Finite Group Databasebeta

← #190778

Subgroups — #190778

Order 512.

892 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#8727C212yesmaximal
5#8594C212yesmaximal, complemented
6#8590C212yesmaximal, complemented
7#8342C212yesmaximal
8#8235C212yesmaximal, complemented
9#8232C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11#2369—22no—
12#2365C414yes—
13#2345—22no—
14Q8×C22⋊C4C414yes—
15#2331—21no—
16#2331—21no—
17#2316—21no—
18C4×C4⋊Q8C2212yes—
19C4×C4⋊Q8C2212yes—
20#2285C2211yescharacteristic
21#2274—21no—
22C4×#117C2211yescharacteristic
23#1924—22no—
24#1917—22no—
25#1842C2211yescharacteristic
26#1840—21no—
27#1831C2212yes—
28#1831C2211yescharacteristic
29#1818—21no—
30#1797C2212yes—
31#1797C2212yes—
32#1767C2211yescharacteristic
33#1767C2212yes—
34C4×C4⋊C8C2211yescharacteristic
35#1750—21no—
36#1746—21no—
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
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×#97—21no—
68C2×#97—22no—
69C2×C4×C8—21no—
70#391—24no—
71#391—24no—
72#391C2×C414yes—
73#391C2×C414yes—
74#389—24no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#387—24no—
80#387—24no—
81#384C2312yes—
82#382—21no—
83#380—21no—
84#380—24no—
85#380C2×C412yes—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—24no—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4D412yes—
98#376D411yescharacteristic
99#376D411yescharacteristic
100C43C2311yescharacteristic
101C43C2311yescharacteristic
102#336—22no—
103#336—22no—
104#336—22no—
105#336—21no—
106#336—21no—
107#336—22no—
108#336—22no—
109#336—22no—
110#333—24no—
111#333—24no—
112#333—24no—
113#333—24no—
114C22×Q8—28no—
115C22×Q8—22no—
116C22×Q8—84no—
117C22×Q8C22×C414yes—
118C23×C4—22no—
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C4⋊Q8—28no—
127C4⋊Q8—24no—
128C4⋊Q8—24no—
129C4⋊Q8—28no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8#4114yes—
133C4⋊Q8#4114yes—
134C4×Q8—84no—
135C4×Q8—84no—
136#117—24no—
137#117—41no—
138#117—41no—
139#117—42no—
140#117—42no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C414yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4C22⋊C412yes—
157C2×C22⋊C4C22⋊C412yes—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—24no—
162C2×C42—24no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C412yes—
165C2×C42C22×C412yes—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2411yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—82no—
184C22⋊C8—44no—
185C22⋊C8—82no—
186#97—42no—
187#97—82no—
188C4×C8—42no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95C22⋊C412yes—
196#95C22⋊C412yes—
197#95C22⋊C412yes—
198#95C22⋊C412yes—
199C24C2×C22⋊C411yescharacteristic
200C2×Q8—28no—
201C2×Q8—24no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—84no—
205C2×Q8—816no—
206C2×Q8#11718yes—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—84no—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4#13714yes—
228C22×C4C23×C411yescharacteristic
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
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—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C22⋊C4—24no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C42—22no—
262C42—21no—
263C42—21no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—42no—
276C42—42no—
277C42—44no—
278C42—84no—
279C42—84no—
280C42C2×#4112yes—
281C42C2×#4112yes—
282C23—22no—
283C23—21no—
284C23—22no—
285C23—21no—
286C23#41912yes—
287C23C22×C22⋊C411yescharacteristic
288Q8—48no—
289Q8—48no—
290Q8—88no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
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—21no—
311C2×C4—21no—
312C2×C4—28no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—88no—
323C2×C4#41912yes—
324C2×C4C2×#11712yes—
325C2×C4#52512yes—
326C2×C4#52511yescharacteristic
327C2×C4#52511yescharacteristic
328C2×C4C2×#13712yes—
329C2×C4#57511yescharacteristic
330C2×C4#57511yescharacteristic
331C2×C4#57712yes—
332C2×C4#57812yes—
333C8—82no—
334C8—82no—
335C8—82no—
336C8—82no—
337C22—22no—
338C22—24no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22#228011yescharacteristic
343C22#288812yes—
344C22C2×#41911yescharacteristic
345C22#289611yescharacteristic
346C22#289812yes—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—41no—
355C4—48no—
356C4—84no—
357C4#311712yes—
358C4#311712yes—
359C4#312112yes—
360C4#312611yescharacteristic
361C4#312711yescharacteristic
362C2—42no—
363C2#1776012yes—
364C2#1776812yes—
365C2#2072911yescharacteristic
366C2#2074111yescharacteristic
367C2#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. #8594C2semidirect
6. #8590C2semidirect
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
#207862
#87272
#85942
#85902
#83422
#82352
#82322