Finite Group Databasebeta

← #190777

Subgroups — #190777

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34209C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#8727C212yesmaximal
5#8589C212yesmaximal, complemented
6#8586C212yesmaximal, complemented
7#8340C212yesmaximal
8#8237C212yesmaximal, complemented
9#8233C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×#519—21no—
12C2×#433—21no—
13#2365—22no—
14#2365C414yes—
15#2344—22no—
16#2344C414yes—
17#2327—21no—
18#2327—21no—
19#2310—21no—
20C4×C4⋊Q8C2212yes—
21C4×C4⋊Q8C2212yes—
22#2285C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1924—22no—
25#1917—22no—
26#1842C2212yes—
27#1842C2211yescharacteristic
28#1832—21no—
29#1831C2211yescharacteristic
30#1798—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1768—21no—
34#1767C2211yescharacteristic
35#1767C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C2×#332C2212yes—
38C2×#332C2212yes—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C23×Q8—21no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2312yes—
52#519—48no—
53C2×C4×Q8—24no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×#117—21no—
57C2×#117—24no—
58C2×#117D411yescharacteristic
59C2×#117D411yescharacteristic
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C412yes—
62C22×C42C2311yescharacteristic
63#433—44no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—21no—
74C2×#97—22no—
75C2×C4×C8—21no—
76#391—24no—
77#391—22no—
78#391—22no—
79#389—22no—
80#389—22no—
81#389C2×C414yes—
82#389C2×C414yes—
83#387—24no—
84#387—24no—
85#387—24no—
86#387—24no—
87#384C2312yes—
88#382—21no—
89#381—24no—
90#380—21no—
91#380C2×C412yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4C2×C414yes—
96C4×C4⋊C4C2×C414yes—
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4D412yes—
100C4×C22⋊C4C2×C412yes—
101C4×C22⋊C4D412yes—
102#376D411yescharacteristic
103#376D411yescharacteristic
104C43C2311yescharacteristic
105C43C2311yescharacteristic
106#336—22no—
107#336—22no—
108#336—21no—
109#336—21no—
110#336—22no—
111#336—22no—
112#332—24no—
113#332—24no—
114#332—24no—
115#332—24no—
116C22×Q8—28no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—44no—
120C22×Q8—42no—
121C22×Q8—42no—
122C23×C4—22no—
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C4⋊Q8—28no—
131C4⋊Q8—24no—
132C4⋊Q8—24no—
133C4⋊Q8—28no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8#4114yes—
137C4⋊Q8#4114yes—
138C4×Q8—44no—
139C4×Q8—44no—
140C4×Q8—416no—
141C4×Q8—44no—
142C4×Q8—44no—
143#117—22no—
144#117—22no—
145#117—24no—
146#117—416no—
147#117—44no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4C22×C414yes—
162C2×C22⋊C4—22no—
163C2×C22⋊C4C22⋊C412yes—
164C2×C22⋊C4C22⋊C412yes—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C412yes—
172C2×C42C22×C412yes—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D411yescharacteristic
181C2×C42C2411yescharacteristic
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—44no—
191C4⋊C8—44no—
192C4⋊C8—82no—
193C22⋊C8—44no—
194C22⋊C8—82no—
195#97—42no—
196#97—82no—
197C4×C8—42no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—24no—
203#95—24no—
204#95C22⋊C412yes—
205#95C22⋊C412yes—
206#95C22⋊C412yes—
207#95C22⋊C412yes—
208C24C2×C22⋊C411yescharacteristic
209C2×Q8—28no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—44no—
213C2×Q8—416no—
214C2×Q8—416no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—44no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—28no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C412yes—
240C22×C4AGL(1,ℤ/8)12yes—
241C22×C4#13712yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C4⋊C4—28no—
258C4⋊C4—28no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—48no—
271C4⋊C4#11718yes—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—44no—
275C42—22no—
276C42—21no—
277C42—21no—
278C42—22no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—21no—
283C42—21no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—42no—
289C42—42no—
290C42—44no—
291C42—48no—
292C42—48no—
293C42AGL(1,ℤ/8)14yes—
294C42C2×#4112yes—
295C42C2×#4112yes—
296C23—22no—
297C23—21no—
298C23—22no—
299C23—21no—
300C23#41712yes—
301C23C22×C22⋊C411yescharacteristic
302Q8—416no—
303Q8—44no—
304Q8—44no—
305Q8—44no—
306Q8—44no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—28no—
328C2×C4—44no—
329C2×C4—416no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4#41712yes—
339C2×C4#41814yes—
340C2×C4C2×#11712yes—
341C2×C4#52512yes—
342C2×C4#52511yescharacteristic
343C2×C4#52511yescharacteristic
344C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
345C2×C4C2×AGL(1,ℤ/8)12yes—
346C2×C4C2×#13711yescharacteristic
347C2×C4#57712yes—
348C2×C4#57812yes—
349C8—82no—
350C8—82no—
351C8—82no—
352C8—82no—
353C22—22no—
354C22—24no—
355C22—41no—
356C22—42no—
357C22—41no—
358C22#228011yescharacteristic
359C22#288912yes—
360C22C2×#41811yescharacteristic
361C22C2×#41711yescharacteristic
362C22#289812yes—
363C4—22no—
364C4—24no—
365C4—22no—
366C4—48no—
367C4—41no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—41no—
372C4—84no—
373C4#310912yes—
374C4#311012yes—
375C4#312112yes—
376C4#312611yescharacteristic
377C4#312711yescharacteristic
378C2—42no—
379C2#1775912yes—
380C2#1776312yes—
381C2#2073011yescharacteristic
382C2#2073811yescharacteristic
383C2#2074511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8589C2semidirect
6. #8586C2semidirect
8. #8237C2semidirect
9. #8233C2semidirect

Maximal subgroups

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

subgroupindex
#342092
#219262
#208122
#87272
#85892
#85862
#83402
#82372
#82332