Finite Group Databasebeta

← #192048

Subgroups — #192048

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#21932C212yesmaximal, complemented
3#21740C211yesmaximal, characteristic
4C2×C4×#99C211yesmaximal, characteristic
5#8852C214yesmaximal, complemented
6#8851C214yesmaximal, complemented
7#7929C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9#2975—21no—
10#2974—24no—
11#2973—21no—
12#2973C2212yes—
13#2972C2212yes—
14D4×M16—24no—
15#2920—21no—
16C22×#99C412yes—
17C2×#407—21no—
18C2×#407C2211yescharacteristic
19C2×#407C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C2×C4×C22⋊C4C2212yes—
22#1870—21no—
23#1870—21no—
24#1869C2212yes—
25#1869C2212yes—
26#1851—24no—
27#1850C2218yescomplemented
28#1817—21no—
29#1797—24no—
30#1779—21no—
31C4×#99C412yes—
32C4×#99C411yescharacteristic
33C4×#99C411yescharacteristic
34C4×#99C2212yes—
35C4×#99C2212yes—
36C2×#342C412yes—
37#1282C2218yescomplemented
38C23×D4C2×C411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42C22×M16—21no—
43C22×M16D414yes—
44C23×C8C2312yes—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4Q811yescharacteristic
54C2×C4×D4Q811yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C22×C4⋊C4Q811yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C414yes—
60C22×C22⋊C4Q811yescharacteristic
61C22×C42—22no—
62C22×C42D411yescharacteristic
63#436—28no—
64#435—28no—
65#435—28no—
66C8×D4—28no—
67#432—22no—
68#432—22no—
69#431—22no—
70#431—22no—
71#423—22no—
72C2×C4⋊C8—28no—
73C2×#99—28no—
74C2×#99C2×C412yes—
75C2×#99C2×C414yes—
76C2×#99C2×C412yes—
77#407—22no—
78#407—22no—
79#407—22no—
80#407—22no—
81#407—22no—
82#407—22no—
83#407—22no—
84#407—22no—
85#407—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8D414yes—
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2314yes—
98C4×M16—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4C2×C412yes—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4C2314yes—
104C2×#95—22no—
105#342C2×C412yes—
106#342C2×C414yes—
107#342C2×C412yes—
108#336—28no—
109#336—22no—
110#336—22no—
111C25C4⋊C411yescharacteristic
112C25C22×C411yescharacteristic
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—24no—
116C22×D4—42no—
117C22×D4C4⋊C411yescharacteristic
118C22×D4C4⋊C411yescharacteristic
119C22×D4C22×C411yescharacteristic
120C22×D4C22×C411yescharacteristic
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—24no—
126C23×C4—82no—
127C23×C4—82no—
128C23×C4C4⋊C411yescharacteristic
129C23×C4C4⋊C411yescharacteristic
130C23×C4C4⋊C411yescharacteristic
131C23×C4C4⋊C411yescharacteristic
132C23×C4C4⋊C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×Q811yescharacteristic
141C23×C4C2×Q811yescharacteristic
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4C2411yescharacteristic
145C2×M16—24no—
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—84no—
150C2×M16—82no—
151C2×M16—84no—
152C22×C8—24no—
153C22×C8—22no—
154C22×C8—24no—
155C22×C8—22no—
156C22×C8—82no—
157C22×C8C2×D418yes—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C4⋊C411yescharacteristic
177C2×C4⋊C4C4⋊C411yescharacteristic
178C2×C4⋊C4C2×Q811yescharacteristic
179C2×C4⋊C4C2×Q811yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4C4⋊C411yescharacteristic
188C2×C22⋊C4C4⋊C411yescharacteristic
189C2×C22⋊C4C4⋊C411yescharacteristic
190C2×C22⋊C4C4⋊C411yescharacteristic
191C2×C22⋊C4C22×C414yes—
192C2×C22⋊C4C22×C411yescharacteristic
193C2×C22⋊C4C22×C411yescharacteristic
194C2×C22⋊C4C22×C414yes—
195C2×C22⋊C4C2×Q811yescharacteristic
196C2×C22⋊C4C2×Q811yescharacteristic
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—84no—
205C2×C42C4⋊C411yescharacteristic
206C2×C42C4⋊C411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8—88no—
212#99—24no—
213#99—28no—
214#99M1618yescomplemented
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8—41no—
223C22⋊C8—42no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—82no—
