Finite Group Databasebeta

← #106941

Subgroups — #106941

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34196C211yesmaximal, characteristic
2C2×#1763C211yesmaximal, characteristic
3C2×#1491C211yesmaximal, characteristic
4#10851C211yesmaximal, characteristic
5#8034C211yesmaximal, characteristic
6#8029C211yesmaximal, characteristic
7#8021C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C22×#103—21no—
11C2×C4×M16C2211yescharacteristic
12C22×#97C2211yescharacteristic
13#2355—22no—
14#2355C412yescomplemented
15Q8×C4⋊C4C412yescomplemented
16#2344—22no—
17#2344—21no—
18Q8×C22⋊C4—21no—
19#2320—22no—
20#2320—21no—
21#2285—21no—
22#2285C2211yescharacteristic
23#2283—21no—
24#2283C2211yescharacteristic
25#2276—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1763—22no—
28#1763C414yescomplemented
29C2×#340—21no—
30C2×#339—22no—
31#1491—22no—
32#1491C414yescomplemented
33C2×Q8⋊C8—21no—
34C2×Q8⋊C8—21no—
35#1295—22no—
36#1295—22no—
37#1294—21no—
38#1294—21no—
39C23×Q8D411yescharacteristic
40C22×M16—21no—
41C23×C8—21no—
42C2×C4×Q8—22no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×Q8Q812yes—
57C2×#117—22no—
58C2×#117D412yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65#432—21no—
66#432—21no—
67#432—21no—
68#432C2×C412yes—
69C2×C4⋊C8—21no—
70C2×#103—28no—
71C2×#103—24no—
72C2×C22⋊C8—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16C2×C412yes—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97—21no—
81C2×#97—21no—
82C2×#97—21no—
83C2×#97—21no—
84C2×#97C2×C412yes—
85C2×C4×C8—21no—
86#391—24no—
87#391—22no—
88#389—22no—
89#389—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#384—22no—
94#384C2×C412yes—
95#383—22no—
96#383—21no—
97#382—22no—
98#382—22no—
99#382D412yes—
100#382D412yes—
101#381—22no—
102#381—21no—
103#380—22no—
104#380—21no—
105#380—21no—
106#380D411yescharacteristic
107#380D411yescharacteristic
108C4×C4⋊C4—22no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—21no—
122#376—22no—
123#376—21no—
124C2×#95—21no—
125#340—24no—
126#339—44no—
127Q8⋊C8—24no—
128Q8⋊C8—24no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—24no—
133C22×Q8—42no—
134C22×Q8C22⋊C411yescharacteristic
135C22×Q8C22⋊C412yes—
136C22×Q8C22⋊C411yescharacteristic
137C22×Q8C4⋊C412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—41no—
148C2×M16—41no—
149C2×M16—41no—
150C22×C8—24no—
151C22×C8—22no—
152C22×C8—42no—
153C22×C8—41no—
154C22×C8—41no—
155C22×C8—41no—
156C22×C8—41no—
157C4×Q8—21no—
158C4×Q8—21no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—44no—
174C4×Q8—44no—
175C4×Q8C4212yes—
176C4×Q8C4212yes—
177C4×Q8C22⋊C412yes—
178C4×Q8C22⋊C412yes—
179C4×Q8C4⋊C412yes—
180C4×Q8C4⋊C412yes—
181C4×Q8C4⋊C412yes—
182C4×Q8C4⋊C412yes—
183#117—42no—
184#117—42no—
185#117—82no—
186#117—82no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C414yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C4⋊C412yes—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—41no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×Q811yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C4⋊C8—22no—
245C4⋊C8—41no—
246#103—216no—
247C22⋊C8—22no—
248C22⋊C8—41no—
249#97—22no—
250#97—22no—
251#97—22no—
252#97—22no—
253#97—41no—
254#97—41no—
255#97—41no—
256#97—41no—
257C4×C8—22no—
258C4×C8—41no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—22no—
264#95—21no—
265#95—22no—
266#95—21no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—41no—
272C24#12011yescharacteristic
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—44no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8#9512yes—
288C2×Q8#9512yes—
289C2×Q8#10414yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—82no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C4⋊C411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12011yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4C22⋊Q811yescharacteristic
333C22×C4C22⋊Q811yescharacteristic
334C22×C4#12311yescharacteristic
335C22×C4#12311yescharacteristic
336M16—24no—
337M16—42no—
338M16—42no—
339M16—42no—
340C2×C8—24no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—82no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—84no—
373C4⋊C4#9512yes—
374C4⋊C4#9512yes—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—42no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—48no—
390C42—44no—
391C42—42no—
392C42—42no—
393C42—82no—
394C42—82no—
395C42—84no—
396C42C2×C4211yescharacteristic
397C42C2×C22⋊C411yescharacteristic
398C42C2×C4⋊C411yescharacteristic
399C42C2×C4⋊C411yescharacteristic
400C42#11711yescharacteristic
401C42#11711yescharacteristic
402C42C4×D411yescharacteristic
403C42C4×D411yescharacteristic
404C42C4×D411yescharacteristic
405C42C4×D411yescharacteristic
406C42#12111yescharacteristic
407C42#12111yescharacteristic
408C42C22⋊Q811yescharacteristic
409C42C22⋊Q811yescharacteristic
410C42#12311yescharacteristic
411C42#12311yescharacteristic
412C23—22no—
413C23—22no—
414C23#37911yescharacteristic
415C23#38511yescharacteristic
416C23#38611yescharacteristic
417C23#45311yescharacteristic
418C23#45411yescharacteristic
419C23#45511yescharacteristic
420C23#45611yescharacteristic
421Q8—24no—
422Q8—22no—
423Q8—22no—
424Q8—22no—
425Q8—44no—
426Q8—44no—
427Q8—42no—
428Q8#33914yescomplemented
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—82no—
484C2×C4C2×#9511yescharacteristic
485C2×C4C4×C22⋊C411yescharacteristic
486C2×C4C4×C22⋊C411yescharacteristic
487C2×C4#38011yescharacteristic
488C2×C4#38011yescharacteristic
489C2×C4#38111yescharacteristic
490C2×C4#38511yescharacteristic
491C2×C4#38511yescharacteristic
492C2×C4#38511yescharacteristic
493C2×C4#38611yescharacteristic
494C2×C4#38611yescharacteristic
495C2×C4#38611yescharacteristic
496C2×C4C2×#10412yes—
497C2×C4#42112yes—
498C8—44no—
499C8—44no—
500C8—82no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—21no—
508C22—21no—
509C22C22⋊#9511yescharacteristic
510C22#178912yes—
511C22#179211yescharacteristic
512C22#179311yescharacteristic
513C22#179612yes—
514C22#185911yescharacteristic
515C22#186011yescharacteristic
516C22#188211yescharacteristic
517C22#188311yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—24no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—84no—
533C4—82no—
534C4—84no—
535C4—82no—
536C4C2×#33911yescharacteristic
537C4#173311yescharacteristic
538C4#182612yes—
539C2—21no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2#841712yes—
544C2#841912yes—
545C2#1073311yescharacteristic
546C2#1077711yescharacteristic
547C2#1078111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2355C4semidirect
15. Q8×C4⋊C4C4semidirect
28. #1763C4semidirect
32. #1491C4semidirect
428. Q8#339semidirect

Maximal subgroups

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

subgroupindex
#341962
C2×#17632
C2×#14912
#108512
#80342
#80292
#80212