Finite Group Databasebeta

← #759873

Subgroups — #759873

Order 512.

3374 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Q8C212yesmaximal
3C22×#384C211yesmaximal, characteristic
4#16563C214yesmaximal, complemented
5#15184C214yesmaximal, complemented
6#15133C214yesmaximal, complemented
7#14821C214yesmaximal
8#14151C211yesmaximal, characteristic, complemented
9#14139C214yesmaximal
10#13926C212yesmaximal, complemented
11#13907C214yesmaximal
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2212yes—
14C23×C4⋊C4C2214yes—
15C23×C4⋊C4C2212yes—
16C23×C42C2211yescharacteristic
17#2582—28no—
18C2×#398C2214yes—
19C2×#398C2214yes—
20C4×C22⋊Q8—216no—
21C2×#391C2214yes—
22C2×#391C2212yes—
23C2×#391C2214yes—
24C2×#389C2212yes—
25C2×#389C2214yes—
26C2×#389C2214yes—
27C2×#387C2214yes—
28C2×#385C2214yes—
29C2×#384—28no—
30C2×#384C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2212yes—
33C2×#384C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×C4×C4⋊C4C2214yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39#1441C2218yes—
40#1441C2218yes—
41#1441C2218yes—
42#1441C2218yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2218yes—
46#1441C2218yes—
47C4⋊#95C2214yes—
48C4⋊#95C2214yes—
49C4⋊#95C2214yes—
50C4⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2218yes—
53C23×Q8C2312yes—
54C24×C4C2311yescharacteristic
55C24×C4C2314yes—
56C24×C4C2312yes—
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8—28no—
59C2×C22⋊Q8Q8116yescomplemented
60C2×C4×Q8—28no—
61C2×C4×Q8—28no—
62C2×C4×Q8—28no—
63C22×C4⋊C4—216no—
64C22×C4⋊C4—216no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—48no—
67C22×C4⋊C4—44no—
68C22×C4⋊C4—44no—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2318yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C42—24no—
86C22×C42—24no—
87C22×C42D418yes—
88C22×C42C2314yes—
89C22×C42C2314yes—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#398—216no—
94#398—48no—
95#391—216no—
96#391—28no—
97#391—48no—
98#389—216no—
99#389—28no—
100#389—48no—
101#387—216no—
102#385—216no—
103#384—216no—
104#384—28no—
105#384—24no—
106#384D418yes—
107#380—28no—
108#380—48no—
109C4×C4⋊C4—216no—
110C4×C4⋊C4—28no—
111C4×C22⋊C4—216no—
112C2×#95C2318yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2318yes—
120C2×#95C2314yes—
121C2×#95C2318yes—
122C2×#95C2318yes—
123C2×#95C2318yes—
124C2×#95C2318yes—
125C25C2411yescharacteristic
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8C2×Q818yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—216no—
144C23×C4—28no—
145C23×C4—28no—
146C23×C4—48no—
147C23×C4—44no—
148C23×C4C2×D414yes—
149C23×C4C2×D418yes—
150C23×C4C2×Q818yes—
151C23×C4#4118yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2414yes—
156C23×C4C2414yes—
157C23×C4C2414yes—
158C23×C4C2411yescharacteristic
159C23×C4C2412yes—
160C23×C4C2411yescharacteristic
161C23×C4C2412yes—
162C22⋊Q8—264no—
163C4×Q8—432no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—216no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—216no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—464no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—432no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C2×D4116yes—
204C2×C4⋊C4C2×D418yes—
205C2×C4⋊C4C2×Q818yes—
206C2×C4⋊C4C2×Q8116yes—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4C2×Q8116yes—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—232no—
218C2×C42—28no—
219C2×C42—28no—
220C2×C42—28no—
221C2×C42—28no—
222C2×C42—28no—
223C2×C42—28no—
224C2×C42C2×D418yes—
225C2×C42C2×D414yes—
226C2×C42C2×D414yes—
227C2×C42C2×D414yes—
228C2×C42#4114yes—
229#95—28no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—216no—
235#95—28no—
236#95—48no—
237#95—48no—
238#95—48no—
239#95—48no—
240#95—44no—
241#95—48no—
242C24—21no—
243C24—21no—
244C24—22no—
245C24—21no—
246C24—22no—
247C24—21no—
248C24C22×D412yes—
249C24C22×Q814yes—