228C22⋊C8—82no—
229#97—24no—
230C4×C8—24no—
231#95—22no—
232#95—22no—
233#95—24no—
234C24—22no—
235C24—22no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—41no—
240C24—42no—
241C24—42no—
242C24—81no—
243C24—81no—
244C24C2×C4⋊C411yescharacteristic
245C24C2×C4⋊C411yescharacteristic
246C24C2×C4⋊C411yescharacteristic
247C24C2×C4⋊C411yescharacteristic
248C24C4×D412yes—
249C24C4×D412yes—
250C24C23×C411yescharacteristic
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—48no—
263C2×D4—48no—
264C2×D4C2×M1614yes—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—48no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—82no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—81no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4—82no—
304C22×C4—84no—
305C22×C4—82no—
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4C2×C4⋊C411yescharacteristic
308C22×C4C2×C4⋊C411yescharacteristic
309C22×C4C2×C4⋊C411yescharacteristic
310C22×C4C2×C4⋊C411yescharacteristic
311C22×C4C2×C4⋊C411yescharacteristic
312C22×C4C2×C4⋊C411yescharacteristic
313C22×C4C2×C4⋊C411yescharacteristic
314C22×C4C4×D412yes—
315C22×C4C4×D412yes—
316C22×C4#12011yescharacteristic
317C22×C4#12012yes—
318C22×C4#12011yescharacteristic
319C22×C4C22⋊Q812yes—
320C22×C4C22⋊Q812yes—
321C22×C4C22⋊Q812yes—
322C22×C4C22⋊Q812yes—
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4#12312yes—
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×Q811yescharacteristic
329C22×C4C2×#4112yes—
330M16—48no—
331M16—84no—
332M16—84no—
333M16—84no—
334C2×C8—28no—
335C2×C8—48no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—48no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—84no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—24no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—48no—
360C22⋊C4C2×M1618yescomplemented
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—48no—
370C42—82no—
371C42—82no—
372C23—24no—
373C23—22no—
374C23—21no—
375C23—21no—
376C23—24no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—24no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—41no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23—84no—
407C23#38512yes—
408C23#38511yescharacteristic
409C23#38511yescharacteristic
410C23#42312yes—
411C23C22×C4⋊C411yescharacteristic
412C23C2×C4×D412yes—
413C23C22×M1612yes—
414C23#56812yes—
415D4—28no—
416D4—44no—
417D4—44no—
418D4—44no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—816no—
471C2×C4#38511yescharacteristic
472C2×C4#38511yescharacteristic
473C2×C4#38512yes—
474C2×C4#42312yes—
475C2×C4C2×#12011yescharacteristic
476C2×C4C2×C22⋊Q812yes—
477C2×C4C2×#12311yescharacteristic
478C8—48no—
479C8—82no—
480C8—82no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—21no—
487C22—21no—
488C22—22no—
489C22—22no—
490C22—21no—
491C22—22no—
492C22—21no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—48no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—84no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—84no—
511C22—81no—
512C22—84no—
513C22#186812yes—
514C22C2×#38511yescharacteristic
515C22C2×#42311yescharacteristic
516C22#290511yescharacteristic
517C22D4×M1614yes—
518C4—22no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—22no—
524C4—48no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—84no—
531C4#302612yes—
532C2—24no—
533C2—22no—
534C2—21no—
535C2—21no—
536C2—42no—
537C2—41no—
538C2—42no—
539C2—81no—
540C2#2083711yescharacteristic
541C2#2088911yescharacteristic
542C2C2×#186811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21932C2semidirect
5. #8852C2semidirect
6. #8851C2semidirect
7. #7929C2semidirect
27. #1850C22semidirect
37. #1282C22semidirect
214. #99M16semidirect
360. C22⋊C4C2×M16semidirect

Maximal subgroups

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

subgroupindex
#602302
#219322
#217402
C2×C4×#992
#88522
#88512
#79292