Finite Group Databasebeta

← #360746

Subgroups — #360746

Order 512.

1924 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2C2×#1867C214yesmaximal
3C2×C4×#99C211yesmaximal, characteristic
4#13965C212yesmaximal, complemented
5#13960C211yesmaximal, characteristic
6#11344C214yesmaximal, complemented
7#10854C212yesmaximal, complemented
8C2×#518C2212yes—
9C22×C4×D4C2211yescharacteristic
10C22×#99C2212yes—
11#2349—24no—
12D4×C4⋊C4—24no—
13#2316—21no—
14#2315—21no—
15#2315C2212yes—
16C2×#386C2212yes—
17C2×#385C2218yes—
18#2285—21no—
19C2×#380—21no—
20C2×#380C2211yescharacteristic
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2212yes—
23C22×#95C2211yescharacteristic
24#1867—216no—
25C4×#99C418yescomplemented
26C2×#342C2218yes—
27C2×#342C2212yes—
28#1444—24no—
29#1441—21no—
30#1439—21no—
31C4⋊#95—24no—
32#1434—21no—
33C22⋊#95—21no—
34C22⋊#95C2212yes—
35C23×D4C2311yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2312yes—
38C24×C4C2311yescharacteristic
39#518—28no—
40#518—44no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—24no—
44C2×C4×D4—28no—
45C2×C4×D4C2×C414yes—
46C2×C4×D4Q814yes—
47C2×#117—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—28no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D414yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4D411yescharacteristic
56C22×C22⋊C4C2312yes—
57C22×C22⋊C4C2314yes—
58C22×C22⋊C4C2311yescharacteristic
59C22×C22⋊C4C2314yes—
60C22×C42—22no—
61C22×C42D411yescharacteristic
62C2×#99C2×C418yes—
63C2×#99D418yes—
64C2×#99Q818yes—
65#389—28no—
66#386—28no—
67#385—28no—
68#385—28no—
69#385—216no—
70#385—48no—
71#384—22no—
72#382—22no—
73#380—22no—
74#380—22no—
75#380—22no—
76#380—22no—
77#380—22no—
78#380—22no—
79#380—22no—
80#380—22no—
81#380—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4C2×C418yescomplemented
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—28no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D414yes—
98#342—216no—
99#342C2×C418yescomplemented
100C25C2×D411yescharacteristic
101C25C2411yescharacteristic
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4C22×C412yes—
107C22×D4C2×D414yes—
108C22×D4C2×Q812yes—
109C22×D4C2×Q814yes—
110C23×C4—24no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—44no—
121C23×C4—82no—
122C23×C4—82no—
123C23×C4—82no—
124C23×C4C4⋊C414yescomplemented
125C23×C4C22×C412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D418yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×Q812yes—
135C23×C4#4114yes—
136C4×D4—28no—
137C4×D4—28no—
138C4×D4—28no—
139C4×D4—28no—
140C4×D4—416no—
141C4×D4—416no—
142#117—44no—
143#117—44no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—416no—
152C2×C4⋊C4—416no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—84no—
155C2×C4⋊C4—84no—
156C2×C4⋊C4—88no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4C4⋊C414yescomplemented
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4C4⋊C414yescomplemented
180C2×C22⋊C4C22×C412yes—
181C2×C22⋊C4C22×C418yes—
182C2×C22⋊C4C2×D418yes—
183C2×C22⋊C4C2×Q812yes—
184C2×C22⋊C4C2×Q818yes—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—28no—
190C2×C42—84no—
191C2×C42C4⋊C414yescomplemented
192#99C4⋊C4132yescomplemented
193#95—24no—
194#95—24no—
195#95—28no—
196#95—24no—
197#95—24no—
198#95—416no—
199#95—44no—
200#95—41no—
201#95—42no—
202#95—41no—
203#95—42no—
204#95—44no—
205#95—82no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24C2×C4⋊C412yes—
215C24#12712yes—
216C24C4⋊Q812yes—
217C24C23×C411yescharacteristic
218C24C22×D411yescharacteristic
219C24C22×D412yes—
220C24C22×D412yes—
221C24C22×Q811yescharacteristic
222C24C22×Q812yes—
223C24C2×#4112yes—
224C24H(5,𝔽2)12yes—
225C242−1+412yes—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—28no—
231C2×D4—28no—
232C2×D4—28no—
233C2×D4—28no—
234C2×D4C2×C4⋊C418yes—
