Finite Group Databasebeta

← #106919

Subgroups — #106919

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic
2#20647C211yesmaximal, characteristic
3C2×#1479C211yesmaximal, characteristic
4C2×#1303C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1286C211yesmaximal, characteristic
8C22×C4×Q8—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#419—21no—
14#2351—22no—
15#2351—22no—
16#2351—22no—
17#2351C414yescomplemented
18C2×#389—22no—
19C2×#389—21no—
20#2285C2211yescharacteristic
21C2×#382—21no—
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1805—22no—
26#1803C414yescomplemented
27#1736—21no—
28#1479—22no—
29#1479C414yescomplemented
30C2×Q8⋊C8—22no—
31#1303—22no—
32#1294—21no—
33#1294—21no—
34#1294—22no—
35#1286—22no—
36C23×Q8D411yescharacteristic
37C22×M16—21no—
38C2×C4×Q8—24no—
39C2×C4×Q8—24no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—42no—
47C2×C4×Q8—42no—
48C2×C4×Q8—42no—
49C2×C4×Q8—42no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8Q812yes—
56C2×#117—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42—22no—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431C2×C412yes—
74#431C2×C412yes—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2×C412yes—
83#419—42no—
84#419—42no—
85C2×#103—24no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#389—24no—
91#389—28no—
92#389—24no—
93#389—28no—
94#384C2×C412yes—
95#382—24no—
96#382—21no—
97#382—21no—
98#382—24no—
99#382—21no—
100#382C2×C412yes—
101#382D412yes—
102#380—21no—
103#380—21no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4D412yes—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116C2×#95D412yes—
117#340—24no—
118Q8⋊C8—28no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8C22⋊C412yes—
127C22×Q8C22⋊C412yes—
128C22×Q8C4⋊C412yes—
129C22×Q8D812yes—
130C22×Q8SD1612yes—
131C22×Q8SD1612yes—
132C22×Q8Q1612yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—41no—
142C2×M16—41no—
143C2×M16—41no—
144C2×M16—41no—
145C2×M16—41no—
146C2×M16—41no—
147C22×C8—22no—
148C22×C8—41no—
149C22×C8—41no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—48no—
163C4×Q8—48no—
164C4×Q8—48no—
165C4×Q8—48no—
166C4×Q8C4214yes—
167C4×Q8C22⋊C414yes—
168C4×Q8C4⋊C414yes—
169C4×Q8C4⋊C414yes—
170#117—41no—
171#117—41no—
172#117—41no—
173#117—41no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C4⋊C412yes—
203C2×C22⋊C4—41no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—28no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q811yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—41no—
237C4⋊C8—41no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240#103—48no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—41no—
244C22⋊C8—41no—
245C4×C8—22no—
246C4×C8—22no—
247C4×C8—41no—
248C4×C8—41no—
249#95—28no—
250#95—22no—
251#95—22no—
252#95—21no—
253#95—21no—
254#95—22no—
255#95—41no—
256#95—41no—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259#95C22⋊C414yes—
260C24#12011yescharacteristic
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8#9514yes—
283C2×Q8#10214yes—
284C2×Q8#10214yes—
285C2×Q8#10314yes—
286C2×Q8#10314yes—
287C2×Q8#10614yes—
288C2×Q8#10714yes—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
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
336C22×C4C2×D811yescharacteristic
337C22×C4C2×SD1611yescharacteristic
338C22×C4C2×SD1611yescharacteristic
339C22×C4C2×Q1611yescharacteristic
340C22×C4AGL(1,ℤ/8)11yescharacteristic
341C22×C4AGL(1,ℤ/8)11yescharacteristic
342C22×C4#13711yescharacteristic
343C22×C4#13711yescharacteristic
344M16—81no—
345M16—81no—
346M16—81no—
347M16—81no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—82no—
