Finite Group Databasebeta

← #241353

Subgroups — #241353

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#10581C214yesmaximal, complemented
6#10578C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7897C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2859—21no—
15#2859—21no—
16#2647—22no—
17#2571—21no—
18#2571C2212yes—
19#2552—21no—
20#2552C2212yes—
21#2442—22no—
22#2442—22no—
23#2317—21no—
24#2316—21no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1948—21no—
30#1948C2212yes—
31#1947—21no—
32#1947C2214yes—
33#1947C2212yes—
34#1852—22no—
35#1845—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—
47C2×#125—22no—
48C2×#125—22no—
49C2×#117—21no—
50C2×#117D411yescharacteristic
51C2×#117D411yescharacteristic
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D412yes—
58C22×C42C2311yescharacteristic
59#431—22no—
60#431Q814yes—
61#428—44no—
62C2×#107—22no—
63C2×#107—24no—
64C2×#106—22no—
65#423—22no—
66#423Q814yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C22⋊C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74C2×C4×C8Q814yes—
75C2×C4×C8C2312yes—
76#400—24no—
77#400—24no—
78#398—24no—
79#398—24no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#385—24no—
91#384—22no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384D411yescharacteristic
98#384D412yes—
99#384D411yescharacteristic
100#384C2312yes—
101#384C2312yes—
102#382—22no—
103#382—22no—
104#382—22no—
105#382—21no—
106#380—24no—
107#380—21no—
108#380D412yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4C2311yescharacteristic
113C4×C4⋊C4C2312yes—
114C4×C4⋊C4C2311yescharacteristic
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117#376—21no—
118C43—21no—
119#340—28no—
120#340—24no—
121#340—22no—
122#340—22no—
123C23×C4—22no—
124C23×C4—41no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×M16—24no—
129C2×M16—42no—
130C2×M16—42no—
131C22×C8—24no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—42no—
136C22×C8—42no—
137#125—84no—
138#125—84no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—42no—
144#117—42no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D414yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×Q812yes—
193C2×C42C2×Q812yes—
194C2×C42C2×Q812yes—
195C2×C42#4112yes—
196C2×C42C2411yescharacteristic
197#107—44no—
198#107—84no—
199#106—44no—
200C4⋊C8—24no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8C2×Q814yes—
204C4⋊C8C2×Q814yes—
205C22⋊C8—24no—
206C4×C8—24no—
207C4×C8—24no—
208C4×C8—22no—
209C4×C8C2×Q814yes—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95—21no—
218#95—21no—
219#95—22no—
220C24#12011yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4#12011yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4C22⋊Q812yes—
243C22×C4C22⋊Q812yes—
244C22×C4C22⋊Q812yes—
245C22×C4C4⋊Q812yes—
246C22×C4C4⋊Q812yes—
247C22×C4C4⋊Q812yes—
248C22×C4#13512yes—
249C22×C4#13512yes—
250C22×C4AGL(1,ℤ/8)12yes—
251C22×C4#13712yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255M16—48no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—48no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C22⋊C4—42no—
299C22⋊C4—44no—
300C22⋊C4—42no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42C22⋊Q812yes—
310C42C22⋊Q812yes—
311C42C22⋊Q812yes—
312C42C4⋊Q812yes—
313C42C4⋊Q812yes—
314C42C4⋊Q812yes—
315C42C22×Q811yescharacteristic
316C42C22×Q811yescharacteristic
317C42C22×Q811yescharacteristic
318C42C2×#4111yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23#39512yes—
324C23#44912yes—
325C23#45212yes—
326C23C2×#12011yescharacteristic
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—164no—
360C2×C4—164no—
361C2×C4#39512yes—
362C2×C4#39512yes—
363C2×C4#39512yes—
364C2×C4#44912yes—
365C2×C4#45212yes—
366C2×C4#45412yes—
367C2×C4#45512yes—
368C2×C4#47814yes—
369C2×C4#47914yes—
370C2×C4#49914yes—
371C2×C4#50114yes—
372C2×C4C2×C22⋊Q811yescharacteristic
373C2×C4C2×C22⋊Q811yescharacteristic
374C2×C4C2×C22⋊Q811yescharacteristic
375C2×C4C2×C4⋊Q811yescharacteristic
376C2×C4C2×C4⋊Q811yescharacteristic
377C2×C4C2×C4⋊Q811yescharacteristic
378C2×C4C2×#13511yescharacteristic
379C2×C4C2×#13511yescharacteristic
380C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
381C2×C4C2×#13711yescharacteristic
382C2×C4#57612yes—
383C2×C4#57812yes—
384C8—44no—
385C8—44no—
386C8—44no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22#205914yes—
394C22C2×#39511yescharacteristic
395C22C2×#44911yescharacteristic
396C22C2×#45211yescharacteristic
397C22#300811yescharacteristic
398C22#301111yescharacteristic
399C22C2×#45511yescharacteristic
400C22C2×#45411yescharacteristic
401C4—22no—
402C4—21no—
403C4—21no—
404C4—81no—
405C4—84no—
406C4—81no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4—82no—
411C4—162no—
412C4#205914yes—
413C4#206114yes—
414C4C2×#47811yescharacteristic
415C4C2×#47911yescharacteristic
416C4#308312yes—
417C4C2×#49911yescharacteristic
418C4C2×#50111yescharacteristic
419C4#316412yes—
420C2—24no—
421C2#1863511yescharacteristic
422C2#1863611yescharacteristic
423C2#1864511yescharacteristic
424C2#1864611yescharacteristic
425C2C2×#205911yescharacteristic
426C2#2355511yescharacteristic
427C2C2×#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. #10581C2semidirect
6. #10578C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#219822
#211712
C2×#18342
#105812
#105782
#80432
#78972
#78902