Finite Group Databasebeta

← #188334

Subgroups — #188334

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2352C211yesmaximal, characteristic
2#21953C212yesmaximal, complemented
3C2×#1466C211yesmaximal, characteristic
4C2×#1460C211yesmaximal, characteristic
5#8561C214yesmaximal, complemented
6#8559C212yesmaximal, complemented
7#8557C212yesmaximal, complemented
8#7914C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#117C411yescharacteristic
11C23×C4⋊C4C411yescharacteristic
12C2×#432C2212yes—
13C2×#431C2212yes—
14#2352—22no—
15#2352—22no—
16#2352—24no—
17C2×#388C2212yes—
18C2×#387C412yes—
19C2×#386C412yes—
20C2×#382C2211yescharacteristic
21C2×#380C411yescharacteristic
22C2×#380C411yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25#1940—24no—
26#1939—22no—
27#1939—22no—
28#1797C2214yes—
29#1797C2214yes—
30#1466—24no—
31#1460—24no—
32C2×#323C2211yescharacteristic
33C2×#323C2212yes—
34C2×#323C2211yescharacteristic
35C2×#323C2212yes—
36C2×#323C2212yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C412yes—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4D414yes—
47C2×C4×D4Q814yes—
48C2×#117—21no—
49C2×#117—22no—
50C2×#117—21no—
51C2×#117—44no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—44no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C42C2×C412yes—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#432—24no—
71#432—24no—
72#431—24no—
73#431—24no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2314yes—
81C2×#97—22no—
82C2×C4×C8—22no—
83#388—24no—
84#388—24no—
85#387—24no—
86#387—24no—
87#386—24no—
88#386—24no—
89#382—22no—
90#382—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95C4×C4⋊C4D412yes—
96C4×C4⋊C4D412yes—
97C4×C4⋊C4D414yes—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—24no—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2312yes—
105#336—24no—
106#336—24no—
107#323—22no—
108#323—22no—
109#323—24no—
110#323—24no—
111#323—24no—
112C25C22×C411yescharacteristic
113C25C22×C411yescharacteristic
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—41no—
119C22×D4—41no—
120C22×D4C4⋊C414yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—44no—
136C23×C4—82no—
137C23×C4C4⋊C414yes—
138C23×C4C4⋊C414yes—
139C23×C4C22×C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—24no—
151C4×D4—28no—
152C4×D4—28no—
153C4×D4—28no—
154C4×D4—28no—
155#117—44no—
156#117—48no—
157#117—44no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C4⋊C414yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4C2×D412yes—
193C2×C22⋊C4C2×D412yes—
194C2×C22⋊C4C2×Q812yes—
195C2×C22⋊C4C2×Q812yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—44no—
199C2×C42—82no—
200C2×C42C2×D414yes—
201C2×C42C2×D414yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×Q812yes—
206C2×C42#4114yes—
207C4⋊C8—44no—
208C4⋊C8—44no—
209C4⋊C8—84no—
210C4⋊C8—84no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—24no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—24no—
217#97—44no—
218C4×C8—44no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95C22⋊C414yes—
224#95C22⋊C414yes—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24C2×C4⋊C412yes—
244C24C2×C4⋊C412yes—
245C24#13112yes—
246C24#13112yes—
247C24C23×C411yescharacteristic
248C2×D4—28no—
249C2×D4—28no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—88no—
312C22×C4—88no—
313C22×C4C2×C22⋊C414yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C4⋊C412yes—
320C22×C4C2×C4⋊C412yes—
321C22×C4C2×C4⋊C412yes—
322C22×C4C2×C4⋊C412yes—
323C22×C4#11712yes—
324C22×C4#11712yes—
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×Q811yescharacteristic
328C22×C4C2×#4112yes—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—44no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—44no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—44no—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4#10014yes—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—48no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—84no—
358C42—44no—
359C42—44no—
360C42—44no—
361C42—44no—
362C42#12114yes—
363C42#12114yes—
364C42C22⋊Q814yes—
365C42C22⋊Q814yes—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23#42414yes—
416C23C22×C22⋊C411yescharacteristic
417C23C22×C4⋊C411yescharacteristic
418C23C2×#11711yescharacteristic
419C23C2×#13111yescharacteristic
420C23C2×#13111yescharacteristic
421C23#56811yescharacteristic
422C23#56811yescharacteristic
423D4—416no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—88no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—88no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4#38014yes—
478C2×C4#38012yes—
479C2×C4#38012yes—
480C2×C4C2×#10014yes—
481C2×C4C2×#10012yes—
482C2×C4#41312yes—
483C2×C4C2×#12112yes—
484C2×C4C2×C22⋊Q812yes—
485C8—82no—
486C8—82no—
487C8—84no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—24no—
501C22—44no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22C2×#38011yescharacteristic
523C22#288011yescharacteristic
524C22#288011yescharacteristic
525C22#288311yescharacteristic
526C22#288411yescharacteristic
527C22C22×#10011yescharacteristic
528C22C2×#41311yescharacteristic
529C22C2×#42411yescharacteristic
530C22#290511yescharacteristic
531C22#292411yescharacteristic
532C22#292411yescharacteristic
533C4—24no—
534C4—24no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—42no—
539C4—44no—
540C4—42no—
541C4—42no—
542C4—82no—
543C4—82no—
544C4#180014yes—
545C2—21no—
546C2—21no—
547C2—21no—
548C2—21no—
549C2—44no—
550C2—82no—
551C2#1742111yescharacteristic
552C2#1742311yescharacteristic
553C2#1742411yescharacteristic
554C2#1742511yescharacteristic
555C2#2062811yescharacteristic
556C2#2063411yescharacteristic
557C2C2×#180011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21953C2semidirect
5. #8561C2semidirect
6. #8559C2semidirect
7. #8557C2semidirect
8. #7914C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23522
#219532
C2×#14662
C2×#14602
#85612
#85592
#85572
#79142