354C2×C8—82no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—416no—
365C4⋊C4—416no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4#9514yes—
375C22⋊C4—81no—
376C22⋊C4—81no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—416no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—81no—
387C42—81no—
388C42C2×C4211yescharacteristic
389C42C2×C22⋊C411yescharacteristic
390C42C2×C4⋊C411yescharacteristic
391C42C2×C4⋊C411yescharacteristic
392C42#11711yescharacteristic
393C42#11711yescharacteristic
394C42C4×D411yescharacteristic
395C42C4×D411yescharacteristic
396C42C4×D411yescharacteristic
397C42C4×D411yescharacteristic
398C42#12111yescharacteristic
399C42#12111yescharacteristic
400C42C22⋊Q811yescharacteristic
401C42C22⋊Q811yescharacteristic
402C42#12311yescharacteristic
403C42#12311yescharacteristic
404C23—22no—
405C23—22no—
406C23#37911yescharacteristic
407C23#38511yescharacteristic
408C23#38611yescharacteristic
409C23#44711yescharacteristic
410C23#44811yescharacteristic
411C23#45011yescharacteristic
412C23#45111yescharacteristic
413Q8—24no—
414Q8—24no—
415Q8—24no—
416Q8—24no—
417Q8—24no—
418Q8—24no—
419Q8—24no—
420Q8#34018yes—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—84no—
456C2×C4—82no—
457C2×C4—84no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4C2×#9511yescharacteristic
463C2×C4C4×C22⋊C411yescharacteristic
464C2×C4C4×C22⋊C411yescharacteristic
465C2×C4#38011yescharacteristic
466C2×C4#38011yescharacteristic
467C2×C4#38111yescharacteristic
468C2×C4#38511yescharacteristic
469C2×C4#38511yescharacteristic
470C2×C4#38511yescharacteristic
471C2×C4#38611yescharacteristic
472C2×C4#38611yescharacteristic
473C2×C4#38611yescharacteristic
474C2×C4C2×#10211yescharacteristic
475C2×C4C2×#10211yescharacteristic
476C2×C4C2×#10311yescharacteristic
477C2×C4C2×#10311yescharacteristic
478C2×C4#41711yescharacteristic
479C2×C4#41711yescharacteristic
480C2×C4#41811yescharacteristic
481C2×C4#41911yescharacteristic
482C2×C4C2×#10611yescharacteristic
483C2×C4C2×#10711yescharacteristic
484C2×C4#42811yescharacteristic
485C2×C4#42811yescharacteristic
486C2×C4#44711yescharacteristic
487C2×C4#44811yescharacteristic
488C2×C4#45011yescharacteristic
489C2×C4#45111yescharacteristic
490C2×C4#48011yescharacteristic
491C2×C4#48011yescharacteristic
492C2×C4#48111yescharacteristic
493C2×C4#48111yescharacteristic
494C2×C4#48311yescharacteristic
495C2×C4#48311yescharacteristic
496C2×C4#48411yescharacteristic
497C2×C4#48411yescharacteristic
498C8—82no—
499C8—82no—
500C8—82no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22C22⋊#9511yescharacteristic
510C22#178611yescharacteristic
511C22#178711yescharacteristic
512C22#185211yescharacteristic
513C22#185311yescharacteristic
514C22#187211yescharacteristic
515C22#187411yescharacteristic
516C22#190512yes—
517C22#190612yes—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—28no—
527C4—44no—
528C4—44no—
529C4—84no—
530C4—82no—
531C4—162no—
532C4C2×#34011yescharacteristic
533C4#173611yescharacteristic
534C4#182311yescharacteristic
535C4#182411yescharacteristic
536C4#185211yescharacteristic
537C4#185311yescharacteristic
538C4#187211yescharacteristic
539C4#187411yescharacteristic
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2#917612yes—
545C2#918012yes—
546C2#1073011yescharacteristic
547C2#1077211yescharacteristic
548C2#1078611yescharacteristic
subgroupG/H
center, socleC23#386
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
17. #2351C4semidirect
26. #1803C4semidirect
29. #1479C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
#206472
C2×#14792
C2×#13032
#80262
#80142
C2×#12862