Finite Group Databasebeta

← #208215

Subgroups — #208215

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3C4×#423C211yesmaximal, characteristic
4#9905C212yesmaximal, complemented
5#9702C212yesmaximal, complemented
6#9135C214yesmaximal
7#8235C214yesmaximal, complemented
8C2×#533C412yes—
9C2×#423C2212yes—
10C2×C4×M16—21no—
11#2868C2211yescharacteristic
12#2868C2211yescharacteristic
13#2862C412yes—
14#2859—21no—
15#2859—21no—
16#2857—21no—
17#2857—21no—
18#2857C412yes—
19#2857C2212yes—
20C4×C4⋊Q8—21no—
21C4×C4⋊Q8—21no—
22C4×C4⋊Q8C412yes—
23C4×C4⋊Q8C2212yes—
24#2285—21no—
25#2285—21no—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1831C2212yes—
29#1817—21no—
30#1797C2212yes—
31#1767—21no—
32#1767C2212yes—
33#1767C2212yes—
34C4×C4⋊C8—21no—
35C4×C4⋊C8C2212yes—
36#1749—21no—
37#1745—21no—
38#1728—21no—
39C2×#333C2218yes—
40#1296C2218yes—
41C22×M16—22no—
42#533—48no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2312yes—
49C2×#125—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C414yes—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C2×#117—22no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42C2311yescharacteristic
62#423—28no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2314yes—
65C2×C4⋊C8C2314yes—
66C2×C22⋊C8—22no—
67C4×M16—44no—
68C2×#97—22no—
69C2×#97—22no—
70C2×C4×C8—22no—
71#391—22no—
72#391—22no—
73#391C2×C414yes—
74#391C2×C414yes—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384C2×C412yes—
90#384C2312yes—
91#382—22no—
92#382—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C43C2×C411yescharacteristic
102C43C2×C411yescharacteristic
103C43D411yescharacteristic
104C43D411yescharacteristic
105C43D411yescharacteristic
106C43D411yescharacteristic
107C43C2311yescharacteristic
108C43C2311yescharacteristic
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#333—216no—
116C22×Q8—22no—
117C22×Q8C22×C414yes—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C2×M16—28no—
126C2×M16—44no—
127C2×M16—44no—
128C22×C8—24no—
129C22×C8—24no—
130C4⋊Q8—28no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137#125—48no—
138C22⋊Q8—48no—
139C22⋊Q8—48no—
140C4×Q8—48no—
141C4×Q8—48no—
142#117—48no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4C22×C414yes—
157C2×C22⋊C4—24no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2411yescharacteristic
195C4⋊C8—28no—
196C4⋊C8—24no—
197C4⋊C8—28no—
198C4⋊C8#4118yes—
199C22⋊C8—44no—
200#97—44no—
201#97—44no—
202C4×C8—44no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207C24C2×C22⋊C411yescharacteristic
208C2×Q8—28no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—44no—
213C2×Q8—44no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4#13714yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252M16—416no—
253C2×C8—28no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—44no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C22⋊C4—48no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—21no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—21no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—41no—
300C42—48no—
301C42#10014yes—
302C42AGL(1,ℤ/8)12yes—
303C42#13712yes—
304C42#13714yes—
305C42C2×#4114yes—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23#41912yes—
313C23C22×C22⋊C411yescharacteristic
314Q8—88no—
315Q8—88no—
316C2×C4—28no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4C2×#10012yes—
357C2×C4C2×#10012yes—
358C2×C4C2×#10012yes—
359C2×C4#41312yes—
360C2×C4#41712yes—
361C2×C4#41712yes—
362C2×C4#41912yes—
363C2×C4#41912yes—
364C2×C4#41912yes—
365C2×C4C4×#4112yes—
366C2×C4#52512yes—
367C2×C4#53212yes—
368C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
369C2×C4C2×#13712yes—
370C2×C4C2×#13711yescharacteristic
371C2×C4C2×#13712yes—
372C2×C4#57512yes—
373C8—48no—
374C8—84no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22#229711yescharacteristic
383C22C22×#10011yescharacteristic
384C22C2×#41311yescharacteristic
385C22C2×#41911yescharacteristic
386C22C2×#41911yescharacteristic
387C22C2×#41711yescharacteristic
388C22#289611yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—84no—
402C4—84no—
403C4#311012yes—
404C4#311312yes—
405C4#317214yes—
406C2—42no—
407C2#1774611yescharacteristic
408C2#1774811yescharacteristic
409C2#1774911yescharacteristic
410C2#1775011yescharacteristic
411C2#2139411yescharacteristic
412C2#2140411yescharacteristic
413C2#2142411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9905C2semidirect
5. #9702C2semidirect
7. #8235C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#207782
C4×#4232
#99052
#97022
#91352
#82352