Finite Group Databasebeta

← #106953

Subgroups — #106953

Order 512.

1052 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34182C211yesmaximal, characteristic
2C2×#1805C211yesmaximal, characteristic
3C2×#1479C211yesmaximal, characteristic
4#8039C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7#7893C211yesmaximal, characteristic
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#103—21no—
14#2351—22no—
15#2351—22no—
16Q8×C4⋊C4—22no—
17Q8×C4⋊C4C414yescomplemented
18#2316—22no—
19#2316—21no—
20#2285—22no—
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#382C2211yescharacteristic
24#1805—22no—
25#1805C414yescomplemented
26C2×#340—21no—
27#1479—22no—
28#1479C414yescomplemented
29C2×Q8⋊C8—22no—
30#1296—22no—
31#1294—22no—
32#1294—22no—
33#1279—22no—
34C23×Q8D411yescharacteristic
35C23×C8—21no—
36#519—42no—
37#519—42no—
38#519—42no—
39#519—42no—
40C2×C4×Q8—24no—
41C2×C4×Q8—24no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8D412yes—
52C2×C4×Q8D412yes—
53C2×C4×Q8Q812yes—
54C2×#117—24no—
55C2×#117—21no—
56C2×#117—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431C2×C412yes—
69#431C2×C412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2×C412yes—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#389—24no—
86#389—28no—
87#389—24no—
88#389—28no—
89#384—24no—
90#384C2×C412yes—
91#384D412yes—
92#382—24no—
93#382—21no—
94#382—21no—
95#382—21no—
96#382—21no—
97#382—21no—
98#382C2×C412yes—
99#380—24no—
100#380—21no—
101#380—21no—
102#380D412yes—
103#380D412yes—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4D412yes—
110C2×#95—21no—
111C2×#95—21no—
112#340—24no—
113Q8⋊C8—28no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8C22⋊C412yes—
122C22×Q8C22⋊C412yes—
123C22×Q8C4⋊C412yes—
124C22×Q8D812yes—
125C22×Q8SD1612yes—
126C22×Q8SD1612yes—
127C22×Q8Q1612yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C22×C8—22no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—48no—
156C4×Q8—48no—
157C4×Q8—48no—
158C4×Q8—48no—
159C4×Q8C4214yes—
160C4×Q8C22⋊C414yes—
161C4×Q8C4⋊C414yes—
162C4×Q8C4⋊C414yes—
163#117—48no—
164#117—48no—
165#117—41no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170#117—41no—
171#117—41no—
172#117—41no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C414yes—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C4⋊C412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—28no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×Q811yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—41no—
227C4⋊C8—41no—
228C4⋊C8—41no—
229C4⋊C8—41no—
230#103—48no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234C22⋊C8—41no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—41no—
238C4×C8—41no—
239#95—28no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—41no—
247#95—41no—
248C24#12011yescharacteristic
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
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#9514yes—
271C2×Q8#10214yes—
272C2×Q8#10214yes—
273C2×Q8#10314yes—
274C2×Q8#10314yes—
275C2×Q8#10614yes—
276C2×Q8#10714yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C4⋊C411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4C22⋊Q811yescharacteristic
312C22×C4C22⋊Q811yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4C2×D811yescharacteristic
316C22×C4C2×SD1611yescharacteristic
317C22×C4C2×SD1611yescharacteristic
318C22×C4C2×Q1611yescharacteristic
319C22×C4AGL(1,ℤ/8)11yescharacteristic
320C22×C4AGL(1,ℤ/8)11yescharacteristic
321C22×C4#13711yescharacteristic
322C22×C4#13711yescharacteristic
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—82no—
329C2×C8—82no—
330C2×C8—81no—
331C2×C8—81no—
332C2×C8—81no—
333C2×C8—81no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—416no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4—88no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4#9514yes—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—416no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42C2×C4211yescharacteristic
370C42C2×C22⋊C411yescharacteristic
371C42C2×C4⋊C411yescharacteristic
372C42C2×C4⋊C411yescharacteristic
373C42#11711yescharacteristic
374C42#11711yescharacteristic
375C42C4×D411yescharacteristic
376C42C4×D411yescharacteristic
377C42C4×D411yescharacteristic
378C42C4×D411yescharacteristic
379C42#12111yescharacteristic
380C42#12111yescharacteristic
381C42C22⋊Q811yescharacteristic
382C42C22⋊Q811yescharacteristic
383C42#12311yescharacteristic
384C42#12311yescharacteristic
385C23—22no—
386C23—22no—
387C23#37911yescharacteristic
388C23#38511yescharacteristic
389C23#38611yescharacteristic
390C23#44711yescharacteristic
391C23#44811yescharacteristic
392C23#45011yescharacteristic
393C23#45111yescharacteristic
394Q8—24no—
395Q8—24no—
396Q8—24no—
397Q8—24no—
398Q8—24no—
399Q8—24no—
400Q8—24no—
401Q8#34018yes—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—162no—
435C2×C4—162no—
436C2×C4C2×#9511yescharacteristic
437C2×C4C4×C22⋊C411yescharacteristic
438C2×C4C4×C22⋊C411yescharacteristic
439C2×C4#38011yescharacteristic
440C2×C4#38011yescharacteristic
441C2×C4#38111yescharacteristic
442C2×C4#38511yescharacteristic
443C2×C4#38511yescharacteristic
444C2×C4#38511yescharacteristic
445C2×C4#38611yescharacteristic
446C2×C4#38611yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4C2×#10211yescharacteristic
449C2×C4C2×#10211yescharacteristic
450C2×C4C2×#10311yescharacteristic
451C2×C4C2×#10311yescharacteristic
452C2×C4#41711yescharacteristic
453C2×C4#41711yescharacteristic
454C2×C4#41811yescharacteristic
455C2×C4#41911yescharacteristic
456C2×C4C2×#10611yescharacteristic
457C2×C4C2×#10711yescharacteristic
458C2×C4#42811yescharacteristic
459C2×C4#42811yescharacteristic
460C2×C4#44711yescharacteristic
461C2×C4#44811yescharacteristic
462C2×C4#45011yescharacteristic
463C2×C4#45111yescharacteristic
464C2×C4#48011yescharacteristic
465C2×C4#48011yescharacteristic
466C2×C4#48111yescharacteristic
467C2×C4#48111yescharacteristic
468C2×C4#48311yescharacteristic
469C2×C4#48311yescharacteristic
470C2×C4#48411yescharacteristic
471C2×C4#48411yescharacteristic
472C8—82no—
473C8—82no—
474C8—82no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22C22⋊#9511yescharacteristic
484C22#178611yescharacteristic
485C22#178711yescharacteristic
486C22#179412yes—
487C22#179612yes—
488C22#185211yescharacteristic
489C22#185311yescharacteristic
490C22#187211yescharacteristic
491C22#187411yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—28no—
501C4—44no—
502C4—84no—
503C4—84no—
504C4—162no—
505C4C2×#34011yescharacteristic
506C4#173611yescharacteristic
507C4#182311yescharacteristic
508C4#182411yescharacteristic
509C4#185211yescharacteristic
510C4#185311yescharacteristic
511C4#187211yescharacteristic
512C4#187411yescharacteristic
513C2—21no—
514C2—21no—
515C2—21no—
516C2—21no—
517C2#842912yes—
518C2#843312yes—
519C2#1073011yescharacteristic
520C2#1077711yescharacteristic
521C2#1077911yescharacteristic
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
17. Q8×C4⋊C4C4semidirect
25. #1805C4semidirect
28. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#341822
C2×#18052
C2×#14792
#80392
#80262
#78932