Finite Group Databasebeta

← #241191

Subgroups — #241191

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic
3#21938C211yesmaximal, characteristic
4#13908C211yesmaximal, characteristic
5#10572C214yesmaximal, complemented
6#10571C214yesmaximal, complemented
7C2×#1294C211yesmaximal, characteristic
8C2×#1294C211yesmaximal, characteristic
9#7907C211yesmaximal, characteristic
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2851—22no—
13#2646C2214yes—
14#2569—22no—
15#2569—22no—
16#2442—22no—
17#2442—22no—
18#2315—21no—
19#2315—21no—
20C2×#380—21no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C22×#95C2211yescharacteristic
26#1853—22no—
27#1853C2214yes—
28#1852—22no—
29#1852C2214yes—
30#1808—22no—
31#1807—22no—
32#1444C2212yes—
33#1442C2212yes—
34#1436—22no—
35C22⋊#95—21no—
36#1294C2212yes—
37#1294C2212yes—
38#1294C2212yes—
39#1294C2212yes—
40#1280C2214yes—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C23×C8—21no—
44C23×C8—21no—
45C2×#125—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4D411yescharacteristic
55C22×C42D412yes—
56C2×#107—24no—
57C2×#106—24no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8Q814yes—
63C2×C22⋊C8Q814yes—
64C2×C22⋊C8C2312yes—
65C2×C22⋊C8C2312yes—
66#400—24no—
67#400—24no—
68#398—24no—
69#398—24no—
70#389—22no—
71#389—22no—
72#387—24no—
73#387—24no—
74#385—24no—
75#385—24no—
76#385—24no—
77#385—24no—
78#382—22no—
79#380—24no—
80#380—24no—
81#380—21no—
82#380—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86#380—21no—
87#380—21no—
88#380—21no—
89#380—44no—
90#380D412yes—
91#380D412yes—
92#380C2312yes—
93#380C2312yes—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95Q814yes—
107C2×#95C2312yes—
108C2×#95C2311yescharacteristic
109C2×#95C2311yescharacteristic
110#340—24no—
111#340—24no—
112C25C2×D411yescharacteristic
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—24no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—44no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—82no—
123C23×C4—82no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×Q812yes—
132C23×C4C2×Q812yes—
133C23×C4C2×Q812yes—
134C23×C4#4112yes—
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—24no—
139C22×C8—24no—
140C22×C8—82no—
141C22×C8—82no—
142#125—84no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C2×D414yes—
158C2×C4⋊C4C2×D414yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—82no—
182#107—84no—
183#106—84no—
184C22⋊C8—24no—
185C22⋊C8—24no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8C2×Q814yes—
189C22⋊C8C2×Q814yes—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—24no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—21no—
202#95—21no—
203#95—44no—
204#95—44no—
205#95—41no—
206#95—41no—
207#95—82no—
208#95C2×Q812yes—
209#95C2×Q812yes—
210C24—42no—
211C24—42no—
212C24#12011yescharacteristic
213C24#12011yescharacteristic
214C24C22⋊Q812yes—
215C24C4⋊Q812yes—
216C24#13512yes—
217C24#13512yes—
218C24AGL(1,ℤ/8)12yes—
219C24#13712yes—
220C24C22×D411yescharacteristic
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—48no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—48no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—84no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—88no—
270C22×C4—88no—
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4C22⋊Q812yes—
274C22×C4C22⋊Q812yes—
275C22×C4C22⋊Q812yes—
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4C4⋊Q812yes—
279C22×C4C4⋊Q812yes—
280C22×C4C4⋊Q812yes—
281C22×C4C4⋊Q812yes—
282C22×C4C4⋊Q812yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C22×Q811yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—84no—
294C2×C8—84no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—88no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—44no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—44no—
319C42—82no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—44no—
333C23—44no—
334C23#39512yes—
335C23#39512yes—
336C23#44912yes—
337C23#44912yes—
338C23#45212yes—
339C23#45212yes—
340C23#47814yes—
341C23#47914yes—
342C23#49914yes—
343C23#50114yes—
344C23C2×#12011yescharacteristic
345C23C2×C22⋊Q811yescharacteristic
346C23C2×C4⋊Q811yescharacteristic
347C23C2×#13511yescharacteristic
348C23C2×#13511yescharacteristic
349C23C2×AGL(1,ℤ/8)11yescharacteristic
350C23C2×#13711yescharacteristic
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—816no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—164no—
404C2×C4—164no—
405C2×C4#39512yes—
406C2×C4#39512yes—
407C2×C4C2×C22⋊Q811yescharacteristic
408C2×C4C2×C22⋊Q811yescharacteristic
409C2×C4C2×C4⋊Q811yescharacteristic
410C2×C4C2×C4⋊Q811yescharacteristic
411C8—44no—
412C8—44no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—24no—
424C22—24no—
425C22—22no—
426C22—22no—
427C22—24no—
428C22—24no—
429C22—44no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22#205914yes—
434C22#205914yes—
435C22C2×#39511yescharacteristic
436C22C2×#44911yescharacteristic
437C22C2×#45211yescharacteristic
438C22#301812yes—
439C22#302612yes—
440C22C2×#47811yescharacteristic
441C22C2×#47911yescharacteristic
442C22C2×#49911yescharacteristic
443C22C2×#50111yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—22no—
449C4—22no—
450C4—44no—
451C4—82no—
452C4—82no—
453C4—84no—
454C4—82no—
455C4—84no—
456C4—162no—
457C2—21no—
458C2—21no—
459C2—24no—
460C2—24no—
461C2—21no—
462C2—21no—
463C2#1838412yes—
464C2#1838512yes—
465C2C2×#205911yescharacteristic
466C2#2356011yescharacteristic
467C2#2356511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10572C2semidirect
6. #10571C2semidirect

Maximal subgroups

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

subgroupindex
#350202
#219402
#219382
#139082
#105722
#105712
C2×#12942
#79072