250C24C2×#4114yes—
251C24H(5,𝔽2)12yes—
252C242−1+412yes—
253C24C2511yescharacteristic
254C2×Q8—216no—
255C2×Q8—216no—
256C2×Q8—216no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—48no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—28no—
279C22×C4—216no—
280C22×C4—28no—
281C22×C4—216no—
282C22×C4—216no—
283C22×C4—216no—
284C22×C4—216no—
285C22×C4—216no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—28no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—28no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—432no—
322C22×C4—416no—
323C22×C4—416no—
324C22×C4—816no—
325C22×C4—88no—
326C22×C4C22⋊Q8132yes—
327C22×C4C22×D418yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D414yes—
331C22×C4C22×D412yes—
332C22×C4C22×D414yes—
333C22×C4C22×D414yes—
334C22×C4C22×Q818yes—
335C22×C4C22×Q818yes—
336C22×C4C22×Q814yes—
337C22×C4C22×Q814yes—
338C22×C4C2×#4112yes—
339C22×C4C2×#4114yes—
340C22×C4C2×#4114yes—
341C22×C4C2×#4114yes—
342C22×C4H(5,𝔽2)12yes—
343C22×C4H(5,𝔽2)14yes—
344C22×C42−1+412yes—
345C22×C42−1+414yes—
346C4⋊C4—216no—
347C4⋊C4—232no—
348C4⋊C4—232no—
349C4⋊C4—48no—
350C4⋊C4—432no—
351C4⋊C4—416no—
352C4⋊C4—48no—
353C4⋊C4—416no—
354C4⋊C4—48no—
355C4⋊C4—416no—
356C4⋊C4—48no—
357C4⋊C4—416no—
358C4⋊C4—88no—
359C4⋊C4—88no—
360C22⋊C4—232no—
361C22⋊C4—416no—
362C42—28no—
363C42—216no—
364C42—416no—
365C42—416no—
366C42#121116yes—
367C23—22no—
368C23—24no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—24no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23C2×C22⋊Q818yes—
386C23#53714yes—
387C23#55718yes—
388C23Q8214yes—
389C23#55914yes—
390C23C23×D411yescharacteristic
391C23C23×D411yescharacteristic
392C23C23×Q812yes—
393C23C22×#4111yescharacteristic
394C23C22×#4112yes—
395C23C2×H(5,𝔽2)12yes—
396C23C2×H(5,𝔽2)11yescharacteristic
397C23C2×H(5,𝔽2)11yescharacteristic
398C23C2×2−1+412yes—
399C23C2×2−1+411yescharacteristic
400C23C2×2−1+411yescharacteristic
401Q8—432no—
402C2×C4—232no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—264no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—216no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—832no—
461C2×C4C2×#12114yes—
462C2×C4C2×#12118yes—
463C2×C4C2×C22⋊Q818yes—
464C2×C4C2×C22⋊Q818yes—
465C2×C4#53714yes—
466C2×C4#53718yes—
467C2×C4#54618yes—
468C2×C4D4×Q818yes—
469C2×C4#55218yes—
470C2×C4#55418yes—
471C22—24no—
472C22—28no—
473C22—24no—
474C22—28no—
475C22—24no—
476C22—28no—
477C22—24no—
478C22—24no—
479C22—28no—
480C22—24no—
481C22#258218yescomplemented
482C22C22×#12111yescharacteristic
483C22C22×C22⋊Q812yes—
484C22C2×#53712yes—
485C22C2×#53711yescharacteristic
486C22C2×#53711yescharacteristic
487C22C2×#54612yes—
488C22C2×D4×Q812yes—
489C22C2×#55212yes—
490C22C2×#55412yes—
491C22C2×#55712yes—
492C22C2×Q8211yescharacteristic
493C22C2×#55911yescharacteristic
494C22#348411yescharacteristic
495C22#348611yescharacteristic
496C22#349211yescharacteristic
497C22#349211yescharacteristic
498C22#350312yes—
499C22#350312yes—
500C22#350812yes—
501C22#350912yes—
502C22#354311yescharacteristic
503C22#354411yescharacteristic
504C22#355212yes—
505C4—28no—
506C4—24no—
507C4—24no—
508C4—48no—
509C4—48no—
510C4—48no—
511C4—84no—
512C4—84no—
513C4—84no—
514C4#2417116yes—
515C2—28no—
516C2C2×#241712yes—
517C2C2×#258211yescharacteristic
518C2#3522311yescharacteristic
519C2#3528012yes—
520C2#3762011yescharacteristic
521C2#5914311yescharacteristic
522C2#5914412yes—
523C2#5915111yescharacteristic
524C2#5932112yes—
525C2#5932211yescharacteristic
526C2#5932911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2582C2semidirect
4. #16563C2semidirect
5. #15184C2semidirect
6. #15133C2semidirect
8. #14151C2semidirect
10. #13926C2semidirect
59. C2×C22⋊Q8Q8semidirect
481. C22#2582semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×C4×C22⋊Q82
C22×#3842
#165632
#151842
#151332
#148212
#141512
#141392
#139262
#139072