Finite Group Databasebeta

← #409262

Subgroups — #409262

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C212yesmaximal
2#14340C211yesmaximal, characteristic
3#13795C211yesmaximal, characteristic
4#11357C212yesmaximal, complemented
5#11325C212yesmaximal, complemented
6#11257C212yesmaximal, complemented
7#11256C212yesmaximal, complemented
8#10730C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C2×#384C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#380C2212yes—
19#1854—28no—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2211yescharacteristic
29#1441—22no—
30#1439—22no—
31C4⋊#95—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C22⋊#95—22no—
36C24×C4—21no—
37C24×C4C2311yescharacteristic
38C23×C8C2312yes—
39C23×C8C2312yes—
40C2×#117—22no—
41C2×#117—24no—
42C2×#117—24no—
43C2×#117—41no—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117—41no—
47C2×#117—41no—
48C2×#117—41no—
49C2×#117D414yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D412yes—
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#107—24no—
68C2×#107—42no—
69C2×#107—42no—
70C2×#106—24no—
71C2×#106—42no—
72C2×#106—42no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8Q818yes—
76#384—24no—
77#382—41no—
78#382—41no—
79#380—24no—
80#380—24no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—42no—
94#340—42no—
95#336—24no—
96#336—41no—
97#336—41no—
98C25C2×D411yescharacteristic
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—82no—
113C23×C4—82no—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×Q814yes—
119C23×C4#4112yes—
120C23×C4#4114yes—
121C23×C4C2411yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8C2×Q814yes—
135C22×C8C2×Q814yes—
136#117—48no—
137#117—48no—
138#117—44no—
139#117—44no—
140#117—44no—
141#117—44no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4#4114yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42#4114yes—
197#107—48no—
198#106—48no—
199C22⋊C8—216no—
200#95—44no—
201#95—44no—
202#95—82no—
203#95—82no—
204#95—82no—
205#95—82no—
206#95—82no—
207#95—82no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24#12112yes—
217C24#12312yes—
218C24C4⋊Q812yes—
219C24C22×D411yescharacteristic
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—48no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4—88no—
283C22×C4—88no—
284C22×C4#12112yes—
285C22×C4C22⋊Q818yes—
286C22×C4#12314yes—
287C22×C4#12312yes—
288C22×C4#12518yes—
289C22×C4C4⋊Q812yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×Q812yes—
292C22×C4C2×#4112yes—
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C22×C4H(5,𝔽2)11yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—88no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—88no—
336C42—24no—
337C42—24no—
338C42—48no—
339C42—42no—
340C42—42no—
341C23—22no—
342C23—24no—
343C23—24no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—24no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—24no—
352C23—24no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23#39814yes—
364C23C2×#12111yescharacteristic
365C23C2×#12311yescharacteristic
366C23C2×#12311yescharacteristic
367C23#52511yescharacteristic
368C23C2×C4⋊Q811yescharacteristic
369C23#53411yescharacteristic
370C23#53511yescharacteristic
371C23#57612yes—
372C23#57712yes—
373C23#57712yes—
374C23#57812yes—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—28no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—816no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#39814yes—
429C2×C4C2×C22⋊Q812yes—
430C2×C4C2×#12512yes—
431C2×C4#52912yes—
432C2×C4#53212yes—
433C2×C4#53411yescharacteristic
434C2×C4#53411yescharacteristic
435C2×C4#53811yescharacteristic
436C2×C4#53811yescharacteristic
437C2×C4#55112yes—
438C2×C4#55112yes—
439C2×C4#55112yes—
440C2×C4#55112yes—
441C2×C4#55412yes—
442C2×C4#55412yes—
443C2×C4#55412yes—
444C2×C4#55412yes—
445C8—44no—
446C8—44no—
447C22—28no—
448C22—24no—
449C22—44no—
450C22—44no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22C2×#39811yescharacteristic
459C22#248211yescharacteristic
460C22#248311yescharacteristic
461C22#250211yescharacteristic
462C22#258311yescharacteristic
463C22#261711yescharacteristic
464C22#285411yescharacteristic
465C22#304512yes—
466C22#304612yes—
467C22#309612yes—
468C22#309612yes—
469C22#309712yes—
470C22#309712yes—
471C22#312211yescharacteristic
472C22#312311yescharacteristic
473C22#312611yescharacteristic
474C22#312711yescharacteristic
475C22#316312yes—
476C22#316412yes—
477C22#320312yes—
478C22#320412yes—
479C22#322412yes—
480C22#322512yes—
481C4—24no—
482C4—44no—
483C4—84no—
484C4—84no—
485C4—82no—
486C4—162no—
487C4—162no—
488C4#250512yes—
489C4#250512yes—
490C4#256412yes—
491C4#256412yes—
492C2—44no—
493C2#1501411yescharacteristic
494C2#2366212yes—
495C2#2759111yescharacteristic
496C2#2759211yescharacteristic
497C2#2760712yes—
498C2#2795912yes—
499C2#2930611yescharacteristic
500C2#2930711yescharacteristic
501C2#2972912yes—
502C2#3311611yescharacteristic
503C2#3311711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11357C2semidirect
5. #11325C2semidirect
6. #11257C2semidirect
7. #11256C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
#143402
#137952
#113572
#113252
#112572
#112562
#107302
#107132