Finite Group Databasebeta

← #190779

Subgroups — #190779

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#8726C212yesmaximal
5#8590C212yesmaximal, complemented
6#8587C212yesmaximal, complemented
7#8342C212yesmaximal
8#8235C212yesmaximal, complemented
9#8232C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×#433—21no—
12#2369—22no—
13#2365C414yes—
14#2345—22no—
15Q8×C22⋊C4C414yes—
16#2331—21no—
17#2331—21no—
18#2316—21no—
19C4×C4⋊Q8C2212yes—
20C4×C4⋊Q8C2212yes—
21#2285C2211yescharacteristic
22#2274—21no—
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×#333C2212yes—
38C2×#333C2212yes—
39C2×#333C2212yes—
40C2×#333C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C23×Q8C2×C412yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2312yes—
52C2×C4×Q8C2×C414yes—
53C2×C4×Q8C2×C414yes—
54C2×#117—21no—
55C2×#117D411yescharacteristic
56C2×#117D411yescharacteristic
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C42C2311yescharacteristic
60#433—44no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—21no—
71C2×#97—22no—
72C2×C4×C8—21no—
73#391—24no—
74#391—24no—
75#391C2×C414yes—
76#391C2×C414yes—
77#389—24no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#387—24no—
83#387—24no—
84#384C2312yes—
85#382—21no—
86#380—21no—
87#380—24no—
88#380C2×C412yes—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4D412yes—
98C4×C22⋊C4—24no—
99C4×C22⋊C4C2×C412yes—
100C4×C22⋊C4D412yes—
101#376D411yescharacteristic
102#376D411yescharacteristic
103C43C2311yescharacteristic
104C43C2311yescharacteristic
105#336—22no—
106#336—22no—
107#336—21no—
108#336—21no—
109#336—22no—
110#336—22no—
111#333—24no—
112#333—24no—
113#333—24no—
114#333—24no—
115C22×Q8—28no—
116C22×Q8—22no—
117C22×Q8—84no—
118C22×Q8C22×C414yes—
119C23×C4—22no—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C4⋊Q8—28no—
128C4⋊Q8—24no—
129C4⋊Q8—24no—
130C4⋊Q8—28no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8#4114yes—
134C4⋊Q8#4114yes—
135C4×Q8—84no—
136C4×Q8—84no—
137#117—22no—
138#117—22no—
139#117—24no—
140#117—42no—
141#117—42no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C414yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4C22⋊C412yes—
158C2×C22⋊C4C22⋊C412yes—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C412yes—
166C2×C42C22×C412yes—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D411yescharacteristic
175C2×C42C2411yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—44no—
185C4⋊C8—44no—
186C4⋊C8—82no—
187C22⋊C8—44no—
188C22⋊C8—82no—
189#97—42no—
190#97—82no—
191C4×C8—42no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—22no—
197#95—22no—
198#95C22⋊C412yes—
199#95C22⋊C412yes—
200#95C22⋊C412yes—
201#95C22⋊C412yes—
202C24C2×C22⋊C411yescharacteristic
203C2×Q8—28no—
204C2×Q8—24no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—84no—
208C2×Q8—816no—
209C2×Q8#11718yes—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—84no—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C412yes—
229C22×C4AGL(1,ℤ/8)12yes—
230C22×C4#13712yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—84no—
260C4⋊C4—84no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—44no—
264C42—22no—
265C42—21no—
266C42—21no—
267C42—22no—
268C42—21no—
269C42—21no—
270C42—22no—
271C42—21no—
272C42—21no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—42no—
278C42—42no—
279C42—44no—
280C42—84no—
281C42—84no—
282C42#13714yes—
283C42C2×#4112yes—
284C42C2×#4112yes—
285C23—22no—
286C23—21no—
287C23—22no—
288C23—21no—
289C23#41712yes—
290C23C22×C22⋊C411yescharacteristic
291Q8—48no—
292Q8—48no—
293Q8—88no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—28no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—88no—
325C2×C4#41712yes—
326C2×C4#41914yes—
327C2×C4C2×#11712yes—
328C2×C4#52512yes—
329C2×C4#52511yescharacteristic
330C2×C4#52511yescharacteristic
331C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
332C2×C4C2×#13712yes—
333C2×C4C2×#13711yescharacteristic
334C2×C4#57712yes—
335C2×C4#57812yes—
336C8—82no—
337C8—82no—
338C8—82no—
339C8—82no—
340C22—22no—
341C22—24no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22#228011yescharacteristic
346C22#288812yes—
347C22C2×#41911yescharacteristic
348C22C2×#41711yescharacteristic
349C22#289812yes—
350C4—22no—
351C4—24no—
352C4—22no—
353C4—41no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—41no—
358C4—48no—
359C4—84no—
360C4#311212yes—
361C4#311312yes—
362C4#312112yes—
363C4#312611yescharacteristic
364C4#312711yescharacteristic
365C2—42no—
366C2#1776012yes—
367C2#1776412yes—
368C2#2072911yescharacteristic
369C2#2074011yescharacteristic
370C2#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. #8590C2semidirect
6. #8587C2semidirect
8. #8235C2semidirect
9. #8232C2semidirect

Maximal subgroups

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

subgroupindex
#342162
#219262
#208122
#87262
#85902
#85872
#83422
#82352
#82322