Finite Group Databasebeta

← #393056

Subgroups — #393056

Order 512.

2224 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic
3C2×#1890C214yesmaximal
4C2×#1799C211yesmaximal, characteristic
5#13904C212yesmaximal, complemented
6#10936C214yesmaximal, complemented
7#10846C212yesmaximal, complemented
8C22×#123C2212yes—
9C2×#518C2212yes—
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×#99C2212yes—
13#2486—24no—
14#2486C418yescomplemented
15C2×#398—21no—
16C2×#397—21no—
17C2×#389—21no—
18C2×#381C2218yes—
19C2×#381C2212yes—
20C2×#380—21no—
21C2×#380C2212yes—
22C2×#380C2211yescharacteristic
23#2275—24no—
24#2274—21no—
25#2273—21no—
26#2273C2212yes—
27C42×D4—24no—
28C4×#117—21no—
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4C2212yes—
31#1890—216no—
32#1799—24no—
33C2×#342C2218yes—
34C2×#342C2212yes—
35#1445—24no—
36C4⋊#95—24no—
37C23×D4C2311yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4C2312yes—
40C24×C4C2311yescharacteristic
41C2×#123—24no—
42C2×#123—28no—
43C2×#123—28no—
44C2×#123C2×C418yes—
45#518—24no—
46#518—28no—
47#518D418yes—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—28no—
51C2×C4×D4—24no—
52C2×C4×D4C2×C414yes—
53C2×C4×D4D414yes—
54C2×#117—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C414yescomplemented
65C22×C4⋊C4D411yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4C2314yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2311yescharacteristic
72C22×C22⋊C4C2314yes—
73C22×C42—22no—
74C22×C42D411yescharacteristic
75C22×C42D414yes—
76C2×#99—24no—
77C2×#99—28no—
78#398—28no—
79#397—28no—
80#389—28no—
81#381—24no—
82#381—216no—
83#381—216no—
84#381C2×C418yescomplemented
85#380—28no—
86#380—22no—
87#380—24no—
88#380—24no—
89#380C2×C414yescomplemented
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—28no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—28no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—28no—
103#376—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—216no—
109C2×#95—22no—
110C2×#95D414yes—
111C43—22no—
112#342—24no—
113#342—216no—
114C25C2×D411yescharacteristic
115C25C2411yescharacteristic
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—24no—
119C22×D4C22×C412yes—
120C22×D4C2×D412yes—
121C22×D4C2×D414yes—
122C22×D4#4114yes—
123C23×C4—24no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—28no—
135C23×C4—24no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4C22⋊C414yes—
139C23×C4C22×C414yes—
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D418yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4#4114yes—
151#123—232no—
152#123—48no—
153#123—48no—
154C4×D4—232no—
155C4×D4—28no—
156C4×D4—28no—
157C4×D4—28no—
158C4×D4—28no—
159C4×D4—232no—
160#117—28no—
161#117—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22⋊C414yes—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4C22⋊C414yes—
204C2×C22⋊C4C22×C414yes—
205C2×C22⋊C4C22×C412yes—
206C2×C22⋊C4C2×D412yes—
207C2×C22⋊C4C2×D418yes—
208C2×C22⋊C4#4118yes—
209C2×C42—24no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—48no—
218C2×C42—84no—
219C2×C42C22⋊C414yes—
220#99—48no—
221#95—24no—
222#95—24no—
223#95—28no—
224#95—24no—
225#95—28no—
226#95—24no—
227#95—24no—
228#95—44no—
229#95—432no—
230C24—21no—
231C24—21no—
232C24—22no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24C2×C22⋊C412yes—
238C24#12114yes—
239C24C23×C411yescharacteristic
240C24C22×D411yescharacteristic
241C24C22×D412yes—
242C24C22×D411yescharacteristic
243C24C22×D412yes—
244C24C2×#4112yes—
245C24C2×#4112yes—
246C24C2×#4114yes—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—28no—
250C2×D4—24no—
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—28no—
254C2×D4—24no—
255C2×D4C4×D418yes—
256C2×D4#12414yes—
257C2×D4#12714yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—28no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—28no—
