Finite Group Databasebeta

← #350129

Subgroups — #350129

Order 512.

1121 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2#33147C211yesmaximal, characteristic
3#33109C211yesmaximal, characteristic
4#25811C211yesmaximal, characteristic
5#9059C214yesmaximal, complemented
6#8923C214yesmaximal, complemented
7#8707C211yesmaximal, characteristic
8#8707C211yesmaximal, characteristic
9#8340C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8—21no—
12C2×#498C2212yes—
13C2×#484—21no—
14#2857C2211yescharacteristic
15#2365—22no—
16#2365—22no—
17#2365—22no—
18#2365C2214yes—
19C2×#391—21no—
20C2×#389—21no—
21C2×#387—22no—
22C2×C4×C4⋊C4—21no—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#2058—22no—
26#1990C2212yes—
27#1988C2212yes—
28#1981C2212yes—
29#1948C2212yes—
30#1947C2212yes—
31#1928—22no—
32#1919—22no—
33#1872—21no—
34#1853—21no—
35#1853—22no—
36#1824—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40C2×#332C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—21no—
44C2×C4⋊Q8—28no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8D412yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2312yes—
50C2×C22⋊Q8—21no—
51C2×C4×Q8—24no—
52C2×C4×Q8—24no—
53C2×C4×Q8—48no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C42—22no—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65#498—28no—
66#484—44no—
67C2×#107—22no—
68C2×#106—24no—
69C2×#106—22no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#103—22no—
73C2×#103—22no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C4×C8C2312yes—
78#391—24no—
79#391—24no—
80#391—21no—
81#389—24no—
82#389—24no—
83#389—21no—
84#389—21no—
85#387—28no—
86#387—28no—
87#384C2311yescharacteristic
88#380—21no—
89#380—21no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D412yes—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4C2312yes—
99C4×C4⋊C4C2312yes—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104#340—24no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#336—21no—
110#336—21no—
111#336—21no—
112#332—28no—
113C22×Q8—28no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—44no—
118C22×Q8—42no—
119C22×Q8—42no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C4⋊Q8—28no—
132C4⋊Q8—28no—
133C4⋊Q8—28no—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8#4118yes—
137C22⋊Q8—42no—
138C22⋊Q8—42no—
139C4×Q8—416no—
140C4×Q8—48no—
141C4×Q8—48no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4SD1614yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C22⋊C4—41no—
171C2×C42—24no—
172C2×C42—28no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—48no—
178C2×C42—44no—
179C2×C42—44no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2411yescharacteristic
184#107—44no—
185#106—48no—
186#106—44no—
187C4⋊C8—24no—
188C4⋊C8—24no—
189#103—44no—
190#103—44no—
191C22⋊C8—24no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C4×C8—24no—
195#95—28no—
196#95—28no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—41no—
202#95—41no—
203C24#12011yescharacteristic
204C2×Q8—28no—
205C2×Q8—28no—
206C2×Q8—44no—
207C2×Q8—416no—
208C2×Q8—416no—
209C2×Q8—42no—
210C2×Q8—44no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—42no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—28no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4C2×SD1612yes—
250C22×C4#13512yes—
251C22×C4#13712yes—
252C22×C4#13712yes—
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—24no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—24no—
269C4⋊C4—28no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—416no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—82no—
287C4⋊C4#12318yes—
288C4⋊C4C2×SD1614yes—
289C4⋊C4AGL(1,ℤ/8)14yes—
290C22⋊C4—82no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—44no—
295C42—44no—
296C42—416no—
297C42—44no—
298C42—44no—
299C42C2×#4114yes—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23#44812yes—
305C23#45212yes—
306C23#45612yes—
307C23C2×#12011yescharacteristic
308Q8—416no—
309Q8—48no—
310Q8—48no—
311Q8—82no—
312Q8—82no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—28no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—416no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4#44812yes—
354C2×C4#45212yes—
355C2×C4C2×#12312yes—
356C2×C4#52512yes—
357C2×C4#52512yes—
358C2×C4C22×SD1611yescharacteristic
359C2×C4C2×#13511yescharacteristic
360C2×C4C2×AGL(1,ℤ/8)12yes—
361C2×C4C2×#13711yescharacteristic
362C2×C4C2×#13711yescharacteristic
363C2×C4#57512yes—
364C8—44no—
365C8—82no—
366C8—82no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22#165314yes—
374C22#284911yescharacteristic
375C22C2×#44811yescharacteristic
376C22C2×#45211yescharacteristic
377C22#300311yescharacteristic
378C22#300511yescharacteristic
379C22C2×#45611yescharacteristic
380C22#302011yescharacteristic
381C4—22no—
382C4—22no—
383C4—22no—
384C4—44no—
385C4—44no—
386C4—41no—
387C4—48no—
388C4—82no—
389C4—82no—
390C4—161no—
391C4—161no—
392C4#310312yes—
393C4#310512yes—
394C4#311012yes—
395C4#311712yes—
396C2—24no—
397C2C2×#165311yescharacteristic
398C2#1853911yescharacteristic
399C2#1875811yescharacteristic
400C2#1878111yescharacteristic
401C2#3299411yescharacteristic
402C2#3300511yescharacteristic
403C2#3300911yescharacteristic
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. #9059C2semidirect
6. #8923C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
#331472
#331092
#258112
#90592
#89232
#87072
#83402