Finite Group Databasebeta

← #241351

Subgroups — #241351

Order 512.

990 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#10582C214yesmaximal, complemented
6#10575C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7897C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C2×#544—21no—
11C2×#431C2211yescharacteristic
12C2×#428—21no—
13C2×#423C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2868—21no—
16#2859—21no—
17#2660—22no—
18#2552—21no—
19#2552C2212yes—
20#2551—21no—
21#2551C2212yes—
22#2440—22no—
23#2440—22no—
24#2317—21no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1948—21no—
30#1948C2214yes—
31#1948C2212yes—
32#1947—21no—
33#1947C2212yes—
34#1853—22no—
35#1844—21no—
36#1834—22no—
37#1834—21no—
38#1834C2214yes—
39#1736—21no—
40C2×#340—22no—
41#1296C2214yes—
42#1279C2212yes—
43#1279C2214yes—
44#1279C2212yes—
45C22×M16—21no—
46C23×C8—21no—
47#544—44no—
48#544—44no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×C22⋊Q8—24no—
56C2×#117—21no—
57C2×#117D411yescharacteristic
58C2×#117D411yescharacteristic
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431Q814yescomplemented
68#428—44no—
69C2×#107—22no—
70C2×#106—22no—
71C2×#106—24no—
72#423—22no—
73#423Q814yescomplemented
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C22⋊C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81C2×C4×C8Q814yescomplemented
82C2×C4×C8C2312yes—
83#398—24no—
84#398—24no—
85#395—24no—
86#395—24no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#385—24no—
94#384—22no—
95#384—22no—
96#384—22no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384D411yescharacteristic
103#384D412yes—
104#384D411yescharacteristic
105#384C2312yes—
106#384C2312yes—
107#382—22no—
108#382—21no—
109#380—21no—
110#380D412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4C2311yescharacteristic
115C4×C4⋊C4C2311yescharacteristic
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C43—21no—
121#340—28no—
122#340—24no—
123#340—22no—
124#340—22no—
125C22×Q8—24no—
126C23×C4—22no—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C2×M16—24no—
132C2×M16—42no—
133C2×M16—42no—
134C22×C8—24no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—42no—
139C22×C8—42no—
140C4⋊Q8—48no—
141C4⋊Q8—44no—
142C4⋊Q8—44no—
143C4⋊Q8—84no—
144C4⋊Q8—84no—
145#125—48no—
146#125—44no—
147#125—44no—
148C22⋊Q8—416no—
149#117—22no—
150#117—22no—
151#117—42no—
152#117—42no—
153#117SD1614yes—
154#117SD1614yes—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D414yes—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q812yes—
204C2×C42C2×Q812yes—
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208#107—44no—
209#106—44no—
210#106—84no—
211C4⋊C8—24no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8C2×Q814yes—
215C4⋊C8C2×Q814yes—
216C22⋊C8—24no—
217C4×C8—24no—
218C4×C8—24no—
219C4×C8—22no—
220C4×C8C2×Q814yes—
221#95—24no—
222#95—24no—
223#95—21no—
224#95—21no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229C24#12011yescharacteristic
230C2×Q8—48no—
231C2×Q8—84no—
232C2×Q8—84no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4C22⋊Q812yes—
255C22×C4C22⋊Q812yes—
256C22×C4C22⋊Q812yes—
257C22×C4C4⋊Q812yes—
258C22×C4C4⋊Q812yes—
259C22×C4C4⋊Q812yes—
260C22×C4C2×SD1612yes—
261C22×C4C2×SD1612yes—
262C22×C4AGL(1,ℤ/8)12yes—
263C22×C4#13712yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267M16—48no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—48no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4C2×SD1612yes—
308C4⋊C4C2×SD1612yes—
309C4⋊C4C2×SD1612yes—
310C4⋊C4C2×SD1612yes—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—44no—
314C42—24no—
315C42—24no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42C22⋊Q812yes—
323C42C22⋊Q812yes—
324C42C22⋊Q812yes—
325C42C4⋊Q812yes—
326C42C4⋊Q812yes—
327C42C4⋊Q812yes—
328C42C22×Q811yescharacteristic
329C42C22×Q811yescharacteristic
330C42C22×Q811yescharacteristic
331C42C2×#4111yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23#39512yes—
337C23#44812yes—
338C23#45012yes—
339C23C2×#12011yescharacteristic
340Q8—168no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—164no—
374C2×C4—164no—
375C2×C4#39512yes—
376C2×C4#39512yes—
377C2×C4#39512yes—
378C2×C4#44812yes—
379C2×C4#45012yes—
380C2×C4#45412yes—
381C2×C4#45512yes—
382C2×C4#47514yes—
383C2×C4#47614yes—
384C2×C4#49814yes—
385C2×C4#50114yes—
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4C2×C22⋊Q811yescharacteristic
388C2×C4C2×C22⋊Q811yescharacteristic
389C2×C4C2×C4⋊Q811yescharacteristic
390C2×C4C2×C4⋊Q811yescharacteristic
391C2×C4C2×C4⋊Q811yescharacteristic
392C2×C4C22×SD1611yescharacteristic
393C2×C4C22×SD1611yescharacteristic
394C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4#57612yes—
397C2×C4#57812yes—
398C8—44no—
399C8—44no—
400C8—44no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22#205814yes—
408C22C2×#39511yescharacteristic
409C22C2×#45011yescharacteristic
410C22C2×#44811yescharacteristic
411C22#300811yescharacteristic
412C22#301111yescharacteristic
413C22C2×#45511yescharacteristic
414C22C2×#45411yescharacteristic
415C4—22no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—84no—
423C4—82no—
424C4—82no—
425C4—162no—
426C4#205814yes—
427C4#206114yes—
428C4C2×#47611yescharacteristic
429C4C2×#47511yescharacteristic
430C4#308312yes—
431C4C2×#49811yescharacteristic
432C4C2×#50111yescharacteristic
433C4#316412yes—
434C2—24no—
435C2#1863111yescharacteristic
436C2#1863211yescharacteristic
437C2#1864111yescharacteristic
438C2#1864211yescharacteristic
439C2C2×#205811yescharacteristic
440C2#2355511yescharacteristic
441C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10582C2semidirect
6. #10575C2semidirect
67. #431Q8semidirect
73. #423Q8semidirect
81. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#219812
#211712
C2×#18342
#105822
#105752
#80432
#78972
#78902