235C2×D4C4⋊Q818yes—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—28no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—48no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—84no—
286C22×C4—84no—
287C22×C4—88no—
288C22×C4—88no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4C2×C4⋊C412yes—
293C22×C4C2×C4⋊C412yes—
294C22×C4C2×C4⋊C412yes—
295C22×C4C2×C4⋊C412yes—
296C22×C4C2×C4⋊C412yes—
297C22×C4C4×D414yes—
298C22×C4C4×D414yes—
299C22×C4#12011yescharacteristic
300C22×C4#12012yes—
301C22×C4#12011yescharacteristic
302C22×C4C22⋊Q814yes—
303C22×C4C22⋊Q814yes—
304C22×C4#12312yes—
305C22×C4#12312yes—
306C22×C4#12712yes—
307C22×C4C4⋊Q812yes—
308C4⋊C4—28no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—832no—
315C4⋊C4—84no—
316C22⋊C4—24no—
317C22⋊C4—28no—
318C22⋊C4—24no—
319C22⋊C4—28no—
320C22⋊C4—28no—
321C22⋊C4—28no—
322C22⋊C4—48no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—48no—
326C22⋊C4—48no—
327C22⋊C4—88no—
328C22⋊C4C2×C4⋊C4116yescomplemented
329C42—24no—
330C42—24no—
331C42—48no—
332C42—88no—
333C23—24no—
334C23—22no—
335C23—24no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—41no—
344C23—44no—
345C23—41no—
346C23—42no—
347C23—44no—
348C23—41no—
349C23—41no—
350C23—44no—
351C23—42no—
352C23—44no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23—42no—
358C23—42no—
359C23—81no—
360C23—81no—
361C23—82no—
362C23—82no—
363C23#38414yes—
364C23C22×C4⋊C411yescharacteristic
365C23C22×C4⋊C412yes—
366C23C2×C4×D412yes—
367C23#52012yes—
368C23C2×#12011yescharacteristic
369C23C2×C22⋊Q812yes—
370C23C2×#12311yescharacteristic
371C23C2×#12711yescharacteristic
372C23C2×C4⋊Q811yescharacteristic
373C23C2×C4⋊Q812yes—
374C23#53511yescharacteristic
375C23#53611yescharacteristic
376C23#54412yes—
377C23D4214yes—
378C23#54714yes—
379C23D4×Q814yes—
380C23#55414yes—
381D4—216no—
382D4—216no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—216no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—416no—
406C2×C4—48no—
407C2×C4—416no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—416no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—44no—
418C2×C4—48no—
419C2×C4—84no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—88no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—88no—
428C2×C4—88no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—168no—
437C2×C4#38414yes—
438C2×C4#38512yes—
439C2×C4#38512yes—
440C2×C4#38514yes—
441C22—22no—
442C22—21no—
443C22—21no—
444C22—22no—
445C22—22no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—28no—
450C22—22no—
451C22—22no—
452C22—28no—
453C22—22no—
454C22—22no—
455C22—24no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—42no—
464C22—42no—
465C22—44no—
466C22—44no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22C2×#38411yescharacteristic
471C22C2×#38511yescharacteristic
472C22#231711yescharacteristic
473C22D4×C4⋊C414yes—
474C22#243311yescharacteristic
475C22#243411yescharacteristic
476C22#244014yes—
477C22#250111yescharacteristic
478C22#250211yescharacteristic
479C22#302512yes—
480C22#302612yes—
481C4—24no—
482C4—22no—
483C4—22no—
484C4—24no—
485C4—416no—
486C4—42no—
487C4—42no—
488C4—84no—
489C4—88no—
490C4—162no—
491C2—24no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—24no—
497C2—44no—
498C2#1388011yescharacteristic
499C2#2088811yescharacteristic
500C2#2088911yescharacteristic
501C2#2577312yes—
502C2#2585412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13965C2semidirect
6. #11344C2semidirect
7. #10854C2semidirect
25. C4×#99C4semidirect
85. C4×C22⋊C4C2×C4semidirect
99. #342C2×C4semidirect
124. C23×C4C4⋊C4semidirect
158. C2×C4⋊C4C4⋊C4semidirect
179. C2×C22⋊C4C4⋊C4semidirect
191. C2×C42C4⋊C4semidirect
192. #99C4⋊C4semidirect
328. C22⋊C4C2×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341812
C2×#18672
C2×C4×#992
#139652
#139602
#113442
#108542