265C22×C4—21no—
266C22×C4—216no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—28no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—48no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—48no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—48no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—48no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—48no—
298C22×C4—42no—
299C22×C4—48no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—84no—
308C22×C4—82no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C4×D414yes—
317C22×C4C4×D414yes—
318C22×C4#12011yescharacteristic
319C22×C4#12012yes—
320C22×C4#12011yescharacteristic
321C22×C4#12114yes—
322C22×C4#12114yes—
323C22×C4#121116yes—
324C22×C4#12114yes—
325C22×C4#12312yes—
326C22×C4#12312yes—
327C4⋊C4—216no—
328C4⋊C4—28no—
329C4⋊C4—48no—
330C4⋊C4—44no—
331C4⋊C4—48no—
332C4⋊C4—44no—
333C4⋊C4—416no—
334C4⋊C4—88no—
335C22⋊C4—216no—
336C22⋊C4—24no—
337C22⋊C4—28no—
338C22⋊C4—28no—
339C22⋊C4—24no—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—28no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—416no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—48no—
350C42—216no—
351C42—24no—
352C42—24no—
353C42—48no—
354C42—84no—
355C42—84no—
356C42—84no—
357C23—24no—
358C23—22no—
359C23—24no—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—24no—
364C23—22no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—41no—
369C23—44no—
370C23—41no—
371C23—48no—
372C23—44no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—48no—
377C23—42no—
378C23—42no—
379C23—81no—
380C23—84no—
381C23—81no—
382C23#39014yes—
383C23C22×C22⋊C411yescharacteristic
384C23C2×C4×D412yes—
385C23C2×C4×D412yes—
386C23C4×#4112yes—
387C23C2×#12011yescharacteristic
388C23C2×#12112yes—
389C23C2×#12114yes—
390C23C2×#12112yes—
391C23C2×#12311yescharacteristic
392C23#52514yes—
393C23#52512yes—
394C23C2×#12411yescharacteristic
395C23#52914yes—
396C23C2×#12711yescharacteristic
397C23#53212yes—
398C23#53214yes—
399D4—28no—
400D4—28no—
401D4—216no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—216no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—416no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—416no—
434C2×C4—84no—
435C2×C4—88no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—816no—
439C2×C4—88no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—816no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4#38612yes—
449C2×C4#38614yes—
450C2×C4#38612yes—
451C2×C4#39014yes—
452C2×C4#45814yes—
453C22—28no—
454C22—24no—
455C22—28no—
456C22—24no—
457C22—21no—
458C22—22no—
459C22—21no—
460C22—22no—
461C22—21no—
462C22—22no—
463C22—21no—
464C22—22no—
465C22—24no—
466C22—42no—
467C22—42no—
468C22—41no—
469C22—44no—
470C22—41no—
471C22—44no—
472C22—42no—
473C22—44no—
474C22—48no—
475C22—42no—
476C22—42no—
477C22—84no—
478C22—82no—
479C22C2×#38611yescharacteristic
480C22C2×#39011yescharacteristic
481C22#229711yescharacteristic
482C22C4×#12114yes—
483C22#284712yes—
484C22#285012yes—
485C22#285412yes—
486C22#285612yes—
487C22C2×#45812yes—
488C22#302812yes—
489C4—24no—
490C4—22no—
491C4—24no—
492C4—416no—
493C4—42no—
494C4—42no—
495C4—84no—
496C4—84no—
497C4—164no—
498C4#189014yescomplemented
499C2—21no—
500C2—22no—
501C2—22no—
502C2—21no—
503C2—22no—
504C2—24no—
505C2—44no—
506C2#1469811yescharacteristic
507C2C2×#189011yescharacteristic
508C2#2096211yescharacteristic
509C2#3292714yes—
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
5. #13904C2semidirect
6. #10936C2semidirect
7. #10846C2semidirect
14. #2486C4semidirect
64. C22×C4⋊C4C2×C4semidirect
84. #381C2×C4semidirect
89. #380C2×C4semidirect
498. C4#1890semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C4×#5182
C2×#18902
C2×#17992
#139042
#109362
#108462