4C2×C411yescharacteristic
42C24×C4—21no—
43C22×M16—22no—
44#525—48no—
45C2×#123—22no—
46C2×C22⋊Q8—22no—
47C2×#121—22no—
48C2×#121—22no—
49C2×#120—22no—
50C4×#41—22no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4—24no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C412yes—
65C2×C4×D4D412yes—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2312yes—
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117C2×C412yes—
71C2×#117D411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#446—28no—
77#436—28no—
78#435—216no—
79#432—24no—
80#432—24no—
81#431—24no—
82#431—22no—
83#431—22no—
84#431D414yescomplemented
85#431C2314yes—
86#423—24no—
87#423—22no—
88#423—22no—
89#423—22no—
90#423C2314yes—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94#413—44no—
95C2×#101—22no—
96C2×#100—22no—
97#407—24no—
98C2×C22⋊C8—22no—
99C4×M16—24no—
100C2×C4×C8D414yescomplemented
101#398C2×C414yes—
102#397C2×C414yes—
103#396—24no—
104#394—24no—
105#394C2×C414yes—
106#392C2×C414yes—
107#390—22no—
108#389—22no—
109#388—24no—
110#386—24no—
111#384—21no—
112#382C2312yes—
113#381—22no—
114#380—22no—
115#380—21no—
116#380—22no—
117#380—22no—
118#380C2×C412yes—
119#379—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C412yes—
122C4×C22⋊C4—24no—
123#343—22no—
124#343—22no—
125D4⋊C8—216no—
126C25—21no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—24no—
130C2×#41—22no—
131C2×#41—22no—
132C22×Q8C2×D411yescharacteristic
133C22×D4—24no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—44no—
138C22×D4C22⋊C412yes—
139C22×D4C22×C412yes—
140C22×D4C2×D411yescharacteristic
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—44no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×M16—28no—
156C2×M16—24no—
157C2×M16—42no—
158C2×M16—44no—
159C2×M16—44no—
160C22×C8—24no—
161C22×C8—42no—
162#123—48no—
163C22⋊Q8—48no—
164#121—48no—
165#121—48no—
166#120—48no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—24no—
181C4×D4—22no—
182C4×D4—24no—
183C4×D4—416no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—41no—
189C4×D4—41no—
190#117—22no—
191#117—22no—
192#117—24no—
193#117—24no—
194#117—44no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22×C412yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42C22⋊C412yes—
225C2×C42C22⋊C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235C4⋊C8—28no—
236C4⋊C8—24no—
237C4⋊C8—24no—
238C4⋊C8—24no—
239C4⋊C8—24no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242#101—44no—
243#100—44no—
244C22⋊C8—28no—
245C22⋊C8—24no—
246C22⋊C8—24no—
247#97—28no—
248C4×C8—24no—
249C4×C8C2×D418yes—
250#95—24no—
251#95—24no—
252#95—22no—
253#95C22×C414yes—
254C24—22no—
255C24—44no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—82no—
260C24C2×C22⋊C412yes—
261C24C2×C22⋊C411yescharacteristic
262#41—44no—
263#41—44no—
264#41—44no—
265#41—44no—
266C2×Q8—22no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—21no—
270C2×D4—28no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—21no—
278C2×D4—22no—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—48no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—84no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—81no—
313C22×C4—82no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C4×D412yes—
325C22×C4C4×D412yes—
326C22×C4#12012yes—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4#12312yes—
330C22×C4AGL(1,ℤ/8)12yes—
331C22×C4AGL(1,ℤ/8)14yes—
332C22×C4#13712yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336M16—44no—
337M16—44no—
338M16—88no—
339M16—84no—
340C2×C8—28no—
341C2×C8—44no—
342C2×C8—84no—
343C2×C8—84no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—48no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—82no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—44no—
365C22⋊C4—84no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—44no—
373C42C4×D412yes—
374C42C4×D412yes—
375C42#12011yescharacteristic
376C42#12011yescharacteristic
377C42#12112yes—
378C42#12112yes—
379C42#12311yescharacteristic
380C42#12311yescharacteristic
381C42C22×D412yes—
382C42C2×#4112yes—
383C23—24no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—48no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—84no—
399C23—88no—
400C23—88no—
401C23—84no—
402C23#38612yes—
403C23#41712yes—
404C23#41812yes—
405C23C22×C22⋊C411yescharacteristic
406Q8—42no—
407Q8—42no—
408D4—28no—
409D4—44no—
410D4—44no—
411D4—48no—
412D4—44no—
413D4—42no—
414D4—42no—
415D4—816no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4#38611yescharacteristic
449C2×C4#38612yes—
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#38611yescharacteristic
453C2×C4#41712yes—
454C2×C4#41812yes—
455C2×C4#46814yes—
456C2×C4#46814yes—
457C2×C4#46914yes—
458C2×C4#46914yes—
459C2×C4C2×C4×D412yes—
460C2×C4C2×#12011yescharacteristic
461C2×C4C2×#12112yes—
462C2×C4C2×#12311yescharacteristic
463C2×C4C2×AGL(1,ℤ/8)12yes—
464C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
465C2×C4C2×#13711yescharacteristic
466C8—48no—
467C8—84no—
468C22—21no—
469C22—22no—
470C22—21no—
471C22—22no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—44no—
480C22—42no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—816no—
485C22—82no—
486C22—82no—
487C22#189514yes—
488C22C2×#38611yescharacteristic
489C22C2×#41811yescharacteristic
490C22C2×#41711yescharacteristic
491C4—21no—
492C4—21no—
493C4—22no—
494C4—21no—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—84no—
500C4—82no—
501C4#189112yes—
502C4#189512yes—
503C4#189512yes—
504C4C2×#46812yes—
505C4C2×#46912yes—
506C2—21no—
507C2—21no—
508C2—24no—
509C2—82no—
510C2—84no—
511C2C2×#189111yescharacteristic
512C2C2×#189511yescharacteristic
513C2#2101211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#34935C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60277C2semidirect
3. #20637C2semidirect
4. C2×#1486C2semidirect
5. #9490C2semidirect
6. #9489C2semidirect
7. #7957C2semidirect
12. #2983C22semidirect
13. #2975C22semidirect
32. #1801C22semidirect
34. #1800C22semidirect
39. #1286C22semidirect
84. #431D4semidirect
100. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
#602772
#349352
#206372
C2×#14862
#94902
#94892
#79572