Finite Group Databasebeta

← #220090

Subgroups — #220090

Order 512.

1015 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36151C211yesmaximal, characteristic
2#22648C212yesmaximal
3C2×#1550C212yesmaximal, complemented
4#9708C212yesmaximal, complemented
5#9139C212yesmaximal, complemented
6#9129C212yesmaximal, complemented
7#8587C211yesmaximal, characteristic, complemented
8#8343C212yesmaximal, complemented
9#8232C211yesmaximal, characteristic, complemented
10C2×#533C2212yes—
11C2×#435—22no—
12C2×#423C2212yes—
13C22×C4⋊C8C2212yes—
14#2862C2211yescharacteristic
15#2857C2212yes—
16#2536C412yes—
17#2489—22no—
18#2453—22no—
19#2421C414yes—
20#2415—22no—
21#2362—22no—
22#2351C412yes—
23C4×C4⋊Q8C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1940—22no—
26#1851—22no—
27#1850—22no—
28#1842C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1767C2212yes—
32#1550—28no—
33C2×#333C2211yescharacteristic
34C2×#333C2212yes—
35C2×#333C2211yescharacteristic
36C2×#332C2212yes—
37C2×#332C2212yes—
38C2×#332C2212yes—
39C2×#332C2212yes—
40#1296C2212yes—
41#1296C2211yescharacteristic
42#1296C2211yescharacteristic
43C22×M16—22no—
44C23×C8—22no—
45#533—24no—
46#533—24no—
47#533—42no—
48#533—42no—
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2312yes—
51C2×#125—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8C2×C414yes—
55C2×C4×Q8—22no—
56C2×C4×Q8C2×C414yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D412yes—
60C2×#117—22no—
61C22×C42C2312yes—
62#435—48no—
63#423—24no—
64#423—42no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—22no—
78#400—24no—
79#398—22no—
80#397—24no—
81#394—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391C2×C412yes—
86#389—24no—
87#389—22no—
88#389—22no—
89#389C2×C414yes—
90#388—24no—
91#388—24no—
92#387—24no—
93#384C2311yescharacteristic
94#382—22no—
95#382C2×C412yes—
96#382C2×C412yes—
97#380—22no—
98#380C2×C414yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4C2×C411yescharacteristic
102C4×C4⋊C4D412yes—
103C4×C22⋊C4C2×C412yes—
104C4×C22⋊C4D412yes—
105C4×C22⋊C4D412yes—
106C43C2311yescharacteristic
107#336—22no—
108#336—22no—
109#336—22no—
110#333—24no—
111#333—22no—
112#333D414yescomplemented
113#332—24no—
114#332—24no—
115#332—42no—
116#332—42no—
117C22×Q8—22no—
118C22×Q8C22×C412yes—
119C22×D4C22⋊C411yescharacteristic
120C23×C4C22×C412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C2×M16—44no—
124C2×M16—44no—
125C2×M16—82no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—44no—
131C22×C8—44no—
132C22×C8—82no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8C2×D412yes—
139C4⋊Q8#4112yes—
140#125—44no—
141#125—44no—
142C22⋊Q8—28no—
143C22⋊Q8—44no—
144C22⋊Q8—44no—
145C22⋊Q8—44no—
146C22⋊Q8—44no—
147C22⋊Q8—44no—
148C4×Q8—28no—
149C4×Q8—44no—
150C4×Q8—44no—
151C4×Q8—44no—
152C4×Q8—44no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—42no—
156#117—44no—
157#117—44no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22×C414yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4C22⋊C412yes—
172C2×C22⋊C4C22⋊C412yes—
173C2×C22⋊C4C22⋊C412yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—22no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C412yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2411yescharacteristic
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—44no—
196C4⋊C8—44no—
197C4⋊C8—44no—
198C4⋊C8—82no—
199C4⋊C8C2×D414yes—
200C22⋊C8—44no—
201C22⋊C8—44no—
202#97—44no—
203#95—22no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—22no—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95C22⋊C411yescharacteristic
212#95C22⋊C411yescharacteristic
213C24C2×C22⋊C412yes—
214C2×Q8—22no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8C4×D414yes—
224C2×D4—24no—
225C2×D4—41no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4AGL(1,ℤ/8)14yes—
250C22×C4#13714yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D412yes—
253M16—88no—
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—88no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—41no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—48no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—48no—
286C22⋊C4—48no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42—48no—
300C42—44no—
301C42C22×D411yescharacteristic
302C42C2×#4111yescharacteristic
303C42H(5,𝔽2)12yes—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23#41714yes—
311C23C22×C22⋊C411yescharacteristic
312Q8—44no—
313Q8—44no—
314Q8—82no—
315Q8—82no—
316Q8—82no—
317Q8—82no—
318D4—44no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4#41312yes—
354C2×C4#41714yes—
355C2×C4C2×C4×D411yescharacteristic
356C2×C4#51811yescharacteristic
357C2×C4D4212yes—
358C2×C4#54612yes—
359C2×C4C2×AGL(1,ℤ/8)12yes—
360C2×C4C2×#13712yes—
361C2×C4#57712yes—
362C2×C4#57812yes—
363C8—44no—
364C8—82no—
365C8—82no—
366C8—82no—
367C22—24no—
368C22—24no—
369C22—24no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22D4×C22⋊C411yescharacteristic
375C22C2×#41311yescharacteristic
376C22#288911yescharacteristic
377C22C2×#41712yes—
378C22#289812yes—
379C4—22no—
380C4—21no—
381C4—21no—
382C4—24no—
383C4—41no—
384C4—42no—
385C4—42no—
386C4—44no—
387C4—42no—
388C4—82no—
389C4—84no—
390C4#326512yes—
391C4#327012yes—
392C4#330812yes—
393C4#331612yes—
394C2—44no—
395C2#1775211yescharacteristic
396C2#1776312yes—
397C2#1778011yescharacteristic
398C2#2241111yescharacteristic
399C2#2242112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1550C2semidirect
4. #9708C2semidirect
5. #9139C2semidirect
6. #9129C2semidirect
7. #8587C2semidirect
8. #8343C2semidirect
9. #8232C2semidirect
112. #333D4semidirect

Maximal subgroups

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

subgroupindex
#361512
#226482
C2×#15502
#97082
#91392
#91292
#85872
#83432
#82322